handout%20solids%20&%20surfaces%202009-2010

Handout

Solids & Surfaces

Surfaces part Version 1.4 2009-2010 Dr. ir. M.B. de Rooij Laboratory for Surface Technology and Tribology University of Twente

Handout Solids & Surfaces 2009-2010

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Table of Contents List of symbols...........................................................................................................4 Introduction ...............................................................................................................6 Chapter 1 Contact mechanics: Contact between smooth surfaces .................................8

1.1 Introduction ................................................................................................8 1.2 Contact types; contraform and conformal contacts ......................................8 1.3 Line loading of an elastic halfspace.............................................................8 1.4 *3D loading of an elastic halfspace ...........................................................22 1.5 Hertzian contacts ......................................................................................26 1.6 Reduced modulus of elasticity and reduced contact radius ........................33 1.7 Excersises.................................................................................................34 1.8 Summary ..................................................................................................35 1.9 References ................................................................................................35

Chapter 2 Contact mechanics: Contact between rough surfaces .................................37 2.1 Introduction ..............................................................................................37 2.2 Elastic contact and wavy surfaces .............................................................37 2.3 Contact mechanics: Random rough surfaces .............................................42 2.4 The sum surface........................................................................................44 2.5 Plasticity index .........................................................................................45 2.6 Elastic contact...........................................................................................45 2.7 Plasticity...................................................................................................46 2.8 Adhesion & Surface roughness .................................................................52 2.9 Excersises.................................................................................................53 2.10 Summary ..................................................................................................53 2.11 References ................................................................................................53

Chapter 3 Contact mechanics: Elastic & plastic contact for a sphere..........................57 3.1 Introduction ..............................................................................................57 3.2 Elasticity...................................................................................................57 3.3 Plasticity...................................................................................................59 3.4 Example ...................................................................................................69 3.5 Effect of sliding on plastic contact ............................................................71 3.6 Shakedown ...............................................................................................72 3.7 Summary ..................................................................................................77 3.8 References ................................................................................................77

Chapter 4 Surface forces............................................................................................80 4.1 Introduction ..............................................................................................80 4.2 The origin of adhesive forces ....................................................................80 4.3 Surface energy & Surface tension .............................................................96 4.4 Surface energy & Work of adhesion..........................................................98 4.5 Measurement techniques for fluids characterization ................................101 4.6 Measurement techniques for solids characterization................................105 4.7 Adhesion between solids and liquids: Wetting......................................... 112 4.8 Spreading over time ................................................................................ 113 4.9 Adhesion & Contact mechanics: Adhesion between spheres ................... 115 4.10 Adhesion in biological systems ...............................................................127 4.11 Summary ................................................................................................134 4.12 Excersises...............................................................................................135 4.13 References ..............................................................................................135

Chapter 5 Surface roughness ...................................................................................136 5.1 Introduction ............................................................................................136


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5.2 Form, waviness and roughness................................................................137 5.3 Characterization of rough surfaces ..........................................................139 5.4 *Random process model .........................................................................153 5.5 Fractal characterization...........................................................................157 5.6 Parameters related to rough surface contact.............................................161 5.7 Processing 3D roughness data.................................................................165 5.8 Example .................................................................................................167 5.9 Excersises...............................................................................................172 5.10 Summary ................................................................................................173 5.11 References ..............................................................................................174

Chapter 6 Mechanical and geometrical properties of surfaces ..................................177 6.1 Introduction ............................................................................................177 6.2 Mechanical properties of surfaces ...........................................................179 6.3 Measurement techniques of mechanical properties..................................179 6.4 Excersises...............................................................................................197 6.5 References ..............................................................................................197

Chapter 7 *Damage mechanisms .............................................................................199 7.1 Introduction ............................................................................................199 7.2 Mechanical damage ................................................................................200 7.3 (Electro)chemical damage: Aqueous corrosion........................................217

Chapter 8 Appendix C: Explicit equations for stresses beneath a sliding contact ......243 8.1 Caused by an applied normal load FN......................................................243 8.2 Caused by a tangential load FT ................................................................245 8.3 References ..............................................................................................247


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List of symbols Symbol Unit Contact radius a [m] Half contact of line contact b [m] Separation d [m] Current density i [A cm-2] Exchange current density iex [A cm-2] Yield strength in shear k [m] Slope m [-] Width of line contact l [m] Mean contact pressure p [MPa] Maximum contact pressure pmax [MPa] Pressure distribution p(x) [MPa] Tangential stress distribution q(x) [MPa] Summit height s [m] Mean summit height sz [m] Load per unit width (line contact) w [N m-1] Surface height z [m] Contact area A [m2] Real contact area Ar [m2] Nominal contact area An [m2] Electrical potential E [V] Modulus of elasticity E [GPa] Overpotential E* [V] Reduced modulus of elasticity E* [GPa] Normal force FN [N] Tangential force FT [N] Hardness H [GPa] Enthalpy H Electrical current I [A] Kurtosis Ku [-] Radius R [m] Equivalent radius / reduced radius R* [m] Average roughness value Ra [m] Root mean average roughness value Rq [m] Entropy S Skewness Sk [-] Energy U Roughness bandwidth parameter α [-] Summit radius of curvature β [m] Deformation /indentation depth δ [m] Deformation /indentation at the onset of plastic deformation δ1 [m] Deformation /indentation at fully plastic deformation δ2 [m] Strain ε [-] Probability density function φ(s) [-] Curvature κ [m-1] Summit density ηs [m-2] Poisson constant ν [-] Standard deviation of the heights σ [m] Standard deviation of the slopes σm [m]


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Standard deviation of the curvatures σκ [m] Stress σ [MPa] Surface tension σ [J m-2] Yield strength σy [GPa] Shear stress τ [MPa] Contact angle θ [-] Surface energy ∆γ [J m-2] Critical surface tension γc [J m-2] Work of adhesion Γ [J m-2]


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Introduction The ‘surfaces’ part of the ‘solids & surfaces’ course will discuss three related topics: Firstly, some important surface properties will the discussed. Some properties that will be discussed are mechanical properties like hardness and elastic modulus as well as surface energy and surface roughness. Besides a discussion on surface properties, also methods to measure these properties will be discussed. Secondly, the important topic of contact mechanics will be discussed. In many technical applications two surfaces are in contact. The mechanics of smooth as well as rough contacts will be treated in several lectures Thirdly, some damage mechanisms related to surfaces will be treated. In this course, the focus will be on chemical damage (corrosion) and mechanical damage (wear) of surfaces. This handout is an almost complete overview of the content of the course. The only subject that is not incorporated (yet) in the handout is a discussion on analytical techniques. Information about this topic can be found in the sheets. During the course, several exercises will be handed out. After completion of these exercises, an appointment can be made for an oral examination. The paragraphs labeled with a star * are not part of the course

M.B. de Rooij


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Chapter 1 Contact mechanics: Contact between smooth surfaces

1.1 Introduction The chapter will discuss the case of two surfaces contacting each other and can considered to be a summary of chapters 2, 3 and 4 of [3]. If not mentioned otherwise, more details can be found in [3]. Contact pressures acting in the contact between two surfaces are very important to known in many practical situations. The material close to the surface will also deform under the influence of these contact stresses. This chapter will be about elastic contact between two smooth surfaces. In later chapters, more details about plasticity and surface roughness will be discussed.

1.2 Contact types; contraform and conformal contacts Two bodies can contact each other in different configurations. The first distinction in contacts is the difference between contraform and conform contacts as illustrated in the following figure.

Figure 1.1: Contraform and conform contacts

Examples of contraform or concentrated contacts are the contact between two teeth of a gear wheel and the contact between a wheel and rails of a train. Conformal contacts or disperse contacts can be found in sliding bearings where the clearance between shaft and bearing is in the order of micrometers. Conformal contacts are generally characterized by a relative more extensive contact area and relatively lower contact pressures than concentrated contacts. Contact pressures in concentrated contacts can reach very high values up to several GPa. Typical pressures in conform contacts are in the order of MPa.

1.3 Line loading of an elastic halfspace This section will describe the case of a relatively simple contact situation, where the contact between two surfaces is represented by a distributed normal load p(x) and a distributed tangential load q(x) acting in the contact area. In order to describe the situation of an elastic halfspace loaded by distributed loads, first the case of a concentrated normal load will be studied in section 1.3.2 and the case of a concentrated tangential load in section 1.3.3. The principle of superposition will subsequently be used to study distributed loads. For special cases for certain distributed loads the reader is referred to [3]. If two bodies are in contact and (one of) the surfaces will deform elastically, a contact area will be formed. In this section, it will be assumed that the contact area is small compared to the radii of curvature of the undeformed surfaces, as is often the case in realistic situations.

Contraform contact

Conform contact


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Assuming that the dimensions of the bodies, like the radius of curvature, are large compared to the contact area, means that the bodies can considered to be semi-infinite. The consequence of semi-infinite contacting bodies is that the dimensions of the body will not influence the contact behavior. Besides the assumption of semi-infinite bodies, in this chapter the deformation will considered to be elastic without any effects of plasticity. As mentioned above, in many engineering situations small contact areas are formed. A small contact area means that, in particular in concentrated contacts, contact pressures can reach high values in the order of GPa. The first case that will be discussed is the situation of an elastic half space loaded along a line with a certain distributed pressure p(x) and a tangential traction q(x). p(x) and q(x) are only dependent on the x-direction.

Figure 1.2: Subsurface stress field for an arbitray p(x) and q(x)

In the following it will be assumed that the semi-infinite body is in a state of plane strain, so εy=0. This means that the thickness of the body is large compared to the width of the loaded region, which is a reasonable assumption in many cases.

1.3.1 Basics elasticity theory, plane strain Force equilibrium in the x-direction and z-direction of an infinitesimal volume element in the solid means that:

0

0

=∂

∂+

∂∂

=∂

∂+

∂∂

xz

zxxzz

xzx

τσ

τσ

(1.1)

From elasticity theory it can also be shown that the strains εx, εz and γxz have to satisfy the following compatibility condition:

zxxzxzzx

∂∂∂

=∂∂

+∂

∂ γεε 2

2

2

2

2

(1.2)

In this equations, the strains are defined as the derivative of the displacements in x, y and z-direction, respectively δx, δy and δz. In the case of plane strain:

zxzxzx

xzz

zx

x ∂∂

+∂

∂=

∂∂

=∂

∂=

δδγ

δε

δε ,, (1.3)

and: 0=yε (1.4)

σx σx σz

σz τxz

τxz

τxz τxz

p(x)

q(x) x

z


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In the case of plane strain, the stress σy is a function of σx and σz and is given by: ( )zxy σσνσ += (1.5)

Hooke’s law in plane strain gives the following expressions for the relation between the stress components and the strains:

( ) ( )( )

( ) ( )( )( )

xzxzxz

xzz

zxx

EG

E

E

τν

τγ

σννσνε

σννσνε

+==

+−−=

+−−=

121

111

111

2

2

(1.6)

In these equations E is the modulus of elasticity and G is the shear modulus of elasticity. To solve the problem, stresses and strains throughout the solid have to be found which satisfy everywhere the equations of equilibrium (equations (1.1)), the equation of compatibility (equation (1.2)) and Hooke’s law (equations (1.6)). It can be shown that all these equations are satisfied if a stress function φ(x,z) can be found with the second derivatives equal to the stresses perpendicular to the direction of the derivatives of φ(x,z):

zx

x

z

xz

z

x

∂∂∂

−=

∂∂

=

∂∂

=

φτ

φσ

φσ

2

2

2

2

2

(1.7)

With φ(x,z) satisfying

02

2

2

2

2

2

2

2

=

∂∂

+∂∂

∂∂

+∂∂

zxzxφφ (1.8)

Besides the equations related to elasticity theory given above, some boundary conditions have to be satisfied at the boundaries. Outside the loaded area there are no normal and tangential stresses acting on the surface. This means the following boundary conditions have to be satisfied at the surface, so z=0, outside the loaded region:

00

==

xz

z

τσ

(1.9)

Inside the loaded region there are stresses acting on the surface. The normal and tangential stresses acting on the surface in the loaded region are equal to the imposed loads p(x) and q(x):

( )( )xqxp

xz

z

−=−=

τσ

(1.10)

With increasing distance from the contact area, stresses in the material will become smaller. At a distance very far away from the contact area the stresses will become zero.

0,,, =zxzx στσ (1.11) In the following, we will also need expressions of equation (1.6), equation (1.7) and equation (1.8) in terms of polar coordinates. Hooke’s law is terms of polar coordinates is given by:


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rrrr

rr

r

rrr

r

rr

δδδγ

θδδ

ε

δε

θθ

θθ

−∂

∂+

∂∂

=

∂∂

+=

∂∂

=

1

1 (1.12)

The stress function φ(r,θ) has to satisfy equation (1.7). In polar coordinates this equation is given by:

011112

2

22

2

2

2

22

2

=

∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

θφφφ

θ rrrrrrrr (1.13)

The stress function was defined in terms of the stresses, see equation (1.8). This equation reads in terms of stresses were given as derivatives σr and σθ reads:

∂∂

∂∂

−=

∂∂

=

∂∂

+∂∂

=

θφ

τ

φσ

θφφ

σ

θ

θ

rr

r

rrr

r

r

1

11

2

2

2

2

2

(1.14)

In the beginning of this section, a figure was shown with a contact pressure p(x) and tractions q(x). These pressures and tractions will cause displacements in x-direction δx(x) and the displacements in z-direction δz(x). In many realistic contact problems with two elastic contacting bodies, just a total normal load FN is given and contact pressure distributions and deformations are not known beforehand. However, there are iterative techniques available to calculate both contact pressures and deformations for contact problems. In principle, contact pressures can be calculated from deformations or vice versa. So, if contact pressures p(x) are given, then normal deformations δx can be calculated using the equations shown above and vice versa. The same is (of course) true for tractions q(x) and tangential deformations δz Depending on the boundary conditions given, different classed of contact problems can be distinguished. Problems were pressures and tractions are given are generally easier to solve than cases were displacements are given as boundary conditions. If a rigid shape, like a sphere, is pressed into an elastic solid, displacements in z-direction are prescribed by the geometry of the rigid shape. If Coulomb friction forces are acting in the contact, then the value of q(x) within the loaded region is equal to q(x)=µ∙p(x). If displacements or pressures are known as boundary conditions, makes a difference in the solution of the problem. In this paragraph some general equations have been presented. Solving the equations above is outside the scope of this handout. For some cases, analytical solutions have been known. For the plane strain case with a known pressure or load with unknown deformations δx and δy, the following cases have analytical solutions:

• A concentrated normal force FN • A concentrated tangential force FT • A contantly distributed pressure p • A constantly distributed traction q • A triangular pressure p(x) • A triangular traction q(x)


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Analytical equations are also known for the case of a rigid punch indenting an elastic solid, which is a case where the deformation is prescribed and the pressures are unknown. There are many elastic loading cases for which there are no simple analytical solutions. In many cases the load known and both the pressure and deformation unknown. In such cases the equations Other cases are complicated geometries or the case of contacting rough surfaces. To solve such problems, use can be made of the principle of superposition. Numerical methods, based on superposition of basic analytical solutions have been developed. An arbitrary pressure distribution could for example be approximated by a piecewise constant pressure distribution and the deformations could be superimposed. Not all cases mentioned above will be discussed in this chapter. For an overview, the reader is referred to [2]. In the following section, using the theory discussed above, the case of a semi-infinite solid loaded by a concentrated normal force FN will be studied. In section 1.3.3 the case of loading by a concentrated tangential force FT will be studied. In section 1.3.4 and 1.3.5 the case of distributed loads will be discussed and in section 1.3.8 the case of a rigid punch.

1.3.2 Concentrated normal force FN The situation of a semi-infinite solid loaded with a concentrated normal force FN is shown in the figure below. This contact problem has been first solved by Flamant in 1892.

Figure 1.3: A semi-infinite solid loaded with a concentrated normal force FN

The solution for stresses and deformations comes from the following stress function φ(r,θ) in polar coordinates with r the radial distance from the center of the contact and θ the angle between the normal through FN and the vector r:

( ) θθθφ sin, Arr = (1.15) In this equation, A is a constant. From this equation stresses can be derived by calculating second derivatives, see equation (1.14). By taking second derivatives it can be shown that in this case the following stresses will exist in the solid:

0

cos2

==

=

θθ τσ

θσ

r

r rA

(1.16)

So, in this case only radial stresses originating from the point where FN acts are present in the solid and the other stress components are zero throughout the solid.

FN

θ r

r0 r0

σr

d δθ

δr

δr|θ=π/2


13

The constant A can be calculated from force balance between the vertical components of the radial stresses σr and the applied force FN:

πθθθθσ

ππ

π

ArdArdF rN ===− ∫∫−

22

20

2cos4cos (1.17)

So, substitution gives:

rFNr

θπ

σcos2

−= (1.18)

So, far we have discussed the contact problem in terms of polar coordinates for reasons of simplicity. In rectangular coordinates stresses are given by:

( )

( )

( )222

2

222

32

222

22

2cossin

2cos

2sin

zxxzF

zxzF

zxzxF

Nrzx

Nrz

Nrx

+−==

+−==

+−==

πθθστ

πθσσ

πθσσ

(1.19)

Using Hooke’s law in polar coordinates given by equation (1.12) gives expressions for the strains εr, εθ and rrθ.

( )

01

cos211

cos21 2

===−∂

∂+

∂∂

+−==

∂∂

+

−−==

∂∂

Grrrr

rF

Err

rF

Er

rr

rr

Nr

Nr

r

θθ

θ

θθ

τγ

δδδ

θπ

ννε

θδδ

θπ

νε

δ

(1.20)

If the strains are known, displacements can be calculated by integration. This will not be further discussed, but results in:

( )( )

( ) ( )( )

( )( ) rCCCFE

FE

FE

rFE

CCFE

rFE

N

NNN

NNr

321

2

21

2

sincossin121

cos121sin21lnsin21

cossinsin121lncos21

+−++−

++−

−+

+−

=

+++−

−−

−=

θθθπ

νν

θθπ

ννθ

πνν

θπ

νδ

θθθθπ

ννθ

πν

δ

θ (1.21)

Now, general expressions for the displacements have been calculated from a given concentrated normal force FN. However, still some integration constants are present in the equations. Boundary conditions can be used to solve the integration constants. Because it is a rotation symmetric problem, points on the z-axis only displace in z-direction. This means that the surface will not tilt and C1= C3=0. The displacements on the surface can be found by substitution of θ=±π/2.

( )( )

CrFE

FE

Nrr

Nrr

+−

=−=

+−−==

−==

−==

ln212

121

2

,,

,,

22

22

πν

δδ

ννδδ

ππ

ππ

θθ

θθ

(1.22)

The constant C is determined by a point on the surface, r0, where the radial displacements are assumed to be zero. Then:


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rrF

E Nrr0

2

,, ln2122 π

νδδ ππ θθ

−−=−= −== (1.23)

This equation is plotted in the following graph. The stress singularity at the origin causes infinite displacements at that location.

normal displacement at the surface for a concentrated normal force

-0.000000001-9E-10-8E-10-7E-10-6E-10-5E-10-4E-10-3E-10-2E-10-1E-10

0-15 -10 -5 0 5 10 15

r

δθ

Figure 1.4: Normal displacement at the surface for a concentrated normal force FN

1.3.3 Concentrated tangential force FT A similar case is a semi-infinite solid loaded with a tangential force FT. A concentrated tangential force will result in the same stress field as a concentrated normal force, but rotated 90o. The angle θ is defined in such a way as shown in the following figure.

Figure 1.5: The situation of a semi-infinite solid, loaded with a concentrated tangential force

If θ is defined as the angle between the horizontal and the vector r, the expressions are the same as the case of a semi-infinite solid loaded with a concentrated normal force FN, see equation (1.13):

0

cos2

==

−=

θθ τσ

θπ

σ

r

Tr r

F (1.24)

In rectangular coordinates the stresses are:

FT

θ r2

d d

σr,2

r1

σr,1

x θ1

z

θ2

δr δθ


15

( )

( )

( )222

2

222

22

222

32

2cossin

2cos

2sin

zxzxF

zxxzF

zxxF

Trzx

Trz

Trx

+−==

+−==

+−==

πθθστ

πθσσ

πθσσ

(1.25)

Again, subsequent substitution of these stresses in Hooke’s law of equation (1.6) gives expressions for the strains εx, εz and γxz and displacements can be calculated using equation (1.3). Without rigid body rotation of the solid and vertical displacement along the z-axis, the surface displacements are given by:

( )( )T

Trr

FE

rFE

2121

ln21

0,,

2

0,,

ννδδ

πν

δδ

θθπθθ

θπθ

+−==−

−−==−

==

==

(1.26)

Results are similar to the case of a concentrated normal load, taking into account the different definition of θ, see equation (1.22). The integration constant C can again be determined by assuming that the deformation is zero at a certain distance from the origin O. The cases of a concentrated normal load and of a concentrated tangential load will be used in the following two cases of a distributed normal load and a distributed tangential load by using the principle of superposition.

1.3.4 Distributed normal and tangential loads Now the case is considered of an elastic half-space loaded against a line over –b<x<a by a normal pressure p(x) and a tangential load q(x). Of interest are now the stresses in the body as well as the displacements of points at the surface of the solid. As is known from elasticity theory, deformations caused by different loads simultaneously acting on a body can be calculated by superposition of the deformations caused by the different load components separately.

Figure 1.6: The situatoin of a semi-infinite solid, loaded with a distributed normal load

p(s)

q(s) x

z

ds

s b a


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The situation of a distributed normal load acting on a surface can considered to be a superposition of an infinite number of point loads equal to p(s)ds acting on this surface. The case of a concentrated normal load was discussed in section 1.3.2. Stresses can be calculated by integration over the loaded region of the results from section (1.3). In the equations, FN=p(s)ds and x has been substituted with (x-s).This gives the following expressions for the stresses caused by a distributed normal load:

( )( )( )( )

( )( )( )

( )( )( )( )∫

∫

∫

−

−

−

+−

−−=

+−−=

+−

−−=

a

bzx

a

bz

a

bx

dszsx

sxspz

dszsx

spz

dszsx

sxspz

222

2

222

3

222

2

2

2

2

πτ

πσ

πσ

(1.27)

Similar expressions can be found for the case of a distributed tangential load. The general expressions for the stresses σx, σz and τzx given a certain normal pressure p(x) and tangential traction q(x) are:

So, suppose p(x) and q(x) are known, stresses can be calculated using the equations shown above. However, an analytical solution of the integrals shown could be difficult and therefore analytical expressions for the stresses are generally not known. Again, given the expressions for the stresses in the body, subsequently expressions for the strains εx, εz and γxz can be found by Hooke’s law and displacements can be calculated by integration. This results in the following expressions for the displacements:

The radial displacements from a concentrated normal force, see equation (1.22)a, and from a concentrated tangential force at the surface, see equation (1.30)b, are both constant and independent of r. However, the sign of the displacements changes of the place where the surface is loaded. This means that the integrals shown above have a discontinuity and have to be splitted into two parts because of these step changes in the displacements at the place where the forces act. In practice equations for the displacements δ are more useful in terms of derivatives of the displacements instead if the displacements themselves. It will turn out that in particular the following expressions for the derivative of the deformations at the surface. are useful:

( )( )( )( )

( )( )( )( )

( )( )( )

( )( )( )( )

( )( )( )( )

( )( )( )( )∫∫

∫∫

∫∫

−−

−−

−−

+−

−−

+−

−−=

+−

−−

+−−=

+−

−−

+−

−−=

a

b

a

bzx

a

b

a

bz

a

b

a

bx

dszsx

sxsqzdszsx

sxspz

dszsx

sxsqzdszsx

spz

dszsx

sxsqdszsx

sxspz

222

2

222

2

222

2

222

3

222

3

222

2

22

22

22

ππτ

ππσ

ππσ

(1.28)

( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 2

2

1

2

2121ln12

ln122

121

CdssqdssqE

dssxspE

CdssxsqE

dsspdsspE

a

x

x

b

x

bz

x

b

a

x

x

bx

+

−

+−+−

−−=

+−−

−

−

+−−=

∫∫∫

∫∫∫

−−

−−

ννπ

νδ

πννν

δ

(1.29)


17

The first expression is in fact the tangential strain εx at the surface. The second expression is the the x-derivative of the normal strain εz at the surface and is in fact the slope of the deformed surface after loading. Above equations will be illustrated with the example of an uniform normal load in the next section, where an uniformly distributed load will be studied.

1.3.5 Uniform distributed normal load The simplest case is perhaps a constant uniformly distributed normal load p with q(x)=0. The constant normal load is acting over the area –a<x<a. This problem is called the Flamant problem.

Figure 1.7: The situation of a semi-infinite solid, loaded with a constant distributed normal load

From equation (1.28) it follows that:

Displacements at the surface will be obtained from the first equation of equation (1.30). For a surface point in the loaded region –a<x<a:

This results in the following expressions for the displacement of a point at the surface in x-direction in the loaded region, if it is assumed that the origin does not displace in lateral direction:

The second equation of equation (1.34) gives:

( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )xqE

dssx

spEx

dssx

sqE

xpEx

a

b

z

a

b

x

ννπ

νδ

πνννδ

+−+

−−

−=∂

∂

−−

−+−

−=∂

∂

∫

∫

−

−

12112

12121

2

2

(1.30)

( ) ( )( )

( ) ( )( )

( )

axz

p

p

p

xz

z

x

m=

−=

−−−−=

−+−−=

2,1

21

2121

2121

tan

2cos2cos2

2sin2sin22

2sin2sin22

θ

θθπ

τ

θθθθπ

σ

θθθθπ

σ

(1.31)

( )( ) pEx

x ννδ +−−=

∂∂ 121 (1.32)

( )( ) pxEx

ννδ

+−−=

121 (1.33)

x

a a

θ1 θ2

z

p


18

This integral is singular at s=x, so the integration has to be done in two separate parts. This can be done using a trick with a very small ε:

The result is known as the Cauchy Principal Value of the integral. So it can be derived that the displacement in z-direction and the displacement in x-direction for a point at the surface inside the loaded region is given by:

And for a point outside the loaded region, so |x|>a:

Calculation of δz outside the loaded region gives no further difficulties because of singularities. The deformations δz of equations (1.33) (1.36) and (1.37) are plotted in the following figure. In the calculations shown, The constant C is chosen in such a way that δz is zero at a distance 3a from the center of the contact.

-3.0E-07

-2.0E-07

-1.0E-07

0.0E+00-4.0E+00 -2.0E+00 0.0E+00 2.0E+00 4.0E+00

x/a [-]

δ(x)

[m]

p(x)

C=δz(-3a)

Figure 1.8: The surface deformation δz, caclulated for the situation of a semi-infinite solid loaded

with a uniform distributed load Stresses and deformations for the case of a uniformly distributed tangential load will be discussed in the next section.

( ) ( )∫− −

−−=

∂∂ a

a

z dssx

spEx πνδ 212 (1.34)

( )[ ] ( )[ ]

( ) ( )xaxa

xssxdsxs

dssx

dssx

ax

xa

a

x

x

a

a

a

−−+=

=−−−=−

−−

=− +

−−

+

−

−−∫∫∫

lnln

lnln111ε

ε

ε

ε

(1.35)

( )( )

( ) ( ) ( ) Ca

xaxaa

xaxapE

pxE

z

x

+

−

−+

+

+−

−=

+−−=

222

lnln1

121

πν

δ

ννδ

(1.36)

( )( )

( )( )

( ) ( ) ( ) Ca

xaxaa

xaxapE

paE

axpaE

z

x

+

−

−+

+

+−

−=

>+−

−

−<+−

=

222

lnln1

a xif 121

if 121

πν

δ

νν

νν

δ

(1.37)


19

1.3.6 Uniform distributed tangential load Now, a uniformly distributed tangential traction is acting over –a<x<a. If p(x)=0 in equation (1.28), then:

The surface displacements in the case of a uniformly distributed tangential load are related to the surface displacements in the case of a uniformly distributed normal load in the following way:

It can be shown that σz shows stress singularities at the edges, so at x=–a and x=a. In reciprocating movement this will cause high stress variations at the edges and will play a role in failure by surface fatigue, as [2] points out.

1.3.7 *The inverse route Up to now, we have discussed the case of a given pressure or traction and calculation of surface deformations from these pressures. Practical cases often involve the calculations of pressures and tangential loads from given deformations. Besides describing pressures or deformations, also so called ‘mixed boundary value problems’ are possible, were combinations of pressures and deformations are given. A starting point for solving pressures are the equation (1.30) which are shown here again:

The integrals are singular at s=x, which complicate things more as we have discussed before. One important result is the following. If q(x)=0, then:

Hooke’s law for plane strain gives the following expression for εx:

( ) ( )( )zxx Eσννσνε +−−= 111 2 (1.42)

Because σz=-p(x) at the surface, this gives: ( )xpzx −== σσ (1.43)

So, whatever the distributions p(x) and q(x), tangential and normal stresses at the surface in the material are compressive and equal. As Johnson [3] points out, this will restrict a surface to yield plastically under normal contact pressure.

( )

( )

( ) ( )( )

( ) 222,1

2121

21

212

1

2sin2sin22

2cos2cos2

2cos2cosln42

zaxr

q

q

rrq

xz

z

x

+=

−+−−=

−=

−−

=

m

θθθθπ

τ

θθπ

σ

θθπ

σ

(1.38)

pxqz

pzqx

δδ

δδ

−=

= (1.39)

( )( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )xqE

dssx

spEx

dssx

sqE

xpEx

a

b

z

a

b

x

ννπ

νδ

πννννδ

+−+

−−

−=∂

∂

−+−

−+−

−=∂

∂

∫

∫

−

−

12112

121121

2 (1.40)

( )( ) ( )xpEx

xx

ννδε

+−−=

∂∂

=121 (1.41)


20

Equations (1.40) can be rewritten as:

If displacements δz(x) and δx(x) are given as boundary conditions, solution (e.q. calculation of p(x) and q(x)), involves the solution of two coupled singular integral equations. If δz(x) and p(x) are given as boundary conditions, or δx(x) and q(x) in the case of ‘mixed boundary value problems’, these integral equations become uncoupled. It can be shown that then the equations get the following shape:

With g(x) known and is either the known component of tangential and normal load or the known component of the displacement gradient. In this equation, F(x) is the unknown function. Such equations are called singular integral equations of the first kind. It can be shown that the general solution is of the following form, with the origin taken at the center of the loaded region:

The constant C comes from the total normal or tangential load:

In order to solve these equations, the Cauchy Principal Value is needed because of the singularity at x=s:

Principal values of a number of common integrals have been tabulated, see [3]. Solution of the inverse case is outside the scope of this handout. However, o ne practical special case will be discussed in the following: Example Suppose a stamp which is pressed into another (flat) piece of material. This shape has the following form:

Then normal displacements at the surface are:

This means that:

If the stamp is frictionless, q(x)=0. If we substitute this in equation (1.44)b, then:

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( ) x

Exqdssx

sp

xExpds

sxsq

za

b

xa

b

∂∂

−−

−−

=−

∂∂

−−

−−

−=−

∫

∫

−

−

δν

πννπ

δν

πννπ

2

2

121221

121221

(1.44)

( ) ( )xgdssx

sFa

b

=−∫

−

(1.45)

( )( )

( ) ( )222

22

222

1xa

Cdssx

sgsa

xaxF

a

a −+

−−

−= ∫

− ππ (1.46)

( )∫−

=a

a

dxxFC π (1.47)

( ) ( ) ( )∫∫∫+

−

−− −+

−=

−

a

x

x

b

a

b

dssx

sfdssx

sfdssx

sf

ε

ε

(1.48)

1+= nBxz (1.49)

( ) ( ) 10 +−= nzz Bxx δδ (1.50)

( ) ( ) nz Bxxdx

xd1+−=

δ (1.51)

( ) ( )( ) ( ) ( )

( )( )( ) n

a

b

za

b

BxnEdssx

spx

Exqdssx

sp 1121212

2122 +

−=

−⇒

∂∂

−−

−−

=− ∫∫

−− νπδ

νπ

ννπ (1.52)


21

This equation is of the general form of equation (1.45).

1.3.8 A rigid punch The simplest case is where pressures have to be calculated from given deformations within the loaded region is the case of a frictionless rigid punch:

Figure 1.9: The situation of a semi-infinite solid, loaded with a rigid puntch with load FN

In this case, the boundary conditions within the loaded area are given by:

Then, referring to the last paragraph:

So this is again a general equation of the type as already discussed in (1.45) and shown here again below. Only in this case g(x)=0.

The general solution to this problem was given by equation (1.46), shown again here:

Now g(x)=0, so F(x) is simply given by:

The constant C could be determined from the following equation, see equation (1.47):

So, it can be shown that the following pressure distribution satisfies the equation:

( ) 0==

xqz δδ

(1.53)

( ) ( )( ) ( ) ( )

( ) 01212

212 =

−⇒

∂∂

−−

−−

=− ∫∫

−−

a

b

za

b

dssx

spx

Exqdssx

sp δν

πννπ (1.54)

( ) ( )xgdssx

sFa

b

=−∫

−

(1.55)

( )( )

( ) ( )222

22

222

1xa

Cdssx

sgsa

xaxF

a

a −+

−−

−= ∫

− ππ (1.56)

( )222 xa

CxF−

=π

(1.57)

( ) N

a

a

FCdxxFC ππ =⇒= ∫−

(1.58)

( )22 xa

Fxp N

−=

π (1.59)

FN

a a

δz

x

z


22

Pressure distribution rigid punch

0.0E+002.0E+074.0E+076.0E+078.0E+071.0E+081.2E+081.4E+081.6E+081.8E+08

-1.5E+00 -1.0E+00 -5.0E-01 0.0E+00 5.0E-01 1.0E+00 1.5E+00

x/a [-]

p(r)

Figure 1.10: Pressure distribution for a semi-infinite solid loaded with a rigid punch and normal

load FN Plotting this pressure distribution shows that large pressure peaks with infinite stress appear at the edge of the punch, see the figure above. So far we have discussed line loading of an elastic half space with the example of a uniform pressure and the example of a frictionless rigid punch. In the next section we will consider point loading of an elastic halfspace.

1.4 *3D loading of an elastic halfspace In a manner similar to the case of plain strain line loading, pressure distributions, strains and deformations can be found for the case of loading on an 3D elastic halfspace. This also means that the expressions for Hooke’s law are more complicated. Because the derivation of the point loading case is similar to the line loading case, this section will be brief about the theory and only results will be presented Also for the 3D case analytical solutions are known for some simple cases. Examples with a known pressure or load are:

• A concentrated normal force FN • A concentrated tangential load FT • Constantly distributed normal pressure p on a circular region, square region, elliptical

region. • Constantly distributed tangential traction q on a circular region, square region,

elliptical region. • A very important case are the Herzian equations

The list above is not complete, but gives some relatively simple cases. Also expressions are also known for, for example, torsional loading and layered solids. The case of a concentrated normal force FN will be discussed in section 1.4.1, the case of a concentrated tangential force FT in section 1.4.2 and the case of a uniformly loaded square patch in section 1.4.4

1.4.1 Concentrated normal force FN The stress state is rotation symmetric in this case, In this 3D situation all six stress components will play a role.


23

Figure 1.11: A concentrated normal force FN on a 3D surface

In a similar way as in the case of a concentrated load in line loading, results for point forces can be superimposed in order to get results for distributed loads. Only some important results will be presented because of their importance for numerical approaches towards contact problems. People who have studied this loading case are Boussinesq (1885), Ceruti (1882) and Love (1952). The probem is commonly called the Boussinesq problem. Here, results will be given for a concentrated normal load FN on an elastic semi-infinite halfspace. It can be shown that for this loading case the stress components are given by: with

222

222

zyx

yxr

++=

+=

ρ (1.60)

These results can be used to calculate the effect of a distributed normal load by superposition of the stresses. Displacements are given by:

( ) ( )

( ) ( )( )

−+=

+

−−=

+

−−=

ρν

ρπδ

ρρν

ρπδ

ρρν

ρπδ

124

214

214

3

2

3

3

zG

F

zyyz

GF

zxxz

GF

Nz

Ny

Nx

(1.61)

In polar coordinates, stresses are given by:

( )

5

2

5

3

322

5

2

22

232

3

1212

31212

ρπτ

ρπσ

ρρν

πσ

ρρν

πσ

θ

rzF

zF

zrz

rF

zrrz

rF

Nrz

Nz

N

Nr

−=

−=

−−−−=

−

−−=

(1.62)

With displacements equal to:

( )

( )

−+=

−−−=

ρν

ρπδ

ρρ

νρπ

δ

124

214

3

2

3

zG

F

rzrz

GF

Nz

Nr

(1.63)

FN

FT


24

1.4.2 Concentrated tangential force FT Similarly, the case will be given of a concentrated tangential load FT on an elastic semi-infinite halfspace. This problem is called the Cerutti problem. It can be shown that for this loading case the stress components are given by:

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( )

5

2

5

32

2

23

2

25

2

5

2

32

2

23

2

235

2

32

3

23

3

235

3

32

32

22132

32

22132

232132

ρπτ

ρπτ

ρρρρρρν

ρπτ

ρπσ

ρρρρρρρν

ρπσ

ρρρρρρρν

ρπσ

zxF

xyzFz

yxz

yxz

yyxF

xzFz

xyz

xyz

xxxyF

zx

zx

zxxxF

Tzx

Tyz

Txy

Tz

Ty

Tx

−=

−=

++

++

+−

−+−=

−=

++

++

+−

−+−=

++

++

+−

−+−=

(1.64)

with

222

222

zyx

yxr

++=

+=

ρ (1.65)

Stresses on the surface can be found by z=0 and ρ=r. These results can also be used to calculate the effect of a distributed normal load by superposition of the stresses. Displacements are given by:

( ) ( ) ( )

( )( )

( ) ( )

+

−+=

+−−=

+−

+−−+=

zxxz

GF

zxyxy

GF

zx

zx

GF

Nz

Ny

Nx

ρρν

ρπδ

ρρν

ρπδ

ρρρν

ρρπδ

214

214

12114

3

23

2

2

3

2

(1.66)

1.4.3 Distributive normal load For distributed loads, the results can be obtained by superposition of the results of the Boussinesq problem. If q(x,y) is the distributed normal load per unit area, then the contribution of q(x’y’)dx’dy’ at (x’,y’,0) is equal to, [3]:

( ) ( )

2222

2

2

)0()'()'('

'012

4'')''(

−+−+−=

−+

−=

zyyxx

zG

dydxyxqd z

ρ

ρρν

πδ

(1.67)

Which results in:

( )

+−= ∫ ∫ 3

2

''')''(

''')''(12

41

ρρν

πδ

dydxyxqzdydxyxqGz (1.68)


25

Similarly, the other displacement components and stresses can be obtained. For exmple, σzz is given by:

∫= 5

3

''')''(

23

ρπσ

dydxyxqzzz (1.69)

For general expressions, there are no analytical solutions for this equation. However, some analytical solutions exist, like a uniformly loaded spherical patch, a uniformly loaded ring and a uniformly loaded square patch, see [2], [3]. The case of a uniformly loaded square patch wil be discussed below.

1.4.4 Uniformly loaded square patch Of particular interest for numerical Boundary Element Methods (BEM) is the case of a uniformly loaded patch. This approach is also of interest for the contact between rough surfaces, where each patch represents a pixel of a measured rough surface.

Figure 1.12: A uniformly loaded square patch

For this case, the following expressions can be found for the displacements δz in the body caused by a uniform pressure p on surface patch with dimensions a∙b, as given by Love (1929):

Stresses are given on the surface, at the center of the rectangle given by:

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )22

22

22

22

22

22

22

22

2

ln

ln

ln

ln1

axbyax

axbyaxby

axbyby

axbybyax

axbyax

axbyaxby

axbyby

axbybyax

pE z

++−++

−+−+−−+

+−++++

−+−+−−+

+−+++−

+++++++

+++−+−

++++++=

−δ

νπ

(1.70)

( )

( )

pbap

abp

z

y

x

−=

−

+−=

−

+−=

−

−

σ

νπ

νσ

νπ

νσ

1

1

tan2122

tan2122

(1.71)

p


26

An arbitrary pressure distribution can be approximated by a superposition a uniformly loaded patches. Because of the superposition principle in elastic deformation, deformations caused by these patches can be superimposed on each other as well.

1.4.5 Numerical contact models In the general case the following equations need to be solved in a numerical contact model: So the total pressures needs to be equal to the applied load.

From equation (1.72) the deformation of each contact spot (x,y) caused by an uniform pressure p is known. By putting all deformations in a column vector [δ] and all local contact pressures in vector [p], an influence matrix [C] can be constructed using equation (1.77). With the following boundary conditions:

So the load carried by all patches together should be equal to the applied load FN. Besides this, tensile stresses are not allowed at the surface:

So, contact pressures should always be larger than zero and if there is no contact, the local contact pressure is zero. Much research has been done to solve the matrix equation of equation (1.72) fast and efficiently for different contact situations. This will not be discussed further in this handout.

1.5 Hertzian contacts In the case of elastically deforming rotation symmetric bodies, small strains and isotropic materials, linear elasticity theory is applicable and the contact can be solved using so-called Hertzian theory.

1.5.1 Contact types Depending on the geometry of the contacting bodies, the shape on the contact area, formed by elastic deformation of at least one of the bodies, has the shape of a circle, a line or an ellipse. These contacts are called respectively a point-, line- or elliptical contact. Point contacts are formed by a sphere contacting another sphere or a plane:

Figure 1.13: Point contacts

[ ] [ ][ ]pC=δ (1.72)

N

n

ii Fp =∑

=1 (1.73)

)0,max( ii pp = (1.74)

Two spheres

2r

Sphere on plane

Contact area


27

Line contacts are formed by a cylinder (a ‘sphere’ with an infinite radius in one direction) and a plane or another cylinder:

Figure 1.14: Line contacts

Finally, elliptical contacts are formed by the following contact situation:

Figure 1.15: Elliptical contacts

In the following, pressure distributions and deformations will be derived for the contacts shown above. These contacts will be called in this handout ‘Hertzian contacts’, after Hertz who found a pressure distribution for this type of contacts [1]. First, the general contact geometry of the contact between two undeformable (rigid) bodies will be described in section 1.5.2. Then, deformation will be taken into account in section 1.5.3.

1.5.2 *General contact between two undeformable bodies A general profile of a certain solid body 1 can be expressed in the following general way:

...12

12

11 +++= xyCyBxAz (1.75) The orientation of the x- and y- axes can be chosen in such a way that the cross term xy vanishes. Then, this expression may also be written as:

21

1''

21

1'1 2

12

1 yR

xR

z += (1.76)

In which R1’ and R1

’’ are the minimum and maximum radii of curvature of all possible directions of the profile. Obviously, this is also the case for the other body in the contact, body nr. 2. The directions are only negative because of the chosen coordinate system:

Cylinder on plane

Two cilinders

2

Contact area

1

Contact area


28

+−= 2

22

''2

22

'2 21

21 y

Rx

Rz (1.77)

The separation between surfaces is from geometry given by the difference in heights z1 and z2. This separation d can also be written in a similar way as z1 and z2 by:

CxyByAxdzzd ++=⇒−= 2221 (1.78)

We would like to express d in a similar way as z1 and z2 as: 2

''2

'22

21

21 y

Rx

RByAxd +⇒+= (1.79)

The set of axes of z1 and z2 are different because the radii R’ and R’’ are in the minimum and maximum curvature directions, which may be present in different directions for each body. Therefore a transformation of the axes is required before substitution.

Figure 1.16: Coordinate transformation for elliptical contacts

It can be shown that substitution of z1 and z2 and transformation of the axes gives for the coefficient of the crossterm C:

αβ 2sin11212sin11

21

''1

'1

''2

'2

−−

−=

RRRRC (1.80)

We want the cross term, and therefore C, to be zero. In order C to be zero, it can be shown that A and B have to satisfy the following two equations:

+++=

+=+ ''

2'

2''

1'

1'''

11112111

21

RRRRRRBA (1.81)

21

2cos11112111121

''2

'2

''1

'1

2

''2

'2

2

''1

'1

−

−+

−+

−=− α

RRRRRRRRAB

(1.82)

These equations can used to determined A and B and therefore R’ and R’’ which are a function of R1

’ ,R1’’ and R2

’ ,R2’’ . R’ and R’’ are called the principal relative radii of curvature of the

two contacting bodies. These equations to determine R’ and R’’ are not easy, but of most interest are some special cases as are shown in the following table: Case R1

’ R1’’ R2

’ R2’’ α A=1/ R’ B=1/R’’

Two axisymmetric bodies

R1 R1 R2 R2 -

+=

21

1121

RRA

+=

21

1121

RRB

Cylinders wth axes parallel

R1 ∞ R2 ∞ 0

+=

21

1121

RRA

0=B

Equal cylinders with axes perpendicular

R ∞ R ∞ 2π

RA 1

21

= R

B 121

=

x1

x2

α


29

So far, we have discussed nothing more than geometry that we put in equations of a special form. Now we will discuss (elastically) deformable surfaces.

1.5.3 *Deformable surfaces Now, a load FN will be applied to the surface. The geometry of the bodies will change from the undeformed geometry as described above to a deformed geometry. Then, from geometry follows:

2121 δδδδ +=++ dzz (1.83) Here, δ1 and δ2 are the macro-displacements of points far away from the contact zone. Using the expressions for the geometry of surface 1 and surface 2 and calling the total macro displacement

21 δδδ += (1.84) It follows that, see the following figure from [3]:

2221 ByAxzz −−=+ δδδ (1.85)

Figure 1.17: The geometry in the case of deformable surfaces

1.5.4 Hertzian equations We had the following general equation for deformable surfaces:

2221 ByAxzz −−=+ δδδ (1.86)

Let us now consider two axisymmetric solids in contact, like spheres. Then, we can define R to be a function of the radii R1 and R2 of body 1 respectively body 2:

RRR111

21

=+ (1.87)

FN

2

1

δ1

δ2

δ

z

z2

δz

δz

δ1

δ2

! S2

S1

a a

S1 and S2 are points on the surface


30

We can also define r as: 222 ryx =+ (1.88)

Then it follows by substitution that when two axi-symmetric solids as described above are brought into contact, the displacement of the surface inside the contact can be written as:

221 2

1 rRzz −=+ δδδ (1.89)

In which R is the reduced radius defined by:

21

111RRR

+= (1.90)

Now we have a contact problem with prescribed displacements. A pressure distribution which causes such displacements has to be found in the way shown in paragraph 1.3.7 and this is general not simple. In the paragraph mentioned the calculation of a pressure distribution from deformations was called the ‘inverse route’. Hertz [1] proposed in 1881 the following elliptical pressure distribution which satisfies the deformations of equation (1.89), so for deformable axisymmetric bodies:

2

max 1

−=

arpp (1.91)

This pressure distribution is elliptical as is shown in the following figure:

Hertzian pressure distribution

-1.0E+00-8.0E-01-6.0E-01-4.0E-01-2.0E-010.0E+002.0E-014.0E-016.0E-018.0E-01

1.0E+00

-1.5E+00 -1.0E+00 -5.0E-01 0.0E+00 5.0E-01 1.0E+00 1.5E+00

r/a [-]

p(r)/

pmax

Figure 1.18: The elliptical Hertzian pressure distribution

That elliptical pressure distribution given by this pressure distribution shown in equation (1.91) satisfied displacements of equation (1.89) is more easy to show than solving the inverse problem itself. Application of Hooke’s law and integration gives that this pressure distribution according to equation (1.91) causes normal displacements according to

( ) ( ) arraa

pE

rz ≤−−

= with 24

1 22max2 πν

δ (1.92)

This equation has the shape of equation (1.89) as it should indeed be. Due to action = -reaction, the pressure acting on body nr.1 is equal to the pressure acting on body nr. 2. If we write E* to be the following and call it the reduced modulus of elasticity E* defined by:


31

2

22

1

21

*

111EEEνν −

+−

= (1.93)

Then substitution of equations (1.89) and (1.93) in equation (1.92) gives the following expression:

( ) 222*

max

212

4r

Rra

aEp

−=− δπ

(1.94)

This gives a radius of contact a of

RE

pa *max

2π

= (1.95)

And an approach of (distant) points of the two contacting solids δ by:

*max

2Eapπ

δ = (1.96)

The total load acting in the contact area is then given by:

( ) 2max

0 322 aprdrrpF

a

N ππ == ∫ (1.97)

This means that:

max32 pp = (1.98)

1.5.5 Point contacts An important case is the case of a sphere with radius R contacting a smooth countersurface. The contact area for a sphere with radius R in contact with a flat countersurface is given by:

δππ RaA == 2 (1.99) Then, the load FN carried by the contact is given by:

23*

34

δREFN = (1.100)

Or vice versa, the contact radius a and the value of δ are given by the following expressions, given a certain load FN:

31

2

2

2 E'R16F9δ

⋅⋅

⋅==

Ra (1.101)

And 31

E'4RF3a

⋅⋅⋅

= (1.102)

The maximum contact pressure pmax is given by: 31

23

2

max RπE'F6p

⋅⋅⋅

= (1.103)

And the mean contact pressure:

max2 32 p

aFp N ==π

(1.104)

The pressures acting on the surface will cause subsurface stresses in both contact materials. An aspect of a contact between a sphere and a flat is that the shear stress occurs at a certain depth below the surface, Expressions for subsurface stresses in a sliding spherical contact, so a sphere contacting a plane and loaded by a normal force FN and a tangential force FT, are given in appendix A.


32

1.5.6 *Elliptical contacts In the above paragraph, where two axisymmetric bodies are in contact, it was clear that the contact area was spherical in shape. In the more general case, the shape of the contact area is not known. We will now assume an elliptical contact area with semi axes a and b. Further analysis will be similar to the spherical case, but will not be discussed in detail. Here, only the general line and the resulting equations will be mentioned. With an elliptical contact area, the pressure distribution will be given with the following expression:

22

max 1

−

−=

by

axpp (1.105)

For elliptical contacts, an equivalent radius R* can be defined as:

yx RRR111

* += (1.106)

With Rx and Ry the minor and the major radius of curvature. This ellipse will be in contact with a smooth countersurface, like in the previous section. Now, the Hertzian contact area is given by:

δγ

ααπ yxRA *2= (1.107)

In this equation, αx and αy are the dimensionless radii of the contact area. The parameter γ is the dimensionless approach between the two bodies. Definitions of αx, αy and g are given by:

( )

( )

( ) ( )

y

x

y

x

αα

κ

κπ

κπ

κγ

κπ

κα

κπ

κα

=

−

−=

−=

−=

−

−

22

2

2

1212

12

12

31

32

31

32

31

31

KE

E

E

(1.108)

In these equations, K and E are the complete elliptic integrals of the first and second kind and κ is the ellipticity ratio of the contact:

( ) ( )

( ) ( )∫

∫

ΨΨ−=

ΨΨ−=−

π

π

2

0

2

2

0

2

21

21

sin1

sin1

dmm

dmm

E

K (1.109)

Evaluation of these elliptic integrals is not straightforward. Moes has given an approximate relation between the ellipticity k and the gap curvature ratio λ by:

y

x

RR

=

+−+=

−

−

λ

λλλ

κ1

16.0ln4ln16ln2

1 (1.110)

Finally, The load carried by the elliptic elastic contact is given by:


33

** 234 2

3

REN

=

γδ (1.111)

1.5.7 Line contacts Another important case is the case of a sphere with radius R contacting a smooth countersurface. Such a contact could, for example, occur in the case of gears. The equations below will be expressed in terms of load per unit length of the line contact:

lFw N= (1.112)

Then, the half-width of the line contact b is equal to

*

4EwRb

π= (1.113)

And the maximum contact pressure pmax is given by:

RwE

bwp

ππ'2

max == (1.114)

The mean contact pressure is given by:

max42p

bwp π

== (1.115)

1.6 Reduced modulus of elasticity and reduced contact radius

1.6.1 Reduced modulus of elasticity In the above discussion the following definition for the reduced modulus of elasticity was used:

2

22

1

21

*

111EEEνν −

+−

= (1.116)

The reader should be aware of the fact that sometimes another definition for the reduced modulus of elasticity is used:

2

22

1

21

*

112EEEνν −

+−

= (1.117)

This definition is not used in this handout.

1.6.2 Reduced contact radius It was mentioned that the radii of two spheres in contact, respectively with radius R1 and R2, can be combined into the reduced radius R:

21

111RRR

+= (1.118)

For elliptical contacts, the following definition for the equivalent radius R* was used:

yx RRR111

* += (1.119)

With Rx and Ry the minor and the major radius of curvature.In the case two that two elliptical bodies are in contact, in the same way an equivalent radius R* can de found as was done for two contacting spheres:


34

22111*

1**

1111111

yxyx RRRRRRR+++=+= (1.120)

If equations are given for a sphere or ellipse contacting a flat, as was done in sections 1.5.4 and 1.5.6, then in fact the equivalent radius R* is half of the radius of the sphere:

spherespherespheresphere

RRRRRR 2

12111 ** =⇒=+= (1.121)

To avoid this problem, sometimes another definition for the equivalent radius R* is used:

spherespherespheresphere

RRRRRR

=⇒=+= **

2112 (1.122)

This definition is not used in this handout.

1.7 Excersises Exercise 1 (in Dutch) De opdracht is om de Hertze vergelijkingen voor een lijncontact te ‘controleren’met behulp van de elastische contacttheorie, zoals in het eerste college is behandeld. De resultaten van de formules voor een lijncontact worden hiertoe vergeleken met de resultaten van een numerieke, stuksgewijs constante benadering van het probleem, volgens de ‘algemene’ methode zoals behandeld op het college. De locale vervormingen worden in het vervolg uitgedrukt als een vector [δ] en de locale contactdrukken als een vector [p]. In het geval van Hertze vervormingen hebben de vervormingen de volgende vorm:

( ) ( )0'2

2

δδ +−=R

rr

Hierin is R’ de gereduceerde straal. Het verband tussen de locale drukken en de locale vervormingen is uit te drukken met de volgende matrixvergelijking: { } [ ]{ }pK=δ In deze vergelijking is [K] de flexibiliteitsmatrix. We willen nu de locale contactdrukken uitrekenen door dit lineaire stelsel op te lossen, gegeven [δ]. Reken de flexibiliteitsmatrix uit voor het geval van een lijncontact. Maak gebruik van de vergelijkingen voor een constante druk over een gebied 2a, zie vergelijking 2.36 van de handout. Stel de constante C in deze vergelijking (voorlopig) op nul. Een mogelijke oplossingsroute voor het uitrekenen van de contactdrukken {p} uit de gegeven

deformatie {δ} is hieronder weergegeven. De verplaatsingen t.g.v. de drukken (zoals blijkt uit [K] van de vorige som) zijn op een constante na bepaald. Omgekeerd betekent dit dat de totale uit te rekenen druk een verplaatsing veroorzaakt welke opgebouwd is uit twee componenten: Een parabolische verplaatsing {δ1} en de (onbekende) constante verplaatsing {δ2}. Het probleem kan dus als volgt geschreven kan worden: { } [ ]{ } [ ]{ } [ ]{ }212121 pKpKppK +=+=+ δδ Hierin is {p1} de druk, samenhangend met de parabolische verplaatsing/deformatie {δ1} en {p2} de druk ten gevolge van de constante verplaatsing {δ2}=δ0. Dus geldt: [ ]{ } { }11 δ=pK En: [ ]{ } { } ( ) { }1022 ⋅== δδpK De drukken {p1} zijn uit te rekenen door het stelsel [ ]{ } { }11 δ=pK op te lossen. Oplossen van het tweede stelsel betekent dat:


35

[ ]{ } { } ( ) { } { } ( ) [ ]{ } { } [ ] ( ) ( )

⇒

=⇔=⇔⋅==00

10

1110 22222 δδδ

δδppKpKpK

Dus door dit stelsel op te lossen kunnen we de drukvector {p2}/C uitrekenen.

Op de rand van het contact geldt de randvoorwaarde dat ter plekke de contactdruk nul moet zijn. Een druk gelijk aan 0 betekent op de rand van het contact: { } { }12 pp −= Dus uit deze randvoorwaarde en de resultaten voor {p1} en {p2}/C geeft ons voldoende informatie om de waarde voor C uit te rekenen. De totale contactdruk is de contactdruk p1 samenhangend met de parabolische verplaatsing/deformatie δ1 en de contactdruk p2 is de druk ten gevolge van de constante verplaatsing δ2. Dus de totale contactdruk is de twee componenten opgeteld: { } { } { }21 ppp += De drukvector {p} is de uiteindelijke druk die we willen weten. De opdracht:

• Kies een waarde voor R en benader deze vervormingen stuksgewijs lineair op een zelf gekozen interval 2a. Dus eigenlijk: Bepaal vector [δ] door bovenstaande vergelijking te discretiseren met intervallen 2a. Kies een vector van minimaal 200 punten.

• Reken de flexibiliteitsmatrix uit voor het geval van een lijncontact. Maak gebruik van de vergelijkingen voor een constante druk over een gebied 2a, zie vergelijking 2.36 van de handout.

• Controleer de resultaten met de analytische Hertze vergelijkingen voor een lijncontact.

• Superponeer een sinus met een kleine amplitude en frequentie op de deformatievector [δ] om het effect van ruwheid te simuleren. Gebruik de ontwikkelde methode om deze situatie op te lossen. Een indicatie van de waarden: Kies in de volgende uitdrukking ongeveer A≈0.01 de ‘macroverplaatsing’ en ω≈0.05 maal de contactbreedte. Wat gebeurt er met de contactdrukken?

( ) ( ) ( )0sin'2

2

δωδ ++−= rAR

rr

Tips:

• Maak gebruik van de ‘\’-operator in Matlab. Dit is een snelle manier om een stelsel Ax=b op te lossen met [A] en {b} bekend.

1.8 Summary In this chapter it is discussed how the contact between two smooth surfaces can be described. Attention is paid to line loading of an elastic halfspace, point loading of an elastic halfspace

and Hertzian contacts.

1.9 References [1] Hertz, H., 1882, Über die Berühring fester elastischer Körper, Journal für die Reine und

Angewandte Mathematik, v. 92, p. 156-171. [2] Johnson, K,L., 1985, Contact Mechanics, Cambridge University Press, Cambridge,

United Kingdom.


36

[3] Ling, F.F, Lai, W.M., Lucca, D.A., 2002, Fundamentals of surface mechanics with applications, Mechanical Engineering Series, Springer – Verlag, New York USA


37

Chapter 2 Contact mechanics: Contact between rough surfaces

2.1 Introduction In the previous chapters the contact of smooth surfaces was discussed. However, as also has been discussed, in reality surfaces are not smooth. There are only few possibilities to obtain very smooth surfaces. An example is mica, which can be cleaved to obtain atomically smooth planes. Surfaces which are initially rough, but are very elastic can also result in a continuous contact area, like in the case of smooth surfaces contacting. An example is soft rubber, where small asperities at the surface are squeezes if a high enough load is applied on the surface A consequence of contacting non-smooth surfaces is that the contact area is not continuous, will be take place on (many) discrete spots. Therefore the real area of contact is much smaller than the apparent area of contact in the case of rough contacting surfaces and is composed of many microcontacts. Another factor that causes contact to occur on many discrete points in realistic cases is the fact that it is very difficult to flatten initially rough surfaces by plastic deformation. The reason for this is that interacting stress fields below the asperities make the asperities more difficult to deform than the bulk material. In the next section, the methods used in chapter 2 will be applied to simplest model for a rough surface possible: a wavy surface with relatively small amplitude compared to the wavelengths. The consequence of this is that only normal displacements can be taken into account. This means that for this case the method developed in chapter 2 are still applicable to rough surfaces. Details of this analysis can be found in [5].

2.2 Elastic contact and wavy surfaces First, an elastic surface is considered with sinusoidal normal pressure acting on the surface. This means that with a certain periodicity the material at the surface is pushed down and pulled up. Although it is difficult to thick of a practical case where such a load would be acting on a surface, is serves as a good introduction for the following analysis of a wavy surface. Such a sinusoidal contact pressure can be expressed as follows

=

λπxpp 2cosmax (2.1)

Here, pmax is the amplitude of the alternating contact pressure and λ is the wavelength. If a pressure was known, the stresses, strains and displacements could be calculated by using Hooke’s law and integration over the strains to obtain the deformations as was discussed in chapter 2. It was also discussed that although general expression for the displacements can be found from t an arbitrary distributed contact pressure, in many cases the derivatives of the displacements are more practical. They are shown again in the following equations:

( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )xqE

dssx

spEx

dssx

sqE

xpEx

a

b

z

a

b

x

ννπ

νδ

πνννδ

+−+

−−

−=∂

∂

−−

−+−

−=∂

∂

∫

∫

−

−

12112

12121

2

2

(2.2)


38

The first expression is in fact the tangential strain εx at the surface. The second expression is the x-derivative of the normal strain εz at the surface and is in fact the slope of the deformed surface after loading. If tangential traction is not present and the expression for p is substituted in equation (1.30), then the following equation is obtained after substitution of x-s=ξ:

Integration gives:

So this gives for the displacement δz:

It is obvious that the wavelength of the deformation is the same as the wavelength of the original surface. As [5] points out, subsurface stresses can be found from the stress function

Stresses can be calculated from this stress function by taking derivatives, see chapter 2

zx

x

z

xz

z

x

∂∂∂

−=

∂∂

=

∂∂

=

φτ

φσ

φσ

2

2

2

2

2

(2.7)

It can be calculated that the maximum principal shear stress τ1 occurs at depth z=λ/2π and has a value of pmax/e. Now let us consider a case where displacements are prescribed. An example is a wavy surface in contact with a flat countersurface, as shown in the following figure which is taken from [5]. Figure (a) shows the initial contact situation.

( ) ( ) ( )

( )( )

∫

∫∫

∞

∞−

∞

∞−

∞

∞−

−

−−

=−

−−=

−−

−=∂

∂

ds

xp

E

dssx

sp

Eds

sxsp

Exz

ξλ

ξπ

πν

λπ

πν

πνδ

2cos12

2cos1212

max2

max22

(2.3)

( )( )

( )

−

−=∂

∂

⇒

−

−−=

∂∂

∫∞

∞−

λπνδ

ξξ

λξπ

πνδ

xpEx

d

xp

Ex

z

z

2sin12

2cos12

max

2

max2

(2.4)

( ) CxpEz +

−

=λπ

πλν

δ2cos1

max

2

(2.5)

( ) ( ) ( ) ( )

λπ

α

αααα

φ

2

cosexp1, 2max

=

−+= xzzpzx (2.6)


39

Figure 2.1: Initial, partial and full contact between a sinusoidal surface and a flat

In this case the geometry of the wavy surface is given by:

( )

=

λπxAxz 2

cos (2.8)

So, the wave has (small) amplitude A and wavelength λ.Then the (undeformed) gap width is simply given by:

( )

−=

λπxAxh 2cos1 (2.9)

In this equation A is the amplitude of the wavy surface and λ is again the wavelength. If the surface is deformed in such a way that everywhere there is contact, so at relatively ‘heavy’ loads, then the displacements at the surface are periodic. So, the surface is completely flattened. Periodic displacements were also found from a periodic pressure distribution in equation (2.5). This means that in this case (so for ‘heavy’ loads) the pressure distribution is given by:

λπ

λπ

EAp

xppp

=

+=

max

max2cos

(2.10)

In order to have continuous contact, so everywhere a positive value for the contact pressure, the average contact pressure has to be higher than the amplitude of the periodic component pmax. This means that the minimum average contact pressure p for the above analysis is:


40

λπEApp =≥ max (2.11)

It is clear that the fraction A/λ is important. If A/λ is high, a high average contact pressure is required to flatten the surface. So, at a fixed amplitude A, smaller wavelengths λ will be more difficult to flatten (so require a higher nominal contact pressure) than longer wavelengths. This is also found in numerical simulations of elastic contact conditions where small roughness details are still present at high loads. Besides this, a smaller elastic modulus will make the flattening process easier than a stiff surface. At lower contact pressures than those according to equation (2.11) ,the contact is not continuous. So, no complete flattening will occur if:

λπEApp =< max (2.12)

In this case, the contact pressure is given by the following equation. This will not be further discussed [5]

( )

≤≤

≤≤

−

=

2 if 0

0 if sinsinsin

cos222

2

λ

λπ

λπ

λπ

λπ

xa

axxaa

xp

xp (2.13)

In this expression 2a is the width of a contact area. The contact situation is shown figure (c) of the figure shown above. Such a contact pressure results in the following displacements:

( ) ( )

( ) ( )

λπ

λπ

λ

ψψ

ψψ

πλν

δ

ψπ

λνδ

x

a

xa

CE

px

ax

CE

px

a

a

aa

a

az

az

=Ψ

=Ψ

≤≤

+

Ψ−ΨΨ

−Ψ−Ψ+

Ψ−

=

≤≤

+Ψ

−=

2

sinsinsinsin

lnsin2

sinsinsin22cos

sin1

0

,2cossin

1

222

22

2

2

2

2

(2.14)

The integration constant C is determined from a point where the deformation is chosen to be zero, see chapter 2. Then there will still be positive gap outside the contact area, so partial contact. The fraction of area in contact can be calculated by inverting this relation:


41

max

1sin22p

pa −=πλ

(2.15)

Asperity interaction

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

normalized pressure [-]

frac

tion

of a

rea

in c

onta

ct

[-]

aaHertz

Figure 2.2: Fraction of area in contact versus normalized pressures for Hertizan contacts and

sinusoidal contacts This relation is plotted in the next figure and compared with the case where asperities are modeled as individual Hertzian contacts. The Hertzian equation is given by the blue line and the ‘complete’ equation is given by the red line. It is clear that the effect of interaction between the asperities results in the phenomenon that the total contact area is higher than in the case of individual Hertzian contact. The reason for this is that pressures acting on a microcontact will also press neighboring microcontacts ‘down’, which results in a higher real contact area. If a high enough contact pressure is applied, the entire surface can be in contact. Then, the apparent contact area is equal to the real contact area in the case of elastic contacts. This is contrary to the case of plastic material deformation because it is difficult to flatten initially rough surfaces by plastic deformation because at a certain value of a/λ the deformation fields start overlapping and further deformation is then more difficult to perform. An example is shown below, where a surface with triangular peaks is pushed down by a rigid flat. In this figure, α=65o. Deformation in the tangential direction is not permitted in the following discussion.


42

Figure 2.3: Plastic compression of triangular wedges

It can be shown by slipline models that for α=65o deformation fields start overlapping at 2a/λ=0.36. Above that value the pressure required to deform the asperities increases. Then The mean pressure at the punch is given by:

λa

kp

kLF

kp N 2

222≈= (2.16)

At very high values of 2a/λ, material is back-extruded, so material is pushed up if the punch is moving down. At a certain limiting value of 2a/λ=0.81 the punch will move the ‘rough surface’ down as a whole. Thus by pure normal loading of a surface like the one discussed, it is not possible to flatten asperities totally. If deformation in the tangential direction is allowed, then asperities are flattened by small pressures on the punch. This contact situation, so without limitations in the deformation parallel to the surface, may occur in metal forming operations, like deep drawing. If a tangential force is applied on the rough surface, then this makes an increase in 2a/λ, easier. The above discussion was for pure plastic deformation. In the case of work hardening materials, flattening of asperities will be more difficult than in the case of pure plastic material behavior.

2.3 Contact mechanics: Random rough surfaces In chapter Chapter 5 it is described which properties of a rough surface are of interest for rough surface contact. In this section it was also discussed how summits can be determined from 3D roughness data. Besides this, it is also discussed how properties of these summits can be measured from roughness data. In fact, a surface was represented as ‘a lot of spheres’, with in the simplest case heights s varying according to a Gaussian distribution and a constant radius of curvature β:

FN

α λ 2a


43

Figure 2.4:: A rough surface represented as a set of spherically tipped summits Chapter 3 shows how a single asperity behaves from a mechanical point of view. If we know how asperities can be determined and if we know the mechanical behavior of single asperities, we can extent the analysis to the contact between two rough surfaces by interpreting rough surface contact as the contact of many single asperities. This approach is taken here. The easiest way is to discuss the rough surface contact starting from single asperity contact is to do this in a statistical way. In this approach, all roughness and elasticity is projected onto a single sum surface that is in contact with a rigid flat. The sum surface is discussed in section 2.4. A statistical model will need statistical parameters. Determination of statistical parameters from rough surface data is discussed in section 5.7. In the models described in sections 2.6 and 2.7 we further assume that:

1. All asperities have spherical tips and the tip radii of all asperities are equal and have average radius β.

2. A statistical approach is followed. It is therefore assumed that the summit heights vary randomly according to some height distribution. The most commonly assumed distribution of the summit heights is a Gaussian height distribution. The (measured or model) distribution in this section will be called φ(s).

3. The asperities are in normal contact. This means that all roughness and elasticity can be taken into account in a single sum surface, as will be discussed in section 2.4. Misalignment effects between contacting asperities are neglected in this approach.

4. The only deformation of importance is the deformation of the asperity tips. The bulk of the material is deforming neither elastically nor plastically.

5. The asperities are deforming independently of each other. 6. The material properties are independent of depth.

Although we make the assumptions as described above, the effects neglected will be further discussed in these notes:

1. Because we assume spherical tips on the asperities, in principle isotropy is assumed in the following sections. However, the analysis can be easily extended to anisotropy by assuming ellipsoidal asperities, as was also done in chapter Chapter 3. Some expressions taking into account ellipsoidal asperities will be given throughout the sections 2.6 and 2.7.

2. The summit heights vary randomly. The most common distribution of the summit heights is a Gaussian height distribution. The distribution in this section will be called φ(s). Contrary to the statistical approach followed in this section, a deterministic approach will be discussed in section 0.

3. In particular in non-normal contact situations (e.g. sliding, static friction), it is important that misalignment between asperities are taken into account. For normal contact the effect of misalignment is small. It will therefore not be discussed here.

4. In some applications not all deformation takes place in the asperity tips. Elastic bulk deformation may play a prominent role in for example rough line contact. Plastic bulk deformation is of great importance in metal forming situations.


44

5. The asperities are deforming independently of each other. If bulk deformation is discussed, the asperities will not deform independently of each other anymore.

6. In coated systems the material properties are not constant over the depth, but a function of e.g. coating thickness.

7. The surfaces are nominally smooth. In curved surfaces, asperity interaction is much more important. The contact of curved rough surfaces will therefore be discussed in section 0.

In some cases, for example in cases of high plastic bulk deformation, the summit-based approach described in the above mentioned sections fails. Then, deformation of the whole surface has to be taken into account.

2.4 The sum surface The contact between two rough surfaces depends on the shape of the gap between the surfaces. This means that the contact between two nominally flat, but rough, surfaces can be replaced by an equivalent rough surface with an equivalent modulus of elasticity E*, see equation (3.2) , against a smooth rigid surface.

Figure 2.5: Gaussian height distribution for rough surfaces

In the case of two nicely Gaussian surfaces with standard deviations σ1 and σ2 contacting each other, the sum surface will again be a nicely Gaussian surface with an equivalent roughness σ:

22

21 σσσ += (2.17)

In the case of non-Gaussian surfaces, there is simple general relation for the roughness of the sum surface. Similarly, an equivalent value of the average radius of the summits β can be calculated from the average radius of the summits of surface 1 and the average radius of the summits of surface 2, respectively β1 and β2:

21

111βββ

+= (2.18)

This equation originates from the reduced radius of two spheres contacting exactly at the tips. Misalignment effects between summits on the two contacting surfaces are neglected. The distance between the mean of the surface heights of the (sum) surface and the rigid flat is called the separation d. After loading the surface, the asperities will deform under the influence of the carried load. This will result in a decrease in the separation from the initial value.

-

-

-

0 1 2 30 0.0

0.

0.1

0.

0.2

0.

0.3

0.

s / σs [-]


45

2.5 Plasticity index In many applications it is very important to know if a rough surface contact will be predominantly elastic or plastic. The easiest way to determine if a contact is plastic or elastic based on properties of the summits, without actually performing a calculation with a contact model, is by calculating the plasticity index. Although there are other definitions, the plasticity index was originally defined to be:

βσ s

HE *

=Ψ (2.19)

In which E* is the reduced modulus of elasticity, H the lowest hardness of the contact partners, σs the standard deviation of the summit heights and β the summit radius. If Ψ <0.6, elastic deformation dominates and if Ψ >1, a large part of the contact will be the plastic deformation mode.

2.6 Elastic contact For contact situations, not the surface properties but the properties of the asperities will be of importance. In section (5.2) is has been discussed that three summit parameters are of importance, the standard deviation of asperity heights σs, and mean curvature of the tips of these asperities κs and the asperity density ηs. The mean summit height will generally not be zero and will be called s . If a summit is of sufficient height to be deformed by the counter surface it will carry a load. This results in a deformation ds −=δ and a (circular) contact area with radius a.

Figure 2.6: Deformation of individual summits, based on the separation d

This means that this summit will have a contact area

( ) 2afA πδ == (2.20)

And carry a load f(δ) which function of the deformation δ.

( )δgF = (2.21)

In the case of elastic contact, f(δ) and g(δ) are given by the Hertzian equations as presented in paragraph 0. The total number of asperities in contact will be ηsAn. Combining this with the probability of a summit with height s as given by the summit height distribution φ(s) and integrating over the contacting summit heights d to ∞ gives the following relation for the load carried by the contact:

( ) ( ) ( ) ( )∫∫∞∞

−=−=d

nsd

eNnsN dssdsEAdssdsFAF φβηφη 23*

, 34 (2.22)

Similarly, the real contact area is then given by:

ds −=δs

δ

d


46

And the number of elastically deforming asperities by:

This results in the important observation that the contact area is proportional with the carried load. It can be shown that this is the case, independent of the load-deformation relation f(δ) and the contact area-deformation relation g(δ) that is assumed, see 3.8. Another important aspect is that the real contact area Ar and the mean contact pressure

are both proportional to the number of asperities in contact. If the real contact area is proportional with the number of asperities in contact, this means that the average size of a microcontact on top of an asperity will be independent of load. An existing microcontact will obviously grow with increasing load, but the average size will be compensated by the areas of newly formed microcontacts in such a way that the average size stays the same, independent of load. The model discussed above is in fact the Greenwood & Williamson model, see [6]. A fundamental assumption of this way of describing contact is that the individual asperities are deforming, independent of the deformation of neighboring asperities. This assumption will fail at higher nominal contact pressures, when adjacent asperities will join in order to form one larger asperity.

2.7 Plasticity

Fully plastic deformation In the case of elastic-plastic contact, the highest asperities are deforming plastically. Based on the onset of plasticity, a critical indentation depth δ1 can be defined at which plasticity will occur. The equations for a single asperity contact for this case are given in paragraph 0 Then, the load carried by both the elastically deforming asperities and the plastically deforming asperities FN is given by for the fully plastic deformation assumption:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫∫

∫∫

∞

+

+

∞

+

+

−+−=

=−+−=

1

1

23

1

1

234 *

,,

δ

δ

δ

δ

φβηπφβη

φηφη

dsn

d

dns

plastic

dplNns

elastic

d

deNnsN

dssdsHAdssdsAE

dssdsFAdssdsFAF4444 34444 214444 34444 21

(2.26)

And the real area of contact Ar:

( ) ( ) ( ) ( )∫∫∞∞

−=−=d

snd

ens dssdsAdssdsAAA φβηπφη (2.23)

( )∫∞

=d

sn dssAN φη (2.24)

r

N

AFp = (2.25)


47

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫∫

∫∫

∞

+

+

∞

+

+

−+−=

=−+−=

1

1

1

1

2δ

δ

δ

δ

φβηπφβηπ

φηφη

dsn

d

dsn

plastic

dplsn

elastic

d

desnr

dssdsAdssdsA

dssdsAAdssdsAAA4444 34444 214444 34444 21

(2.27)

And the number of contacting asperities N:

( )∫∞

=1

dsn dssAN φη (2.28)

Elasto-plastic deformation In the case of elastoplastic deformation, there are three deformation regimes of interest. The lowest asperities will be deforming elastically (δ<δ1), the ‘middle-high’ asperities elasto-plastically (δ1<δ<δ2) and the highest asperities will be in the fully plastic deformation regime (δ>δ2). In this section the fully plastic contact behavior is assumed, but in a similar way volume conservation can be assumed in this regime. The equations for the elasto-plastic regime are given in paragraph 0, see [7].

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )∫

∫

∫

∫∫∫

∞

+

+

+

+

∞

+

+

+

+

−+

+

−

−−+

−

−−−−+

+−=

=++=

1

2

1

1

23

2

2

1

1

2

321

34

2

12

1

3

12

1

*

,,,

δ

δ

δ

δ

δ

δ

δ

δ

φηβπ

φδδ

δδδ

δβπη

φβη

φηφηφη

dn

d

dns

d

dns

plastic

dplNns

ticelastoplas

d

depNns

elastic

d

deNnsN

dssdsHA

dssdsdsdsA

dssdsAE

dssFAdssFAdssFAF444 3444 21444 3444 21444 3444 21

(2.29)


48

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( )( ) ( )∫

∫

∫

∫∫∫

∞

+

+

+

∞

+

+

+

+

−−+

−

−−+

−

−−−⋅

⋅−

−−

−−

+

+−=

++=

1

1

1

2

2

1

1

1

2

12

1

3

12

1

12

2

2

321

lnlnlnln1

δ

δ

δ

δ

δ

δ

δ

φδηβπ

φδδ

δδδ

δ

δδδδ

βηπ

φβηπ

φηφηφη

dn

d

dsn

d

dsn

plastic

desn

ticelastoplas

d

desn

elastic

d

desnr

dssdsA

dssdsds

dskHH

A

dssdsA

dssAAdssAAdssAAA44 344 2144 344 2144 344 21

(2.30)

Obviously also elasto-plastic deformation for ellipsoidal asperities can be derived in a similar way as the equations for spherical asperities described above. This will not be done here.

Example In this section, the theory above will be illustrated with the rough surface taken as an example is chapter 2. If this grinded surface is analyzed, the following values of the parameters ηs, β and σs are obtained:

ηs 2.71e11 [m-2] β 8.105e-7 [m] σs 4.72e-7 [m]

These values will be used in the contact calculations shown in this paragraph. Besides geometrical parameters, also material properties will be of importance for the contact calculations. Here, the following properties are used:

Ε∗ 1.1538e+011 [Pa] Η 3.0000e+009 [Pa]

Using these values, some results with the described contact models will be shown for spherical asperities, so an isotropic surface. The results for elliptical asperities will be similar and will therefore not be discussed in this paragraph. In figure results are shown for the elastic-plastic contact model. It is clear from the figure that the (normalized) nominal contact pressure is proportional to the fraction of area in contact. Although not shown here; this proportionality will also be found when other deformation modes are used.


49

Figure 2.7: Fraction of area in contact is linear with the normalized contact pressure

The proportionality shown also means that the applied load FN=pAn is proportional to the real contact area Ar=αAn. This last result is very important and is often used to explain Amonton’s law which says that the friction force FW is proportional to the normal force FN with a load independent shear strength at the interface τ acting on the real contact area Ar.

ppAA

FFf

r

r

N

W ττ=== (2.31)

The calculations shown above have been done with a series of separations with constant step sizes between the values. It is clear from the figure that at low values of a the ‘dots’ are close together, which means that α and p do not change much at an decreasing separation d. At lower separations (higher) loads, this effect is much larger. This is also clear from the next figure.

Figure 2.8: The nominal contact pressure stronlgy increases with decreasing separation

In contact between rough surfaces, only the highest asperities will be in contact, The minimum height of the contacting asperities is equal to the separation d. In the following figure the nominal contact pressure is shown as a function of separation between the contacting surfaces assuming elastic-plastic contact behavior.. At high separations, the


50

nominal contact pressure is relatively low, but with decreasing separation (the surfaces are closer), the nominal contact pressure increases strongly. Taking the contacting asperities, the highest asperities will be in the plastic regime, because the indentation depth δ is the highest for these asperities and therefore the contact pressure is more easily higher than the critical value, which is equal to the hardness H. The lower contacting asperities will be in the elastic regime, because the indentation depth is low, and the pressure will be below the hardness H. Increasing the load carried by the contact will result in a smaller separation d. This means that summits which were previously in an elastic contact mode, will now be elastoplastically or plastically deforming. The highest asperities will also form the largest microcontacts and the lower summits will form smaller microcontacts. This means that according tot his approach the larger microcontact areas will be in the plastic deformation mode and the smaller contact areas, which are caused by lower summits, will be in the elastic deformation mode. When the load is increased, the number of microcontacts will increase. However, the average area of a microcontact will stay the same and more or less independent of the load, as is shown in the next figure.

1.5 2 2.5 3 3.51.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6x 1012

aver

age

size

of m

icro

cont

act [

m-2]

separation [-]

Figure 2.9: The average area of a microcontact is more or less independent of the load

In the next figure the difference between elastic and plastic behavior is shown. Because the contact pressures in the case of elastic deformation reach higher values than in the plastic case where p=H, the nominal contact pressures are much higher in the elastic case.

Figure 2.10: Difference between elastic and plastic contact behavioor. In both cases lineair

relations are found between the fraction of area in contact and the normalized contact pressure


51

*Two contacting rough spheres Greenwood & Tripp [7] describe the contact between a rough sphere and a flat, which, by using a sum surface, can also describe the contact between two rough spheres. In a rough sphere contacting a flat it is not immediately clear what the separation d is. If the surface was rigid as well as the flat and this flat covered with elastically deforming asperities, The pressure distribution can be obtained. However, this pressure distribution will deform the sphere and the flat which will in turn change the separation between the (deformed) surfaces. If it is assumed that the single asperity equations for elastically deforming spheres equations are valid (equations). To start with: The profile of the undeformed sphere is given by the following relation, taking a parabolic approximation for the sphere [7]:

( )R

rdry2

2

−= (2.32)

In this equation r is the radial position, d is the separation in the middle of the contact, taking the mean line of the rough flat as a reference and R the reduced radius of the spheres:

21

111RRR

+= (2.33)

If the surfaces deform, then geometry y(r) changes. The deformation has two components, the deformation of the bulk δb (the spheres) and the deformation of the summits δs. Including deformation, the separation over the contact area dradial(r) can be expressed as follows:

( ) ( ) ( )rR

rdryrd bbradial δδ ++−=−=2

2

(2.34)

Then, the asperity displacement ddisplacement (not deformation) will be

( ) ( )rdsrd radialntdisplaceme −= (2.35)

Besides this ‘macro’ displacement, the asperity will deform on a ‘micro’ level because of (elastically assumed) deformation with an amount δs. Then, the load carried by the asperity is given by the Hertzian equation:

23*

, 34

seN EF δβ= (2.36)

The load carried by a set of summits is in general given by:

( ) ( )∫∞

−==d eNsN dssdsFAApF φη , (2.37)

Using these equations, the contact pressure p(r) carried by the total rough sphere will be:

( ) ( )( ) ( )∫∞

−=d eNs dssrdsFErp φβη 2

3

,*

34 (2.38)

The bulk deformation δb is not known until now. However, the equations for the axi-symmetric deformation of an elastic half space give the following relation between the pressure p(r) and the bulk deformation δb:


52

( ) ( ) ( )

trrtk

dtkKtprt

tE

ra

b

+=

+= ∫

2

40*π

δ (2.39)

In this equation K(k) is the complete elliptic integral of the first kind with argument k. Because p(r) is dependent on δb and δb is vice versa dependent of p(r), this problem needs to be solved in an iterative way, as is done in [7]:

Figure 2.11: Iteration loop for the Greenwood & Tripp model

*Plastic bulk deformation Generally plastic bulk deformation and the influence of asperities on neighboring asperities is generally a more complicated to handle than elastic bulk deformation as described in the sections 0. The reason for this is that the superposition principle of stresses and strains is not valid in plastic deformation as it is in elastic deformation. At a certain extent of plastic deformation the summit approach towards the contact of rough surface fails. Then, surface based models will be a more suitable description of the contact.

2.8 Adhesion & Surface roughness From the adhesion theory discussed in chapter is was found that according to the JKR model a force was required to separate two smooth contacting elastic spheres according to:

RFa γπ∆=23 (2.40)

From experiments it is known that surface roughness has a large influence on the adhesion force between two surfaces. By using the same approach as discussed in the chapter about rough surface contact it is possible to calculate the pull-off force in the case of rough surfaces. According to the adhesion model, each sphere will adhere to its contact partner until it is separated by a distance δc from its (flat) countersurface. Besides this, asperities can be in tension at when the applied force is lower than the adhesive forces in the contact area. Equation (4.141) can be rewritten in the form:

==

cc

N fFF

Fδδ (2.41)

Elastic deformation of the asperities δs

Pressure distribution p(r)

Elastic bulk deformation of sphere and plane δb

Separation dradial(r)


53

In the case of a multiasperity contact the dimensionless force roughF can therefore be written as:

( )∫∞

−

=

cd ccrough dssfFF

δ

φδδ

η (2.42)

2.9 Excersises Exersise 1 Let us assume the following roughness parametersη=1010 m-2, β=5·10-6 µm en σs=1·10-6. Let us assume that this rough surface is in contact with a smooth countersurface with is rigid. Other material properties: E=210 GPa, ν=0.3 en H=3 GPa.

• Plot the following plots for a elastic-perfectly plastic material behaviour: 1. Mean contact pressure versus separation 2. Fraction of area in contact versus separation 3. Fraction of area in contact versus nominal contact pressure

• Does the assumed deformation mechanism influence the contact behaviour when compared to elastic contact behaviour? Why?

• Is the contact mainly plastic or mainly elastic at a contact pressure of 1MPa, 10 MPa en 100 MPa?

• What is the fraction of area in contact at the contact pressures mentioned above? • For a certain application, the contact behaviour should be elastic at 100 MPa. Which

values for σ, σm en σκ are required, assuming a constant bandwidth parameter α, (see also the surface roughness chapter). What does this mean for the roughness of the surface? Do you think these values are realistic?

• In another application, the heat conduction is important. What contact property do you expect to be of importance for heat conduction? What increases the heat conduction more: Decreasing the hardness with a factor of 2 or increasing the roughness with a factor of 2? Why?

2.10 Summary In this chapter it is discussed how the contact between two rough surfaces can be described in

terms of the contact between single asperities. The contact between rough surfaces is described both in a statistical as in a deterministic way. So, based on the situation of a single

asperity contact described in the previous chapter, rough surface contact is discussed.

Besides the static contact case, sliding is also briefly discussed. Besides this, effects like the effective material properties of layered solids and asperity interaction are also discussed.

2.11 References [4] Hertz, H., 1882, Über die Berühring fester elastischer Körper, Journal für die Reine und

Angewandte Mathematik, v. 92, p. 156-171. [5] Johnson, K,L., 1985, Contact Mechanics, Cambridge University Press, Cambridge,

United Kingdom [6] J.A. Greenwood and J.B.P. Williamson, 1966, Contact of nominally flat surface,

Proceedings of the Royal Society of London A., vol. 295, p. 300-319.


54

[7] J.A. Greenwood and J.H. Tripp, 1967, The elastic contact of rough spheres, Journal of applied mechanics, p. 153-159.

[8] Zhao, Y. and Chang, L., 2001, A model of asperity interactions in elastic-plastic contact of rough surfaces, v. 123, p. 857-864.

[9] Gelinck, E.R.M and Schipper, D.J., 1999, Deformation of rough line contacts, Journal of Tribology, v. 121, p. 449-454


55


56


57

Chapter 3 Contact mechanics: Elastic & plastic contact for a sphere

3.1 Introduction In chapter Chapter 5, it has been discussed how the asperities and summits can be extracted from measured roughness data. In this chapter, it will be discussed how the mechanical behavior of a single asperity can be modeled. This is done by application of models developed for smooth surfaces, see chapter , to single asperities. One of the reasons that typically spherical or elliptical shapes are assumed for asperities, which makes application of these models, like Hertz, to single asperity contacts possible if such a spherical asperity is in contact with a flat (one rough surface contacting a smooth surface), or if two spherical asperities are in contact (two contacting rough surfaces). In the case of anisotropic surfaces, the asperities will not be spheres, but will be elliptical in shape. In this case, the contact area will be ellipsoidal in shape. Also such contacts can be calculated using the theory of Hertz as was discussed in chapter 2

3.2 Elasticity Hertz theory has been discussed in detail in chapter 2. Here, only a summary of the resulting equations will be given.

Summary circular and line contact A summary of the equations for elastic spherical contact is given in this section. Besides this, limiting values for elasticity will also be given. The background of these values will be discussed later in this chapter. The reduced radius R is given by:

21

111RRR

+= (3.1)

R1 is the (equivalent) radius of body 1 and R2 is the radius of body 2. The reduced modulus of elasticity E* is given by:

2

22

1

21

*

111EEEνν −

+−

= (3.2)

Table of equations for a sphere contacting a flat, see [3] Table 3.1: Overview of relevant equations for point contacts and line contacts

Parameter Circular contact Line contact lFw N /=

contact radius a contact with

δππ RaA == 2

lF

w

EwRb

N=

= *

4π

Carried load FN

23*

34

δREFN =

approach δ 31

2

22

E'R16F9

⋅⋅

⋅==

Ra

δ


58

mean contact pressure p maximum contact pressure

maxp

max2 32 p

aFp N ==π

31

23

2

max RπE'F6p

⋅⋅⋅

=

RwE

awp

ππ'2

max ==

max42p

bwp π

==

pressure distribution p

2

max 1

−=

arpp

2

max 1

−=

brpp

Maximum tensile stress

( )3

21 maxpν−

at r=a zero

Maximum shear stress

0.31pmax at r=0, z=0.48a for v=0.3

0.30pmax at r=0, z=0.78a for all v

Limit of elasticity

Tresca yp σ6.1max =

Von Mises yp σ6.1max =

Tresca yp σ67.1max =

Von Mises yp σ79.1max = (ν=0.3). von Mises depends on

ν. Table 3.2: Equations for circular elastic contact (Hertzian equations) This gives the following expressions in terms of indentation depth, which will be used in the rough surface contact models:


23*

34

δREFN = (3.4)

*Summary elliptical contact In this paragraph, a summary of these equations will be given for an elastic elliptical contact situation. First we need again the reduced modulus of elasticity, see equation (1.93):

2

22

1

21

*

111EEEνν −

+−

= (3.5)

The equivalent radius of the elliptical asperity is presented as follows:

yx RRR111

* += (3.6)

With Rx and Ry the minor and the major radius of curvature. Now, the Hertzian contact area is given by:

δγ

ααπ yxRA *2= (3.7)

In this equation, αx and αy are the dimensionless radii of the contact area. The parameter γ is the dimensionless approach between the two bodies. Definitions of αx, αy and g are given by:


59

( )

( )

( ) ( )

y

x

y

x

αα

κ

κπ

κπ

κγ

κπ

κα

κπ

κα

=

−

−=

−=

−=

−

−

22

2

2

1212

12

12

31

32

31

32

31

31

KE

E

E

(3.8)

In these equations, K and E are the complete elliptic integrals of the first and second kind and κ is the ellipticity ratio of the contact. The elliptic integrals are given by:

( ) ( )

( ) ( )∫

∫

ΨΨ−=

ΨΨ−=−

π

π

2

0

2

2

0

2

21

21

sin1

sin1

dmm

dmm

E

K (3.9)

Evaluation of these elliptic integrals is not straightforward. Moes [17] has given an approximate relation between the ellipticity κ and the gap curvature ratio λ by:

y

x

RR

=

+−+=

−

−

λ

λλλ

κ1

16.0ln4ln16ln2

1 (3.10)

Finally, the load carried by the elliptic elastic contact can be calculated by:

** 234 2

3

REFN

=

γδ (3.11)

Tangential loads In many practical situations, tangential loads are present in contacts. However, tangential loads generally have limited effect on the contact properties in the contact of materials with the same elastic properties. However, if the elastic properties of the two contacting bodies are different, tangential displacements will take place in the interface. Then, influence on the contact behavior is expected.

3.3 Plasticity

Failure: The onset of plasticity If a surface is in contact with a countersurface and this contact is loaded with a force, a stress field will develop. If, from an originally elastic contact situation, the load on the asperity couple is increased, at a certain moment plasticity will be develop below the surface. The place where the plasticity starts is dependent on the normal force FN and the tangential force FT. If only a normal force is acting on the surface, the place of maximum von Mises stress will be subsurface in the middle of the contact. In the case also a tangential force FT is acting in the surface, this will not be the case, and the place of maximum von Mises stresses will be at the


60

backside of the contact, at a lower depth below the surface than in the case when only a normal force FN is acting on the surface.

J2

J2

Figure 3.1: Subsurface stress field for a normally loadsed surface with load FN and a normally as well as tangentially loaded surface with loads FN and FT If the maximum von Mises stresses become higher than the maximum stress allowed by the material, plastic flow will occur. From calculations of the subsurface stresses it can be concluded that plasticity in the material will first develop under the surface of the material for a coefficient of friction<0.3. For a larger coefficient of friction, plastic flow will first occur at the surface instead of subsurface. The onset of plasticity will only be discussed for a spherical contact. The analysis for line contacts and elliptical contacts is similar. Results for elliptical contacts which can be obtained for elliptical summits will be given at the end of this section. Spherical contact In the case of circular Hertzian contact, stresses inside the loaded circle at the surface in polar coordinates are given by the following equations [3]:

2

2

2

2

2

2

max

111321 2

3

ar

ar

ra

pr −−

−−

−=

νσ (3.12)

2

2

2

2

2

2

max

1211321 2

3

ar

ar

ra

p−−

−−

−−= ν

νσ θ (3.13)

−−= 2

2

max

1ar

pzσ (3.14)

And outside the loaded circle ( )

2

2

maxmax 321r

app

r νσσ θ −=−= (3.15)

Considering a ring of concentrated force at radius r, the stresses on the z–axis can be calculated from the following formula:

FN FT=0.25FN FN

FT=0


61

( )1

2

21

maxmax

121tan11

−

−

−+

−+−==

az

za

az

ppr ν

σσ θ (3.16)

And 1

2

2

max

1−

+−=

az

pzσ

(3.17)

When increasing the load from an initially elastic contact situation, at a certain moment the yield stress will be exceeded. Then, according to the Tresca criterion, yield will occur if the maximum shear stress, which is half the difference between the maximum and minimum principal stresses, equals the yield stress in pure shear k which equals half the yield strength in pure tension σy. So,

221,

21,

21max 133221

ykσ

σσσσσσ ==

−−− (3.18)

This Tresca criterion is sometimes also expressed as:

( ) ( ) ( )2

2213

232

221 24

1,41,

41max

==

−−− yk

σσσσσσσ (3.19)

In the case of two contacting spheres, the maximum shear stress will occur at the symmetry axis and will have a value of 0.31pmax at a depth of 0.48a, with a the radius of contact (for an assumed value of ν=0.3). On the symmetry axis σr, σθ and σz are the principal stresses. Besides this, on the symmetry axis σr=σθ. This means that at the onset of yield, according to Tresca:

yyyrz ppk σ

σσσσ61.1

231.0

22 maxmax =⇔=⇔==−

(3.20)

It can be shown that according to the von Mises criterion

( ) ( ) ( )( )36

12

2213

232

221

ykσ

σσσσσσ ==−+−+− (3.21)

Yield will occur if pmax=1.67σy (for ν=0.3) which is about the same value as the Tresca criterion. Because for metals H≈2.8σy, and because max3

.2 pp = it follows that initial yield will occur at the contact pressure equal to:

4.04.06.06.0 maxmax

=⇒≈=⇒≈

kHpkHp

(3.22)

To keep things general, a factor k is introduced: kHp ≈ (3.23)

This factor k=0.4 for yield in a point contact, both for the von Mises criterion and the Tresca criterion. The Hertzian equations, which are still just valid at the situation when plastic deformation is just occurring, are just valid at the onset of plastic deformation. The load is given according to Hertz:

23*

34

δREFN = (3.24)

The average contact pressure can be calculated by dividing the load by the contact area: 31

2

2*

2

21

=⇒=

REFp

aFp NN

ππ (3.25)

As discussed above, the transition from elastic to elastoplastic contact takes place when:


62

kHp ≈ (3.26)

These expressions gives an expression for the indentation depth δ1 at first yield: 2

*

2

*

2

*

max1 89.0

43

2

=

=

=

EHRR

EkHR

EHk ππ

δ (3.27)

Sometimes the constant 0.89 is found to be a little lower, depending on the assumed value of k (e.g. [3] gives 0.81). In a similar way it can be shown that for line contacts initial plastic flow will occur at pmax=1.67σy according to the Tresca criterion and at pmax=1.79σy for the von Mises criterion which for line contact also results in initial yield at about pmax≈0.6H. *Elliptical contact Similarly, it can be shown that for elliptical contacts, see [10],[11], using kmax=0.6, so pmax≈0.6H:

2

**222

1 89

=

EkHRyx γααπδ (3.28)

It turns out that there is little influence of the maximum contact pressure on the magnitude of the maximum shear stress where yield occurs. For spherical contacts, line contacts and elliptical contacts with varying eccentricity, this values is always about pmax=1.6σy. Although the shape of the contact is not of much influence, the presence of tangential loads is of influence. This effect will be discussed next. The effect of tangential loads on the onset of plasticity So far, we have only discussed subsurface stresses and the onset of plasticity for contacts with only a normal load FN applied on it. In a more general (and also more realistic) case, there will also act a tangential load FT on the surface. Let us consider a sliding sphere, it turns out that for µ>0.3, first yield does not occur under the surface as discussed above, but on the surface. The effect of tangential traction on the magnitude of pmax at which first yield occurs, is shown in the following figure, see [3] :

Figure 3.2: The ffect of tangential loads on the onset of plasticity


63

It can be concluded from this that the transition from elasticity to fully plastic behavior will be more direct if tangential forces play a role in the contact behavior. For elliptical contacts, it again can be shown that the maximum contact pressure at first yield is not so dependent on the eccentricity e of this ellipse, as is the case for ‘simple’ normal loading.

Full plasticity When plasticity initiates, first there will only be one spot where the maximum shear stress equals de yield stress in pure shear, as discussed above. If the load is increased, more plastic flow will occur at the borders of the plastic zone, which will make the plastic zone larger. This is illustrated by the following picture.

Figure 3.3: Elastic deformation and full plasticity

At the same time, the pressure distribution will become more flat, see the following figure, from [3].

Figure 3.4: The development of the pressure distribution for full plasticity

The surrounding material around the plastic zone is still elastically deformed in the elastoplastic regime. Eventually, the plastic zone will reach the surface. The pressure

FN FN

Elastic Plastic


64

distribution will not be Hertzian anymore. The contact will behave like a rigid-perfectly plastic contact, and its pressure distribution will be close to that of a plastic sphere pressed to a rigid surface. According to this pressure distribution the average contact pressure p0 is almost constant over the contact and approximately equal to the hardness H at full plasticity.

Hp y ≈= σ8.20 (3.29)

At high loads stress fields of adjacent asperities will start interacting. Therefore, it is not sure that pressures equal to the hardness will occur in real tribological contacts. As long as a summit can considered to be deforming independently of its neighbors, the pressure p0 being equal to the hardness H is a good assumption for full plasticity. The contact radius a for a fully plastically deforming asperity follows directly from the assumption of the average pressure being equal to the hardness:

HFa

HFa NN

ππ =⇒=2 (3.30)

In geometrical terms, the contact area is the geometrical intersection of the undeformed asperity with the flat counter surface, as shown in the figure below:

Figure 3.5: Contact geometry for the case of fully plastic deformation

.

δππ Ra 22 = (3.31)

One of the aspects of this model is that volume conservation is violated. In fact, the material of the plastically deforming asperity is disappearing. Therefore, another model for a fully plastic deforming asperity has been developed. This model will be discussed in the following. Summarizing, this gives the following expressions for the carried load FN and the contact area A as a function of δ:

( ) δπδ RA 2= (3.32)

( ) HRFN δπδ 2= (3.33)

Similarly, for elliptical summits [11]: ( )

( ) HRRF

RRA

yxN

yx

δπδ

δπδ

+=

+= 2 (3.34)

Summary fully plastic contact A summary of the equations for spherical fully plastic behaving spherical summits is given in the following table. Table 3.3: Summary of the relevant equations for fully plastic contact of point contacts

Parameter Full plasticity model contact radius a δππ RaA 22 ==


65

contact with Carried load FN δπHRAHFN 2==

approach δ HR

FR

a N

πδ

22

2

==

mean contact pressure p Hp =

And for elliptical asperities in fully plastic deformation in the following table: Table 3.4: Summary of the relevant equations for fully plastic contact of elliptical contacts

Parameter Full plasticity model


( ) δπδ yx RRA += 2

Carried load FN ( ) HRRF yxN δπδ +=

mean contact pressure p Hp =

Elasto-plastic contact The in-between regime between elastic deformation and fully plastic deformation has already been more or less discussed in the previous section, and is shown in the following figure:

Figure 3.6: Development of elastic contact, via elastoplastic contact to fully plastic contact

The in-between region between the elastic regime and the fully plastic regime, called the elasto-plastic region, is shown in the following figure, from [3]

FN FN FN

Elastic Elastoplastic Plastic


66

Figure 3.7: Mean contact pressure in the elastoplastic region, analyzed with several models and

experiments. The A,B, C and D, E, F, and G lines are not of much interest here. They correspond to elastic cones (A), elastic spheres (B) and FEM calculations (solid line), and experiments

(crosses for pyramids and circles for spheres) and other sources and cases Here, the mean contact pressure as a fraction of the hardness H is plotted against the quantity E*a/σyR. This plot will be calculated in section 3.4. If follows from this figure that we can expect full plasticity at E*a/σyR=40 which corresponds to

400yfplasticit oonset ,

plasticfully , =N

N

FF

(3.35)

Not only the contact radius and the pressure are of interest. Also of interest is the indentation depth, which is in the elasto-plastic and the plastic regime given by [3]

Figure 3.8: The normalized ocntact radius versus normalized indentation depth

[7] gives a modeling approach to this important deformation regime and is summarized below. The plots shown above will also be calculated with this model in section 3.4.

elastic plastic

elastoplastic


67

The onset of plasticity has been discussed in section 0. The indentation depth for the onset of plasticity as given in equation (3.27) can be rewritten in the following form:

2

*

2

*

max2

*

max1 89.0

22

=

=

=

EHRR

EHkR

Ep ππ

δ (3.36)

The indentation depth at fully plastic flow δ2 is more difficult to determine. In [7] it is done in the following way: Assume that at δ2 the contact pressure is equal to the hardness H. Then:

HRFN 22 δπ= (3.37)

If the deformation had been elastic, then it would be: 23

34

δREFN = (3.38)

This gives the following inequality: 2

222 434

342 2

3

>⇔<

EkH

kEHR π

δδδπ (3.39)

Reworking these equations and using equation (3.36), this means that for k=0.4 the following inequality is got:

1221

2 254

δδδ

δ >⇔>k

(3.40)

According to Johnson [3], as discussed above


plasticfully , =N

N

FF

(3.41)

Using the equation for the load at the onset of plasticity and the inequality for the load at fully plastic flow, it can be derived that

121

2 5440023

δδδδ

≥⇔≥

(3.42)

Based on a statistical analysis of spherical indentations [8], the following expression for the mean contact pressure p in the in-between region between δ1 and δ2 is obtained:

+=

aaap δln21 (3.43)

The constants a1 and a2 still have to be determined. In the elastic and fully plastic deformation regimes we have seen that:

plastic)(fully for 2

(elastic) for

2

1

δδδ

δδδ

>=

<=

Ra

Ra (3.44)

Therefore a following expression is expected in the elasto-plastic regime: plastic)-(elasto for 21 δδδδ <<= RCa (3.45)

Using equation (3.43), this means that the constants are determined by: ( )

24

2213

43

5.0ln5.0ln

ln

aaRaCaaa

aap

=−+=

+= δ (3.46)

For continuity in contact pressure in the transition elastic - elasto-plastic and elasto-plastic –fully plastic the constants a3 and a4 can be solved:


68

−−

=

−−−=

⇒

=+=+

124

12

23

243

143

lnln)1(

lnlnln

)1(

lnln

δδ

δδδ

δδ

kHa

kHHa

HaakHaa

(3.47)

This gives the following expression for the mean contact pressure p :

12

2

lnlnlnln)1(

δδδδ

−−

−−= kHHp (3.48)

Now, an expression for the contact area in the elasto-plastic regime is derived. In order to guarantee continuity in the function and its first derivative at the transition elastic-elastoplastic and elastoplastic-plastic, the following template curve is used:

23 32 xxy +−= (3.49)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Figure 3.9: The template to model the transition from elastic deformation to fully plastic

deformation in the elastoplastic regime Using the scaled indentation depth as a coordinate:

12

1

δδδδ

−−

=x (3.50)

It follows that ( )( )23

elasticplasticfully elasticplastic-elasto 32 xxAAAA +−−+= (3.51)

This gives after substitution of Aelastic, Afully plastic and x:

−−

+

−−

−=2

12

1

3

12

1plastic-elasto 321

δδδδ

δδδδ

δπRA (3.52)

As models for the fully plastic region, also the model for volume conservation could be used. Summarizing, this gives for the carried load FN and the contact area A as a function of δ for a spherical contact:

( )

−−

+

−−

−=2

12

1

3

12

1plastic-elasto 321

δδδδ

δδδδ

δπδ RA (3.53)

And: ( )

( )

−−

−−

−−

+

−−

−

=

12

2

2

12

1

3

12

1

plastic-elasto

lnlnlnln

1321δδδδ

δδδδ

δδδδ

δπ

δ

kHHR

FN

(3.54)

For elliptical summits similar equations can be derived


69

Using equations (3.48), (3.50) and (3.51),(3.51) as well as (3.52), the equations from section 0 and 0 also area- and equations for elliptical contacts can be derived. In the case of elliptic asperities the indentation depth δ1 for the onset of plastic deformation was given by:

2

**222

1 89

=

EkHRyx γααπδ (3.55)

If again it is assumed that:


plasticfully , =N

N

FF

(3.56)

Then, assuming that transition to fully plasticity takes place at an average contact pressure equal to H the transitions are given by:

( )( )2

2

1

2

11

1400

κκκ

λλ

δδ

−−

+=

KEk (3.57)

Now, the elastic contact regime for elliptical asperities, the fully plastic contact regime for elliptic asperities as well as the transitions have been discussed. An elastoplastic deformation model for elliptical asperities can be relatively simple constructed by following the methodology described for spherical asperities. This will not be further discussed.

Summary elasto-plastic contact spheres Table 3.5: Summary of the relevant equations for point contact s in the elastoplastic regime

Parameter


−−

+

−−

−=3

12

1

3

12

1plastic-elasto 321

δδδδ

δδδδ

δπRA

Carried load FN

( )

( )

−−

−−⋅

⋅

−−

+

−−

−=

12

2

3

12

1

3

12

1plastic-elasto

lnlnln

1

321

δδδδ

δδδδ

δδδδ

δπδ

nkHH

RFN

approach δ

( )

−−

−−

−−

+

−−

−

=

=

−−

+

−−

−

=

12

2

3

12

1

3

12

1

plastic-elasto

3

12

1

3

12

1

plastic-elasto

lnlnln

1321

321

δδδδ

δδδδ

δδδδ

π

δδδδ

δδδδ

π

δ

nkHHR

F

R

A

N

mean contact pressure p

12

2

lnlnlnln)1(

δδδδ

−−

−−= kHHp

3.4 Example In this section the some calculation results with the single asperity models above will be given. The first plot gives the characteristics for normalized contact area A/β as a function of the dimensionless indentation δ/β. From the plot it is clear that the elastoplastic solution (blue


70

line) is a transition regime between the elastic Hertzian solution (green line) and the fully plastic solution (red line). It is also clear from the figure that the elastoplastic deformation regime is large compared to the elastic regime.

Figure 3.10: Transition from elastic deformation to fully plastic deformation for the contact area In the previous plot the normalized contact area A/β as a function of the dimensionless indentation δ/β was shown. The contact pressure p normalized with the hardness H is as a function of the dimensionless indentation δ/β is shown in the next plot. Also with respect to the contact pressures, the elastoplastic equations (blue line) form a transition between the elastic (green line) and the plastic regime (red line). The normalized contact pressure is limited to the hardness of the softest contact partner, so p/H=1. Radii β of asperities on rough surface will generally be smaller than radii β of asperities on smooth surfaces. If a certain load is applied on an asperity with a larger radius β, the indentation δ will be smaller. This means that the asperity will more easily operate in the elastic regime. So, smooth surfaces operate more easily in the elastic regime than rough surfaces, as can be expected. The results for ellipsoidal asperities will show similar behavior, but will not be discussed further in this section.

Figure 3.11: Transition from elastic deformation to fully plastic deformation for the contact

pressure

0 0.005 0.01 0.015 0.02 0.025 0.03 0

0.5

1

1.5

2

2.5

3

elastic

plastic

elastoplastic

δ/R [-]

p/H [-]

elastic

0 0.005

0.01

0.015

0.02 0

0.2

0.4

0.6

0.8

1

1.2

1.4

x 10

-6

plastic

elastoplastic

Contact area

δ/R [-]

Α/β [-]


71

The next plot gives the results for all single asperity contacts occurring in the example surface of paragraph 5.8. at a load of 5 N. Here the contact pressure is normalized with the yield stress σy and on the x-axis the dimensionless number aE*/βσy is plotted, in which a is the radius of the contact area. Thus, results are plotted for different asperities, with, different asperity heights and different radii and therefore in different deformation regimes. Each asperity contact is represented by one dot in the plots. The scaling at the axes is chosen so that all summits fall on a nice line, see [3]. From this figures it follows that the general shape is similar to these graphs, which are based results from experiments, simple analytical models and FEM-calculations, see also the first picture in paragraph 0 which has the same scaling. Here the contact pressure is limited to p=2.8σy. The reason for this is that for metallic surfaces H=2.8σy. In this case, at a load of 5N on the total set of asperities, only two asperities turn out to be acting in the elastic regime (green), while most asperities are acting in the elastoplastic (blue line) or the fully plastic regime (red line).

Figure 3.12: Results for all single asperity contacts occurring in the example surface of

paragraph 3.8. at a load of 5 N.

3.5 Effect of sliding on plastic contact Up to now, we have only discussed static contact, i.e. there is no sliding in the interface. However, properties we have calculated, like the real area of contact and the contact pressure on top of asperities, are of importance for friction and wear. Now we will discuss in qualitative terms the contact conditions under conditions of sliding, so in the contact conditions of most friction and wear processes. If wear occurs, the following will occur in the case of a hard spherical asperity (top) contacting a soft flat material (bottom), loaded with a normal force FN and a tangential force FT:

10 -1

10 0

10 1

10 20

0.5

1

1.5

2

2.5

3

p/σy

elastic elastoplastic plastic

aE*/Rσy


72

Figure 3.13: Effects of plasticity in the case of a sliding contact, where a hard body a ploughing

through a softer body Initially, at low FT, the situation is the static contact situation. No deformation in tangential direction occurs and the load is carried by the real contact area. The initial indentation depth is δ1. When FT is increased, the trailing edge of the (spherical) asperity will not carry any load anymore. Then, after a while, the entire load is carried by the front half of the contact area. In order to be able to carry the load FN the same contact area is needed, assuming that the contact pressure is still equal to the hardness of the softest material H. So, the indentation depth will be larger and δ2> δ1. If in this stage the ploughing processes is continued and material build-up occurs at the front of the asperity, then the front half of the contact area will be able to carry a larger load, and this will cause the asperity to rise a bit. If multiple asperity contact is considered, all asperities will be in a different stage of this sliding process. Therefore an asperity might, for example, temporarily loose contact if the load carried by another asperity gets higher. This means that the above discussed mechanism of an asperity being in these three stages may occur many times for a certain asperity during sliding.

3.6 Shakedown Initial yield will occur if the contact pressures are higher than the onset of yield, as discussed above. Subsurface stresses are a function of both the normal load and the tangential load, which means that also tangential (friction / traction) forces will influence the onset of yield. In the case of higher tangential forces, the point where initial yield occurs will be closer to the surface. In many practical applications contacts are designed in such a way that multiple passes take place over the same area. Rolling contacts, like train wheels and roller bearings are examples of contact elements where many passes take place over the same surface caused by repeated rolling. In some cases an initially plastic contact will not stay plastic in repeated contacts. During the first pass, some (subsurface) plastic deformation may take place. If plastic deformation takes place, residual stresses will be introduced in the surface. If the second pass takes place, the original surface is altered in such a way that to cause ongoing plastic deformation, the combined action of residual stresses and contact stresses determines whether plastic deformation will take place or not. In most cases, residual stresses introduced into the surface are protective which makes plastic deformation after the first pass less likely than at the first pass. It is possible that after some passes the residual stresses built up in the surface reach high enough values in order not to cause plastic deformation in further passages. If this is the case, only elastic deformation will take place, although initially deformation may be plastic. This process is called shakedown. If shakedown takes place, is dependent on three factors:

1. The build-up of residual stresses 2. Geometrical changes of the surface, resulting in lower contact pressures because the

contact is getting more conformal 3. Work-hardening resulting a ‘harder’ material

In the next sections, the first factor will be discussed

δ1

δ2

δ3

FN FN FN

v v v

FT FT FT


73

If shakedown occurs, can be stated by the so-called Melan’s theorem: If any time dependent distribution of residual stresses can be found which, together with the elastic stresses due to the load, constitutes a system of stresses within the elastic limit, then the system will shakedown [2]. The stress state under a (rolling) surface is generally a three dimensional stress state as shown in the picture below.

Figure 3.14: General 3D stress state

Rolling cylinder without traction A simple case is the case of a rolling cylinder. In this case it can be assumed that no deformation takes place perpendicular to the direction of rolling. If this is the case, stress components under the surface will be independent of y, and the residual stresses (τxy)r, (τyz)r will be zero. Further, if deformation is considered to be continuous, the residual stresses developing in the material will be independent of x. If this is the case, the subsurface stress state has become much simpler. Melan’s theorem states that if a subsurface stress distribution can develop which will prevent yield, than this will happen. Potential residual stresses in the case of a rolling cylinder discussed above are therefore:

( ) ( )( ) ( )

( )

( )( )( )

===

===

000

02

1

rzx

ryz

rxy

rz

ry

rx

zfzf

τττ

σσσ

(3.58)

Subsurface stresses under the surface will be dependent on both stresses caused by contact stresses as well as residual stresses. From the subsurface stress state, principal stresses can be calculated by Mohr’s circle. These principal stresses for the stress state under consideration are given by:

( )( ) ( )( )( )( )( ) ( )( )( )

( ) ( )ryzx

zxzrxxzrxx

zxzrxxzrxx

σσσνσ

τσσσσσσσ

τσσσσσσσ

++=

+−+−++=

+−++++=

3

222

221

21

21

421

21

421

21

(3.59)

Now, according to Melan’s theorem, a residual stress state will develop which prevents yield. So, (σy)r can be chosen to develop in such a way that σ3 is the intermediate principal stress. Then yield will be avoided, according to the Tresca criterion, by:

σx σx

σy

σz τzx

τzy τxy

τxz

τyx τyz


74

( ) 22214

1 k≤− σσ (3.60)

With k the yield stress in simple shear. Substitution of the equations for the principal stresses results in

( )( ) 222

41 kzxzrxx ≤+−+ τσσσ (3.61)

If τzx>k, the equation shown above will never be true. So if τzx=k, then (σx)r=σz-σx. So if τzx anywhere in the solid is equal to k, then the shakedown limit of the system is reached. Subsurface stresses in the case of a rolling cylinder are given by the following equations at the symmetry axis:

( ) ( )

( ) ( )

+−−++−=

+−+++−=

+−

−=

++

−−=

−

++

+−=

2222222222

2222222222

22

22max

22

22max

22

22max

421

421

1

21

zxbzxzxbn

zxbzxzxbm

nmzmn

bp

nmnzm

bp

znmnzm

bp

xz

z

x

τ

σ

σ

(3.62)

In this equation, b is half the contact width and given by, see above:

lF

w

EwRb

N=

= *2π (3.63)

And the maximum contact pressure pmax is given by:

R

wEbwp

ππ

*

max2

== (3.64)

The shear stress τxz is maximum at points ±0.87b and 0.50b and has a value of 0.25pmax. So for shakedown: kp 00.4max ≤ (3.65)

Then, the residual stresses required at that depth 0.50a required for shakedown are:

( )( ) max

max

213.0134.0

pp

ry

rx

−=

−=

σ

σ (3.66)

The onset of yield for the case without shakedown is given by: kp 1.3max ≤ (3.67)

Comparing equation (3.65) with equation (3.67) shows and calculating the maximum contact pressure back to applied loads using the Hertzian equations shows that the shakedown load is 1.66 times higher than the yield at which initial yield occurs. A qualitative description of shakedown can be given using the following figure:


75

Figure 3.15: Qualitative description of shakedown for a rolling cylinder

At C first yield will occur. Yield will occur in such a way that C is compressed in vertical direction and tries to expand in horizontal direction. Because the cylinder is rolling over the surface, all subsurface elements are going through this deformation cycle at their turn. Residual compressive stresses parallel at the surface will restrict lateral expansion of the subsurface volume elements. If these subsurface stresses (σx)r are fully developed, compression of volume element C will be prohibited. The alternating shear deformation cycle at B and D cannot be prohibited by an residual stress (τxz)r , because the stress state is alternating and therefore changing sign before and after the contact. This means that the stress state at B and D will be determining the shakedown limit.

Rolling sphere without traction In the case of a rolling sphere a simplifying assumption because of symmetry can be assumed. Symmetry means that only a symmetric stress state can develop under the surface which means that at the center of the contact, so at y=0, ( )( ) 0

0

=

=

ryz

rxy

τ

τ (3.68)

The shear stress τxz has equal and opposite maxima at both sides of the rolling contact. This means that just adding a unidirectional residual stress, as in the rolling cylinder case, is not sufficient to prevent yield because both the ‘positive’ and the ‘negative’ peak needs to be protected. The maximum value of τxz is given by: max21.0 pxz ≤τ (3.69)

Following the same reasoning as above, this results in kp 7.4max ≤ (3.70)

Initial yield will occur if kp 8.2max ≤ (3.71)

FN

p(x)

σx σx

σz

τxz

C B A D E


76

From the equations of an elastic line contact it can be shown that the shakedown load is 4.7 times higher than the yield at which initial yield occurs for the case of a rolling sphere without traction.

Shakedown maps A shakedown analysis for relatively simple cases was shown above. More complex cases can also be studied, like a rolling cylinder with traction or three dimensional rolling elements including traction. Such cases are much more complicated because no assumptions which make the stress state simpler can be made, as was done in the rolling-cylinder case by assuming that no deformation takes place perpendicular to the direction of rolling.

Figure 3.16:Shakedown map for a rolling cylinder with traction

Results can be summarized in so called shakedown maps. The figure above shows a shakedown map of a rolling cylinder including traction. The broken lines represent the contact pressure for first yield and the solid line represents the shakedown limit. It is clear from the figure that the shakedown limit decreases at increasing coefficient of friction, so increasing tangential load at the surface. This was also the case for the load at which the onset of yield takes place, as was already known from the beginning of this chapter. It is also clear that the difference between the shakedown load and the load for initial yield is higher for surface yield than for subsurface yield. If the load is higher than the shakedown limit, at each cycle plastic deformation will take place. Then, at B and D, shear will take place. In the elastic case, the elastic stresses at B and D are opposite and equal. In the plastic case the permanent shear takes place at D. When repeated rolling takes place, surface layers are displaced ‘forward’. The stress cycle at B and D is schematically shown in the following figure.


77

Figure 3.17: Stress cycli for shakedown and repreating plasticity

3.7 Summary

In this chapter it is discussed how the contact of a single rough asperity can be described. Most attention is paid to spherical asperities, but also elliptical asperities are discussed.

Three deformation regimes are distinguished: elastic, elasto-plastic and plastic deformation.

Besides the static contact case, repeating contacts resulting in shakedown phenomena are also discussed.

3.8 References [1] Hertz, H., 1882, Über die Berühring fester elastischer Körper, Journal für die Reine

und Angewandte Mathematik, v. 92, p. 156-171. [2] Johnson, K,L., 1985, Contact Mechanics, Cambridge University Press, Cambridge,

United Kingdom [3] Bhushan, B., Contact mechanics of rough surfaces in tribology: single asperity

contact, 1996, Applied Mechanics Review, v. 49, n. 5, p. 275-298. [4] Bhattacharya, A.K. and Nix, W.D., 1988, Analysis of elastic and plastic deformation

associated with indentation testing of thin films on substrate, International Journal for Solids and Structures, vol. 24, p. 1287-1298.

[5] Chang, W.R., Etsion, I and Body, D.B., 1987, An elastic-plastic model for the contact of rough surfaces, 1987, Journal of Tribology, v. 109, p.257-263.

[6] Chang, W.R., An elastic-plastic contact model for a rough surface with an ion-plated soft metallic coating.


78

[7] Zhao, Y., Maietta, D.M., Chang, L., 2000, An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow, Journal of Tribology, v. 122, p. 86-93.

[8] Francis, H.A., 1976, Phenomenological analysis of plastic spherical indentation, Journal of Engineering Materials and Technology, July, p. 272-281.

[9] Visscher, H., On the friction of thin film rigid disks, 2001, PhD Thesis University of Twente, Enschede, The Netherlands

[10] Horng, J. H, An elliptic elastic-plastic asperity microcontact model for rough surfaces, 1998, Journal of Tribology, v. 120. p. 82-88.

[11] Jeng, Y.R. and Wang, P.Y., An elliptical microcontact model considering elastic, elastoplastic, and plastic deformation, 2003, Journal of Tribology, v. 125, p. 232-240.

[12] Hamilton, G.M., 1983, Explicit equations for the stresses beneath a sliding spherical contact, Proceedings of the institution of mechanical engineers, Vol 197 C, p. 53-59

[13] Holmberg, K. and Matthews, A., 1994, coatings tribology: properties, techniques and applications in surface engineering, Elsevier, Amsterdam, NL.

[14] Puchi-Cabrera, E.S., Berrios, J.A., Teer, D.G., 2002, On the computation of the absolute hardness of thin films, Surface and Coatings Technology, vo. 157, p. 185-196.

[15] Michler, J., Blank, E., 2001, Analysis of coating fracture and substrate plasticity induced by spherical indentors: diamond and diamond-like carbon layers on steel substrates

[16] Gao, H.E.A., 1992, Elastic contact versus indentation modeling of multi-layered materials, International Journal of Solids and Structures, vol. 29, no. 20, p. 2471-2492,

[17] Moes, H., **naam**


79


80

Chapter 4 Surface forces

4.1 Introduction Surface forces are important in the case of large contact areas compared to the size of the objects. This means that these forces can become even dominant over other forces like body forces for cases involving small objects or smooth surfaces. An application where adhesive forces between solid surfaces are of importance is micromechanics. Here, applications with very smooth surfaces may be contacting each other under light loads. A characteristic of applications in vacuum is that oxide layer, p[resent on surfaces under ambient conditions, are absent. This means that bare metal may be in contact with bare metal, which is another reason that adhesive forces in vacuum applications are very important. Real-life examples of strong attractive forces between solids and fluids are the sticking of chewing gum, clay or glue. Adhesive interaction between a fluid and a solid surface is also related to ‘wetting of liquids on solid surfaces’ and to capillary forces, lubrication and coating technology. If attractive forces are present between bodies, a force will be required to separate the two surfaces. The force perpendicular to the surface, required to break the adhesion forces, is called the adhesion force Fa.

4.2 The origin of adhesive forces The origin of adhesion forces lie in atomic and molecular interactions between the two solids as well as the medium separating the solids. To start with, in the next section several bonds occur between atoms will be described:

4.2.1 Types of bonds A distinction can be made between strong bonds and weak bonds between atoms. Within the class of ‘strong bonds’, the following kinds of bonds can be distinguished:

• The ionic bond or electro covalent bond. This bond is caused by attraction forces between ions of opposite charges. The ionic bond is of importance for the cohesion within ionic crystals like NaCl

• The covalent bond is based on sharing one or more pairs of electrons. An example where covalent bonds are found is a diamond crystal.

• The metallic bond is caused by free electrons moving between positive ions. This force is the basis of cohesion of metals and their alloys both in the solid as well as in the liquid state.

Also within the class of ‘weak bonds’ several types of bonds can be distinguished: • The hydrogen bond. The hydrogen bond is formed by bonding two molecules with an

‘intermediate’ hydrogen atom. The hydrogen atoms, bounded to a molecule, will form a positive pole to the molecule. This positive pole attracts a nearby negative molecule and forms a bond. The hydrogen bond is therefore of electrostatic nature.

• The van der Waals forces. The van der Waals force is caused by electromagnetic interactions which are caused by continuous fluctuations in the electron distribution of atoms or molecules. An example: If such attractive forces exist between gas molecules, the perfect gas law PV=RT is not valid anymore. Van der Waals forces are generally responsible for attrictive (or repulsive) forces between bodies.

In most cases the van der Waals forces are the most important contribution to surface forces. Other contributions are present in solutions, like the electrostatic double layer force and at separations below a few molecular diameters solvation forces [13]. These forces will not be discussed here.


81

In section 4.2.2 the background of van der Waals forces between molecules will be discussed. Based on this, interaction energies between macroscopic bodies can be determined by pairwise addition of interaction energies between individual molecules, see section 4.2.3. An import aspect of the adhesion theory is that the interaction energy can be separated into a geometry dependent part and a geometry independent part. The geometry independent part is called the Hamaker constant. This Hamaker constant can, as an alternative to pairwise addition of interaction energies of individual molecules, also be determined using a macroscopic theory, called the Lifschitz theory, see section 4.2.4. For simple geometries interaction energies can be calculated using analytical expressions. Such cases are summarized in Figure 4.1. For other cases without a simple geometry, interaction energies can be obtained by intergration using the Derjaguin approximation, see section 4.2.8. In section 4.3, the basics of surface energy and surface tension will be discussed. In section 4.4 surface energy will be related to Hamaker constants as discussed in the beginning of chapter Chapter 4. In section 4.5 and 4.6 measurement techniques for surface tension of fluids and surface energy of solids will be discussed. Finally, in section 4.7 wetting is sdicussed as an application of solid-fluid contacts. In section 4.9 solid-solid contacts will be studied and applied to biological systems in section 4.10.

4.2.2 *Van der Waals forces between molecules In this section, the basic mechanisms which contribute to the van der Waals force between molecules will be discussed. As we will, see, three contributions can be distinguished: 1. Dipole- dipole interactions (Keesom interactions) 2. randomly induced dipole interactions (Debeye interactions) 3. Interation between non-polar molecules (London or dispersion interactions) The basis for understanding intermolecular forces is the Coulomb force [12]. The Coulomb force is the electrostatic force between charge Q1 and charge Q2:

20

21

4 DQQF tricelectrosta πεε

= (4.1)

Here, ε is the dielectric permittivity. The corresponding potential energy between Q1 and Q2 is the distance between the two charge is D is:

DQQW tricelectrosta

0

21

4πεε= (4.2)

If Q1 and Q2 have an opposite charge, Welectrostatic is negative and if the charges have equal sign, W is positive. In the last case, Welectrostatic decreases is D is increased. Molecules are not charged as a whole, but the electric charge within a molecule is not distributed equally over the whole molecule. For example water is a so called polar molecule with a positive and a negative side. The electric properties of polar molecules can be desribed by the dipole moment. For Q and –Q with a distance d in between the charges, the dipole moment m is m = Qd with units Coulomb m [Cm]. In practice, often the unit Debeye is used. 1 Debeye (D) is equal to a positive and a negative unit charge with a distance d of 0.21 Å, so 1D = 3.336 10-30 Cm. If there are more than two charges, the following general definition of the dipole moment is true:

( )∫= rdVreρµ (4.3)

In this equation, ρe is the charge density, which has to be integrated over the volume to get the dipole moment. As we can also see from the equation above, the dipole moment for more than two charges can be calculated based on superposition. It can be shown that the potential energy between a dipole and a single charge is equal to the following expression, assuming that D is large compared to the size of the dipole, and the angle between the dipole vector and D is equal to θ:


82

204cos

DQW

πεθµ

−= (4.4)

Real molecules do not have a fixed position, but are for example able to rotate. In this case, the molecule will orient in such a way that the negative side will be close to the positive side of the neighboring charge. Besides this, thermal fluctuations will distort the perfect position. However, on the average, a certain orientation will exist. This results in the following expression for the potential energy between a freely rotating dipole and a fixed charge Q:

( ) 420

22

46 TDkQW

Bπεµ

−= (4.5)

Here, kB is the Bolzmann constant (1.381 x 10-23 J K-1) and T is the absolute temperature is the Bolzmann constant times the temperature. The quanity kBT is a measure for the kinetic energy of a moving molecule. Equation (4.5) desribed the interaction energy between a fixed charge and a dipole. Equation (4.6) gives the potential energy between two freely rotating dipoles. This randomly oriented dipole – dipole interaction is called the Keesom energy:

( ) 620

22

21

6 43 TDkDC

WB

orient

πεµµ

−=−= (4.6)

As follows from the equation, Corient is strongly dependent on distance. For example, two water molecules with a distance of 1 nm apart result in a Keesom energy of -9.5 10-24 J. In the case a molecule is not polar, so does not have a dipole moment, the above expression will result in zero energy. How also for this case a attractive force can be present and is caused by the fact that presence of a charge induces a charge shift in the non-polar molecule if the molecule is polarisable. So, an induced dipole is the result in the molecule which originally did not have a dipole moment. The expression for this randomly induced dipole interaction is called the Debeye interaction and the energy is given by:

( ) 420

2

6 42 DQ

DCW oind

πεα

−=−= (4.7)

With α the polarizability of the molecule in C2m2J-1. The induced dipole moment µind is defined as αE whith E the electric field strength. For the case of a static dipole interacting with a polarisable molecule, the Helmholz free energy is given by the following expression, assuming that the molecules are different. If the molecules are identical, a factor of two has to be inserted in the equation below.

( ) 620

2

6 4 DDCW ind

πεαµ

−=−= (4.8)

The third type of interaction is the interaction between non-polar molecules. The so called London or dispersion force is responsible for this interaction between non-polar molecules. This interaction can be understood by visualizing a molecule as a positively charged nucleus with electrons circulating around it with a high frequency of 1015 to 1016 Hz. At every time instant, the atom is polar. Further, attractive orientations turn out to have higher probabilites than repulsive interactions, resulting in a net attractive force. The Helmholz free energy between two molecules with ionization energy hν1 and hν2 can be approximated by:

( ) 21

2162

0

216 42

3νννν

πεαα

+−=−=

hDD

CW disp (4.9)

Here h is the Planck constant van v the orbiting frequency of the electron. Planck's constant = 6.626068 × 10-34 J. s.


83

The ionizing energy can be further understood by the Bohr atom [13], where an electron is orbiting around a proton. Here, the smallest distance between an electron an a proton is also known as the first Bohr radius a0. At this radius, the Coulomb energy is equal 2hv:

hva

e 24 00

2

=πε

(4.10)

For a Bohr atom, v = 3.3 1015 m s-1, so hv= 2.2 10-18 J. This amount of energy needs to be supplied to the atom in order to ionize the atom. Dispersion interactions increase with the polarizability of two molecules. The total van der Waals force is the sum of all three dipole – dipole interactions, so:

dispindorienttotal CCCC ++= (4.11) Usually the London dispersion term is dominating. Polar molecules do not only interact via de Keesom and Debeye force, but also via the London dispersion force. The following table, also from [12] summarizes properties of some gases. µ α/4πε0 hν Corient Cind Cdisp Ctotal Cexp (D) 10-30 m3 eV 10-79 Jm6 10-79 Jm6 10-79 Jm6 10-79 Jm6 10-79 Jm6 He 0 0.2 24.6 0 0 1.2 1.2 0.86 Ne 0 0.40 21.6 0 0 4.1 4.1 3.6 Ar 0 1.64 15.8 0 0 50.9 50.9 45.3 CH4 0 2.59 12.5 0 0 101.1 101.1 103.3 HCl 1.04 2.7 12.8 9.5 5.8 111.7 127.0 156.8 HBr 0.79 3.61 11.7 3.2 4.5 182.6 190.2 207.4 HI 0.45 5.4 10.4 0.3 2.2 364.0 366.5 349.2 CHCl3 1.04 8.8 11.4 9.5 19.0 1058 1086 1632 CH3OH 1.69 3.2 10.9 66.2 18.3 133.5 217.9 651.0 NH3 1.46 2.3 10.2 36.9 9.8 64.6 111.2 163.7 H2O 1.85 1.46 12.6 95.8 10.0 32.3 138.2 176.2 CO 0.11 1.95 14.0 0.00012 0.047 64.0 64.1 60.7 CO2 0 2.91 13.8 0 0 140.1 140.1 163.6 N2 0 1.74 15.6 0 0 56.7 56.7 55.3 O2 0 1.58 12.1 0 0 36.2 36.2 46.0 For distances larger than 10 nm, the van der Waals energy decreases more steeply (for molecules with 1/D7 for smaller distances. This effect is known as retardation

4.2.3 Microscopic approach: van der Waals forces between macroscopic solids

As explained above, the potential energy for the interaction between two molecules due to the van der Waals forces is proportional the inverse of the distance between the molecules to the 6th power:

6DCW AB−= (4.12)

Here, CAB is the summation of the contributions of the three dipole-dipole interactions. This means that this energy is very strong dependent on the distance, and will typically become negligible at distances in the order of nanometers. For macroscopic solids, the van der Waals energy between a molecule and an infinite solid plane can be obtained by integration over the volume of the interactions between molecule A and all molecules in solid B, using the molecular density ρb [molecules m-3], see the following figure:


84

So, in the analysis below, pairwise additivity between molecular interactions is assumed, and influence of neighboring molecules on the interactions are neglected:

( )( )( )

( )( )∫ ∫

∫ ∫∫∫∫∞ ∞

∞ ∞

−=++

−

=++

−=−=

0 03322

2

0 03226/

6

2'

DCdx

rxD

rdC

rxD

rdrdxCdVD

CW

BABBAB

BABB

ABplaneMol

ρπρπ

πρ

ρ

(4.13)

It is clear from this equation that instead of a proportionality of the van der Waals energy between molecules with D-6, now the energy is proportional to D-3. For the case of two infinitely extended flat solids with a distance D, the following result can be obtained, see het equation and the figure below:

( ) ( )dzdydx

xDCdV

xDCW ABABABAB

planeplane ∫ ∫ ∫∫∫∫∞ ∞

∞−

∞

∞− +−=

+−=

033/ 66

ρρπρρπ (4.14)

This integral is infinite due to the infinite area of the solids. The Work per unit area is given by:

( ) ( )

2

002

03

/

12

21

66

DC

xDC

xDdxC

AW

ABAB

ABABABABplaneplane

ρρπ

ρρπρρπ

−

=+

−−=+

−=∞∞∞

∫∫ (4.15)

Now it can be seen form the equation that for the case of two contacting solids this a proportionality of the van der Waals energy between decreases proportional with D-2. In many calculations involving adhesive interactions, the so called Hamaker constant is used. The Hamaker constant AH is defined as:

BAABH CA ρρπ 2= (4.16) Then equation (4.15) reads:

D’ D

x r

y

z

x


85

212 DAW H

π−= (4.17)

Similar to the calculation above, the van der Waals energy can be calculated for other geometries. For two spheres with radii R1 and R2 the van der Waals energy can be shown to be given by:

( ) ( )( )( )

21

212

221

2

221

221

221

221 ln

226

RRdD

RRdRRd

RRdRR

RRdRR

DAW H

−−=

−−+−

+−−

++−

−= (4.18)

It can be seen that also the Hamaker constant appears in calculations for other geometries. The only difference is the factor involving the geometry of the contacting bodies. In the equations above, only attractive forces are taken into account. At very short distances electron orbitals overlap, causing a repellent force. For D<<R1,R2, the equation above can be simplified to:

21

21

6 RRRR

DAW H

+−= (4.19)

With the force as the negative derivative:

21

2126 RR

RRDA

dDdWF H

+−−= (4.20)

The Hamaker constant describes the dependence of geometry on the separation between the bodies D. Figure 4.1 gives and overview for several Hamaker constants for several common geometries [13]. Typical values for Hamaker constants of condensed solid or liquid phases are around 10-19 J under vacuum conditions. As an example, take two spheres with a radius of 1 cm = 10-2 m at a distance D=0.2 nm in vacuum or air. Then the adhesion force and energy will be, see [13]

( )mN

DRAF H 2

102121010

12 310

219

2 =⋅

==−

−−

(4.21)

At D = 10 nm, the calculation will result in F= 10-6 N, which is much lower. Then the interaction energy is

JD

RAW H 142 10

12−−≈

−= (4.22)

Another example: For two planar surfaces in contact at D=0.2 nm, the adhesive pressure is:

GPaD

Ap H 7.06 3 ≈=π

(4.23)


86

Which is a high pressure! So, it can be seen that adhesive interactions can result in high forces and/or pressures Because W=2γ, this calculation results in a surface energy of 33 mJ m-2, which is the expected value for surface energy of solids or surface tension of liquids.

Figure 4.1: Interaction energy for several geometries [13]

4.2.4 Lifshitz theory: van der Waals forces between macroscopic solids

In the analysis above in section 4.2.3,, pairwise additivity of molecular interactions was assumed. However, the van der Waals force between two molecules will change due to the presence of a third molecule. This means that the assumption of pairwise additivity of molecular interactions is not valid. An example is the change of the polarity. In gases where there are relatively few molecules present interactions effects are small resulting in a validity of the pairwise additivity. But in the case of condensed media (solids and liquids) or bodies interacting in a medium, pairwise additivity does not hold. The Lifshitz theory neglects discrete atomic structures and treats solids as continuous materials with bulk properties like dielectric permittivity and refractive index. Properties as the molecular polarizablity and ionization frequency are replaced by static and frequency


87

dependent dielectric permittivity. For material 1 interacting with material 2 across medium 3 the non-retarded Hamaker constant then becomes:

( ) ( )( ) ( )

( ) ( )( ) ( )

hTk

iiii

iiiihTkA

B

BH

πν

νενενενε

νενενενε

πεεεε

εεεε

ν

2

43

43

1

32

32

31

31

32

32

31

31

1

=

+−

+−

+

+−

+−

= ∫∞

(4.24)

The first term, with static dielectric permittivity of material 1,2 and 3 of respectively ε1, ε2 and ε3 represent the Keesom plus the Debeye energy. These terms are important in water because water molecules have a strong dipole moment. Usually the term with the integral dominates. Here, the dielectric permittivity is not constant, but depends on the frequency of the magnetic field. 7ε1(iν), ε2(iν) and ε3(iν) are dielectric permittivities at imaginary frequencies. The value of ν1 is = 3.9 1013 Hz at 25oC. This means that wavelengths of 760 nm contribute to the Hamaker constant. For frequencies starting in the visible range, the dielectric permittivity can be described by:

( )2

2

2

1

1

11

e

ni

νν

νε+

−+=

(4.25)

If the absorption frequencies of all three materials are assumed to be the same, then the following expression for the non-retarded Hamaker constant is obtained, for two macroscopic phases 1 and 2 interacting across medium 3, with νe= 3 1015 Hz and n the refractive indices

( )( )( )2

322

23

21

23

22

23

21

23

22

23

21

32

32

31

31

283

43

nnnnnnnnnnnnh

TkA

e

BH

+++++

−−

+

+−

+−

≈

ν

εεεε

εεεε

(4.26)

In this equation, kB is the Bolzmann constant1 (kB=1.380 10-23 J K-1) and h is the Planck constant2 (h=6.626 10-34 J K-1). Here, it is assumed that the major contribution of the Hamaker constant comes from frequencies in the visible wavelengths of UV. If there are two identical phases 1 interacting across medium 3 this reduces to:

( )( )2

323

21

223

21

2

31

31

2163

43

nn

nnhTkA eBH

+

−+

+−

≈ν

εεεε

(4.27)

The expressions above are valid for all geometries summarized in Figure 4.1. The following table gives an overview of dielectric permittivity ε, refractive indices n and the main absorption frequency in the UV νe for various solids, liquids and polymers, from [12] Table 4.1: dielectric permittivity, refractive index and main absorption frequency of various solids, liquids and polymers [12] Material ε

[−] n [-]

νe [1015 Hz]

Al2O3 (alumina) 9.3-11.5 1.75 3.2 C (Diamond) 5.7 2.40 2.7 CaCO3 (calciumcarbonate) 8.2 1.59 3.0 CaF2 (fluorite) 6.7 1.43 3.8 KAl2Si3AlO10(OH)2 (muscovite mica) 5.4 1.58 3.1 KCl (potassium chloride) 4.4 1.48 2.5

1 The Bolzmann constant kB is a that relates the average energy of a molecule to its absolute temperature according to E=kBT 2 The Planck constant h is the constant of proportionality relating the energy of a photon to the frequency η of that photon according to E=hη


88

NaCl (sodium chloride) 5.9 1.53 2.5 Si3N4 (amorphous silicon nitride) 7.4 1.99 2.5 SiO2 (quartz) 4.3-4.8 1.54 3.2 SiO2 (silica, amorphous) 3.82 1.46 3.2 TiO2 (titania, average) 114 2.464 1.2 ZnO (zinc oxide) 11.8 1.910 1.4 Acetone 20.7 1.359 2.9 Chloroform 4.81 1.446 3.0 n-Hexane 1.89 1.38 4.1 n-Octane 1.97 1.41 3.0 n-Hexadecane 2.05 1.43 2.9 Ethanol 25.3 1.361 3.0 1-Propanol 20.8 1.385 3.1 1-Butanol 17.8 1.399 3.1 1-Octanol 2.38 1.430 3.1 Toluene 2.38 1.497 2.7 Water 78.5 1.333 3.6 Polyethylene 2.26-2.32 1.48-1.51 2.6 Polystyrene 2.49-2.61 1.59 2.3 Poly(vinylchloride) 4.55 1.52-1.55 2.9 Poly (tetrafluorethylene) 2.1 1.35 4.1 Poly(methyl methaacrylate) 3.12 1.50 2.7 Poly (ethylene oxide) 1.45 2.8 Poly (dimethyl siloxane) 2.6-2.8 1.4 2.8 Nylon 6 3.8 1.53 2.7 Bovine serum albumin 4.0 2.4-2.8 Using the equation above, Hamaker constants can be calculated for several materials, see the table below for calculated values using the simplified equation (6.27) compared with measurements [13]. Table 4.2: Non retarded Hamaker constants for two identical phases interacting Medium 1,2 Medium 3 Dielectric

constant ε Refractive index n

Absorption frequency νe

Equation (4.27)

Exact solution

Experimental value

Water Vacuum 80 1.333 3.0 3.7 3.7,4.0 n-Pentane Vacuum 1.84 1.349 3.0 3.8 3.75 n-Octane Vacuum 1.95 1.387 3.0 4.5 4.5 n-Dodecane Vacuum 2.01 1.411 3.0 5.0 5.0 n-Tetradecane Vacuum 2.03 1.418 2.9 5.0 5.1,5.4 n-Hexadecane Vacuum 2.05 1.423 2.9 5.1 5.2 Hydrocarbon (crystal)

Vacuum 2.25 1.5 3.0 7.1

Cyclohexane Vacuum 2.03 1.426 2.9 5.2 Benzene Vacuum 2.28 1.501 2.1 5.0 Carbon tetrachloride

Vacuum 2.24 1.460 2.7 5.5

Acetone Vacuum 21 1.359 2.9 4.1 Ethanol Vacuum 26 1.361 3.0 4.2 Polystyrene Vacuum 2.55 1.557 2.3 6.5 6.6,7.9 Polyvinyl chloride

Vacuum 3.2 1.527 2.9 7.5 7.8

PFTE Vacuum 2.1 1.359 2.9 3.8 3.8 Fused quartz Vacuum 3.8 1.448 3.2 6.3 6.5 5-6 Mica Vacuum 7.0 1.60 3.0 10 10 13.5 CaF2 Vacuum 7.4 1.427 3.8 7.0 7.2 Liquid He Vacuum 1.057 1.028 5.9 0.057 Alumina Al2O3 Vacuum 11.6 1.75 3.0 (est) 14 Iron Oxide Fe3O4

Vacuum - 1.97 3.0 (est) 21

Zirconia n-ZrO2 Vacuum 20-40 2.15 3.0 (est) 27 Rutile TiO2 Vacuum - 2.61 3.0 (est) 43 Silicon carbide Vacuum 10.2 2.65 3.0 (est) 44 Metals Au, Ag, Cu

Vacuum ∞ - 3-5 25-40 (equation (4.29)

Other data can be found in table 6.2 and table 6.3 of [12]. With the equation (4.26), the Hamaker constant can be calculated for different situations, Further, it cal be calculated whether attraction or repulsion is expected. An attractive van der


89

Waals force corresponds to a positive Hamaker constant, while a negative AH corresponds to a repulsive action. It can be derived from the equation that:

• Van der Waals forces for similar materials are always attractive. It can be seen that substitution of ε1=ε2 and n1=n2 the Hamaker constant AH is always positive

• If two media interact through vacuum (ε3=n3=1), the van der Waals force is also attractive.

• Van der Waals forces between different materials through a condensed phase can be repulsive if medium 3 is more strongly attracted to medium 1 than to medium 2.

The above equation (4.26) is valid for insulating (dielectric or non-conducting) materials. For electrically conductive materials like metals, the static di-electric constant is infinite and the equation above cannot be used. Then, the dielectric permittivity can be approximated as:

( ) 2

2

1νν

νε ei += (4.28)

Here, νe is the so called plasma frequency of the free electron gas and has typically a value of 3-5 1015 Hz. This leads to the following expression for the Hamaker constant for the case of two metals interacting across a vacuum:

JhA eH19104

2163 −⋅≈≈ ν (4.29)

Hamaker constants calculated from spectroscopic data can be found in many publications. It can be seen from the values above that Hamaker constants of metals and metal oxides can be up to an order of magnitude higher than those for non-conducting media. In the case of aqueous media, dissolved ions will affect the Hamaker constant because ion hinder the water molecules in their hydration shell form orienting in an external electric field. This results in a change in the first term of the Hamaker constant.. Further, the salt concentration is often much higher at surfaces than in the ‘bulk’ of the solution. So, the dielectric constant can be smaller than in the bulk phase.

4.2.5 Working with Hamaker constants Approximate values for unknown Hamaker constants can be obtained in terms of known values [13]. If A132 is the (non retarded Hamaker constant for media 1`and media 2 interacting across medium 3, then it can be shown to known values of A131 (the Hamaker constant of material 1 interacting through medium 3 with itself) and A232 (material 2 across medium 3 with itself) by

232131`132 AAA ±≈ (4.30) This also gives that

2211`12 AAA ≈ (4.31) Where A12 is the Hamaker constant for media 1 and 2 interacting through vacuum (no medium 3) and A22 is the Hamaker constant for Medium 2 interacting with itself through vacuum. Other useful relations are:

( )2

3311133311313131 2 AAAAAAA −≈−+≈≈ (4.32)

Combining equation (4.30) with equation (4.32) gives: ( )( )33223311132 AAAAA −−≈ (4.33)

These (approximate) relations are only applicable for materials where dispersion forces are dominant. For media with high di-electric constants like water, these approximation relations are not inaccurate. It is recommended to use equation (4.26) directly in order to avoid erroneous results.


90

4.2.6 Surface energy The (reversible) amount of energy required per unit area of the created surface under isothermal conditions is called the surface energy and is represented by the symbol ∆γ and is typically expressed in [J m-2]. The separation of two solids (this may be two different solids or the formation of a surface in a solid, for example by a crack) comes together with the formation of a surface. This can be illustrated with the following figure.

Figure 4.2: Surface energy as the increased energy of surface atoms

The figure shows that two surfaces that are brought into contact until the point that the atoms of the two interacting surfaces come into each other sphere of influence. At the first stage, surface atoms of the lower body in the picture only ‘feel’ attractive surfaces from all sides except from the surface. This means that atoms at the surface have an increased energy compared to atoms in the bulk of the solid. The increased energy of atoms at a surface compared to atoms within the bulk of the material is represented by the surface energy γ. If in the picture two surfaces are getting closer, at a certain point attractive forces will develop between atoms of both bodies as discussed above. This means that atoms of the counter surface will tend to perform work. This work is equal to maximally equal to the work of adhesion Γ. If the surfaces ‘touch’, the surface atoms will be in more or less the same situation as atoms within the bulk of the solid. However, the surface may still have a certain energy per unit area if both solids are dissimilar. This energy is called the interfacial energy and is represented by the symbol γab. Although the above figure suggests that the structure of the solid remains intact if both bodies are separated, in reality several phenomena will take place: • Real surfaces are not atomically flat but have a certain surface roughness. So, when two

bodies are separated, generally not the whole surface will be in actual contact. Surface roughness is very important for adhesion phenomena. Besides this, many finishing processes result in more or less amorphous surface layers instead of the ‘nice’ crystal structure on the figure above

• Besides the effects above also the following atomistic processes take place at the solid surface:

o Surface relaxation: Because of the ‘missing’ interaction forces for the atomic layers at the surface, the distance between the atomic planes will be slightly higher for these layers.

o Surface reconstruction. For the same reason, the uppermost atoms will leave their positions according to the crystal structure and find new positions with a low


91

energy level, which will deviate from the positions according to the crystal structure.

• In a non-vacuum environment, also interaction will the environment will take place: o Oxide layers may be formed on the surface o Gas molecules may become attached to the surface by physical adsorption o Gas molecules may become attached to the surface by chemical adsorption. o Condensation, a special case of physical adsorption, may take place

So, generally the interface will have a certain amount of energy Besides the phenomenon of surface energy, in the following we will also encounter surface tension. Before we will discuss the phenomenon of surface energy and surface tension, first a ‘formal ’definition will be given and some remarks will be made: The surface energy γ of a solid or a liquid in the presence of a gas is defined as the work γdA needed to create reversibly and isothermally an elemental area dA of new surface in equilibrium with the medium [5] and has units [J m-2]. Among others, the following comments on surface energy are made in [5] and are repeated here:

• Only the work needed to create the surface is relevant, and work in the bulk is not taken into account.

• Surface energy is related to the creation of a new surface with equilibrium inte-atomic distances. Surface tension is related to stretching of an already formed surface.

• Atomic planes close to the surface has a modified distance between the atomic planes. This is caused by the fact that atoms at the surface ‘feel’ attractive forces from all sides except from the surface. A modified distance between the atomic planes minimizes the energy of the system and is therefore energetically favorable..

• The surface remains in equilibrium with the gas. This means that any creation of the surface is followed by adsorption of gas molecules at the surface.

In this handout, the term surface tension will be reserved for fluids and the term surface energy will be reserved for solids.

4.2.7 Hamaker constant, Work of adhesion and surface energy Based on the Hamaker constant, an estimate for the work of adhesion can be made for the case of two molecular crystals. If a crystal is cleaved in two parts and separated by an infinite distance, the work required per unit area can be obtained by pairwise summation of interaction energies of individual atoms with the atoms of the other medium. This gave the following expression for the interaction energy of two identical media, see section 4.2.3:

2012 D

AW H

π= (4.34)

With D0 the distance between two planes. Suppose the interface is formed by breaking a crystal into two parts. Then the work to separate the two surfaces (the interaction energy) is equal to the energy of the two surfaces which are formed.

2024

2D

AW H

πγγ =⇒= (4.35)

Often for D0, a value of 0.165 nm is taken, which results in values in agreement with experiments [13]. In this calculation it is assumed that energy is not consumed by e.g. visoelastic processes. The following table gives an overview of theoretic AH values and measured versus experimental values for the surface energy γ, using the above mentioned value for D0


92

Table 4.3: Experimental and theoretical values for the surface energy for several metals [12]

Material Theoretical AH

2024 D

AH

πγ =

D0=0.165 nm

Experimental

Liquid helium 0.057 0.28 0.12-0.35 n-Pentane 3.75 18.3 16.1 n-Octane 4.5 21.9 21.6 Cyclohexane 5.2 25.3 25.5 n-dodecane 5.0 24.4 25.4 n-Hexadecane 5.2 25.3 27.5 PFTE 3.8 18.5 18.3 CCl4 5.5 26.8 29.7 Benzene 5.0 24.4 28.8 Polystyrene 6.6 32.1 33 Polyninyl chloride 7.8 38.0 39 Acetone 4.1 20.0 23.7 Ethanol 4.2 20.5 22.8 Methanol 3.6 18 23 Glycol 5.6 28 48 Glycerol 6.7 33 63 Water 3.7 18 73 H2O2 5.4 26 76 It can be seen that calculated and measured surface energies are in good agreement using the approach mentioned above. Only for high polar H-bonding liquids, which is the lower part of the table, the theoretical value is an underestimation. Further, the equation seems to be reliable within 10-20 %. So, the following approximate relation is valid for the Hamaker constant and the surface energy for the case of non H-bonding liquids:

γ21101.2 −⋅=HA (4.36) With γ in mJ m-2. Using values of Table 4.2, surfaces energy of metals around 200 mJ m-2 would be calculated. This value is much higher than non-metallic compounds but is an order of magnitude lower than typically measured values of 400 to 4000 mJ m-2, see the following table: Table 4.4: Surface energy of metals [13]

Material Transition temperatures Surface energy Boiling point

TB [K] Melting point TM [K]

Just above TM [mJ m-2]

Just below TM [mJ m-2]

At 300 K [mJ m-2]

Aluminium 2543 931 700 800 1100 Silver 2223 1233 1000 1200 1500 Copper 2603 1356 1300 1600 2000 Iron 2773 1803 1500 1800 2400 Tungsten 5273 3653 2500 3600 4400 Silicon 2623 1683 750 1100 1400 Ice 373 273 75 110 71 This means that attractive forces between metals are not only caused by van der Waals forces. The strong adhesion is explained by electron exchange between the contact partners, so the formation of metallic bonds. For two similar metallic surfaces the following phenomenological expression for the interaction potential per unit area is given [13]:

( ) ( ) MDD

M

eDD

DW λ

λγ /0 012 −−

−−−= (4.37)

Here, λM is a characteristic decay length for metals.


93

4.2.8 The Derjaguin approximation for van der Waals forces In equation (4.15) the van der Waals force between two spheres was calculated and in equation (4.18) the van der Waals force between two planar surfaces was calculated by integration. These calculations gave relatively simple expressions for these geometries. For other geometries with analytical expressions for the interaction energy, see Figure 5.1. However, in general, volume integrations can lead to difficult relations. The Derjaguin approximation is a relation between the energy per unit area w between two planar surfaces separated with a gap of width x and the energy between two bodies of arbitrary shape W which are at distance D according to [12]:

( ) ( )∫∞

=D

dAxwDW (4.38)

Here, the integration is over the entire surface of the solid. In cylindrical coordinates, which are useful for rotation symmetric situations, this equation reads:

( ) ( )( )∫∞

=D

rdrrxwDW π2 (4.39)

Or in terms of the x- derivative of the cross sectional area

( ) ( )∫∞

=D

dxdxdAxwDW (4.40)

This approximation is only valid is the decay length of the surface force is small in comparison with the curvature of the surface. The approximations above are often called the Derjaguin approximation, who used this approach to calculate the interaction between two ellipsoids. Similarly, the force can also be calculated:

( ) ( ) ( )∫∫∞∞

==DD

dxdxdAxfdAxfDF (4.41)

Example 1: van der Waals force between a cone and a planar surface

Figure 4.3: Interaction between a cone and a planar surface, from [12]

The area of the cross section A is given by: ( )( )2tanαπ DxA −= (4.42)

This gives for dA/dx:

( )DxdxdA

−= απ 2tan2 (4.43)

For the force F, a similar equation as the equation for the interaction energy W, equation (4.41), is valid. Substitution in gives:

D

x

r

α


94

( ) ( ) ( )

∫

∫∫∞

∞∞

−=

=−−==

D

H

D

H

D

dxx

DxA

dxDxx

AdxdxdAxfDF

3

2

23

3tan

tan26

α

αππ

(4.44)

Integration gives:

( )

DA

DDA

xD

xADF

H

H

D

H

6tan

211

3tan

21

3tan

2

2

2

2

α

αα

−

=

+−=

+−=

∞

(4.45)

Example 2: van der Waals force between two identical spheres For the case of two identical spheres separated with distance D, radius R and coordinates x and r, the situation is shown below:

Then, the geometrical relation x(r) is given by

( )

dxrRrdr

drrR

rdxrRRDrx

22

22

22

2

222

−=

⇒−

=⇒−−+= (4.46)

If the range of the interaction forces is much smaller than R, then only the outer two caps of the interacting spheres needs to be taken into account, because only interactions with a small r are effective. Simplification of the equation above leads to

RdxdxrRrdr ≈−= 222 (4.47) And substitution in equation (4.39) gives:

( ) ( )( ) ( ) ( )∫∫∞∞

=⇒=DD

dxxwRDWrdrrxwDW ππ2 (4.48)

Then, the force is given by the following equation, because w(∞)=0

( ) ( )DRwdxxwdDdR

dDdWF

D

ππ =

−=−= ∫

∞

(4.49)

Because the van der Waals energy per unit area between two infinite flat solids with gap D is given by:

212 DAw H

π−=

(4.50)

This gives:

R

R

x

r D


95

212DRAF H−=

(4.51)

A fundamental consequence of the approximation of Derjaguin is that the force or energy between two bodies depends on the shape, on material properties and on the distance between the bodies. However, the force, or the energy, can be divided into a geometrical factor and a material- and distance dependent term w(x). So, the interaction can be described independent of the geometry by w(x). This w(x) can be of van der Waals nature as described above. However, also other forces can contribute to this, like the electrostatic double layer force.

4.2.9 Attractive and repulse forces between atoms Van der Waals forces generally vary inversely to the 7th power with the distance between the atoms because f(D)=-dW/dD and W(D) is proportional with D to the 6th power. So:

7

1r

F svanderWaal ∝ (4.52)

The adhesive force will not tend towards infinity according to the relation mentioned above with decreasing distance between the atoms or molecules. It turns out that besides a bonding effect by van der Waals forces, repulsion can be observed if electronic clouds of atoms or molecules which come very near start overlapping. It can be derived that these repulsion forces:

nrepulsion rF 1

∝ (4.53)

With n>10. Repulsion forces between the surfaces is a result of compressive forces between the solids. With increasing distance between the surfaces, the repulsive forces will decrease in strength. This means that the attractive forces will ‘win’ from the repulsive forces at a certain interatomic distance. At a certain moment the attractive and the repulse forces will be in balance and the resulting force on the solids will be equal to zero. This distance, often called zo, is then an equilibrium separation between the atoms. If the distance is larger than zo, the atoms will attract each other. The distance zo has about the same value as the interatomic distance. It can be calculated that at a distance of 1.11z0 the attractive force has its maximum. The result of competing attraction and repulsive forces can be represented in a so called force-separation curve. An example of a force-separation curve is shown in the picture below. An empirical equation representing this force separation curve is called the Lennard Jones potential, or more accurately, the force which can be derived from the Lennard Jones potential. As told above, a force is required to separate the surfaces if adhesion is taking place. So if two solids are being separated work has to be performed, remember that ‘work’=force times distance. The work that has to be performed in order to separate two surfaces is generally called ‘work of adhesion’ and is represented by the symbol Γ.


96

Force separation relation

-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3 3.5

z/z0 [-]

F/Fm

ax

Figure 4.4: A typical force - separation curve

4.3 Surface energy & Surface tension

4.3.1 Surface tension & Surface energy Consider the case of a membrane of liquid, drawn out of soapy water by a ring-shape wire. The force required to support the membrane per unit width is (by definition) called the surface tension σ and has units [N m-1].

Figure 4.5: A liquid membrane

So, for a circular wire with radius r, the total ‘length’ of the soap membrane is equal to two times the circumference of the wire (the membrane has a font- and a backside), so

rrL ππ 422 =⋅= (4.54) This means that the force to sustain the membrane is, by definition:

σπrF 4= (4.55) If the surface is extended by moving the wire from the surface of the liquid with amount ∆x, then the length increase is given by the following expression, assuming that the circumference of the membrane stays the same:

xrL ∆=∆ π4 (4.56)


97

Now, a surface has been ‘generated’. This surface has a certain surface energy γ, so the energy of the generated surface is 4πrγ∆x. For this case, the work performed (per unit length) is

γπ

γπ=

∆∆

=∆

xrxr

LxF

44 (4.57)

So, this means that the surface tension σ and the surface energy per unit area γ are identical for the soap membrane considered. Soap bubbles can exist because the surface layer of a liquid—in this case water—has a certain surface tension, which causes the layer to behave as an elastic sheet. A common misconception is that soap increases the water's surface tension. Actually soap does the exact opposite, decreasing it to approximately one third the surface tension of pure water. Soap does not strengthen bubbles, it stabilizes them, via an action known as the Marangoni effect. As the soap film stretches, the concentration of soap decreases, which causes the surface tension to increase. Thus, soap selectively strengthens the weakest parts of the bubble and tends to prevent them from stretching further. In addition, the soap reduces evaporation so the bubbles last longer [6]. For solids, the situation is different. Here, the surface can be stressed because of elastic strain. This means that not only energy is used to create the surface, but also energy is stored to strain the material. The reasons is that in the case of solids, atoms are less able to ‘take a place’ at the surface because the bonds between the atoms are stronger. Or in other words, liquids cannot sustain elastic strains because atoms rearrange to relax elastic strains. If motion of atoms in the condensed phase is sufficient to allowing minimalization of surface energy, the system will try to minimize the product γA. For the liquid soap membrane, γ is fixed and A will be minimized. This is why at further lifting of the wire from the soap surface, the membrane snaps to the ring wire. When blowing a bubble, the area is minimized by the formation of a spherical bubble. In the case of solids, surface energy tries to minimize itself by surface diffusion. For example in the case of growth of a coating on the surface, both A, which depends on the surface topography and γ can be varied. In the case of solids, also they will be characterized by a crystal structure, which may for example be face-centered-cubic, body-centered cubic or hexagonal. Then, the situation of a so called anisotropic γ exists. For example, in the case of an fcc lattice, the (111) plane is close packed and has the lowest surface energy. In he case of a solid with ionic bonds (e.g. NaCl), a low surface energy will be present if an equal number of Na and Cl at the surface because then the surface is nonpolar, In the case of layered solids’ (graphite, MoS2), there are no chemical bonds between the layers. Then, the layers have lowest surface energy. This is the principle on which dry lubricants are working. In real solid surfaces the surface energy will be lowered by so called reconstruction. This means that the atoms at the surface have distorted positions compared to their bulk lattice positions in order to relax their energetic situation. Besides this, also adsorption of ‘passivating’ monolayer of element, resulting in dangling bonds become terminated bonds will lower the surface energy. This last phenomenon is more effective than reconstruction, because it causes less strain in lattice. An important application involving capillarity and drop formation is inkjet printing technology. Here, ink is present in a small square chamber. At the ‘outlet’, a meniscus is balanced by the pressure in the chamber and the capillary force. In the wall of the chamber, a heating device is present which can reach temperature gradients op to 108 oC s-1. Then, very quickly a vapor bubble is grown which ejects a drop through the ‘outlet’. After that, the chamber is refilled. This process is repeated at a frequency of 8000 Hz.

4.3.2 Young-Laplace equation Now, let us consider a spherical soap bubble. This soap bubble has amount γ energy per unit area. The bubble is spherical because this is the minimum area for a certain amount of enclosed volume. The total surface energy of the sphere is therefore equal to 4πr2γ. If the


98

radius of the bubble would become larger with dr, then the total surface energy would be 4π(r+dr)2γ, which is higher. This means that, if higher order terms are neglected, the surface energy has increased with 8πrγdr. Similarly, decreasing r will decrease the surface energy. This means that a pressure difference over the membrane should be acting in such a way that, when decreasing the bubble with dr, a work against the pressure difference, which is equal to ∆p4πr2dr, is performed which is just equal to the decrease in surface energy. This means that:

rpdrrdrrp γ

γππ284 2 =∆⇒=∆ (4.58)

So, a smaller bubble will mean a higher pressure inside the bubble. The expression above is a special case of the so called Young-Laplace equation which reads:

+=∆

21

11RR

p γ (4.59)

In large systems, also the hydrostatic pressure has to be taken into account. Then, the equation is extended by an extra term:

ghRR

p ργ +

+=∆

21

11 (4.60)

The Young-Laplace equation has some implications [12]: • If the shape of the liquid surface is known, then the pressure difference can be

calculated • If gravity is neglected: then the pressure in the liquid is everywhere the same. So, Dp

is constant. Then the Young-Laplace equation shown that the curvature is the same everywhere.

• Using the Young-Laplace equation it is possible to calculate the equilibrium shape of the surface. If the pressure difference is known as well as some boundary conditions, then the geometry of the liquid surface can be calculated.

This important equation is the basic equation of capillarity and is used to measure the surface energy of fluids as will be discussed later.

4.4 Surface energy & Work of adhesion

4.4.1 Work of adhesion Adhesion is the phenomenon by which two contacting materials form a region of adhesive bonds. This region is able to sustain, to a certain amount, tensile stresses. This means that a force is required to separate the two surfaces. This force is called the adhesion force, and the work that has to be performed is called the work of adhesion Γ. In the next figure two surfaces are shown, respectively surface a and surface b. Both surface can be considered to have a surface energy γa respectively γb. If these to surfaces are brought into contact, the interface will still have a certain interfacial energy. This interfacial energy is called γab. The work that has to be done in order to separate the two surfaces will be called Γab and can be expressed in terms of surface and interfacial energies:

abbaab γγγγ −+=∆=Γ (4.61) So, the work that has been done to separate the solids is the energy needed to create the two surfaces, which is γa +γb, minus the energy of the original interface, which was γab. In the case of identical surfaces, this equation gets the following form:

γγ 2=∆=Γab (4.62) The work of adhesion Γab has units [Nm/m2] or [N/m].


99

Figure 4.6: Surface energy, interfacial energy and the work of adhesion

4.4.2 Surface energy of solids As discussed above, Γ is the work required to separate two solids. Unfortunately, Γ is not easy to measure for the case of two contacting solids because if two solids are separated also elastic recovery of the two bodies will ‘release’ energy and influence eventual measurements. But how can we measure the work of adhesion between to solids? A possibility would be to measure the work of adhesion by measuring the components which together form the work of adhesion, see equation (4.61). However, also the surface energy of solids is also not so easy to measure directly. The surface energy is strongly related to the type of bonds between the atoms by which the solid is formed. The general expression for the work of adhesion was, as discussed above:

abbaab γγγγ −+=∆=Γ (4.63) Metallic materials and materials with covalent bonds like diamond have a high surface energy of 1000-3000 mJ m-2. Ionic bonds are weaker and materials with ionic bond have also a lower surface energy in the order of 100-500 mJ m-2. Molecular crystals experience van der Waals forces and still have lower surface energies in the order of 100 mJ m-2. If adsorption occurs at the surface, the short range forces caused by short range bonds like covalent or ionic bonds are ‘covered’. So a metallic surface in air will have a very low surface energy corresponding to van der Waals interactions between the adsorbed surfaces. Typical values for the surface energy of some molten metals elements are given in the following table. Obviously, these materials are characterized by metallic bonds in the solid state. Table 4.5: Surface energy of metal elements in the molten state

Metallic elements T [oC]

γ [ mJ m-2]

E [GPa]

σy [GPa]

Al 660 900 63 1.1 Melting 660 oC 800 850 1000 830 1830 680 Cu 1150 1370 120 3.2 Melting 1083 oC 1550 1265 Au 1063 1120 81 2.1 Melting 1063 oC Fe 1534 1500 204 2.5 Melting 1534 oC 1700 1760 Mg 650 560 44 1.5 Melting 650 oC 700 542 740 528 Ni 1053 1700 208 3.2 Melting 1053 oC 1850 1620 Ag 961 920 78 2.0 Melting 961 oC 1200 876

bγ

aγabγ


100

Sn 232 570 44 0.15 Melting 232oC 600 525 Zn 420 790 91 1.3 Melting 420oC Hg 25 465 150 439 300 402 Pb 350 440 800 412

The table below gives surface energies measured of solid metals Table 4.6: : Surface energy of metals in the solid state

Metallic elements

T [oC]

γ [ mJ m-2]

Athmosphere

Al 180 1140 Vacuum Cu 1006 1720 Helium,

Hydrogen Au 970 1450 Vacuum 1025 1400 Helium 968 1390 Air Fe 1460 1910 Argon 1410 2320 Hydrogen Zn 480 830 Helium Pb 309 560 Vacuum,Argon

And the surface energies of some non-metallic, characterized by ionic bonds, are given by: Table 4.7: Surface energy of some ceramics, characterized by ionic bonds

Non-metals γ [mJ m-2]

ZrO2 530 TiC 900 Al2O3 740

The above values are for pure materials. Typical values for interactions involving van der Waals forces are 10-50 mJ m-2.

Figure 4.7: Mututal solubility of metallic elements

W Mo Cr Co Ni Fe Nb Pt Zr Ti Cu Au Ag Al Zn Mg Cd Sn Pb InInPbSnCdMgZnAlAgAuCuTiZr Same metalsPt Extensive solid solubilityNb Limited solid solubilityFe Some liquid v. lowsolid solubilityNi No liquid or solid solubilityCoCrMoW


101

Such values for the surface energy could be expected for polymeric surfaces or metallic surfaces covered with absorbed layers etc. As discussed above, surface energy is related to inter-atomic forces in solids. This is the reason that relations between the surface energy and mechanical properties of solids (like hardness and elastic modulus) can be expected because mechanical properties are also dependent of inter-atomic forces. Rabinowicz found an empricial relation between the hardness of fully deformed metals and the surface energy:

31

HC ⋅=γ (4.64)

4.4.3 Interfacial energy: Bondi rules Adhesive forces between metals are dependent on the mutual solubility of a metal couple in the case that two dissimilar metals are contacting. If two metals have high mutual solubility, adhesion forces between the metal couple will be strong. This means that for such a metal couple the interfacial energy is low and the work of adhesion, needed to separate the surfaces, high. If the mutual solubility of a metals couple is low in the solid state, the interfacial energy is high and the work that has to be performed in order to separate the surfaces will be low. A graphical overview of the mutual solubility of metal coupled is given in the next figure. Rabinowicz gave in 1972 a semi-quantitative expression of the Bondi-rules discussed above. According to Rabinowicz, the interfacial energy can be expressed in terms of the surface energy of two metals in the following way:

( )baab γγαγ += (4.65) This given for the work of adhesion:

( )( )ba γγα +−=Γ 1 (4.66) For metals with a high mutual solubility, α<0.1, for limited solid solubility 0.4<α<0.6 and for non-metals generally α>0.7.

4.5 Measurement techniques for fluids characterization For the measurement of surface tension in fluids some methods are available. Many methods are based on the Young-Laplace equation discussed above and shown again below:

+=∆

21

11RR

p γ (4.67)

Capillary rise This equation can be applied to the case of capillary rise of a fluid. If the two radii of curvature are equal, so a hemispherical meniscus, then 1/R1=1/R2. Then, ∆p in that equation should equal the hydrostatic pressure drop in the liquid column.

Figure 4.8: Capillary rise

This means that:

h


102

Rhg

a

Rgh

=∆

=

⇒=∆

ργ

γρ

2

2

2 (4.68)

With ∆ρ the difference in density between the gas and the liquid. The factor a is called the capillary constant or the capillary length. In reality, the meniscus will deviate from the hemispherical shape. Then at each point at the meniscus, the meniscus will have such a shape that

gyp ρ∆=∆ (4.69) With y the local height of the meniscus above the liquid surface. By writing the Young –Laplace equation for each point, it can be shown that:

( )( )

( )( )( ) ( )

=+

+=∆⇒

∆=∆

+=∆

=

+=

+

+ 2

23

2

2

2

23

2

2

122

21

12

21

1

111

1

1

1

dxdydx

dy

x

dxdy

dxdy

dxyd

x

dxdy

dxdy

dxyd

Rgy

gyPRR

P

R

R

σρ

ρ

σ

(4.70)

So, the shape of a liquid surface is determined by a differential equation, which can be solved using boundary equations. By solving the equation above, it can be shown that the weight of the liquid column in the capillary is given by:

θσπ cos2 RW = (4.71) Physically, the equation above can be understood by realizing that the supported weight of the ‘hanging fluid’ is equal to the vertical component of the surface tension σ, so σcosθ, multiplied with the circumference of the capillary cross section 2πr. Based on capillarity, many measurement methods have been developed to determine σ. Such methods are called capillary rise methods. The capillary rise method is generally considered to the most accurate method to measure the surface energy γ (or in fact surface tension σ) of a fluid. measure the surface energy or the surface tension:

• The maximum bubble pressure method • Detachment methods • Methods based on the shape of static drops and bubbles • Dynamic methods of measuring surface tension

These measurement methods will be briefly described below, see also [11]. The maximum bubble pressure method. By the maximum bubble pressure method, bubbles of an inert gas are blown under the surface of the liquid under consideration. The pressure required to ‘push the bubble out’ of the capillary is used to calculate the surface tension by the Young-Laplace equation.


103

Figure 4.9: Maximum bubble pressure method If the pressure in the tube is increased, the bubble is pushed out more and more, until bubbles are blown which are shaped according to a section of a sphere. If the liquid does not wet the material of the tube, then the the radius r will be equal to the inner radius of the tube. If the liquids wet the tube, the r will be equal to the outer radius of the tube. If the gas pressure is increased, at a certain moment the bubbles are unable to grow and break away from the tube. At this moment, the pressure difference over the surface of the liquid is given by:

hppp −=∆ maxmax (4.72) With pmax the maximum pressure measured and ph the hydrostatic pressure of the water column with height h. This gives:

ghp ρ∆=∆ max (4.73) Then, using equation (4.68), the surface tension can be calculated:

2maxpr∆

=σ (4.74)

The expressions above are only valid for sufficiently small tubes. Detachment methods The surface tension of liquids can also be measured by detachment of a solid surface from a liquid surface. There are different configurations possible, like measurement of the force to detach a ring, thin plate or wire from the surface of the liquid. Another possibility is the weight in a drop falling from a capillary tube. The last method will be discussed in the next section. The drop weight method In this method, drops of liquid are formed at the end of a capillary tube. In this way several drops are generated, and grown until they detach from the tube. As long as the drop hangs at the end op the capillary tube, the weight of the drop is more than balanced by the surface tension. At the moment that the drop detaches from the capillary, then the gravity is equal to the surface tension times the circumference of the drop, so:

σπrmg 2= (4.75) This method has been first used by Tate (1864) and the equation above is also called Tate’s law. So by measuring the mass of the fallen drop (or the average mass of multiple fallen drops) and with a known value for the radius r, the surface tension σ can be calculated. In experimental practice, falling drops can be collected and the weight is measured. In this way, the weight of a single drop can be measured if the number of drops is counted. The accuracy of the method is limited. For example, the drop will not be exactly spherical, but shows a cylindrical neck at the top. The accuracy of this method can be increased by correction factors f.

frmg σπ2= (4.76)

Not wetting,

Wetting


104

The methods above are not commonly used for measurement of surface tension liquids. Common devices are the ring tensiometer, also called Du-Noüy tensiometer and the Wilhelmy slide method. Contrary to the methods above, these methods are based on the measurement of forces instead of the measurement of pressures. Such methods involving the measurement of forces are called tensiometric methods. Ring tensiometer In the ring tensiometer, the force required to detach a ring from the surface of a liquid is measured. The force required to detachment is given by:

( )σπ ai rrF += 2 (4.77)

Figure 4.10: Ring tensiometer method

A requirement for the ring tensiometer is that the ring is completely wetting. For this, often a platinum wire is used. Also this method is typically applied including correction factors. Wilhelmy slide method A widely used method to measure surface tension of fluid is the Wilhelmy slide method. Here, a thin plate of glass, platinum or filter paper is placed vertically with the bottom hanging in the fluid to be measured.

Figure 4.11: Wilhelmy slide method

In this method, the weight of a thin plate, like a glass cover plate for a microscope, is measured when a liquid meniscus is supported. The weight can be measured by a static force measurement of the force required to prevent that the plate is drawn in the liquid or by detachment as in the cases above. The total weight of the place including the meniscus is given by the following equation, assuming zero contact angle:

LWW plate σ+= (4.78) In this equation, L is the circumference of the meniscus, so for a plate with length l, L=2l. In the equation above, the weight of the liquid directly under the plate is neglected. It can be shown that this measurement method is about 0.1% accurate. No correction factors are required for the Wilhelmy slide method.

ra

ri


105

Alternatively, it is possible to raise the level of a liquid gradually until a plate vertically hanging on a balance is reached. The increase in weight is noted, Values for the surface tension of some common fluids and other materials are summarized in the following table, see also [11]: Table 4.8: Surface tension of some common fluids

Fluid Temperature T σ [mN m-1] Water 20 oC 72.94 Ethanol 20 oC 22.39 Toluene 20 oC 28.52 Glycerine 20 oC 48.09 Ether 25 oC 20.14 N2 75 K 9.41 H2 20 K 2.01 Hg 20oC 486.5 CH4 110 K 13.71

The surface tension is typically decreasing for increasing temperature. The next table gives an indication for water. Table 4.9: Surface tension of water at different temperatures

Fluid Temperature T σ [mN m-1] Water 0 oC 75.50 Water 10 oC 74.40 Water 20 oC 72.88 Water 30 oC 71.20 Water 40 oC 69.48 Water 50 oC 67.77 Water 100 oC 58.91 Water 200 oC 37.77 Water 300 oC 14.29

4.6 Measurement techniques for solids characterization

4.6.1 Young-Dupré equation The practical way by which the work of adhesion is measured is by measuring the contact angle of a drop of fluid with a known surface energy on a substrate. Suppose we would like to measure the work of adhesion to separate interfacial energy between two metals. If a drop of molten metal or fluid a is put on metal b, the drop will get a certain shape. Then, the contact angle β of a drop of fluid is measured. This drop can be a drop of liquid or a drop of molten metal. The surface energy of metals can be used (by approximation) by using molten metal drops as will be explained below. With metals, the error that is made by measuring the surface energy just above the melting point instead of at room temperature is relatively small.


106

Figure 4.12: Graphical representation of the Young - Dupré relation

From force balance in the horizontal direction between the drop and the interface the following it follows that:

0cos0 =−−⇒=Σ abbaxF γθγγ (4.79) Or more precisely, the surface energy terms γa and γb are in fact also interfacial energies. Property γa is the interfacial energy between the solid and the vapor denoted by γsv and γb is the interfacial energy between the liquids and the vapor, denoted by γlv. This gives the following equation:

0cos =−− sllvsv γθγγ (4.80) This equation is called the Young’s equation or the Young-Dupré equation (1805). As is shown, this simple equation can be derived from force balance in the horizontal direction. It however also is related to minimum free energy of the drop but this will not be further discussed. Combining the Young-Dupré equation with the equation of the work of adhesion gives the following relation between the surface energy of the drop and the work of adhesion Γab

)cos1(0cos

θγγθγγ

γγγ+=Γ⇒

=−−

−+=Γbab

abba

abbaab (4.81)

So, if the surface energy of the drop is known and the contact angle θ is measured, the work of adhesion between the substrate and the liquid of the droplet can be calculated. A series of liquids with different liquid-vapor interfacial energies are used to measure several values for θ. In practice, if the work of adhesion between various liquids and solids is known, then surface energy can be estimated if the solid-liquid interfacial energy is assumed to be small by extrapolating the results to θ = 0, which is also referred as the critical surface tension σc, Using this method, initially performed by Zisman and coworkers many contact angle measurements are performed using a set liquids with different values for the surface tension σ on several organic polymers. When the (measured) values for the contact angle θ where plotted against the (known) values of the surface tensions σ of the liquid, a nearly linear plot was obtained. Based on these measurement, a critical surface tension for the solid can be defined, when the line fitted in the graph crosses cos θ=1, see Figure 4.13. Zisman found that σc varies systematically with the chemical constitution of the solid substrate.

θ

γb

γa γab


107

Figure 4.13: Critical surface tension

However, this technique cannot be extended to the measurement of interfacial energies between two solids. As is clear from this equation: The work of adhesion is only a function of the surface energy of the drop and the contact angle β. Knowledge of the surface energy of the solid γa and the interfacial energy γab is not necessary for obtaining an expression for the work of adhesion. However, the work of adhesion is composed of the surface tension of the liquid, the surface energy of the solid and the interfacial energy. The surface energy and the interaction energy are impossible to measure using the Young-Dupré equation alone. The picture below shows two situations: In the left figure β is high and in the right picture β is low. So, the left case is related to a high interfacial energy and the right case to low interfacial energy between bodies a and b. In the case of water, the first case would be a hydrophobic surface like wax and the second case a hydrophilic surface like a contact lens

Figure 4.14: A non-wetting solid and a wetting solid

It has been illustrated above how the work of adhesion can be measured using the Young-Dupré equation. For measuring the interfacial energy, the surface energy of one of the bodies needs to be known, and it was explained that the surface energy of solids is generally not easy to measure. There are several ways to measure the surface energy of a body. Besides the use of the Young-Dupre equation, there are not many methods to measure the surface energy of solids that are generally applicable. Three methods will be discussed in the following: Semi-empirical relations. Semi-empirical relations exist where the interfacial energy is related to other components. Such equations generally have limited applicability. An example of a semi-empirical relation between the surface energy of the liquid γL, the surface energy of the solid γS and the interfacial energy γSL is given by the following relation from Li and Neumann:

2)(2 SLeSLSLSLγγβγγγγγ −−⋅⋅−+= (4.82)

θ θ

cos θ

σ [J m-2]

1


108

In this equation β = 0.0001247 [m2/mJ]2 This equation relates the interfacial energy of an interface between a liquid and a solid to the surface energy of the liquid and the surface energy of the solid. Combined with the Young’s equation, this gives:

2)(21cos SLeL

S γγβ

γγ

θ −−⋅+−= (4.83)

So by measurement of the contact angle with a known fluid, the surface energy can be estimated. Further, several measurement methods exist where the surface energy is divided into several components. These will be discussed in the next section. As a third alternative, Based on the assumption of a contact model including adhesion, called the Johnson, Kendall, Roberts (JKR) theory. This model will be explained later in the next chapter. Using these techniques, the contact area (by means of contact radius or contact length) and the applied load are measured. Several devices have been developed for this purpose, namely the so called JKR apparatus and the surface force apparatus (SFA). One of the main drawbacks of these methods is that the JKR theory should be valid for the material combination studied. Measurement methods based on drops of liquids on a solid surface are often called sessile drop methods

4.6.2 Analysis using contact angle measurements Good-Girifalco-Fowkes combining rule Girifalco, Good and Fowkes considered solids and liquids where molecules are held together by van der Waals forces. The Good-Girifalco-Fowkes combining rule for two apolar materials is only given by van der Waals, or Lifschitz-van der Waals forces (LW):

LWLWLWLWLWLWLW2121

2

2112 2 γγγγγγγ −+=

+= (4.84)

Which gives for the relation between contact angle θ and the surface energy of the solid γs the following equation:

LWL

LWs γγθ 2cos1 =+ (4.85)

This equation is sometimes called the Young-Good-Girifalco-Fowkes equation. In the case of polar interactions next to the non-polar interactions described above, extra components have to be added. Now the surface energy is expressed as:

ABLW γγγ += (4.86) Where the LW component represents the apolar or Lifshitz-van der Waals electrodynamic forces (LW), in its turn composed of randomly oriented permanent dipole–permanent dipole (orientation) interactions (Keesom), randomly oriented permanent dipole–induced dipole (induction) interactions (Debeye) and fluctuating dipole–induced dipole (dispersion forces, London), which have already been discussed. AB (Acid Base) forces are always asymmetrical because they comprise the electron donating as well as the electron accepting properties of a surface. Thus, the AB surface free-energy component, γAB, consists of two non-additive parameters

−+= 111 2 γγγ AB (4.87)

Where, γ− is the electron donor contribution and γ+ is the electron acceptor contribution. Further, the following expression for the polar interaction between two different solid materials is valid:

−−+= −++−−+−+

2121221112 2 γγγγγγγγγ (4.88)

This gives for the total interfacial energy:


109

−−++

−= −++−−+−+

21212211

2

2112 2 γγγγγγγγγγγ LWLWAB (4.89)

Equation 11 shows that where γ12AB> γ12

LW and γ12AB<0, the total interfacial tension is a

negative value and repulsive interaction occurs Here, at least three liquids with known values of γL

D, γL+ and γL

- which are respectively the dispersive, acid and base components of the surface energy of the fluids are used. Then the following equation is applied:

( )

++=+ +−−+

LSLSLW

LLW

SL γγγγγγθγ 2cos1 (4.90)

Three or more liquids must be used, of which at least two must be polar. Contact angles obtained with apolar liquids (for which γL

+ and γL- are zero) so that γL=γL

LW give directly the value L of the LW surface tension component, γS

LW for the solid. Contact angles obtained with polar liquids yield values for γS

+ and γ S - by the solution of a set of simultaneous

Young’s equations, with the number of equations equal to the number of polar liquids. With these techniques, contact angles may be measured resulting in errors for the γ values of 2%. Recommended test liquids are methylene iodide or bromonaphthalene for the apolar liquid and a polar liquid pair of either water and glycerol or water and formamide. Geometric mean: Owens and Wendt method As discussed in the section about the theoretical background of van der Waals forces, the total surface energy of a solid, or the surface tension in the case of a liquid, can be decomposed into a dispersion polar and induction component for both the solid surface as well as the liquid drop

K

K

+++=

+++=I

LD

LP

LL

IS

DS

PSs

γγγγ

γγγγ (4.91)

The induction components are generally negligible, resulting in:

DL

PLL

DS

PSs

γγγ

γγγ

+=

+= (4.92)

The work of adhesion between a solid and a liquid ΓSL can be estimated by the geometric mean of the components

+=Γ D

LD

SP

LP

SSL γγγγ2 (4.93)

The interfacial energy is then given by:

+−+= D

LD

SP

LP

SLSSL γγγγγγγ 2 (4.94)

Together with the Young- Dupré equation, this gives:

( )

+=+ D

SD

LP

SP

LL γγγγθγ 2cos1 (4.95)

The equation above can be considered as a generalization for the Young-Dupre equation and was proposed by Owens and Wendt. This equation is widely used to estimate the surface energy of solids based on contact angle measurements of (for example) three liquids (at least two liquids) with γL

D and γLP known.

The equation above can also be written as follows with γL=γLP+γL

D the surface energy of the liquid and γS=γS

P+γSD the surface energy of the solid

( ) DSD

L

PLP

SDL

L γγ

γγ

γ

θγ+=

+ cos1 (4.96)

The equation above has the shape of a linear equation y=ax+b. So, plotting the results as shown below results in the two components of the surface energy of the solid surface


110

Figure 4.15: Geometric mean of the polar component of surface energy and the dispersive component

So, based on the measurement of contact angles with known values for γLD and γL

P, the surface energy of a solid can be calculated. The geometric-mean method uses two pure liquids denoting their dispersive and non-dispersive values. Water and methylene iodide are a convenient choice for test liquids. Different liquid pairs tend to give different results. The surface energies and polarities of some low-energy solids obtained by this method are often much lower than those calculated by other methods. Harmonic mean Another approximate generalization of the Young equation is shown in equation (4.97). This equation if based on the harmonic mean of surface energy components and is similar to the geometric mean method, see equation (4.95). Because two components of the van der Waals forces are taken into account, also heare two liquids with γL

D and γLP known are used. In order

to measure the surface energy of the solid. However, now the equation below is used:

( )

+++

+=+ P

SP

L

PS

PL

DS

DL

DS

DL

Lγγ

γγ

γγ

γγθγ 4cos1 (4.97)

When two liquids are used, 2 equations with 2 unknowns γSD and γS

P are obtained, which can be solved. The above treatment of contact angles assumes that everything is in equilibrium. In principle, this requires letting the drop sit on the surface for a long period of time. Nowadays measurement of a single static contact angle are no longer thought to be adequate to characterize the surface energy of a solid surface. Experience showed that many values for the contact angle are possible, depends on recent history interaction. Often one measures advancing and receding contact angles. In this case, we can measure the contact angle as the drop is expanding (advancing contact angle) or contracting (receding contact angle). When the receding contact angle is different from the advancing contact angle, this is called contact angle hysteresis. There are at least three possible reasons for contact angle hysteresis: o Contamination. The drop may become contaminated as it moves across the surface,

which will change the surface tension of the liquid. This may also clean or contaminate the surface.

DL

PL

γ

γ

( )D

L

L

γ

θγ cos1+

DSγ

PSγ


111

o Surface roughness. On a rough surface, the drop may spread over different portions of the surface. The less polar portions may affect advancing angles while the receding angle may be affected by polar regions

o Surface reconstruction. The surface itself may change in the presence of the liquid. For example, the hydrophobic group of a monolayer may become slightly buried when using water to measure contact angles.

Given the above reasons, a small difference (<5 degrees) between advancing and receding angles suggests that the surface is free of contamination, well organized, and smooth. Therefore, often two angles are measured, namely the receding and advancing contact angle in the dynamic situation of the advanced and receded angle in the static situation. Often these values are different, which is called contact angle hysteresis. The receding and advancing contact angles measured in this way are termed Dynamic Contact Angles. The difference between static advanced/receded contact angles and advancing/receding dynamic contact angles is that in the static case motion is incipient while in the dynamic case motion is actual. Dynamic contact angles may be assayed at various rates of speed. Dynamic contact angles measured at low velocities should be equal to properly measured static angles.

4.6.3 Receded and advanced contact angles The method of receding and advancing angles is schematically shown below. Here, a drop of liquid is put on a solid surface and removed again. The contact angle is measured both in the ‘advancing’ stage of the drop as well as in the ‘receding’ stage of the drop. An advancing contact angle is measured when the drop has the maximum volume allowable for the liquid-solid interfacial area: any addition will make the drop expand and increase the liquid-solid interfacial area. This can be thought of as the "wetting angle"θa because the drop is ready to wet additional area. The receding angle is the opposite: if any liquid is removed from the drop, the liquid-solid interfacial area will decrease. This is the "de-wetting angle." θr. The advancing angle is the largest possible angle and the receding is the smallest possible contact angle. Both are presumed to be measured at thermodynamic equilibrium. Alternatively, a drop can be placed on a tilted plane. This technique is useful to measure both the receding and advancing contact angles at the same time. Here, the sample is mechanically tilted so the drop wants to run downhill. This requires more hardware but is a well-respected technique. Conversely, there are two requirements on the tilting plate experiment: The tilt rate must be a compromise between tilting slowly, to minimize any vibration in the apparatus, and tilting quickly enough that evaporation of the drop is not an issue. Rates like 1°/s are normally satisfactory. Important is to determine when drop motion is just ready to begin (incipient motion) so you can use the angles at that time. The graph of the time variation of measured base diameter is helpful here.

4.6.4 Receding and advancing contact angles Measurement of the advancing angle θa and the receding angle θr gives enough information, together with the known surface energy of the liquid γL, to calculate the surface energy of the


112

solid γs. The method is illustrated by the picture below and the equation is given by equation (4.98).

Figure 4.16: Receding and advancing contact angles

22

2

)cos1()cos1()cos1(

)cos(cosar

aarLS θθ

θθθγγ

+−++

⋅−= (4.98)

After we have discussed various aspects of adhesion in the paragraphs above, in the following paragraphs we will discuss some more practical issues related to adhesion. Another possibility to measure receding and advancing contact angles is to use a dynamic Wilhelmy slide method, where the hanging plate makes a harmonic movement.

4.7 Adhesion between solids and liquids: Wetting Adhesive interaction between fluids and solids can be measured in the same way as between the discussed example of a drop of metal b on a solid metal a. In this case a drop of fluid is put on the solid and Young’s equation is used to study if the fluid shows wetting of the solid. The following table gives an indication of contact angles typically measured when a drop of water is put on some plastics:

Material Contact angle Nylon 0o Perspex 0o Polythene 89o Teflon 126o

Figure 4.17: Contact angle measured between some plastics and a drop of water A surface is said to be hydrophobic if it tends to repel water, which corresponds to a large value for the contact angle. This is the case for teflon. A hydrophilic surface is a surface which has an affinity for water which results in a small contact angle as is the case for nylon and Perspex in the table shown above. Complete wetting will occur if:

SLSLSSLL

SSLL

LSLS

,,

,

,

00 0

0cos

γγγγγγ

γγγβ

βγγγ

−≤⇒=−+

⇒=+−−⇒=

=−−

(4.99)

Partial wetting will occur if:

SLSLSSLL

SSLL

LSLS

,,

,

,

00 0

0cos

γγγγγγ

γγγβ

βγγγ

−>⇒>−+

⇒<+−−⇒>

=−−

(4.100)

θa θr γL

γS γSL

γS

γL

γSL

solid liquid

vapor


113

The surface energy of a fully wetted surface is the energy required to create the solid-liquid interface, so the interfacial energy γSL plus the energy to create the liquid surface γL:

LLStot γγγ += , (4.101) In wetting problems sometimes a spreading parameter is defined:

( )LLSVSS γγγ +−= ,, (4.102) The spreading parameter is in fact the difference in surface energy of a dry solid and the surface energy of a solid wetted by a thick film of liquid. If the Young-Dupré equation is used, we get the following expression for the spreading parameter:

( )VLLS

LSVLVSLS

VLLSVStottotS,,

,,,,

,,,12

2γγγγ

γγγγγ−Γ=

−+=Γ

+−=−= (4.103)

2γL is also the energy of cohesion of the liquid. So if the energy of adhesion is higher than the energy of cohesion of the liquid, then the fluid will spread over the surface. In the case of a negative S, the dry surface has the lowest surface energy and the liquid does not spread over the surface. In the case that S is zero or positive, complete wetting will take place. The Young’s equation is only usable in the case that S is negative, because then only lateral force equilibrium can be maintained.

4.8 Spreading over time When a droplet of volume V is deposited on a dry surface, this droplet spreads by the flow of a very thin film of a few tens of a nanometer thick over the surface by a hydrodynamic process. The apparent contact angle βa and the visible radius of the droplet R vary with time. These variations with time are given by co called Tanner’s law:

( )

( )101

103

∆∝

∆∝

−

ttR

tta

ηγ

ηγ

β

(4.104)

Or if gravity effects become dominant over adhesion effects:

( )101

83

∝ tgVtR

ηρ (4.105)

Here, η is the viscosity of the liquid and V the volume of the drop. If a liquid does not wet a surface (S<0), then it is still possible to deposit a film on a horizontal surface because at a certain moment gravitational forces stabilize the film. For this, a critical film thickness, typically in the order of a millimeter, is required:

gS

hc ρ

2= (4.106)

If the film has a smaller thickness then the film is not stable. De-wetting and the nucleation of droplets can occur. An example where wetting influences the physical behavior of a surface is skiing, where the adhesion between the bottom of the ski and solid ice is important. In the case that the bottom of the ski is hydrophilic, the water formed on an ‘ice-asperity’ will become attached to the ski and spread over the ski in time. The water layer on the ski will then freeze. Now we have in fact two similar surface contacting and seizure of the contact may occur because snow sticks to the ski.


114

In the case of a hydrophobic ski the water on the snow asperity will stay on the snow asperity and not spread over the ski surface in time. Surface tension effects will then cause repulsion between the surfaces and result in low friction forces. Teflon is a material with a high contact angle when in contact with water as has shown above. Teflon is therefore commonly used on skis.


115

4.9 Adhesion & Contact mechanics: Adhesion between spheres

As already discussed in the beginning so called force-separation curves are important when modeling adhesion, Although such curves are often called force-separation curves, in reality the are ‘pressure-separation curves with units [N/m2] or [Pa] along the vertical axis. Force-separation curves can have different shapes, depending on the type of potentials that cause the compressive or tensile forces between atoms. The work that has to be performed in order to separate the two bodies is the ‘work of adhesion Γ’ as already discussed in the beginning of this chapter. The work of adhesion can also be interpreted as the work required to move surfaces from ‘contact’ to infinity’. If we look again at the force-separation plot, Γ is the area ‘under the negative part’ of the force-separation curve and is given for two similar surfaces, so ∆γ=2γ, by:

γγ 20

==∆=Γ ∫∞

zpdz (4.107)

The unit of the work of adhesion is [m]∙ [N/m2]= [Nm/m2]. In the case of two dissimilar surfaces the work of adhesion is given by:

12210

γγγγ −+==∆=Γ ∫∞

zpdz (4.108)

If these force-separation curves are known, they can be used to model adhesive forces between bodies. Because of it special relevance for contact mechanics applications, in the next paragraph several cases of adhesion between spheres will be discussed.

4.9.1 Rigid spheres: Bradley model The first case is the case of two contacting rigid spheres and is shown in the next figure. This case was already studied by Bradley in 1932, see also [1].

Figure 4.18: The Bradley model: Adhesion between rigid spheres

From geometry the separation between the two spherical surfaces can be expressed as, see chapter 2:

0

2

2z

Rrz += (4.109)

With r the radial distance from the center of the contact. The force-separation model that is taken here is based on the Lennard-Jones potential, which is a common choice for a potential.

0z

Rrz2

2

=


116

The force (‘pressure’)-separation curve originating from a Lennard-Jones potential is given by:

( )

−

∆

=93

038

zzzz εεγ

σ (4.110)

In this equation ∆γ is the surface energy, ε is the interatomic spacing and z0 is the equilibrium separation between the surfaces. Integration of the contact pressures caused by the tensional forces between the surfaces over the ‘contact area’ gives an expression for the total (attractive) force Fa between the two rigid spheres. Remember that there is not a real contact area because there is deformation of the surfaces. The ‘contact area’ is here the area over which interatomic forces are present in the contact. The attractive force Fa in this case is given by:

( ) ( ) ( )∫∫∞∞

=⋅=00

220zz

a dzzRdrzrzF σπσπ (4.111)

Substitution of the expression for the pressure-separation curve gives the following:

( ) ( )

−

∆== ∫

∞ 8

0

2

00 3

13422

0zz

RdzzRzFz

aεε

γπσπ (4.112)

As mentioned before, at the equilibrium position zo is equal to the interatomic distance, so ε=z0. Substitution of this value in the equation gives the value for the pull-off force Pc for the case of two rigid spheres:

( ) γπ ∆== RzFP ac 20 (4.113) Remember that no elastic deformation or ‘contact area’ caused by deformation is present in the Bradley model.

4.9.2 Elastic spheres: Johnson, Kendall & Roberts (JKR) Model In the case of elastic spheres, the situation is different. Then, a contact area is spherical and formed by elastic deformation of one or both of the bodies. Again Lennard-Jones interaction will be assumed, as was done in the case of rigid surfaces. The contact situation is schematically shown in the next figure:

Figure 4.19: The Johson, Kendall and Roberts (JKR) model: Adhesion between elastic spheres

The Hertzian contact area for the case of a sphere with radius R contacting a flat was given by:


23*

34

δREFN = (4.115)

2a

δ

2a

δ


117

The contact area a is now given by Hertz:

*433

ERF

a N= (4.116)

And the mutual approach δ is given by:

Ra 2

=δ (4.117)

For clarity, a brief recapitulation of the Hertzian equations will be given below. When discussing Hertz, the geometry of both bodies was described is the following way:

2

2

21

2

1 2,

2 Rrz

Rrz == (4.118)

The displacements between the bodies 1 and 2 were then given by:

( ) arR

rr <−=+= ,2

2

21 δδδδ (4.119)

The following elliptical pressure distribution satisfies deformations of the above type:

( )2

max 1

−=

arprp (4.120)

Also in the case of elastic deformation according to Hertz + adhesive interaction between the two bodies, the total deformation should also be elastic. So we are looking for a combination of two pressure distributions resulting in the same deformations. One pressure distribution is the result if the interfacial attraction and the other pressure distribution is the result of elastic deformation. Displacements according to equation (4.119) can also be ‘generated’ by a combination of the following pressure distributions pI(r) and pII(r):

( )2max,

1

1

−

=

ar

prp II (4.121)

Which results in a uniform displacement:

*max,

Eap I

I

πδ = (4.122)

And:

( )2

2

max,

1

−

=

ar

ar

prp IIII (4.123)

Which results in a displacement according to:

( )

+= 2

2

*max,

421

ar

Eap

r IIII

πδ (4.124)

In order to obtain displacements by a combination of pI and pII according to equation (4.119), the term with r2 should have the same coefficients in equation (4.119) and equation (4.124), so this results in:

RaEp II π

*

max,2−

= (4.125)

Besides this, we also need to determine pI,max. For this, we can use the conditions at the edge of the contact area. The first condition is that interference of the surfaces outside the contact area is impossible. This means that, an infinite compressive pressure, like in the case of a flat


118

punch, see chapter 1, is impossible. If pI was acting as the only pressure distribution then this will cause infinite pressures at the edge, as can be seen from the pressure distribution, see equation (4.121). The second condition is that, if there is no adhesion present, no tension at the edge can be sustained. These two conditions together means that

max,max, III pp −= (4.126) Then, superposition of pI and pII gives the Hertzian pressure distribution as should be the case. If adhesion is acting, tensile forces can act at the edge of the surface and

max,max, III pp −≠ (4.127) The total free energy content of the system composed of two contacting solids is an addition of energy stored originating from elastic strain and surface energy. If the system is in equilibrium, no further elastic deformation will take place and the contact radius a will not grow. This means that if the system is in equilibrium:

0=∂∂

+=

aU

UUU setot

(4.128)

The surface energy is, in case the contact radius is a, equal to: 22 aU s γπ−= (4.129)

Here it is assumed that two similar surfaces are in contact. Otherwise 2aU s γπ∆−= (4.130)

At equilibrium also no increase in surface energy will take place, so

aa

U s γπ∆−=∂

∂2 (4.131)

It can be shown that the energy because of elastic strain is given by:

++= 2

max,max,max,2

max,*

22

157

34

IIIIIIe ppppE

aU π (4.132)

With a relative displacement of the spheres:

+= max,max,* 2

1III pp

Eaπ

δ (4.133)

Then:

( )2max,max,*

22

IIIe pp

Ea

aU

+=∂

∂ π (4.134)

This gives:

aEpp III π

γ *

max,max,2∆

±= (4.135)

Stresses at the edge of the contact are only allowed to be tensile. This means that:

aEpp III π

γ *

max,max,2∆

−= (4.136)

The top line in the next graph is the sum of of pI and pII. In the middle of the contact area there are compressive stresses and at the edge of the contact there will be infinite tensile stresses. This means that at the edge of the contact the situation will be similar to the opening of a crack in elastic material by tensile forces, see also [2]. In the following figure a Hertzian pressure distribution is shown together with a pressure distribution including adhesion. The values are in this graph chosen such that the maximum Hertzian pressure (in the center of the contact) is chosen to be equal to the maximum JKR pressure.


119

Hertzian and JKR pressure distribution

-1.0E+00-8.0E-01-6.0E-01-4.0E-01-2.0E-010.0E+002.0E-014.0E-016.0E-018.0E-011.0E+00

-1.5E+00 -1.0E+00 -5.0E-01 0.0E+00 5.0E-01 1.0E+00 1.5E+00

r/a [-]

p(r)

/pm

ax

Figure 4.20: Hertizan and JKR pressure distributions

The total contact force can be obtained by integration of the pressure distribution over the contact area. This can be shown to be equal to:

+= max,max,

2

342 IIIN ppaF π (4.137)

Substitution of equations (4.125) and (4.136) gives:

( )23

3633*4

γπγπγπ ∆+∆+∆+= RFRRFR

aENN (4.138)

This equation gives the relation between applied load FN, interfacial energy ∆γ and contact radius a. Because of adhesion, two extra terms appear in the equation compared with the Hertzian case. When the equation above is considered, it can be seen that at a certain load, the contact area will be larger than in the Hertzian case. If ∆γ=0, then the original Hertzian expression is recovered, see equation (4.116). This theory of elastic + adhesive interaction is called the Johnson, Kendall, Roberts (JKR) theory [3]. This theory is very important for modeling the contact between highly elastic surfaces, like rubber materials. If equation (4.138) is taken, and FN=0 is substituted, we will still calculate a value for the contact radius a:

31

*29 2

0

∆=

ERa γπ (4.139)

So even without an externally applied load, deformation by the adhesive forces will cause the formation of a contact area. If a=0 is substituted, we have the moment that the two bodies will loose contact (‘snap-off’). As was the case of the Bradley model (adhesion between rigid spheres), we will now get a value for force required to break the contact, so the adhesion force. A tensile force is required to break the contact, so the adhesion force will have a negative sign. For the JKR model the adhesion force is given by:

γπ ∆−= RFa 23 (4.140)

This pull-off force is independent of the reduced elastic modulus E*, although the model is valid for elastic contacts! A dimensionless load can be defined by the applied load divided by the adhesion force Fa:


120

γπ ∆−==

RF

FFF N

a

N

23

(4.141)

And in the same way a normalized contact radius can be defined by dividing it by the contact radius just before snap-off ac:

31

*89 2

∆==

ER

aaaa

c γπ

(4.142)

And the normalized separation is defined by the separation divided by the maximum dislacement before the adhesive bond breaks δc:

31

2

22

*823

∆==

ERc γπ

δδδ

δ (4.143)

Now, the relation between load and indentation depth can be expressed as: ( )( )3

1

32

1223

1223

FF

FF

+++

+++=δ (4.144)

And the relation between indentation depth and contact radius can be expressed as:

−= − 2

331

3413 2 aaδ (4.145)

The total behavior can be summarized in the following plot, where the normalized load is plotted against the normalized contact area.

Figure 4.21: The dimensionless load versus the dimensionless contact area for Hertz and the JKR model The normalized load plotted against the normalized separation gives the following plot:

1 0.5 0 0.5 1 1.5 2

0.5

1

2

2 1 −

caa

compression tension

Instable part ‘snap-off’

Snap-off

‘Equilibrium’

Hertz

Dimensionless load

Dim

ensi

onle

ss c

onta

ct a

rea


121

Figure 4.22: The dimensionless load versus the dimensionless separation for Hertz and the JKR model The Bradley and the JKR models are two extreme cases of adhesion modeling. In the Bradley case elastic deformation is neglected and adhesion forces have no influence on the contact area. In the JKR case elastic deformation is significant and results in a larger contact radius compared to the case where there are no adhesion forces. Although it will not be discussed further here, it can be found that the following parameter is important for intermediate cases:

( ) 31

30

2*

2

∆=

zER γ

µ (4.146)

This parameter µ is a measure of the importance of elastic deformation compared to the surface forces. If µ< 0.1: Elastic deformation is negligible and the Bradley model is valid. If µ>5: Elastic deformation if significant, then the JKR theory is a suitable model. In the intermediate regime another theory is applicable: The Maugis-Dugdale theory.

4.9.3 Adhesion maps All adhesion regimes can be plotted together in a so called adhesion map. The adhesion map is shown below and is taken from [4]. In this adhesion map, the parameter µ is put on the horizontal axis. A dimensionless load parameter has been plotted on the vertical axis, which is defined by:

force Adhesiveload Applied

=∆

=R

FP N

γπ (4.147)

The dimensionless load parameter can be interpreted as a measure for the applied load compared to the adhesive force in the contact.

0.5 0 0.5 1 1.5 2 2.5

1

0.5

0.5

1

1.5

2

1 − 0.481 −

Zero force

compression

tension

Hertz

JKR

cPcP

Instable part

dim

enei

onle

ss lo

ad

Dimensionless separation


122

Figure 4.23: Adhesion map representing the different adhesion regimes for tension and

compression

4.9.4 Intermediate regime: Maugis –Dugdale model Now a model for the intermediate regime will be discussed. In this model a Dugdale model for the adhesive forces will be assumed, which is a simplification for the Lennard Jones interaction between atoms. The Dugdale force-separation curve is shown, together with the curve representing Lennard-Jones interaction, in the following plot:


123

Figure 4.24: The Dugdale representation for the force - separation curve

The Dugdale approximation is shown in more detail in the next figure. The work of adhesion is again the ‘negative area’ under the curve.

Figure 4.25: The Dugdale approximation for the force - separation curve in more detail

At separations a little higher than the equilibrium separation z0, the surface will ‘feel’ a constant adhesive stress σ0 which is taken to be the maximum adhesive stress from the Lennard-Jones model. The maximum range until the stresses act of the surfaces has been taken in such a way that the total work adhesion will be ∆γ is equal for the Lennard Jones and the Dugdale model. This means that h0=0.971z0. Like was done in the JKR model, also for the Maugis-Dugdale model the pressure distribution is composed of two terms. The first term is a Hertzian pressure distribution and contact stresses caused by the Dugdale approximation for the adhesive stresses. An important difference with the JKR model is the assumption that adhesive stresses are also acting outside the contact area. The Hertzian pressure is given by:

( )2

2 123

−=

ar

aFrp N

Hertz π (4.148)

According to Hertz, the total carried load is given by:

1 1.5 2 2.5 34

2

0

2

45

2.396−

F LJ r( )

F DD r( )

31 r

distance

Tension C

ompression

0h

0σLennard-Jones

Dugdale

1 1.5 2 2.5 3

2

0

22

3−

F DD r( )

31 r

σ0

z0 z/z0

h0


124

( )RaEdrrprF

a

HertzHertzN

3*

0, 3

42∫ =⋅= π (4.149)

And the displacement in the center of the contact is given by:

Ra

Hertx

2

=δ (4.150)

The displacement at r=c by The Hertzian stresses is given by:

( ) ( )

−+

−= − 22122

, sin21 acacaca

RcHertzz π

δ (4.151)

And the gap between the surfaces at the edges is given by:

( ) ( )cR

cch zHertzHertz δδ +−=2

2

(4.152)

The Dugdale approximation for the adhesive stresses is given by:

( )

≤≤−

≤

−

−−−

=cra

arrc

rcarpDugdale

,

,2arccos

0

22

2220

σπσ (4.153)

Contact stresses are shown in the next figure. The adhesive force, resulting from the Dugdale approximation of the stresses, is given by:

( ) ( )

−+

−=

=⋅+⋅=

−

∫∫

22120

0,

cos2

22

acacac

drrprdrrprFc

aDugdale

a

DugdaleDugdaleN

σ

ππ

(4.154)

And the compression by: 22

*02 ac

EDugdale −−=σ

δ (4.155)

The gap between the surfaces at r=c is given by:

( )

−+

−−= − ca

caac

EchDugdale

122*0 cos4 σ

π (4.156)

The net stresses on the surface are given by

( ) ( ) ( )rprprp DugdaleHertz += (4.157) The results are plotted in the next figure. The red line is the Hertzian pressure distribution and the blue line the pressure including adhesion


125

Dugdale + Hertzian pressures

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-1.5 -1 -0.5 0 0.5 1 1.5

r/a

p(r)

pHertzpDugdalepHertz+pDugdale

σ0

c/a

Figure 4.26: Pressure distribution for a Hertizan contact, the Dugdale model and the sum of the two pressure components

The total contact force is given by: DugdaleNNtotN FFF ,, += (4.158)

The adhesive forces on the surface should be zero if the separation (‘gap’) between the surfaces is equal to h0, because in the Dugdale model it is assumed that adhesive stresses only act at distances smaller than h0. This means that:

( ) ( ) ( ) 0hchchch DugdaleHertz =+= (4.159) The work of adhesion ∆γ is equal for a Lennard-Jones approximation for the adhesive stresses and for the Dugdale approximation. This means that:

( ) ( ) ( )0

000 σγ

σγ∆

==+=⇒=∆ hchchchh DugdaleHertz (4.160)

After definition of the following dimensionless parameters:


126

2

2

2

20

2

2

31

31

31

31

3*4

3*4

16.1*19

92

9*16

cA

REcc

REaa

ER

RuE

RF

F NN

π

γπ

γπ

µγπ

σλ

γδδ

γπ

=

∆

≡

∆

≡

=

∆

=

∆

=

∆=

(4.161)

The parameter µ is the parameter shown in equation (4.146). If expressions for hHertz(c), see equation (4.152), and hDugdale(c), see equation (4.156), are substituted in equation (4.159), then equation (4.159) may be written as:

( )

acm

mm

ma

mm

ma

=

=

+−

−

+

−+

−

111arccos134

11arccos221

22

222

λ

λ

(4.162)

And substitution of equation (4.149) and equation (4.154) in equation (4.158) gives for the total load

+−−=+=

mmmaaFFF DugdaleNHertzNN

1arccos1 2223,, λ (4.163)

Equation (4.150) and equation (4.155) gives for the total deformation:

134 22 −−= maa λδ (4.164)

Results are presented schematically in the following figures:


127

Maugis Dugdale

0

0.5

1

1.5

2

2.5

3

3.5

-3 -2 -1 0 1 2 3 4 5

Dimensionless load [-]

Dim

ensi

onle

ss r

adiu

s c0

[-]

λ=0.3

λ=0.5

λ=1

λ=5

Figure 4.27: Dimensionless radius versus dimensionless load for the Dugdale model and different values of λ

Maugis Dugdale

-5

-4

-3

-2

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4 5

Dimensionless Load [-]

dim

ensi

onle

ss s

epar

atio

n de

lta0

[-]

λ=0.3

λ=0.5

λ=1

λ=5

Figure 4.28: Dimensionless separation versus dimensionless load for the Dugdale model and different values of λ This ‘intermediate model’ is in the limit, when the separation zone is small to the size of the contact, the JKR model is recovered again. This is the case if λ>5 corresponds to the results of the JKR model.

4.10 Adhesion in biological systems It is well known that many animals, like spiders, flies and gecko’s can move vertically along a wall. The reason that this is possible is because the feet of these animals adhere to the wall


128

surface. On the feet of these animals attachment ‘devices’ are present in order to cause the strong adhesion forces. In particular long range forces, so van der Waals forces play an important role. Very typical is that small animals, like beetles, have µm terminal elements of there feet, while larger animals like gecko’s have much smaller elements on their feet, in the order of sub-µm size see [8] and the following figure:

Figure 4.29: Animal size and the size of the attachment elements

In this section it will be shown that indeed adhesion forces are larger for a large number of small attachment elements compared with a smaller number of larger attachment elements. Adhesion, for example between a feet and a wall, is often obtained by a combination of molecular attraction forces, capillary attractive forces with secretions as a fluid or van der Waals forces. Insects produce secretion fluids, but spiders and gecko’s have a dry contact between their feet and the wall. There is strong evidence that for the case of gecko’s van der Waals forces are responsible for adhesion. The feet of animals often have fine structures down to the µm and sub-µm’s. From experiments it has been found that the density of the terminal elements n [m-2] is related to the body mass m [kg] of the animals studied, according to:

mn log699.08.13log += (4.165) The measurements are shown in the following graph.


129

Figure 4.30: Density of attachment elements and body mass of different animals

So, it seems like that smaller contacting elements increases the adhesion force acting in the contact between an animal feet and the wall. This phenomenon can be explained as follows: According to JKR theory, the full off force is given by the following equation for a sphere in contact with a flat, with ∆γ the adhesion energy per unit area.

γπ ∆= RFa 23 (4.166)

So, a spherically tipped leg with a radius of R=200 µm would require a ∆γ of 1 J m-2 in order to support the weight of a fly (80 mg) when hanging on one leg. A value of ∆γ= J m-2 is unrealistic for van der Waals forces, which are typically 10-50 mJ m-2 Now suppose the spherical contact is slit up into n smaller spherical contacts with radius R/√n, so a self similar scaling law. Then the total adhesion force of a set of small spheres would be √n times higher than the previous case. Now, the weight of the fly can be supported if the total number of contacts would be would be 103 to 104 contacts per fly, which is about true from measurements. So, a higher body mass of an animal needs to be compensated by a higher number of attachment elements on the feet of the animal. In the case of self similar scaling as discussed above, the expected density of attachment elements n should scale with body mass according to:

γπρ

κ

κ

∆=

=

32

432

32

322

gkp

mn (4.167)

With k a safety factor and p a shape factor. Also other scaling laws can be considered. For example if the contact radius R is assumed to be independent of the size of the attachment element, then:

31

mR

n κ= (4.168)

Both cases are schematically illustrated by the following figure:


130

Figure 4.31: Self similar scaling and radius invariant scaling

In this figure a. shows self similar scaling and b. shows radius invariant scaling. When compared to experimental data, the self similar scaling law (red line) seems to explain the results better than the radius invariant law (blue dotted line). However, in the graph also lines of constant R are shown. It seems that each animal type have a constant R, where the R of large animals like lizards is lower (0.3 µm) than for flies (1.6 µm).

Figure 4.32: Density of attachment elements and body mass of different animals and the scaling laws for the radius of the attachment elements Besides the advantage of higher adhesion forces, an additional advantage of a surface composed of many small contacts is the reliability in the sense that defects at individual contacts, for example by dust or mechanical damage, will not strongly influence the total adhesion force. Not in all cases animal attachment elements have spherical shapes [9]. Some examples are shown in the following figure. Most of the following is taken from [9].


131

Figure 4.33: Different shapes of attachment elements found in nature Therefore, some shapes will be studied theoretically. The theoretically studied geometries are shown in the following figure [9].

Figure 4.34: Models of some shapes of attachment elements found in nature Horizontal cylinder or torus It can be shown that the pull of force for a cylinder r(or torus) of length L and radius R is given by


132

LREFa

31

223 2*

−=

γπ (4.169)

So, for a slender torus with radius of the cylinder r is much smaller than the length of the torus L=2πR,

RrEFa πγπ 3

1

23

2*

−= (4.170)

So, contrary to the spherical case, the elastic modulus is important Vertical cylinder or flat puch For the case of a vertical cylinder with radius R, then the pull-off force is given by:

γπ 3*8 REFa −= (4.171) Assuming that the contact radius a=R. This assumption is only valid if the punch has a higher modulus of elasticity that the substrate. General axisymmetric profiles A general axisymmetric profile given by:

( ) 1−= q

q

Rrrh (4.172)

Gives the following expression for the pull-off force: ( ) ( ) ( )

121

122

*1213

−+

−−

−−

∝ qq

qq

qq

a ERF γ (4.173)

So, The dependence of Fa on R is always positive for q>1. The dependence of E* is always negative for 1<q<2. For q→∞, the relation reduces to the flat punch expression and for q=2, which is a parabolic shape, the expression of JKR is obtained. Only in the last case, the pull-off force is independent of the elastic modulus. Film peeling: contact of an elastic tape In the case of peeling of an elastic tape from a surface with width 2R and thickness h, then the vertical component of the force required for peeling is given by:

( )λαγ ,2 RgFa −= (4.174) With the function g given by:

( )( ) ( )αλα

αλα

cos12cos1

sin2,2 −++−

=g (4.175)

In this equation, a is the peel-off angle and:

Ehγ

λ = (4.176)

Suction cup A suction cup is not dependent on van der Waals forces, but on a pressure difference between the inside of the hollow cup and the outside. For this, a pull-off force for a cylindrical cup is given by:

2RpFa π∆= (4.177) So, for a suction cup, the radius R has the largest influence compared to the other attachment mechanisms. This also means that small suction cups will not be effective. A comparison for the different geometries is shown in the following figure, taken from [9]. It can be seen figure a) that a flat punch and the torus have the highest pull-off forces for µm-sized R and a small E= 1 MPa. At R<100 nm, the differences between the different


133

attachment devices is small. At a higher value of E=1 GPa,, the differences between the set of attachment geometries is larger. It is seen that at these higher values of E, the type of contact element is still important at R=100 nm.

Figure 4.35: Pull-off forces for single contact for various shapes as a function of feature size If the different shapes are compared, then the following table can be composed. In this table, m is the mass that has to be carried by the attachment mechanism. A high value for s results in a high pull-off force at a large value of R. An example is the suction-cup mechanism with s=2. It can also be seen for the exponent r that a spherical contact, assuming radius in variant splitting, results in the highest increase in pull-off force when the contact is splitted (r=1). So a spherical attachment element will increase the pull-off force the most when the contact is splitted. Table 4.10: Splitting efficiency of several attachment elements

spherical torus Elastic tape

Flat punch Suction cup

General

sa RF ∝ 1 4/3 1 3/2 2 ( )

1213

−−

qq

ma MF ∝ 0 1/3 0 ½ 0

122

−−

qq

kaF γ∝ 1 2/3 1 ½ 0

121

−+

qq

ar

na FnF ∝, ½, 1a 1/3,1/2 a ½ ¼ 0

( )1221−

+q

q

pmn ∝ 2/3, 1/3 a 1,2/3 a 2/3 4/3 ∞ ( )( )13

122+−

qq

a: Curvature invariant splitting In the following figure, the total pull-off force is shown as a function of the number of contact elements in an apparent contact area of 100 cm2. The same E and γ are used as in the previous figure a) and b). It is clear that for contacts of about 100 nm, the lines for the different geometries converge. The reason for this is that shapes with lower pull-off forces of the individual elements generally have a higher splitting efficiency.


134

This also means that at sizes below 100 nm the adhesion force is insensitive for the exact shape of the attachment element. So, a robust attachment mechanism can be designed with sufficiently small fibers. This can also be seen in nature where hairy attachment mechanisms [10]. For carrying a human being, a flat punch or a torus would be a preferred shape, as can be seen from the figure. The reason that a flat punch performes so well as that the contact area is the largest. However, in practice this shape would also be more sensitive to roughness effects, dirt or other factors which disturb the contact. The flat punch is therefore not considered as a ‘robust’ contact geometry.

Figure 4.36: Comparison of pull-off forces for different shapes in terms of contact splitting efficiency The main observations from the calculations can also be found in nature: The flat punch configuration of attachment devices is rarely found in nature. Only the contacting elements of a grasshopper are similar. Suction cups are found for water beetles and mites. The smallest suction cups found in nature are not below 10 µm, which is a value for water beetles. This is also reflected in the fact that in the calculations that suction cups are not effective at small sizes. Suction cups are however useful for the macro-world, as can be seen from their application in household items, like hooks. Torus-like structures are found as attachment devices of the fly and the beetle. This type combines high absolute forces with an appreciable scaling potential, as shown in the figure above. For very small contact elements, the size of the contact element is not so important. Then, shapes like the elastic tape and the sphere become efficient. And indeed very small spherical and tape-like elements are indeed found for spiders and lizards.

4.11 Summary


135

In this chapter an overview is given about adhesive interaction between surfaces. First, surface energy, interfacial energy, surface tension and work of adhesion are discussed. Also, methods to measure these properties are discussed. After that, some models with respect to

the modeling of forces due to adhesive interaction between surfaces are presented as well as a discussion about adhesion in biological systems.

4.12 Excersises Excersise 1 The excersise is about the contact between a rubber ball and a steel plate. Some material properties are gin in the following table. The load on the ball is 5 N.

Rubber Steel Water

Elasticiteitsmodulus E 500 MPa 210 GPa γL 72.8 mJ/m2

Dwarscontractiecoefficient ν

0.3 0.29

Radius R 10-2 m ∞ (plane) 1. Calculate the deformation δ and the contact radius a using Hertz Plot the pressure

distribution. • We would like to measure the surface energy of the rubber ball. With the method of

the advancing and receding contact angle we measure an advancing angle θa= 92o and a receding angle θr= 56o. Calculate the surface energy of rubber.

• Calculate the work of adhesion Γ, assuming a Rabiniwicz assumption for the interfacial energy with α=0.2.

2. In the contact operating in the JKR regime? 3. Use the JKR model to calculate:

• The ‘snap-off’ contact radius ac. • Contact radius a0 at a FN=0. • The pull-off force Fa. • Plot de dimensionless load FN/Fa versus the dimensionless separation δ/δc. • Plot de dimensionless load FN/Fa versus the dimensionless contact radius a/ac. • Compare the results with the Hertzian situation. Under which conditions (load,

material properties) are the adehion forces becoming important? • What do you think that the influence of roughness on adhesion will be?

4.13 References [1] K.L. Johnson, Mechanics of adhesion, 1998, Tribology International, vol. 31, no. 8, p.

413-418. [2] K.L. Johnson, 1976, Adhesion at the contact of solids, from ‘Theoretical and applied

mechanics’, North-Holland publishing company. [3] K.L. Johnson, K. Kendall, A.D. Roberts, 1971, Surface energy and the contact of elastic

solids, Proceedings of the Royal Society of London A, vol. 324, p. 301-313. [4] K.L. Johnson, A continuous mechanics model of adhesion and friction in a single asperity

contact, from Micro/nanotribology and its applications, ed. B. Bhushan, Kluwer Academic Publishers.

[5] D. Maugis, 2000, Contact, Adhesion and Rupture of Elastic Solids, Springer-Verlag, ISBN 3540661131

[6] http://en.wikipedia.org/wiki/Soap_bubble#Surface_tension_and_shape

http://en.wikipedia.org/wiki/Soap_bubble#Surface_tension_and_shape


136

[7] B.N.J. Persson, 2003,On the mechanism of adhesion in biological systems, Journal of chemical physics, vol. 118.no. 6., p. 7614-7621

[8] E. Arzt, S. Gorb, R. Spolenak, 2003, From micro to nano contacts in biological attachment devices, PNAS, vol. 100, no. 9, p. 10603-10606

[9] R. Spolenak, S. Gorb, H. Gao, E. Arzt, Effect of contact shape on the scaling of biological attachments, Proceedings of the Royal Society of London A, vol 461., p. 305-319.

[10] H. Gao, H. Yao, 2004, Shape insensitive optimal adhesion of nanoscale fibrillar structures, PNAS, vol. 101, no. 21, p. 7815-7856..

[11] A.W. Adamson, A.P. Gast, 1997, Physical chemistry of surfaces, Wiley-Interscience ISBN 0-471-14873-3

[12] H-J Butt, K. Graf, M. Kappl, 2003, Physics and chemistry of interfaces, Wiley, ISBN3-527-40413-9

[13] J. Israelachvili, 1991, Intermolecular & surface forces, Academic Press, ISBN 0-12-375181-0

Chapter 5 Surface roughness

5.1 Introduction Surface roughness is a complicated phenomenon. Some aspects of surface roughness will be discussed in these sections by showing some examples. In the following sections of this chapter the problem of characterising surface roughness will be treated into more detail. Most technical surfaces can be considered as a random process in the sense that that most surfaces have mainly random characteristics, originating from the finishing operation (like grinding) or a combination of finishing operations (line grinding followed by polishing). This random structure is caused by the random interactions of the finishing tool with the surface. The grains of a grinding stone for example will randomly contact the workpiece, resulting in random properties of the workpiece surface. Some finishing operations show more deterministic characteristics than random characteristics. Some examples of more or less deterministic finishing operations are turning and EDT (Electro Discharge Texturing) of sheet materials. Turing will result in a periodic grooved appearance of the surface and EDT will result in a deterministic cratered surface.

Figure 5.1: Surface roughness profiles. Typically the z-scale is enlarged compared to the x-axis

In this chapter most attention will be paid to random surfaces. In the simplest case a random surface can be considered as having the same random properties, independent of the direction, local spot or area. However, most technical surfaces are not ideally random but show more complicated behaviour, like

• Inhomogenity. The surface looks different at different spots.


137

• Scale dependency. The properties of the surface are dependent on the size of the measured surface or dependent on the resolution of the instrument that is used to measure the surface.

• Anisotropy. The properties of the random surface are different is different directions. An example of a strongly anisotropic surface is a grinded surface.

In the next section, some characteristics of rough surfaces will be illustrated with images of measurements.

Figure 5.2: High surface heights are represented by red and low surface heights are represented by blue

In these images, high surface heights are represented by red and low surface heights by blue colors, as is shown in the color bar above.

5.2 Form, waviness and roughness Generally three properties or rough surface can be distincted:

• Form. the general shape of the surface • Waviness. The long frequency details of the surface • Roughness. The surface that remains after subtraction of form and waviness.

In the picture below a cylindrical, grinded surface is shown. The measurement is filtered in such a way that the bottom picture only shows the high frequency details, so roughness details. The middle picture shows the roughness details and waviness. The top picture shows the complete surface, including the cylindrical shape.

Figure 5.3: Form, waviness and roughness

When roughness is discussed in this chapter, shape and waviness aspects are not taken into account. Before discussing roughness values, first some general aspects of surface roughness will be illustrated below

5.2.1 General aspect of surface roughness Anisotropy

50 100 150 200 250 300 350 400 450

100

200

300

400

500

600

50 100 150 200 250 300 350 400 450

100

200

300

400

500

600

50 100 150 200 250 300 350 400 450

100 200 300 400 500 600


138

From the grinded surface below two profiles are taken and shown at the right side and at the bottom of the surface. It is clear from these profiles that the surface shows different characteristics in the two perpendicular directions. This is called anisotropy.

Figure 5.4: Anisotropy

Scale and roughness The surface shown below is a piece of a metal sheet. The metal sheet has been made by a rolling operation. The rolling operation has formed plateau-like features at the surface. In the figure below the measurement is shown together with smaller local areas taken from the surface. It is clear that if only a single plateau is considered, the surface looks totally different than if one is looking at a larger area of the sheet material.

Figure 5.5: Roughness and scale: The effect of measurement area

The above pictures illustrate the influence of measurement area. When a measurement apparatus is used, this apparatus will have a certain pixelsize or lateral resolution which forces an upper limit on the details one is able to measure with the equipment. In the set of images below the grinded surface from section 5.2.1 is shown again, only measured with the orginial 480 x 480 pixels, sampled down to 120 x 120 pixels and again sampled down to 30 x 30 pixels. Is will be clear from these images that the pixelsize is of influence on the properties of the surface. In this case, 480 x 480 pixels was the maximum the apparatus was able to measure with the chosen measurement area. However, it should not be forgotten that the characteristics of an 480 x 480 image is in the same way an approximation of the real surface as the 120 x 120 pixel image.

20 40 60 80 100 120 140 160 180 200 220

50

100

150

200

250

300

10 20 30 40 50 60

10

20

30

40

50

60

70Size measurement area

5 10 15 20 25 30

5

10

15

20

25

Limit longest wavelengths

20 40 60 80 100 120 140 160 180 200 220

50

100

150

200

250

30050 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5x 10-6

50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5x 10

-6


139

Figure 5.6: Roughness and scale: The effect of sampling frequency or pixelsize

Although homogeneity and isotropy have an obvious meaning, the most obvious characteristic of a rough surface is the ‘amount’ of roughness on surface, or the amplitude of the roughness.

5.3 Characterization of rough surfaces The most common parameters describing surface roughness or roughness of a profile, like the Ra value and the Rq value, are indeed parameters describing amplitude characteristics of surface roughness. However, amplitude properties are not sufficient to characterize a surface. Besides amplitude characteristics, also parameters related to the wavelengths present in a surface will be required. For contact problems, defining roughness in terms of wavelengths and amplitudes is not sufficient. In order to model contact between rough surfaces, the first step is to look at which characteristics of these surfaces are of interest for contact. In this section, some properties of rough surfaces which are of interest for rough surface contact are also mentioned. Besides this, also some ways to measure these parameters from rough surface data are also discussed.

5.3.1 Height characteristics In this section the most common roughness parameters indicating height characteristics are discussed. Roughness parameters indicating height characteristics are the most common roughness parameters. How these parameters can be measured from measured rough surface height data is discussed in section 5.7.1. Rough surface parameters are illustrated with an example in section 5.8.1. Special attention will be paid to the contact behavior of a rough surface and how roughness parameters are related to this. The average roughness value (Ra) A common statistical measure for the roughness of a surface that is often encountered is the Ra value that is defined by:

dxzL

RL

a ∫=0

1 (5.1)

5 10 15 20 25 30

5

10

15

20

25

30

20 40 60 80 100 120

20

40

60

80

100

120

20 40 60 80 100 120 140 160 180 200 220

50

100

150

200

250

30050 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

450

480x480

120x120

30x30

Limit smallest wavelengths


140

In this equation, z is the height above the meanline of the surface and L is the sample length. It is important to realize that all statistical values are defined relative to the mean of the surface. In case there is an underlying shape of the surface (like a sphere or a cylinder), this shape should be removed from measured roughness data before calculating statistical roughness properties. Roughness values are measured according to fixed DIN or ISO standards. An example is a fixed measurement length of typically 0.8 mm. The Ra value can be interpreted as the average absolute distance of the surface points from the mean line. The root mean square value (Rq , RMS or σ) The root mean square value is another roughness parameter than is often used to characterize surface roughness. The Rq roughness value can be interpreted as the average quadratic distance of the surface points from the meanline. The Rq roughness value is the most well known parameter together with the Ra value

dxzL

RL

q ∫==0

222 1σ (5.2)

The peak to valley value (Rt)

minmax zzRt −= (5.3)

The Rt value is the height difference between the highest peak and the lowest valley within a certain measurement. This parameter is not a statistical characteristic of the surface. A consequence of this is that the value is typically not very stable, so it may change a lot when another roughness measurement of the same sample is done. The ‘average peak to valley’ value (Rz)

5

5

1

5

1∑∑

==

−= i

ii

i

z

VPR

(5.4)

The reliability of a peak to valley value can be increased by taking the average of more peaks and valleys. The Rz value is defined as the difference between the average height is 5 peaks and the average height of 5 valleys. The values discussed above are unfortunately not unique. In the picture below several surfaces are shown with the same Rq value, but with a different appearance. The reason for this is that is no spatial information is taken into account in height-related values like the ones shown in this paragraph.

Figure 5.7: Several profiles with the same Rq value

Amplitude Probability density function (PDF)

0 20 40 60 80 100 120 140 160 180 200

2

0

2

3

3−

sinusxrange

blockxrange

randxrange

2000 xrange


141

Another possibility to analyze surfaces is to count ‘how often’ a certain height is present in a surface. This is done by dividing all present heights into equal height intervals ∆z=z2-z1. Then it is possible to count how often a height z2<z<z1 is found in the surface. The results can be presented in a surface height histogram. If this histogram is normalized with the total number of points, the result is the Probability Density Function (PDF). The total area of this curve is 1, and the height of each bin of the PDF represents the probability that such a height is present in the surface. Many surfaces, and in particular unworn surfaces, show a height distribution with shapes close to a Gaussian function. This is shown in the following figure, where a measured height distribution is compared with a Gaussian height distribution.

In mathematical form the probability density of a Gaussian curve is given by:

( )

−= 2

2

2exp

211

σπσφ

zz (5.5)

A Gaussian PDF means that ‘middle high’ spots are common on the surface and that extreme low heights as well as extreme high heights are not much present. The Rq roughness value is in fact the standard deviation of the PDF of the surface. The shape of the height distribution function is important for the functional behavior of the surface. Of particular importance for contact behavior is the shape of the distribution at high surface heights, i.e. the ‘right part’ of the graph shown above because the high points on the surface will mostly contact the countersurface. The shape of the PDF can be influenced by finishing operations. Polishing will for example remove the ‘right tail’of the figure shown above. As is clear from the PDF shown, almost all surface heights will be present within three times the standard deviation of all surface heights, so -3 Rq<z<3Rq. Many surfaces have a clearly non-Gaussian appearance. Examples are worn surfaces or polished surfaces, where the highest asperities have been removed. Turned surfaces , because all ridges and all troughs have about the same height. In such cases some additional parameters can describe the shape of the (measured) PDF. Several parameters can be derived from this curve. The following expression gives the average height of the surface and is also called the first moment of the PDF.

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

Summit Probability Density Distribution

Surface Height/RMS [-]

Probability density Measured Gaussian

Figure 5.8: A Measured probability density function (PDF) compared to a Gaussian probability density function


142

( )∫∞

∞−= dzzzz φ (5.6)

Similarly, also higher moments can be defined. In the next equation mk is the kth order moment of the PDF curve. Later, the 3rd order moment will be discussed as well as the 4th order moment.

( )∫∞

∞−= dzzzm k

k φ (5.7)

Before discussing the moments, the bearing area curve, which is related to the PDF will be presented briefly. Bearing area curve The bearing area curve is also called the Abott-Firestone curve or the material-ratio curve. This curve gives the relation between the cut-off height and the ratio of the surface that is filled with material at this cut-off height. This curve is in fact the cumulative probability distribution function (CDF) and is given by:

( ) ( )∫∞

=Φz

dzzz φ (5.8)

A typical CDF is shown in the figure below. As is clear, the value of the CDF will be zero at the right side (no heights are present above the highest point) and one at the left side because all surface heights are lying above the lowest point present in the surface.

Figure 5.9: The bearing area or Abott curve

It has to be noted that the bearing area curve is not a curve that gives information about the part of the surface that is carrying the load, as the name of the curve may suggest. Skewness (Sk) and Kurtosis (Ks) Two important properties desribing the shape of the PDF are the Skewness Sk and the Kurtosis Ks. The skewness is a measure for the symmetry of the (measured) probability density function and the kurtosis is a measure for the spread of the surface heights around the average. A Gaussian distribution has the values Sk=0 and Ks=3. A larger spread around the average than in the case of a Gaussian height distribution results in a kurtosis Ks<3. A distribution with a concentration of surface heights above the average has a skewness Sk<0. The skewness is defined in the following way:

dzzS k ∫∞

∞−

= 33

1σ

(5.9)

And the kurtosis is defined by:

-5 0 5

0.2

0.4

0.6

0.8


Bea

ring

Are

a

Bearing Area Measured Gaussian


143

dzzK s ∫∞

∞−

= 44

1σ

(5.10)

In these equations, it is assumed that all heights are given relative to the average height. An example of a surface with a negative skewness is a polished surface where the highest surface parts are removed in the polisinh process. A turned surface generally has a positive skewness because of the ridges in between the cuts. How profiles with a negative or positive skewness respectivewly kurtosis look like is shown in the next picture which is taken from the handbook about surface roughness of Whitehouse [1].

Figure 5.10: Illustration of skewness and kurtosis on profile geometry

Up to now we have only discussed characteristics related to the heights of surfaces. Besides heights, also spatial characteristics are of importance. One possibility is to characterize the surface in the form of frequencies. This can be done by the autocorrelation function and in the frequency domain by the power spectrum, and will be discussed shortly. The power spectrum and the autocorrelation function are both statistical parameters giving information about the spatial, or frequency, characteristics of the surface.

5.3.2 Spatial and hybrid properties In the same way as for the height parameters, ‘average’ spatial parameters can be defined. Such parameters will not be discussed here in detail. Possible parameters are for example:

• Number of zero-crossings • Peak density • Peak count in an amplitude discriminating band

Figure 5.11: Spatial roughness properties Properties representing spatial information of a surface similar to the PDF which represents height information are the power spectrum, the autocorrelation function and the structure function.

Power spectrum, autocorrelation function and structure function More insight in the frequency composition of a surface, so the spatial characteristics of the surface, is given by the autocorrelation function and the power spectrum.


144

The autocorrelation function is the most popular parameter for estimation of spatial properties from a profile or a surface. For a profile of length L, the autocorrelation function is the expected product of a height z(x) and a height z(x+τ) ‘a little further’ at position x+τ.

( ) ( ) ( )( ) ( ) ( )dxxzxzL

xzxzEAL

∫ +=+=0

1τττ (5.11)

At τ=0, the autocorrelation value is the expected value of the local surface height squared.

( ) ( ) ( )( ) ( )( ) 220 σ=== xzExzxzEA (5.12)

So, at τ=0, the autocorrelation value is equal to the RMS roughness value squared, which is the variance of the surface heights. When calculated from rough surface data, the autocorrelation function often shows exponential decay characteristics:

In the next figure, the normalized version of the autocorrelation function is shown, so

( ) ( ) ( )x

x eeAAA*

1

2ββτ

στ

τ−

− ==⇒= (5.14)

Figure 5.12: The autocorrelation function

From this graph, a length can be defined, at which the autocorrelation function decays to a fixed value, which is 1/e times the maximum of the autocorrelation function, at τ=0. This distance is called the autocorrelation length β* and has units [m]. The autocorrelation length can for example be calculated by fitting expression (5.13) through a measured autocorrelation function. The meaning of the autocorrelation function can be seen from different ways:

• The autocorrelation in fact calculates to what extent, on average, the height at position x+τ is still similar to the height at the position x. All random effects are averaged out. This is also related to the following point

• The autocorrelation function gives information about the ‘average machining element’. The finishing operation can be interpreted as an ‘average machining element’ that takes place at random spatial places. For example, if an autocorrelation function is measured from a periodic, e.g. turned surface, the autocorrelation function will show periodic decay. In that case the autocorrelation function looks like a ‘conventional’ autocorrelation function with a small sinus wave superimposed on it.

( ) xeA βστ −= 2 (5.13)

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [m]

auto

corre

latio

n

β=5

β=4


145

Then, an autocorrelation length defined as a ‘length at which the autocorrelation function decays to a fixed value’ is not obvious.

When we discussed height properties, the Rq already gave a lot of information about the heights present in the surface. Later it turned out that the Rq roughness values was in fact the standard deviation of the PDF. In a similar way the autocorrelation length τ gives already a lot of information about the spatial properties of the surface. However, full information is given by the complete autocorrelation function. A well known technique originating from the field of signal processing is the power spectrum. This representation of the signal in the frequency domain gives in fact the same information as the original signal, but in frequency domain. As is done is signal processing, the power spectrum P(ω) of height data can be calculated by taking the the square of the Fourier transform of the height data:

( ) ( ) 21ωω F

LP = (5.15)

The power spectrum has in the case of an autocorrelation function as equation (5.13) the following form:

( ) 22

22ωββσ

πω

+=P (5.16)

A plot is this power spectrum is given in the following figure:

Figure 5.13: The power spectrum

As already explained, the power spectrum gives information about the frequency content of the surface. For example, in the case that the slope at the right side of the spectrum is steep, high frequencies will not be prominent in the surface, but the surface will have low frequency, so slowly varying, characteristics. There are relations between the Rq= RMS = σ height characteristics and the power spectrum. Although is will not be discussed further, it can be found that the area under the power spectrum is equal to the roughness value σ2, which is in its turn equal to the autocorrelation at τ=0:

( ) ( ) ωωσ dPA ∫∞

==0

2 0 (5.17)

10

-2 1

0 -1 1

0 0

10

110

210

-4

10

-3

10

-2

10

-1

10

0

ω

power spectrum


146

The power spectrum P(ω) is related to the autocorrelation function A(τ) by the Fourier transform and vice versa. Here the cosine transform is shown which is only the real part of the complex Fourier transform. The cosine transform is sufficient because the autocorrelation function and the power spectrum do not give complex data.

( ) ( ) ( )

( ) ( ) ( ) τωττω

ωωτωπ

τ

dAP

dPA

∫

∫∞

∞−

∞

∞−

=

=

cos

cos21

(5.18)

Most often, the power spectrum is calculated by the square of the Fast Fourier transform, see equation (5.71), and the autocorrelation function is calculated by the inverse Fourier transform of the power spectrum. A function similar to the autocorrelation function is the structure function. The structure function is defined in the following way:

( ) ( ) ( )( )[ ]2ττ +−= xzxzES (5.19)

The structure function can be interpreted as the squared expected height difference in the profile, measured over a distance τ. The structure function has the characteristic that it gives a more accurate description of the surface characteristics, although in principle it contains the same information as the autocorrelation function. The structure function is used as a tool in fractal analysis of surfaces to measure the fractal dimension D. It is related to the autocorrelation function in the following way.

( ) ( ) ( )( )[ ] ( )[ ] ( ) ( )[ ] ( )[ ]( ) ( )( ) ( )( )τστ

ττττ

AAAxzExzxzExzExzxzES

−=−=

=+++−=+−=

12022

2

222

(5.20)

In this equation, ( )τA is the normalized version of ( )τA . At ∞→τ , the SF will reach the value of 2σ2. In the simplest case a surface can be described by an exponential autocorrelation function with an autocorrelation distance β to describe the spatial properties and a Gaussian PDF describing height properties with its standard deviation σ . So, in this case the surface can be described by two parameters: σ and β which is in fact a very simple and convenient description for the characteristics of a random rough surface. Now we will discuss the root-mean-square slope and the root-mean-square curvature. These parameters also give information about the spatial characteristics of the roughness. Root-mean-square slope (σm) and Root-mean-square curvature (σκ) In the section above, properties of the surface related to height characteristics were described as well as parameters related to spatial properties of the surface. However, the functional behavior of surfaces is not influenced by heights of spatial characteristics only.

.

m


147

Figure 5.14: Slopes and curvatures of a roughness profile

Local slopes and curvatures are of importance for the functional behavior of contacting surfaces, like in friction and wear. Schematically this is illustrated by the above pictures. Parameters related to slopes and curvatures are ‘hybrid’ parameters in the sense that they give a combination of height and spatial information. From local slopes and curvatures, similar to the RMS roughness value, the root-mean-square σm slope can be defined in the following way:

( ) dxL

L

dxdz

m ∫=0

22 1σ (5.21)

So, the root-mean-square slope is in fact the standard deviation of all the slopes present in the surface. Similarly, the root-mean-square-curvature σk can be defined in the following way:

( ) dxL

L

dxzd∫=

0

222

21κσ (5.22)

The root-mean-square curvature is the standard deviation of all the curvatures present in the surface. The power spectrum can also be used as a basis for estimating σ, σm and σκ and therefore as a basis for estimation of summit properties (as we will see later) in the following way:

( )

( ) ωωωσ

ωωωσ

κ dP

dPm

∫

∫∞

∞

=

=

0

42

0

22

(5.23)

So, if a power spectrum is measured, σm and σκ could be calculated from this measured spectrum. The parameters σ, σm and σκ can be combined into a single parameter α, which is in fact a measure for the frequency composition of an (isotropic) surface, see [5]:

crossings positive ofdensity maxima ofdensity

2 =

=

mσσσ

α κ (5.24)

A low value for α can be interpreted as ‘high slopes for the roughness value’, so a surface with strongly varying heights. All above described statistical properties of surfaces (like σ, σm and σκ) are dependent on the measurement length L and the sample interval h used to measure the surfaces. The reason for this is that the measurement length L is in fact an upper limit for the wavelengths that can be measured with the equipment. Similarly, the sample interval h is determining the maximum frequency, and therefore a lower limit for the wavelengths that can be measured with this equipment. This limits caused by the measurement equipment has brought the idea of ‘functional filtering’ roughness data in order only to take into account wavelengths of

1/κ


148

importance for the contacting situation. An upper limit for the frequencies of importance could for example be the length of the microcontact and a lower limit could be related to the thickness of the boundary layer present on the surface. The effect of sampling interval and measurement area on the summit properties will be discussed later.

5.3.3 *The effects of a finite sampling interval Unfortunately, surface roughness is not as easy as it seems and one of the main reasons is the effect of a sampling interval h which is inherent on measuring surface roughness in practice. Suppose the sampling interval has a value h. If we now take the value of the (normalized) autocorrelation function at τ=0, τ=h and the value of at τ=2h and call these values ( )0A ,

( ) 1AhA = and ( ) 22 AhA = . ( )hA will probably be higher than ( )hA 2 because the autocorrelation function will generally be decaying. Both values will be between 0 and 1, because we now consider the normalized version of the autocorrelation function. Remember that the meaning of the autocorrelation function: If the height of a surface has a certain value, say 1, the autocorrelation function gives the expected value of the product of the height at x=0 and the height at x=τ distance from that original point. Now we calculate the values of σm and σκ, which we now already have seen several times, using the following numerical estimates for the slope m and the curvature κ:

2201

2,1,,1

01,,1

22h

zzzh

zzzh

zzh

zzm

x

jijiji

jiji

+−=

+−=

−=

−=

−+

+

κ (5.25)

Then the expected values of σm and σκ with assuming the values A1 and A2 at respectively A(h) and A(2h) are [2]: Error! Objects cannot be created from editing field codes. (5.26)

In the case of an exponential autocorrelation function:

( )( ) h

h

eAhAeAhA

β

β

22

1

2 −

−

==

== (5.27)

Then:

( )

( )∞→

+−=

+−

=

∞→−

=

−

=

4212

2

22012

212

201

432

2

12

hAA

hzzzE

hA

hzzEm

σσ

σσ

κ

(5.28)

So, if the autocorrelation function is really exponential and the sampling interval tends to be zero, then two basic surface properties, the standard deviation of the slopes and the standard deviation of the curvatures, go to infinity. Finite values are only measured because of the finite sampling interval of the measurement instrument. So, determination of all roughness properties, and therefore also the summit properties are dependent of two aspects:

• Sampling interval. This characteristic of the measurement equipment limits the highest frequency, or the smallest wavelength, that can be measured. This means that ‘smaller’ roughness details are not visible with this equipment.


149

• Measurement area. The measurement area is the limit to the highest frequency that can be measured with the measurement equipment and limits the lowest frequency, or the longest wavelength, that can be measured with the roughness measurement equipment used.

The previous discussion was based on the assumption of an exponential autocorrelation function A(τ). It can also be shown that as a result of this phenomenon many surface properties are dependent on the sampling interval h. Although very inconvenient, this result is in accordance with measurements. However, there is a large need to be able to describe properties of the surface, for example to use roughness values in contact, friction and wear models. But how can we describe properties of a surface is the results are dependent on the measurement equipment used? There is no good answer yet on this question. Up to now we have described surface properties in the shape of statistical curves like the ACF and the PDF. This has advantages because ‘average’ effects are measured while (local) random variations are canceled out. Not in all cases it is desirable that local information is lost. For example, when one is interested in local extremes, defects or inhomogenities in the surface, statistical parameters are not the best way to look at a surface. There are many techniques available to measure and analyze local properties of surfaces or more generally signals. In the following, one such technique, wavelets, will be discussed briefly.

5.3.4 *Wavelets What is a wavelet? In principle, a wavelet is a localized function with an integral equal to zero. This means that the area under the wavelet is zero. Two of the simplest examples of wavelets are the Mexican Hat wavelet and the Morlet wavelet. The Mexican Hat wavelet is given by [3]:

( ) ( ) )21exp(1 22

2 xxxg −−= (5.29)

And the complex Morlet wavelet is given by:

( ) ( )

6

)21exp(exp

0

20

41

=

−=Ψ

ω

ωπ xxix (5.30)

Both wavelets are plotted below. It is clear from the plots that both wavelets are localized functions with zero integral.


150

Figure 5.15: The Mexican hat wavelet and the Morlet wavelet The duration of the wavelet can be varied by calculating the wavelet with a multiple of x, like:

( )xg 42 (5.31)

The position of the wavelet can be varied by subtracting a constant from x, like for example:

( )342 −xg (5.32)

Both wavelets are plotted with a simple cosine signal in the next figure.

Figure 5.16: A cosine wave and two wavelets shifted in location

It is clear from the figure that at 4x, the ‘local duration’ of the wavelet is approximately equal to the cosine signal. In the next figure, wavelets of multiple durations are multiplied ‘at all times’ with the cosine signal and the results are plotted. For clarity, the wavelet that is used for the calculations is plotted through the result of the calculations.

( )( )342 −xg

( )xg 42

6 4 2 0 2 4 6 5

4

3

2

1

0

1

x

5 0 51

0.5

0

0.50.446

1−

g 2 x( )

55− x

6 4 2 0 2 4 6 2

1

0

1

2 1.331

1.163 −

Re Ψ x ( ) ( )

5 5 − x


151

Figure 5.17: Convolution of Mexcian hat wavelets with different durations with the cosine wave

In mathematical terms, the following operation was performed with the following values for a: a=2, a=4 and a=8:

( ) ( ) ( )( )∫∞

∞−

−= dxtxagxftF 2 (5.33)

It is clear that the response at a=4 is the highest, because the local duration is approximately equal to the wavelength of the cosine wave. The next figure shows a contour plot for many durations a

Figure 5.18: Contour plot of the convolution of the cosine wave with the Mexican hat wavelet for

many durations a

It is clear that the effect of multiplication of the cosine wave with the Mexican Hat wavelet is that local maxima and minima are isolated. So, F(t) will have a high value at the maxima of the cosine wave and will have a low value at the minima of the cosine wave. Let us now consider another wavelet which is called the antisymmetric wavelet. This wavelet is given by the following equation and plotted in the next figure:

Q( )

( )xg 42

( )xg 22

( )xg 82

4 2 0 2 48

6

4

2

0

21

7−

g xrange( )

g 2 4 xrange⋅( ) 4−

g 2 2 xrange⋅( ) 2−

g 2 8 xrange⋅( ) 6−

F t 2,( ) 2−

F t 4,( ) 4−

F t 8,( ) 6−

55− xrange xrange, xrange, xrange, t, t, t,


152

Figure 5.19: Antisymmetric wavelet It can be shown that using this wavelet isolates slopes from the signal, as the Mexican Hat wavelet isolated local maxima and minima. In fact, multiplication of the signal with a wavelet (of different durations, varied by variation of a, highlights features similar to the shape of the wavelet. This is in fact the idea of the continuous wavelet transform, shortly called CWT. In mathematical shape the wavelet transform is defined by:

( ) ( ) ( )( )∫∞

∞−

−= dxtxkgxfktkF 2, (5.35)

So it is the same as the multiplication of equation (5.33). The difference is that this multiplication is performed for many values of a and that each integral is normalized with the normalization factor k .

Up to now, we have only considered simple cosine signals. In the next figure a combination of cosine signals is analyzed . The combination is given by:

( ) ( ) ( )xxxh ππ 5.0cos5.02cos += (5.36)

If this ‘profile’ is analyzed using the CWT by the Mexican Hat wavelet, then this results in:

( )

−= 2

1 21exp xxxg (5.34)

5 0 51

0.5

0

0.5

10.607

0.607−

g 1 xrange( )

55− xrange


153

Profile

-1.5

-1

-0.5

0

0.5

1

1.5

2

-6 -4 -2 0 2 4 6

x-coordinate

heig

ht

Figure 5.20: Superposition of two cosine waves

Figure 5.21: Superposition of two cosine waves analyzed with the antisymmetric wavelet

It is clear from the figure that using the Mexican Hat wavelet ‘isolates’ maxima and minima of different durations. The wavelet theory presented above is taken from [3]. The interested reader is referred to this article for more information about the CWT. The CWT can be used to isolate shapes from a surface, like defects or local maxima or minima. Wavelet theory is much more complex than the CWT shown above. Other wavelet transforms are the DWT, the discrete wavelet transform and wavelet packets. These transforms will not be discussed here.

5.4 *Random process model Up to now we have discussed several methods to analyze properties of rough surfaces. We also would like to construct a model of a rough surface and with this ‘predict’ properties, like slopes, curvatures and maxima, from a calculation. Here, the model will be used to predict properties of local maxima, called ‘peaks’, present in the surface. A further discussion about local maxima in given in section 5.6 One such a model is the random process model, see [4]. One of the starting points for the random process model is the autocorrelation function which was discussed extensively above:

( ) ( ) ( )[ ]ττ += xzxzEA (5.37)

Three values of a (measured) autocorrelation function are considered: A(0), A(h) and A(2h) with h the sampling interval. Then, if the height at τ=0 is called z0, the height at τ=h is called z1 (or z-1).

Q( )


154

( ) [ ] [ ] [ ]( ) [ ] [ ]( ) [ ]11

0110

21

21

20

2

0

zzEhAzzEzzEhA

zEzEzEA

−

−

−

=

==

===

(5.38)

The other starting point of the random process model is the joint probability distribution of a succession of the three heights in the surface profile:

( )

−

∆= jiijnn yyMzzzzp

21exp

2,...,,,

21321

π (5.39)

With ∆ is determinant of C. Matrix C is given by:

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

== −

221

22221

12121

1

nnn

n

n

ijij

yEyyEyyEyyEyEyyEyyEyyEyE

MC (5.40)

Finite difference approximations of slopes and curvatures can be expressed as linear combinations of three successive heights:

2101

2,1,,1

11,1,12

01,,1

2222

hzzz

hzzz

hzz

hzz

m

hzz

hzz

m

x

jijiji

jiji

jiji

−−+

−−+

+

+−=

+−=

−=

−=

−=

−=

κ

(5.41)

Expected values of heights, slopes, curvatures and cross terms can be expressed in terms of the values of the autocorrelation function:

[ ][ ] [ ] ( )

[ ] [ ] ( )

[ ] ( ) ( )[ ][ ]

[ ] ( ) 2012

2222104

22

1022001

212

2

2022111

212

222

022

200

4432

221

2241

2

2

m

mm

m

AAh

zE

mEzmE

hAAA

hE

AAh

zzzzEh

mE

AAh

zzzzEh

mE

AzE

σκ

κ

σσσκ

σ

σ

κ

−=−=

==

−=+−==

−=+−=

−=+−==

==

−−

(5.42)

Next, the joint probability distribution can also be expressed in terms of slopes and curvatures:


155

( ) ( )[ ] [ ]

[ ][ ] [ ]

( )

( )

α

σσσ

σσσσσ

σσσ

σσκ

σσσ

σσ

κκ

κ

κκ

κκκ

κ

κ

=

=−=−=∆

∆∆

∆∆=

⇒

−

−=

==

−

−

2

2222422

22

2

22

22

2

22

2

2

2

1

with 1

0010

0,,

000

0

000

0

,,,,

2

r

rr

mzM

EzEmE

zEzE

mzMmzC

mm

m

m

m

m

m

m

(5.43)

This results in the following expression for the joint probability distribution of heights, slopes and curvatures:

( )

( ) ( )( )

α

σ

κσκσσσσσσσπ

κ

κκκ

=

−−⋅

⋅

++

−−

−

=

−2

2

2

2222222222

2

223

2exp

212

1exp12

1

,,

r

m

zzrr

mzp

m

m

m (5.44)

This distribution looks very complicated. It however shows a Gaussian distribution of heights independent on heights and curvatures. The heights and curvatures are correlated Gaussian variables with correlation coefficient –r. With this distribution it is possible to calculate many properties of a rough surface, like the mean peak height that is given by:

( )

( )∫∫

∫∫=

slopes allcurvatures all

slopes allcurvatures all

,,

,,

κ

κ

mzp

mzpz

z p (5.45)

Or for example the mean peak curvature that is given by:

( )

( )∫∫

∫∫=

slopes allheights all

slopes allheights all

,,

,,

κ

κκ

κmzp

mzp

(5.46)

Analytical expressions can be found for the mean peak height and the mean peak curvature:

σθ

θπκ

sin2

= (5.47)

And


156

σθ

θπ sin2

rz peak = (5.48)

The parameter θ is one of the parameters of importance in the model dependent on the sampling interval h.

= −

m

hσσ

θ κ

2sin 1 (5.49)

This means that the model predicts summit curvatures and other parameters which are

dependent on the sampling interval h! The other parameter is r, which is related to the Nayak parameter α shown in equation (5.24).

α

σσσ

κ

=

=

−2

2

r

r m

(5.50)

It was already discussed there that r (or α) is related to the frequency content of the surface. A calculated summit height probability density distributions is plotted below. It is clear from the figure that the summit height distribution is nearly Gaussian.

Figure 5.22: Gaussian probability density for the surface and the derived probability density for

peaks The probability distribution of the curvatures follows a Raleigh distribution:

Figure 5.23: Probability density for the curvature of peaks

Gaussian surface Nearly Gaussian peaks

6 4 2 0 2 4 60

0.1

0.2

0.3

0.4

0.50.414

7.217 10 8−×

P ξ( )

dnorm ξ 0, 1,( )

55− ξ

low low high

0 1 2 3 40

0.2

0.4

0.6

0.80.726

0

P t t( )

40 t


157

Also the average curvature as a function of summit height can be calculated and the results are shown below:

Figure 5.24: Average curvature as a function of peak height

So far only results for profiles and their local maxima called ‘peaks’ are shown. Such an analysis is also possible for the 3D case of surfaces and local maxima called ‘summits’, see for details [4]. So important conclusions can be drawn from the random process model:

• Roughness properties as well as summit properties are dependent on scale. This means that roughness measurements are dependent on the limitations of the measurement equipment like measurement area and pixelsize. That roughness is scale dependent is also ‘intuitively’ true!

• When presenting roughness data, always give roughness values with – Measurement length – Pixelsize (sampling interval)

5.5 Fractal characterization

5.5.1 Background The consequence of the above discussion is the following: There will never be a sampling interval h, at which we can measure the slope (for example). The smaller we look, so the smaller the sampling interval h, the higher slopes we will measure. The consequence of this is that it is impossible to measure surface roughness parameters, like it is possible to measure for example the mass, thermal conductivity or the hardness of a sample. Although this seems difficult to accept at first sight, there are many examples in the real world where measuring lengths or sizes of irregular items is difficult. What is the area of a snowflake? What is the length of a river? How long is the coastline of Norway? The results of such measurements will depend on the scale one is looking at it. Mandelbrot [10] has studied this phenomenon and showed that when using a ‘smaller’ length measure (like a meter or a millimeter), the length of the coast of Britain is not converging to a certain fixed value, but increases with decreasing length measure. This means that the length of the coast of Britain is dependent on the length measure one is using. Mandelbrot found a power law relation between the length of the coast L and the length measure l:

DlL −∝ 1 (5.51)

The exponent D was called the dimension, or fractal dimension, of the coast.

6 4 2 0 2 40.5

1

1.5

21.858

0.598

t avg ξ( )

45− ξ


158

A curve, like a coastline, is called ‘self similar’ if, after magnification, the curve is still the same. A curve can also be ‘self-affine’. In this case the curve appears the same after magnification, but the magnification factors in the direction of the curve and perpendicular to the curve need not necessarily to be the same values. A fractal surface is a surface that looks the same, or similar, at all scales so when looking at a surface at different magnifications, the details of the surface look ‘similar’. When a roughness profile or a surface is measured at a higher magnification, more details will be visible which are not visible at a lower magnification. However, in the case of rough surfaces, magnification of a certain part of the surface will not exactly give the original surface back again by multiplication of the obtained surface with a certain factor (or by different factors in height direction and in lateral direction). However, one can obtain the same statistical properties of the original surface by the multiplication process. This characteristic is called statistical self-similarity. A curve can be statistical self-similar or statistical self-affine. A explained above, more details will be visible if a rough profile is measured at a higher magnification. A (deterministic) mathematical function which has similar properties to a rough profile is the so-called Weierstrass-Mandelbrot function, given by:

( ) ( )( )∑

∞

=−

−=1

21 2cos

nnnD

nD xGxz

γπγ

(5.52)

In this function, a profile z(x) is calculated, by a summation of several frequencies. There are several parameters of this function: D, G, γ and n1.

• D is the fractal dimension and has a value 1<D<2. • G is a scaling factor and depends on the finishing operation. • γ is a parameter with an arbitrary value larger than 1. However, γn is related to the

inverse of the roughness wavelengths γn=1/λn. γ is determining the density of the (discrete) spectrum, and is also determining for the apparently ‘random’ properties of the Weierstrass-Mandelbrot function. In practice γ=1.5 is a good value to simulate rough surfaces.

• The last parameter in the function is n1. This parameter is related to the length of the sample by γn1=1/L, and therefore related to the lowest frequency that can be measured with the equipment used. This means that n1 can be calculated from:

γln

1ln1

=Ln

(5.53)

The most important parameter is D, which is the fractal dimension (for a profile 1<D<2). In the figure below the Weierstrass-Mandelbrot function is plotted together with a measured profile


159

Figure 5.25: Measured profile and the Weierstrass - Mandelbrot simulation

It is clear from the figure that a plot of this function looks similar to a measured profile at first sight. In principle, the Weierstrass-Mandelbrot function is a summation (or superposition) of different frequencies, as is clear from the equation. This means that the power spectrum of the function will give insight in the roughness properties at different wavelengths.

5.5.2 Surface characterization using fractals The power spectrum of the Weierstrass-Mandelbrot function is not a continuous spectrum like a rough profile. However, it can be derived that the power spectrum of this curve can be modeled by the following expression, see [7][8]:

( ) D

DGP 25

22 1ln2 −

−

=ωγ

ω (5.54)

From this expression is can be seen that the power spectrum plotted on log-log scale as a function of ω will be a straight line with slope 5-2D. It is found from roughness measurements of some surfaces that the power spectrum of a measured profile or a surface, at least at a range of frequencies, is a straight line on log-log scale. If the following form is assumed, which has been found for a wide range of surfaces [6]:

( ) pp CCP 2

2−== ω

ωω (5.55)

Then the fractal dimension from a measured profile can be estimated from the slope of the power spectrum by 5-2D=2p. If a power spectrum of this form is assumed, it can be shown that the roughness parameter RMS or σ is given by the integral of the power spectrum over all wavelengths like is the case with all power spectra, see equation (5.17):

( ) ωωωωσ dCdP p∫∫∞

−∞

==0

2

0

2 (5.56)

0 2 4 6 8 100.5

0

0.50.377

0.305−

z x( )

9.911 x

Profile result

0 50 100 150 200 250 300 350 400 450 500 -1.5 -1

-0.5 0

0.5 1

1.5 x 10 -6 Plot Weierstrass-Mandelbrot

Plot measured profile


160

This value will only be finite if p>0.5. Similarly, σm2 will only be finite for p>1.5 and σκ

2 will only be finite for p>2.5. As discussed before, these parameters can also be calculated from the power spectrum. This means that outside the mentioned borders, no ‘real’ values can be obtained for those roughness values. Then, if the measurement area is increased, we will measure larger values. Of course, measured values of σm(h)2 and σκ(h)2 with a finite measurement length /area and a given finite sampling interval h will be finite. Alternatively, the Structure function can be used for determination of G and D. This relationship is exponential:

( ) kCS ττ = (5.57)

If this function is plotted on a log-log scale, it has the form:

( ) bkS += ττ loglog (5.58)

So, if a measured SF is plotted on log-log scale, the fractal parameters can be obtained by:

( ) ( )

( )122

2/32sin32log

10

22

−

−−−Γ

−

=

−=

DD

DDb

G

kD

π (5.59)

5.5.3 Fractals & contact behaviour If a Weierstrass-Mandelbrot function is assumed to be a model for a rough profile, then also values for statistical properties of the surface, like σ, σm and σκ can be derived, see [7][8].

( )( )

−

−=== −−

−∞

∫ Dh

Dl

D

DGdPz 2424

12

0

22 1124

1ln2 ωωγ

ωωσ

( )( )2222

122

2

221

ln2−−

−

−==

D

lD

h

D

m DG

dxdz

ωωγ

σ

( )( )D

lD

h

D

DG

dxzd 22

122

2

2

2

21

ln2ωω

γσκ −==

−

(5.60)

In these equations ωl is the lowest frequency possible to measure and is related to the size of the sample. The frequency ωh is related to the highest frequency possible to measure and is related to the resolution of the measurement equipment. So:

• From random process theory it followed that statistical parameters based like σ, σm and σκ are dependent on sampling interval h.

• But: D is scale independent parameter. However, ωl amd ωh are still dependent on the limitations of the instrument which again gives instrument dependent roughness parameters.

Also from fractal theory it follows that measured roughness values are dependent on the

measurement area (and the pixel size), so on the limitations of the equipment that is used to measure the surface.

Based on this, also an expression for the α can be derived:


161

( )( )

D

l

h

DDD

242

21

−

−

−=

ωω

α (5.61)

This expression for α can be compared with equation (5.24) Applicability of roughness characterization with fractals A requirement for characterization with fractals is that the power spectrum is linear on log-log scale. Many measured power spectra are not linear on log-log scale, which makes fractal characterization not a good model for the surface in all cases. Fractal characterization has for example been successful on hard disks and magnetic tapes. Some people argue [1][10] that fractal characterization may be possible by surfaces which are formed by fracture processes (like cleaving) or growth processes (like PVD). Because surface formation mechanism of such processes is not fundamentally changing down to atomic scale, self-similarity or self-affinity can be found in such surfaces. Fractal characterization has also been applied to e.g. sheet material, but characterization of surfaces involving plastic deformation is disputable. The reason for this is that such surfaces are mostly formed by manufacturing processes like grinding, turning or polishing. In such processes grains, tools or abrasives at repeatedly hitting/cutting a surface in a deterministic way, but at random spots. This means that phenomena at a scale larger than the grains, tools or abrasives are fundamentally different than the phenomena at scales smaller than the grains, tools or abrasives. So, self-similarity or self-affinity is then not expected from the physical process responsible for the formation of the surface. As has been done with the ‘traditional’ characterization of rough surfaces, contact models based on fractals have also been formulated. However, contact models based on fractals will not be further discussed in this handout.

5.6 Parameters related to rough surface contact In the case of rough surfaces contacting, they will touch each other at the high sports present on the surface. The properties of the spots where they touch, the asperities, will be very important for the aspects one is interested in, like

• Thermal conduction in the contact area • Friction forces • Wear

Because asperities have a generally irregular shape, it is not so easy to find mathematical expressions for the contact behavior of a single microcontact, like it was done in the previous chapter for spheres. Therefore it is attractive to ‘convert’ the irregular asperity to an easier shape with known properties. For this often spheres and paraboloids are chosen, although other shapes are also possible. Calculations have shown that the exact shape that is chosen is in fact not so important for the behavior of a rough surface as a whole. A rough surface is in fact a large collection of microcontacts, and differences are often statistically canceled out. When calculating local phenomena like the average contact pressure on a certain asperity or local thermal conductivity, the exact shape that is chosen is of relevance. Because of the importance of the dimensions of the asperities, attention will be paid in this section to different ways to characterize the asperities. Two approaches about what asperities in reality are will be discussed. The first is to consider asperities as local maxima present in the surface. These asperities we will in the following call ‘summits’ for surfaces and ‘peaks’ for profiles. The next possibility is to take a cut-off height and to look at the microcontacts that are formed by the surface spots above this cut-off height. These features will be called ‘asperities’ in the following. Asperities are generally more ‘macro-scale’ features than summits.


162

In both cases the extremes on the surface are approximated with a easier, equivalent shapes, which makes the application of contact mechanics to such an ‘equivalent’ shape possible. Asperities are local maxima, present in the surface. As discussed above, local maxima present in the surface will be called ‘summits’ or ‘peaks’ from now on. Although many summit definitions are possible, the most obvious summit definitions are the following:

9 point summit definition

5 point summit definition

Figure 5.26: 9 point summit definition and 5 point summit definition

The 9 point summit definition means that a point is considered to be a summit if it is higher than its 9 neighboring points. The five point summit definition means that a point is considered to be a summit if it is found to be higher than its four neighboring points. The 9 point definition is to be preferred because less ‘false’ summits, like saddle points are detected. A first way to characterize the summits present on a rough surface is to calculate the parameters σ, σm and σκ and calculate average summit properties from these values using equations from the random process model from section 5.4. Alternatively, one can search directly for summits present on the surface and calculate properties of these summits from measured height data. Because we can measure heights and radii for each individual summit, we are not limited to the three parameters discussed above, although it is possible to calculate average properties for the measured values of the individual summits. The same approach, so look at individual summits, one can also take for asperities. The theory of summit- and asperity determination is discussed in this section. An example is shown in section 5.8.2. In this case it will turn out that three properties are of importance (in the easiest case) for contacting rough surfaces: the standard deviation of asperity heights σs, and mean curvature of the tips of these summits κs and the summit density ηs which can be calculated from measured roughness data as will be discussed in section 5.7. Now, in fact the contacting part of the surface is approximated by spheres with varying heights (characterized by the standard deviation σs), but with the same radius β. A certain number of these local features are present on the surface and this number is given by the asperity density ηs. Given the values of the parameters σs, σm and σk and the combined parameter α, random process theory gives predictions for the parameters that determine rough surface contact: the standard deviation of the summit heights σs, the average summit radius β, the asperity density ηs. and the mean summit height sz: Table 5.1: Summit properties as derived from the random process model

Parameter Estimation


163

Summit density ηs 2

2

361

ms σ

σ

πη κ=

Mean summit height sz κσ

σ

πασ 2

2568.24 mzs ==

Mean summit curvature 1−= βκ κκ σπ

σκ 5045.1

38

==

Standard deviation of the summit heights σs σα

σ

−=

8468.012s

Using the above relations, theoretical values of the summit parameters can be determined without detection of the summits in the surface data, but by measuring or calculation of the parameters σs, σm and σk. However, it is important to realize that the underlying assumptions are an isotropic random surface with an exponential autocorrelation function and a Gaussian height distribution. Besides this, infinitely small sampling intervals are assumed. The sampling interval h will be of importance for the determination of the summits, because summits are considered to be local, so small-scale, features. The effects of sampling interval on the properties of the summits are illustrated in paragraph 5.8. In short, summits or peaks have the following properties:

• Local maxima present in the surface • Characterized by a height s and a radius β or eventually a radius or curvature in two

perpendicular directions. *Asperities are that what makes a contact. Up to now local maxima on the surface, so summits, where considered as important for contact. Alternatively, one can think of ‘what makes a contact’ as an important property for contact behavior. This is illustrated in the following figure and will be further discussed in this section.

Figure 5.27: Asperties are that what makes contact Asperities according to the definition given above will be called asperities in the remaining part of this handout. This is done to distinguish asperities from summits, which are defined as local maxima present in a surface. This approach was first followed in contact models based on fractal theory. Based on the fractal model of surface roughness, Majumdar and Bushan [9] formulated a contact model and derived that asperities with a certain length l will have a curvature radius of:


164

( ) D

D

lGl

12 −

=π

κ (5.62)

As follows from this equation, an important parameter relating the radius to the length of the asperity is the fractal dimension D. The value of this parameter can for example be an estimation from the power spectrum or the structure function, see section 5.5. Greenwood [2] gives alternative approaches to estimate asperity radii, based on their length. According to Greenwood, a parabolic asperity of length l and area A, see the figure, will have a curvature of

3

12l

A=κ (5.63)

This relation is a general equation originating from the equation of a parabola.

Figure 5.28: Approximation of an asperity with a parabola

The above equation can be directly used to measure asperity radii β=κ-1 from a profile microcontact based on the length l. Similarly, this approach can be extended to the 2D case of approximating a microcontact above a certain cutoff with an elliptic paraboloid. The general equations are the following:

xy

xy

x

yx

LL

LL

AV

κκ

πκ

2

24

=

=

(5.64)

In this equation, V is the volume displaced by the microcontact, A is the area of the microcontact and Lx/Ly is major length of the microcontact Lx divided by the minor length Ly, i.e. a measure for the eccentricity of the microcontact. So, by measuring the displaced volume V, the contact area A and the eccentricity Lx/Ly, the radii of curvature κx and κy can be calculated at a chosen value of δ. In this way an ‘irregularly shaped’ microcontact can be substituted with a ‘regular shaped’ elliptical paraboloid, having the same eccentricity, contact area and displaced volume as the original microcontact. An equal displaced volume and contact area is convenient, in particular for plastic contact situations, as will be discussed in the next chapter. However, there is a price we have to pay for this and that is that properties of individual asperities, like the radius β, have become intrinsically dependent on the value of δ that is taken. For example if the cut-off height is decreased, an individual microcontact will grow and will result in a larger calculated radius β of the microcontact under consideration. Besides this, details of the microcontact at a small scale are assumed not to be of influence on the mechanical behavior.

A

l

δ


165

Compared with summits, asperities have the following properties: • Defined as ‘that what makes contact’. • Also characterized by a height s and a radius β or eventually a radius or curvature in

two perpendicular directions. • Radius b is dependent on the local cut-off height, so on the load carried by the

microcontact Although properties of individual microcontacts will be dependent on δ, average properties of a set of microcontacts will be more or less independent on the value of the separation d (and therefore of the resulting values of δ) of the microcontacts, see also [2].

5.7 Processing 3D roughness data

5.7.1 Roughness parameter estimation The height parameters discussed in section 5.3 can be easily estimated from measured surface height data. An overview of the formulas is given in the following table for a matrix of surface data with size M x N:


Average roughness value (Ra) ∑∑

= =

=M

i

N

jjia z

MNR

1 1,

1

Root mean average value (Rq or σ) ( )∑∑= =

==M

i

N

jjiq z

MNR

1 1

2,

22 1σ

Skewness (Sk) and ( )∑∑= =

=M

i

N

jjik z

MNS

1 1

3,3

11σ

Kurtosis (Ks) ( )∑∑= =

=M

i

N

jjis z

MNK

1 1

4,4

11σ

5.7.2 *Contacting rough surfaces Because the slopes of the surface are in fact the first derivative and the curvatures are in fact the inverse of the second local derivative, some kind of differentiation of the surface is needed for the estimation of slope- and curvature related parameters. Although there are more possibilities, here the slopes can calculated from the surface height data by finite difference approximations using three points:

central difference forward difference backward difference

y

jijiy

x

jijix

hzz

m

hzz

m

1,1,

,1,1

2

−+

−+

+=

−=

y

jijiy

x

jijix

hzz

m

hzz

m

,1,

,,1

+=

−=

+

+

y

jijiy

x

jijix

hzz

m

hzz

m

1,,

,1,

−

−

+=

−=

(5.65)


166

Alternatively, slopes can be determined by taking into account more neighboring points or by differentiating in the frequency domain by

( ) ))(( zFFTiIFFTz

zd kk

k

ω= (5.66)

Because the curvatures of summits are related to the second derivative, the curvatures can also be calculated in the frequency domain. The finite difference expressions for the curvatures are

21,,1,

2,1,,1

2

2

y

jijijiy

x

jijijix

hzzz

hzzz

+−

+−

+−=

+−=

κ

κ

(5.67)

Like was the case with the slopes, the curvatures are calculated in x- and y-direction. After that, these curvatures are combined into an equivalent second derivative κ by taking their mean. Now, the root-means-square-slope and the root-mean-square-curvature parameters σm and σk can be easily calculated by:


σm ( )∑∑= =

=M

i

N

jjim m

MN 1 1

2,

2 1σ

σκ ( )∑∑= =

=M

i

N

jjiMN 1 1

2,

2 1κσ κ

In the following, the central difference option is used, and the slopes in both directions are combined by taking their mean. In order to calculate the properties of the contacting surfaces, summits have to be detected in measured surface data. From the determined summits, the standard deviation of asperity heights σs, and mean curvature of the tips of these summits κs and the summit density ηs will be important in contact models If the curvatures are measured at summits only, then a value for β can easily be obtained in the following way:

∑=

=max

1max

111

i

isummitsi

κβ

(5.68)

Similarly, σs can be measured by calculating the standard deviation, only taking into account the height at the summit positions:


167

( ) ( )∑ ∑− −

=−=max max

1 1

2

max

2

max

2 11 i

i

i

isummitsssummitss s

izz

iσ (5.69)

The summit density ηs can easily calculated by dividing the total number of summits found by the area measured:

NMhhi

Ai

yxs

maxmax ==η (5.70)

Probability density function (PDF) The probability density function (PDF) of the surface φ(z) can be measured from measured surface height data by measuring the histogram and normalizing it. If only the summit heights are taken into account when calculating the histogram, one gets the probability density functions of the summit heights φ(s), as is needed for many contact models. Power spectrum and autocorrelation function The power spectrum and the autocorrelation function by can be calculated by:

( ) ( )( )2zFFTP =ω (5.71)

( ) ( )( )ωPIFFTxA = (5.72)

5.8 Example The following grinded surface will be used as an example. A picture with some characteristics is shown in the following table. It is clear from the picture that this (grinded) surface has an anisotropic appearance. The roughness is measured over a relatively small area, and the Rq=0.46 µm means that the surface is relatively rough. A slightly negative skewness means that there are a ‘lot of high’ spots on the surface.

Surface Parameter Value

Area 188 x 188 µm Ra 0.36

hx x hy 0.39 x 0.39 µm

Rq 0. 46

pixels 480 x480 Sk -0.57

file Handout.sdfa Ku 3.78

Figure 5.29: Roughness parameters estimated from a grinded surface Some statistical diagrams are shown next. The surface probability density is shown, the surface power spectrum down left and the surface autocorrelation function down-right.

x [micrometer] 0 50 100 150 200

50

100

150

y [micrometer]


168

5.8.1 Roughness parameters As is clear from the surface probability density, this surface has a slightly negatively skewed character. Besides this, the frequencies of this grinded surface in x-direction are much more dominant in the low frequencies than the y-direction as follows from the power spectrum and the autocorrelation function in the bottom line of the table. This will also mean that the curvatures of the summits will also have different curvatures in x- and y- direction.

Figure 5.30: Probability density, estimated from a grinded surface

50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

350

400

450

Figure 5.31 a) areal autocorrelation b) areal power spectrum, calculated from a grinded surface More summit properties determined from the surface, using the method as discussed in section 5.7.1, are shown in the next table.

parameters value

ηs [m-2] 2.71*1011

β [m] 8.10*10-7

σs [m] 4.72*10-7

As is clear from this table, the asperity density is typically very large, and the order of magnitude of the asperity radius is around 1 micrometer.

-5 0 5 0

0.1

0.2

0.3

0.4

0.5 Surface Probability Density Distribution


Measured Gaussian

Probability density


169

5.8.2 *Summits & asperities Summits The properties of the summits will influence the contact properties of the surface in contact. The next figures will show more details about the summits. The summit probability density, which is similar to the surface probability density, is shown below. The shape of this distribution is also similar to the shape of the surface probability density. Although in this case they are similar, this is not necessarily the case.

Figure 5.32: Probability density of a surface compared with the probability density of the summits Above, we have shown the probability distribution of the summit heights. In the same Way, it is possible to construct is probability distribution of the curvatures, so the inverse of the tip radii. This probability distribution is shown below. From this probability distribution it follows that there are ‘a lot of’ summits with low curvature (high radius of curvature, so ‘blunt’). Further, there a few summits that are very ‘sharp’, but they could be damaging, for example they could cause abrasion to the counter surface.

Figure 5.33: Probabvility density of the curvatures, estimated from a grinded surface

Finally, the next plot shows the relation between the average curvatures as a function of summit height. What you can see here, is that higher summits in general have higher

-5 0 5

0.1

0.2

0.3

0.4

Summit Probability Density Distribution

Measured Gaussian

s/σs [-]

2 4 6 8 0

1

2

3

4

5

x 10 -12

Curvature [m-1]

Curvature Probability Density Distribution


170

curvature, which is equivalent with a lower radius of curvature. So, higher summits are in general ‘sharper’.

Figure 5.34: The average curvature as a function of summit height, estimated from a grinded surface Detailed summit properties are shown next. In this diagram, each summit is represented by a dot, where the x-coordinate is the scaled height s/σs and the y-coordinate is the scaled radius β/βmean. Here you see again that higher summits in general have lower radius of curvature/higher curvature. Although it is not so clear now, often you roughly see an ‘L-shaped’ curve.

-4 -3 -2 -1 0 1 2 3 410

-1

100

101

102

β / β

avg [-

]

s / σs [-]

Figure 5.35: The normalized radius β/βavg as a function of normalized sumit height s/σ

Finally, a diagram is shown where the location of all the summits located in the surface are plotted. It is clear that most of them are located on the ridges present in the surface.

-2 0 2

0.5 1

1.5

2 2.5

3

3.5 x 10 6 Summit Height-Average Curvature

Summit Height/RMS [-]


171

Figure 5.36: A plot showing the locations of the summits

At light loads, only the highest summits will be in contact, and the properties of these summits will only be of importance for the contact. If the load increases, also lower summits will play a role in the contact behavior because the high summits are the contacting summits. In the case of rough surfaces, we measure radii with small radius of curvature (small β) which are varying a lot in height (high σs). If the summits of interest are relatively sharp (i.e. have small radius of curvature), the situation can be compared to the presence of ‘needles’ on the surface. In this case, already at light loads high pressures will occur in the microcontacts and plastic deformation will dominate. If the variation in height is high, these means that only a small part of the summits present on the surface will be available to carry the load, which are only the highest ones. In the case of very smooth surfaces, we will measure large radii of curvature. Then, the contact is expected to be much more elastic, because the microcontacts are much larger in size and therefore the contact pressures much lower. This means that the microcontacts will probably be predominantly elastic. In the smooth case, the variation in height is also lower (low σs), which will strengthen this effect. Scale effects on summit properties As discussed above, the sampling interval (or the pixel size of the measurement equipment) is of influence on the summit properties. Here, the surface, originally 480x480 pixels wide, is resampled into 60x60 samples in order to illustrate the effect of pixel size. We would have measured this surface if we would have used a magnification using measurement with 64x larger pixels. This could be a stylus with a larger tip radius or a lower magnification lens in case of an optical measurement system. The results are shown in the following table: Table 5.2: Scale effects on summit properties

I II II/I pixels 480x480 60x60 pixels 0.125 pixel size

0.39x0.39 µm

3.12x3.12 µm 64

area 188 x 188 µm 188 x 188 µm 1

ηs 2.75e11 m-2 8.65e9 m-2 32


172

β 8.21e-7 m 1.76e-5 m 21 σs 4.91e-7 m 3.26e-7 m 0.66

As we can see, increasing the pixel size has the most effect on the summit density en the least on the standard deviation of the summit heights σs. This is a general trend for most rough surfaces. It is important to realize this scale effect. In order to compare the contact behavior of different surfaces, it is therefore important to measure them with the same pixel size and the same measurement area. Asperities Below the asperities are shown, if a cut-off height of 1.5σ is chosen. It is clear from the figure that, as expected, the asperities are oriented on the ridges of the original surface. The number of asperities is much less than the number of summits found, as is clear from the figure.

Figure 5.37: Asperites, determined from the surface

Now we have discussed properties of summits and asperities present on a surface, we will further discuss the contact of individual summits or asperities in the next chapter.

5.9 Excersises Excersise 1 (in Dutch) Gegeven is een oppervlak, gemeten met behulp van een interferentiemicroscoop. Gegevens over de meting, zoals de samplegrootte, zijn opgenomen in de bijgevoegde file, ook verkrijgbaar via deze link. Als check: Na inlezen van het oppervlak hoort deze er zo uit te zien:

β [m]

s [m]


173

50 100 150 200

50

100

150

200

250

300

350

Deze opdracht bestaat uit het uitvoeren van een analyse van dit gemeten oppervlak. In opdracht 3 worden de analyseresultaten gebruikt voor het berekenen van het contact tussen dit ruwe oppervlak en een vlak tegenloopvlak.

• Is het oppervlak homogeen? Is het oppervlak isotroop? • Reken de volgende uit ruwheidsparameters uit.

o Ra o Rq o Rt o Rz o Sk o Ku

• Bereken de kansdichtheidsfunctie van de hoogtes van het oppervlak. Vergelijk deze met een Gausse verdeling. Conclusie?

• Bereken een ‘gemiddelde’ autocorrelatiefunctie van het oppervlak in één richting door autocorrelaties van een aantal parallelle profielen te middelen. Een autocorrelatiefunctie is te berekenen rechtstreeks via de definitie of via de inverse fourier transformatie van het vermogensspectrum. Vergelijk deze met een exponentiele autocorrelatiefunctie. Conclusie?

• Bereken σm en σκ : o uitgaande van de A(0), A(h) en A(2h) van de gemeten autocorrelatiefunctie. o uitgaande van een exponentiele benadering van de autocorrelatiefunctie, met

punten A(0), A(h) en A(2h). o rechtstreeks uit de gemeten oppervlaktehoogtes, uitgaande van het berekenen

van locale afgeleiden door centrale differenties. Conclusie? • Reken de bandbreedte parameter α uit. • Gebruik σm en σκ om benaderingen voor η, σs en β uit te rekenen (zie paragraaf 1.2.7

van de handout)

5.10 Summary In this chapter it is discussed how rough surfaces can be described in terms of assuming it to be a random process. In this context, the background for statistical roughness values like the

Ra value and the RMS value, the height distribution and the autocorrelation function are discussed. Parameters related to contact between rough surfaces are also discussed.


174

Besides a background for surface roughness parameters and rough surface contact, also methods to determine these parameters from rough surface data are discussed and illustrated

with an example.

5.11 References [1] D.J. Whitehouse, 1994, Handbook of Surface Metrology, Physics Publishing, London [2] J.A. Greenwood, 2001, Surface roughness and contact: an apology, Meccanica, vol. 36,

no. 6, p. 617-630 [3] J. Lewalle, 1995, Tutorial on continuous wavelet analysis of experimental data, online

version available at http://www.ecs.syr.edu/faculty/lewalle/tutor/tutor.html [4] J.A. Greenwood, 1984, A unified theory of surface roughness, Proceedings of the Royal

Society of London A, vol. 393, p. 133-157. [5] P. R. Nayak, 1971, Random process model for rough surfaces, Journal of lubrication

technology, vol 93, p. 398-407 [6] R.S. Sayles and T.R. Thomas, 1978, Surface topography as a non-stationary random

process, Nature, vol. 271, p. 431-434. [7] A. Majumdar and C.L. Tien, 1990, Fractal characterization and simulation of rough

surfaces, Wear, vol. 136, p. 313-327. [8] A. Majumdar and B. Bhushan, 1991, Role of fractal geometry in roughness

characterization and contact mechanics of rough surfaces, Journal of Tribology, vol. 112, p. 205-216.

[9] A. Majumdar and B. Bhushan., 1991, Fractal model of elastic-plastic contact between rough surfaces, Journal of Tribology, vol. 113., p. 1-11/

[10] D.J. Whitehouse, 2001, Fractal or fiction, Wear, vol. 249, p. 345-353. [11] B. B. Mandelbrot, 1982, The fractal geometry of nature, W.H. Freeman, New York.

http://www.ecs.syr.edu/faculty/lewalle/tutor/tutor.html


175


176


177

Chapter 6 Mechanical and geometrical properties of surfaces

6.1 Introduction Surface properties can be divided into several classes: Mechanical properties, geometrical properties, physical properties and chemical or material properties. The following table gives an overview of the different properties. This chapter will discuss the mechanical and geometrical properties of surfaces and methods to measure these properties. Table 6.1: Types of surface characteristics

Mechanical characteristics

Geometrical characteristics

Physical characteristics

Chemical Material characteristics

Hardness E-modulus Fracture Residual stress

Topography Roughness Defects

Structure Surface energy

Elemental composition Chemical composition Molecular composition

Mechanical properties of surfaces are important for many applications. Examples are wear resistant applications, where the resistance against abrasive wear is proportional with the hardness of the surface. Besides mechanical properties like hardness, also internal stresses are of importance for coated systems. Internal stresses are almost always present in coatings, when they are deposited on a substrate. Residual stresses, which can be both compressive and tensile in coated systems, will affect the adhesion of the coating to the substrate and other aspects related to the mechanical behavior of the coated surface. An example is the possible presence of compressive stresses, which are favorable for the resistance against fatigue, because cracks will be less able to grow in the presence of compressive internal stresses. Besides this, internal stresses in components where the coating is significant with respect to the dimensions of the component, or if the flatness of the component is critical as is the case with optical components, internal stresses in the coating may cause shape distortion of the components. Besides this, the increase of internal stresses with coating thickness in the case of deposition processes like PVD is also of the main factors limiting the maximum coating thickness that can be deposited on a substrate surface. The most important mechanical properties of bulk materials are the elastic properties, represented in the easiest case by the elastic modulus E and the Poisson constant ν. Besides this, also the maximum allowable stress, represented here by the yield stress σy is a basic parameter that has to be known. Knowledge about the mechanical properties of bulk materials are very important in the design stage of a component, because based on these numbers a component bas to be dimensioned with respect to strength or stiffness. The parameters mentioned above are related to slow application of stress on a component, and can be measured by measurement techniques where the stress is applied more or less slowly on a component. The most well known test to obtain the parameters mentioned above is the tensile test, where an unaxial stress state is applied on the component. In the elastic regime, the stress and the strains are proportional to each other with proportionality constant E, according to Hooke’s law:

xx Eεσ = (6.1)


178

In which σx is the stress in x-direction and εx the strain. Also in the shear case a similar relation can be formulated between the shear stress τxy and the shear strain γxy. This results in Hooke’s law for shear:

xyx Gγτ = (6.2) The proportionality constant is in this case the shear modulus G. The shear modulus G is related to the elastic modulus E according to:

( )υ+=

12EG (6.3)

With ν the Poisson constant. Ïn the figure below it is shown how the E and the σy can be obtained from a stress strain curve. From the figure it is clear that plasticity will occur at stresses which are higher than the yield stress σy. In the case of metals, there is a direct proportionality between the hardness and the yield strength of the metal:

yH σ8.2= (6.4) Further details about bulk mechanical properties will not be discussed in this handout.

Figure 6.1: Stress - strain curve

The above discussion was about relatively slow application of stress. In some applications the behavior in the case of a rapidly applied stress is important. The behavior of material under rapid application of stress can be studied by impact testing. In other applications is it important to known how a certain material responds to cracks and flaws in the material. The parameter describing this property is called the fracture toughness. Sometimes is is relevant to now how the material responds to cyclic loads. For these type of loads fatigue tests have to be performed. Most materials may behave differently at higher temperatures than they do at room temperature. This means that also creep properties will be strongly temperature dependent. It will be clear from the above discussion that different material properties will describe the material behaviour at different loading conditions. In the next paragraph we will focus on the mechanical properties of coatings which are relevant in the case of relatively slow application of stress, so the elastic modulus and the yield strength.

ε

σ

σy

α

E=tan α


179

6.2 Mechanical properties of surfaces One of the main difficulties in the determination of mechanical properties of surfaces in general, is that the surface is not separately available from the bulk of the material. Besides this, when a mechanical load is applied to a component by e.g. bending, the difference is bending force may only be slightly influenced by the coating, because the thickness of a coating is generally much lower than the total thickness of the component on which the coating is deposited. Besides this, also the adhesion of the interface, where the coating may become possibly debonded at high loads, plays a role in the resulting mechanical behavior of the coated system. In some cases, the substrate can be removed from the coated system. If this is possible, this gives the possibility to measure on so-called free standing coatings. When the elastic properties of coatings are compared to the elastic properties of the same material in bulk form, often a large resemblance is found. So, the elastic modulus of a material in coating shape is often about equal to the elastic modulus of the same material in bulk shape. Contrary to the elastic properties, the plastic properties of metals (yield strength, hardness) often higher than bulk material. For example for thin metal coatings with a thickness<0.3 µm it is known that the strength is dependent on the thickness according to the following Hall-Petch relation:

thicknessbKstrength += (6.5)

6.3 Measurement techniques of mechanical properties Over the years, several methods have been developed in order to measure mechanical properties over surfaces and more specific coatings. In all cases a mechanical load is applied to either a coating with the substrate removed or to a coating well bonded to the substrate. This load can be a bending load applied on a clamped beam, an indenter normally loaded and indenting a surfaces or a fluid pressure which is applied on the freestanding coating. The mechanical loading conditions and the internal stress state is different for all these cases. Most methods simulate a simple loading situation where the analytical solution is known from basis mechanics of materials courses. Comparing experimental results with analytical solutions gives the desired properties of the coating.

6.3.1 Indentation techniques In the case of indentation techniques, the mechanical properties are obtained in a different way as for example by bending of a beam. While the bending beam case is suitable to obtain elastic properties, indentation techniques are mostly used to obtain plastic properties, like the hardness of the material. In this case, a load is applied on an indenter. After this, either the contact areas is measured by the indentation depth δ or by the surface area A.


180

Figure 6.2: Indentation tests The hardness of a sample is by definition equal to the load devided by the contact area. :

AF

H N≡ (6.6)

Many types of indenters are available, with defined geometries like cones, spheres or three or four sided pyramids. The following figure shows an overview of the types of indenters that are available and the names of the indentation experiments using these indenters.

Figure 6.3: Common indenters used for indentation experiments These indenters following properties according to the table of the following paragraph:

Brinell, Rockwell-B

Vickers

Berkovich Modified Berkovich

Knoop

Sphere

4 sided pyramid

3 sided pyramid

4 sided pyramid

Cone, Spherical tip Rockwell-C, Alpha cone, Ludwik

FN


181

Table 6.2: Properties of some common indenters

Name Geometry indenter

Material Size Application V/δ3 C=A/δ2

Brinell

Sphere (hard steel or WC)

Diameter 2.5, 5 or 10 mm

Metals

−

31

δπ

R δ

πR2

Rockwell B

Sphere hard steel or WC

1/16 or 1/8 inch ≈ 3.2. mm

Soft metals

−

31

δπ

R δ

πR2

Rockwell C

Cone, spherical tip

Diamond Apex angle 120o Spherical tip radius 0.2mm

Hard metals 3.14 9.42

Alpha cone

Cone Diamond Apex angle 120o No spherical tip

3.14 9.42

Ludwik Cone Diamond Apex angle 900o

1.05 3.14

Vickers 4 sided pyramid

Diamond 1360 between opposite faces 148o between edges

Metal 8.17 24.50

Knoop 4 sided pyramid

1720 between opposite faces 130o between edges

Thin plastic sheets Thin metal sheets Brittle materials

21.81 65.44

Berkovich

3 sided pyramid

Diamond 650 between side and perpendicular Spherical tip radius <0.1 µm

Coatings 8.19 24.56

Modified Berkovich

3 sided pyramid

Diamond 900 between faces

Coatings 0.87 24.56


182

The C in the table above is a number representing the relation between the indentation depth δ and the contact area A according to:

2δCA = (6.7) Is is clear from the table above that the C for the Berkovich and the Vickers indenters are equal. This means that the Berkovich indenter is designed in such a way that the ration between the indentation depth and the contact area is equal to the Vickers indenter. The advantage of the Berkovich indenter over the Vickers indenter is that in the case of a three sides pyramid, there will always be a single top on the indenter. In the case of a Vickers indenter, making a single tip where the four sides of the pyramid come together is not easy to do. If the relation between the indentation depth δ and the contact area A is taken into account, then, the hardness H can be expressed as a function of indentation depth δ by:

2

1δ

NFC

H = (6.8)

As expressed above, the hardness is in fact a pressure and has units [Pa]. The first hardness scale was a hardness scale based on scratch experiments and was formulated in 1812 by Frederich Mohs (1773-1839) as the Mohs scale of mineral hardness. This scale was based on the fact that a material is considered harder if the material is able to scratch another softer material. In the Mohs scale, the hardness ranges from 1 for the softest material up to 10 for the hardest material. The Mohs scale, and some modern equivalents of materials with a corresponding hardness is shown below. Table 6.3: The Mohs hardness scale and modern equivalents

Rating Type mineral Everyday equivalent

1 Talc Baby powder 2 Gypsum fingernail 3 Calcite bronze coin 4 Fluorite iron nail 5 Apatite glass 6 Feldspar penknife blade 7 Quartz steel knife 8 Topaz sandpaper

9 Corundum

10 Diamond

Other common hardness scales are summarized in the following table: Table 6.4: Common hardness scales

Type Symbol Indenter

Rockwell B HRB diamond hemisphere-conical

Relatively soft materials

Rockwell C HRC diamond hemisphere-conical

Materials harder than HRB100


183

Brinell HB 10 mm ball Metallic Materials

Knoop HK pyramidal diamond axis 7:1

Thin plastic sheets Thin metal sheets Brittle materials

Vickers Pyramidal diamond General use

Shore A, B HS Rebound of Hammer Elasticity of Rubber

The following figure gives an overview of the relation between the Mohs hardness and the Vickers hardness. The Vickers hardness is the hardness measured with a pyramidal diamond. The Vickers hardness can be converted to the hardness in GPa in the following way:

][80.9][ HvHvMPaHv = (6.9)

Figure 6.4: The relation between the Vickers scal eand the hardness in GPa is non -linear For surfaces the classical measurement techniques using indentations at relatively large loads are not practical because then the hardness of the bulk of the material is measured instead of the hardness of the surface. Therefore, special techniques called nanoindentation techyniques have been developed. The following figure gives an overview of some characteristics of macro-, micro-, adn nanoindentation techniques.


184

Figure 6.5: Nanoindentation, microindentation and macroindentation

6.3.2 Nanoindentation techniques Nanoindentation techniques are specifically used for the measurement of the hardness of surfaces. This means that the applied loads are typically small and in the order of mN. Using nanoindentation techniques, both the hardness as well as the elastic modulus of the surfaces can be measured. Nanoindentation is a very sensitive technique, which means that hose measurements typically are performed in a climatized environment at a vibration-free table. Another issue related to nanoindentation is that the tip shape is very important at low indentation depths. In macro-scale indentation depths, a simple shape, like a cone, pyramid or sphere is often used. Here, the indentation depths are so small that the curvature at the tip of the indenter has to be taken into account, as is shown in the following figure.

Figure 6.6: The round-off radius is important in nanoindentation experiments

For nanoindentation, typically Berkovich or spherical indenters (Brinell) are used. Berkovich indenters are used because fabrication of a 4-sides pyramid with an exact tip (Vickers) is very difficult compared to the fabrication of a 3-sides pyramid (Berkovich). The advantage of Brinell , Bekobch and also Knoop indenters is that much plastic deformation is caused at the surface at relatively low indentation depths. This characteristic is important for measuring a hardness value close to the surface. Another characteristic of nanoindentation is that the indentation depth is measured as a function of load (depth sensing technique). This is different from macro-hardness measurements, where mostly the contact area is measured by measuring depth or the diagonals of the square indentation in the case of the Vickers indentation technique.


185

In the case of indentation experiments on (coated) surfaces, a rule of thumb is that the indentation depth δ should be less than 10% of coating thickness d to avoid that the hardness of the substrate influences the hardness measurements. This means that a small indentation depth related to the contact area is required for the hardness measurements of relatively thin coatings. This also means that the C is the previous table needs to be high for hardness measurements on thin coatings. For thin hard coatings, indentation techniques which are called nanoindentation techniques are often used. In the case of nanoindentation, the indentation depth is measured as a function of load. The unloading part of the load-indentation depth curve is then used to measure coating properties as elastic modulus E and hardness H. A load indentation depth curve is schematically shown below. When the load is increased from zero to the indentation load FN, the upper line of the curve is followed, so from A to B. Then, at B the maximum load is reached. After this the load will be decreased and the curve is followed along B-D. In the fist stage of the unloading curve, the behaviour will be elastic relaxation. This part of the curve shows a linear relation between FN and δ. Then, the curve will deviate from this linear relation. Often, a line is drawn along the linear part of the unloading curve. In this way point C is obtained, which is the point where this line crosses the δ-axis.

Figure 6.7: Typical load - indentation depth curve obtained from nanoindentation experiments

This depth δp that is obtained in this is taken as the plastic indentation depth. From this depth, the hardness can be calculated by the following equation, which has already been shown above:

2

1

p

NFC

Hδ

= (6.10)

As told above, the initial part of the unloading curve is an elastic event. This means that this initial stage gives information about the elastic modulus of the material that is indented. The contact situation is similar to a cylindrical punch which is pressed on the surface and unloaded, as is shown in the following figure. If small strains and isotropic material behavior is assumed, then the problem can be analyzed by linear elastic contact mechanics, as is explained in [3].

Indentation depth δ [nm]

δp

Load [mN]

FN

A

B

C D

α


186

Figure 6.8: Modelling the unloading part of the load - indentation depth curve by a cylindrical

punch In the case of a 2D problem of a flat punch indenting an half infinite solid, the pressure under the indenter has the following form, see also the chapter about the contact of smooth surfaces:

( )222 xa

Cxp−

=π

(6.11)

With the boundary condition that the total pressure integrated over the contact area should be equal to the applied load FN:

( ) N

a

a

FCdxxFC ππ =⇒= ∫−

(6.12)

This gives the following expression for the pressure distribution under the indenter:

( )22 xa

Fxp N

−=

π (6.13)

In the case of a rigid cylindrical punch with a flat end, the relation between load and indentation depth δ is given by:

δν 21

2−

=ERFN (6.14)

With R the radius of the cylindrical punch. The contact stiffness S for this situation can be obtained by taking the derivative of load to indentation depth from the above equation:

212

νδ −==

ERd

dFS N (6.15)

From geometry of the circular cross section is follows for the contact stiffness S:

πν

ππ

AES

ARRA

2

2

12

−=

=⇒= (6.16)

In the case of two contacting elastic materials, the reduced modulus of elasticity has to be taken:

2

22

1

21

*

111EEEνν −

+−

= (6.17)

FN

a a

δz

x

z


187

Using the reduced modulus of elasticity, this can be shown to lead to the following equation for the contact stiffness:

πAES *2= (6.18)

If the contact situation discussed above is taken as a model for initial unloading of indenters, then we expect a relation for the contact stiffness according to:

πAES *2≈ (6.19)

Remember that from the geometry of the indenter it followed that: 2pCA δ= (6.20)

With the plastic depth equal to the total depth – the elastic depth: etp δδδ −= (6.21)

This gives for the inverse of the contact stiffness S:

pN CEdFd

SK

δπδ 11

21

*

1

=

== −

(6.22)

In the unloading curve,

pN CEdFd

δπδ

α11

21tan *== (6.23)

So the slope of the unloading curve gives a possibility to calculate the reduced modulus of elasticity E*. If the elastic modulus of the indenter is known, then the elastic modulus of the sample can be calculated. For thin hard coatings, often Berkovich or spherical (Brinell) indenters are used. The reason for using a Berkovich inter instead of a Vickers indenter has been explained above. In particular at low indentation depths, the exact geometry of the indenter tip is very important. Also important in the case of nanoindentation is a smooth substrate surface. Because the indentations are so shallow, the presence of roughness asperities of the surface may strongly influence the results of the experiment. In a real nanoindentation experiments, extra experimental complications may arise, like that the load deflection curve measured needs to be corrected with the stiffness of the machine. Another practical complication of nanoindentation experiments is that it is, due to the very small loads, difficult to determine contact, so the zero-load situation. As told above, a rule of thumb is that the indentation depth should not exceed 10% of the coating thickness in order to avoid measuring a combined hardness of coating and substrate. Sometimes analytical expressions are used to correct the measured hardness with the substrate properties in order to obtain the coating properties at relatively large indentation depths. An example of an expression for the apparent hardness is:

ψ−

−+= e

HH

HH

s

c

s

11 (6.24)

With

hEE

ssy

scy δσ

σ

,

,=Ψ (6.25)

In this equation, δ is the indentation depth and h is the coating thickness. Such equations, when used to analyze indentation results, should be applied with care. Because the indentation depth is measured as a function of load, it is also possible to measure hardness as a function


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of depth. In this way, also the hardness of multilayered structures can be measured by indentation techniques.

6.3.3 Measurement techniques for surface roughness Several techniques exist for measurement of surface roughness. The following table gives an overview of the most common techniques and their characteristics. Table 6.5: Overview of measurement techniques for surface roughness

Method Contact or

Contact-less

Point or surface

Characteristics

Stylus Contact Point Damaging possible (higher loads / softer samples) Long ‘track’ possible From workshop tool to very accurate height measurements (MEMS) Long measurement time for measurement of large areas

Laser profilometer

Contactless Point Sufficient light refection required Long ‘track’ possible Long measurement time for measurement of large areas No damaging of samples

Interference-microscope

Contactless Surface Sufficient light refection required Slopes on surface not too high Limited area (3x 4 mm max.) Stichting possble Accurate fast measurement method

AFM ‘Contact-less’ Point Small area (50 mm x 50 mm) Accurate & Versatilr ‘lab equipment’

Confocale microscope

Contactless Point / Surface

Suitable to transparant surfaces (lenses) Less limitation on lacal slopes Less accurate than interference

In the next three sections these roughness measurement systems will be discussed in more detail.


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Figure 6.9: Comparison of length scales and the place of interferometry Interference microscope

Theoretical background Roughness measurements can be done using a Micromap Mirau type white-light type interference microscope. Interference microscopes are non-contact measurement tools that can measure roughness not only in profiles but also in three dimensions. They are comparably faster than other measurement tools and thus suited well for semi-online measurements of surface roughness. An interference microscope utilizes the interference patterns formed during a measurement, to calculate the surface heights and re-construct a surface. These interference patterns are called fringes and are presented in the following figure for an Al2O3 sphere surface. As it can be seen, fringes have dark and light bands positioned next to each other. These bands are at a distance of half the wavelength of the illumination light (as will be derived later in this section) and they form the isometric lines on the areas with similar heights of the surface.

Figure 6.10: Interference fringes from Al2O3 sphere surface A basic setup for a better understanding of how fringes can be used to obtain the height data can be seen in figure 3.3. This setup is called an air wedge setup which is composed of two glass plates positioned with an angle to each other forming the shape of a wedge. The medium of the setup is air. When a light beam from a light source reflects from the first glass plate, a percentage of light will continue its path with a diffraction angle. This beam then reflects


190

from the second glass plate and leaves the wedge again with a diffraction angle. The initially reflected beam follows a path that is shorter than the ray reflected from the second glass plate. Considering that the light motion is in the form of a wave, this path difference causes shift in the positions of the highest and lowest amplitude of light wave with respect to the initial wave. Thus, the light and dark interference patterns form because of a phase shift between the light beam reflected from the reference mirror and the light beam reflected from the surface. In phase beams lead to light fringes (constructive interference) and out-of-phase beams to dark (destructive interference) fringes respectively.

Figure 6.11: Air wedge forming interference fringes In the air wedge setup, the interfering rays are; the one reflected from the first glass plate and one from the second glass plate. If the number of fringes formed in an air wedge setup is represented as n (where n=0,1,2…), considering that at x=0 the phase change is λ/2 (due to reflection of light from air to glass) and the path followed in between the two plates is equal to 2.t (due to very small wedge angle), then the path difference at the nth fringe (constructive interference) at thickness t can be represented as,

22 λ

λ +⋅= tn (6.26)

and for n+1th fringe (destructive interference) it is,

221 λ

λ +⋅=⋅+ t̀)n( (6.27)

Using the previous two equations

2λ

=− t̀t (6.28)

Knowing the wavelength of illumination, one can calculate thickness of a point by counting the light fringes placed between x=0 and the measurement position. Then the equations above can be used for calculating the heights. With similar principles, for a Mirau interference microscope, the lower glass plate in the wedge problem is the surface of interest where a measurement is done and the top glass plate is the reference surface. The fringes are established upon the interference of the rays reflected from these two surfaces as explained in the air wedge example. The operating principle of a basic two beam Mirau interference microscopy system is shown in the following figure (left). A beam splitter is used to split the light from a light source to the surface of interest. Light reflected from the measurement surface and the reference surface within the objective of the microscope in fthe following figure (right) form the fringes as the system is focused on the specimen.


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For a two beam interference at a point, the basic equation for the calculation of the surface height at a phase of Ф is given in

( )yx4λy)h(x ,, φ⋅

= (6.29)

For the calculation of phase within the interference field, the equation for the intensity of the interference fringes (I) is used. This equation, where γ is the fluctuation of the intensity (0< γ<1) is defined as:

( ) ( ) ( ) ( )[ ]yxyxyxIyxI ,cos,1,', φγ ⋅+⋅= (6.30) Using equation the equation above it is possible to calculate the interference intensity at the four phase shifts of 0, π/2, π and 3π/2. These intensities I1, I2, I3, I4 then can be presented as:

( )[ ] ( )φγφγ cos1'0cos1'1 +=++= III

( )φγπ

φγ sin1'2

cos1'2 −=

++= III

( )[ ] ( )φγπφγ cos1'cos1'3 −=++= III

( )φγπ

φγ sin1'2

3cos1'4 +=

++= III

(6.31)

The equations above can be used to calculate the phase at each point as:

( )

−−

= −

31

241tan,IIIIyxφ (6.32)

Substituting Ф in equation (6.29), one can calculate the surface heights. In a Mirau type interference microscope the phase shifts are done by means of the PZT movement of the focus and the explained algebra for the surface reconstruction is done with the developers software. The Micromap interference microscope produces 3D surfaces with height resolutions of approximately 1 nm and an in-plane resolution less than 1 μm depending on the magnification used. The areas that can be measured are presented in the following table for a variety of magnification options. By increasing the magnification, more detail on surface features (such as a wear track geometry) can be obtained.

Magnification Pixel size (μm) Measurement area (for 307200 pixels)

10 0.98 627.20 μm x 470.40 μm 20 0.49 313.60 μm x 235.20 μm 25 0.39 250.88 μm x 188.16 μm

Figure 6.12: Mirau type interference microscope (left), lenses used (right)


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50 0.20 125.44 μm x 94.08 μm 75 0.13 83.63 μm x 62.72 μm

100 0.09 55.75 μm x 41.81 μm As the principle of measurement is based on the light reflected from a surface and its intensity, the specimen on which a topography measurement is done should have reasonable reflectivity. For the current interferometer the required minimum reflectivity of the sample is approximately 1%. The height resolution is independent of the magnification however for rough surfaces the maximum possible slopes that can be measured is dependent on it. For a magnification of x20 the maximum slope that can be measured is 17 degrees. At higher magnifications i.e. x50 or x100, measurements on larger slopes are possible. However, in case slopes are higher than the permissible maximum slope then light is not reflected back to the lens and these spots with high slopes are represented as missing points in the measured surface. A sample measurement which was not processed is presented in figure 3.5. The measurement was done with a 75 times magnification. The white spots show the regions where no valid measurement points are obtained. The data could also involve outliers and measurement noise which should be removed. Methods and procedures for these purposes will be discussed later in this chapter.

Figure 6.13: Surface measured with interfereometry

Confocal microscopy A relatively new optical, so non contact, technique to measure 3D roughness characteristics is confocal microscopy, see [1]. The principle is based on the fact that when light is fed into a pinhole, this results in a point light source. This pinhole is called the illumination pinhole. The light emitted from this pinhole light source is focused by a microscope objective. Light will be reflected from the sample into a second pinhole, after which the detector is placed. If the surface of the sample is positioned in the focus plane of the microscope objective, then a much higher light intensity will be measured by the detector behind the pinhole as the case when the sample is not in the focal plane of the objective, as is illustrated by the following figure, from [1]. This means that the intensity signal from the detector gives height information about the sample.


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Figure 6.14: Basic principle of confocal microscopy

If a 3D surface has to be measured, then the beam resulting from the pinhole needs to be scanned over the surface, so in x- and y- direction. Besides this, a scan in z-direction is required to find the position at which the intensity is maximum, which gives information about the local height at the position of the light beam. If 3D scanning has to be performed over a surface, this would be a very time consuming measurement technique. However, special techniques have been developed to perform the in-plane scanning in an efficient way. A well known technique is Scanning Confocal Microscopy (SCM). Here, a rotating Nipkow disk, which is a disc with multiple pinholes, is used to move the light beam over the surface. A Nipkow disk often consists of pinholes in a spiral shape. Typical values for the pinholes are diameter of 20 µm and a distance between the pinholes of 200 µm. Some basic equations related to confocal microscopy will be discussed below. Firstly, the light intensity as a function of the defocusing distance z, which is the distance in which the specimen is placed out the focal plane, is given by:

( ) ( )( )( ) 0

2

cos1cos1sin I

kzkzzI

−

−=

αα (6.33)

Here α is the aperture angle of the microscope, k is the wavenumber 2π/λ and λ is the wavelength of light. The aperture angle can be interpreted as the angle of the cone of light from specimen plane accepted by the objective. Image-forming light waves pass through the specimen and enter the objective in an inverted cone as illustrated below. A longitudinal slice of this cone of light reveals the angular aperture, a value that is determined by the focal length


194

of the objective. Higher values of α allow increasingly oblique rays to enter the objective front lens, producing a more highly resolved image.

Figure 6.15: Numerical aperture and the angle of the light cone α

If n is the index of refraction of medium between point source and lens, relative to free space, then the relation n sin α is often expressed as NA (numerical aperture). The NA is the lens specification which quantifies the amount of detail that can be seen. The NA takes both the effects of the refractive indices of the media in the light path and the angle over which the objective is capable of receiving light from the specimen these factors into account. The bigger a cone of light that can be brought into the lens, the higher its numerical aperture is. Value of n varies between 1.0 for air and 1.52 for a majority of immersion oils utilized in optical microscopy. Typical height resolutions that can be obtained by confocal microscopy are in the order of 10 nm. The advantage of confocal microscopy is that, compared to interference microscopy, relatively steep slopes can be measured. Another advantage is that the lateral resolution is relatively high. A disadvantage is that some kind of scanning has to be performed in order to obtain a full height map of the surface.

Atomic force microscope

The atomic force microscope (AFM), or scanning force microscope (SFM) was invented in 1986 by Binnig, Quate and Gerber. Like all other scanning probe microscopes, the AFM utilises a sharp probe moving over the surface of a sample in a raster scan. In the case of the AFM, the probe is a tip on the end of a cantilever which bends in response to the force between the tip and the sample.

Most AFM’s employ an optical lever technique to detect the bending of the cantilever.The diagram illustrates how this works; as the cantilever flexes, the light from the laser is reflected onto the split photo-diode. By measuring the difference signal (A-B), changes in the bending of the cantilever can be measured. Since the cantilever obeys Hooke's law for small displacements, the interaction force between the tip and the sample can be found for elementary mechanics of materials. The optical lever operates by reflecting a laser beam off the cantilever. Angular deflection of the cantilever causes a twofold larger angular deflection of the laser beam. The reflected laser beam strikes a position-sensitive photo detector consisting of two side-by-side photodiodes. The difference between the two photodiode signals indicates the position of the laser spot on the detector and thus the angular deflection of the cantilever.


195

Because the cantilever-to-detector distance generally measures thousands of times the length of the cantilever, the optical lever greatly magnifies motions of the tip. The movement of the tip or sample is performed by an extremely precise positioning device made from piezo-electric ceramics, most often in the form of a tube scanner. The scanner is capable of sub-angstrom resolution in x-, y- and z-directions. The z-axis is conventionally perpendicular to the sample.

Figure 6.16: Working principle of the Atomic Force Microscope (AFM)

There are many extensions to this basic AFM principle, like tapping mode, where the cantilever is tapped on the surface of FFM measurements, where also lateral forces can be measured.

Stylus and optical stylus Mechanical stylus The classical tool and the industrial standard for measuring surface profiles is the mechanical stylus. The stylus probe is passed across the surface and its movement, as it follows the surface profile, is measured. This is still the standard surface profiling technique and can offer high precision allowing the measurement of surface roughness. However, the stylus technique has its disadvantages. There are limits in lateral resolution, set by the size of the probe tip; peaks that are narrower than the probe tip will be recorded as being broader than they really are, while the probe will be physically incapable of reaching the bottom of narrow troughs. This reduced lateral resolution results in a loss of surface height information and results depend on the shape of the stylus. In addition, the stylus probe technique measures the profile along a line of the surface, and, if the entire surface is to be mapped, many such lines must be scanned. It can also have difficulties measuring highly curved or convoluted surfaces with steep slopes. However, the major drawback of the technique is that it involves contact with the surface. The stylus probe tip can damage or alter the surface, which for many applications makes it unsuitable as a profiler. An important choice that has to be made in stylus measurements is the radius of the tip that is going to be used. A large tip radius around 1mm neglects the high frequencies at the surface and is therefore advantageous for shape measurements. A small tip radius in the order of 5 µm is advantageous for roughness measurements. Some other remarks related to a mechanical stylus are:

• Because of the size of the tip, it is has to measure small and deep indents in the surface.


196

• There will always be a small load applied on the tip of the stylus. This means that in the case of fragile samples or very soft samples, there is a risk of damaging the sample with a mechanical stylus.

Stylus equipment is also sometimes used for shape measurements, like roundness, cylindricity and flatness. An example of a roundness measurement is shown in the following figure.

Figure 6.17: Stylus used for roundness measurements

Stylus equipment can be obtained in different forms, ranging from portable equipment suitable for a workshop to measure a roughness value to equipment with a high resolution to measure MEMS. In mechanical stylus equipment, the displacement at the tip of the stylus is typically measured by means of an electronic circuit, as is shown in the figure below.

Figure 6.18: Tip an delectronic circuit of a mechanical stylus

Optical stylus The optical stylus as, as is the mechanical stylus, able to make tracks on the surface. The largest advantage of an optical stylus compared to a mechanical stylus is that it is a contactless method, which means that no damage will be caused to the surface. A disdadvatage is the higher price of the equipment and the fact that some reflectivity of the substrate is required because it is an optical system. An optical stylus is in fact device which measures the distance between the frame on which the sensor is mounted and the surface which is passing underneath the sensor. Optical styli are available with different working principles, like auto focus, confocal or interference sensors.

Capacitor plates

Stylus tip Electronic circuit

Small load (mN)


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6.4 Excersises Excersise 1

• Het oppervlak had een (opgegeven) hardheid van 2 GPa. Nu willen de deze hardheid nameten met behulp van nanoindentatie met een Rockwell-indenter, . Welke indentatiediepte reken je uit bij een belasting van 10 mN?

• Hoe verhoudt dit deze indentatiediepte tot de ruwheid van het oppervlak? Wat betekent dit voor de nauwkeurigheid van de meting? Hoe kunnen de hardheid nauwkeuriger bepalen?

• Schets het te verwachten indentatiediepte-belastingsdiagram voor de beschreven meting, uitgaande van een glad oppervlak. Hoe zal deze figuur er ongeveer uitzien in het geval van een ruw oppervlak?

6.5 References [1] M. Ohring, Materials Science of Thin Films, 2002, Academic Press. [2] H.J. Jordan, M. Wegner and H. Tiziani, 1998, Highly accurate non-contact

characterization of engineering surfaces using confocal microscopy, Measurement Science Technology, vol. 9, p. 1142-1151.

[3] Johnson, K,L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, United Kingdom


198


199

Chapter 7 *Damage mechanisms

7.1 Introduction As told in the introduction chapter, there are four main reasons to use coatings and surface treatments on surfaces. One of the reasons is to protect the surfaces from influences of the environment or other factors, like the applied mechanical load. In order to understand the application of surface treatments and coatings for reasons of protection, in this chapter causes for failure or degradation of surfaces will be discussed. The most common degradation mechanisms are shown in the following figure, which looks very similar to a figure in the previous chapter, where different coating techniques and surface treatments were summarized according to the same mechanical, thermo-mechanical, thermal, thermo-chemical, electrochemical, physical and biological mechanisms.

Figure 7.1: Overview of several degradation mechanisms Atmospheric damage mechanisms include the combined effects of humidity, rain, snow, sunlight, sea water, erosion, and includes a combination of mechanisms mentioned above. In section 7.2 mechanical damage mechanisms, in section 7.3 (electro)chemical damage mechanisms will be studied. Thermal, physical and biological damage to surfaces is not extensively discussed, but only briefly mentioned. Almost all degradation mechanisms are system properties and not material properties. For example, a material is corrosion resistant in a certain environment with a certain pH. A system will be wear resistant if it is in contact with a certain countersurface under certain operational conditions, like load, velocity and temperature. If thermal damage is considered, a system will undergo thermal damage above a certain temperature of the environment or in a certain contact situation where the heat is not conducted easily away from the surface.

Degradation mechanisms

Thermo-mechanical

Thermal Mecha-nical

Thermo-chemical

Electro-chemical

and chemical

Physical

Creep

Relaxation

Thermal damage

Melting

wear

Direct flames

Oxidising

gases

Abrasive wear

Adhesive

wear

Fracture

Fatigue

Vibration

High T corrosion

Inorganic aqueous chemical reactions

Oxidative

wear

Attack by acids,

alkalis, salts,

solvents

Radio- activity

X-ray

damage to polymers

Consumption

Bacterial corrosion

Biological Biophysical Biochemical


200

Corrosion (chemical damage), wear (mechanical damage) or melting (thermal damage) will generally be different in another contact situation or in another environment.

Figure 7.2: Degradation mechanisms like wear and corrosion are system properties

Mechanical damage by wear takes place at the interface between two contacting bodies. This means that the following aspects will determine the behavior of the so called tribological system:

• Properties of surface 1 and surface 2 o Material properties (chemical, physical, mechanical etc.) o Geometrical properties (roughness and shape)

• Properties of the environment o Type of environment (air, vacuum, etc.) o The presence of dirt o The relative humidity

• Properties of the lubricant, if present o Viscosity o Chemical composition

• Operational conditions o Load (contact pressure) o Velocity o Temperature

Besides that mechanical damage can occur in the case of two contacting bodies, also thermal damage can occur in such a system. In the case of chemical damage, for example corrosion, the damage takes place at the interface between a solid surface and the environment. This means that the system is, in this case, composed of a single surface and the environment. Thermal damage can also occur in such a system, as it can occur in the case of two contacting bodies.

7.2 Mechanical damage A machine component can fail because in the bulk of the component the stresses reach too high values. This problem can be solved by redesigning the component or by choosing a more suitable (stronger) material. Such (mechanical) failure cannot be solved by a surface treatment, because surface treatments and coatings are thin compared to the dimensions of the machine component, and do not contribute to extra strength and stiffness of the component as a whole. However, surface treatments and coatings can contribute to decrease mechanical

2

1

FN

v1

v2

System

Corrosion

Friction and wear


201

failure of the surface or mechanical failure of the component as a whole, where initiation takes place at the surface. One of the main reasons of mechanical failure taking place at surfaces is wear. In this section, different wear mechanisms will therefore be briefly discussed. After that, other mechanical failure mechanisms will be briefly described.

7.2.1 Contact Contact of smooth bodies In many applications, two bodies will be in contact. If two bodies are in contact, and have to carry a load, then contact pressures will the acting the surfaces of the contacting bodies. Contact situations can be classified according the following figure into single body, double surface contraform contacts, double body conform contacts and three body contacts:

• A double surface contraform contact. Such a contact is characterized by high contact pressures and a small contact area. An example is the contact between a wheel and a road.

• A double surface conform contact. Such a contact is characterized by lower contact pressures and a larger contact area. An example is a sliding bearing.

• A three body contact. Here, solid particles are present in the contact between two bodies. The particles can be present in a lubricant as a carrier. An example is the presence of dirt and dust in a tribological contact.

• The fourth contact type is a single body contact. Here a surface is in contact with a stream of gas, fluid and/ or solid particles.

Figure 7.3: Double surface contraform, double surface conform, three body and single body

contact situations The double body contacts can be further divided into several types. Depending on the geometry of the contacting bodies, the shape on the contact area, formed by elastic deformation of at least one of the bodies, has the shape of a circle, a line or an ellipse. These contacts are called respectively a point-, line- or elliptical contacts. Point contacts are formed by a sphere contacting another sphere or a plane:

double surface conform

double surface contraform

three body single body


202

Figure 7.4: Point contacts

Line contacts are formed by a cylinder (a ‘sphere’ with an infinite radius in one direction) and a plane or another cylinder:

Figure 7.5: Line contacts

And elliptical contacts are formed by the following contact situation:

Figure 7.6: Elliptical contacts

1

Contact area

Cylinder on plane

Two cilinders

2

Contact area

Two spheres

2r

Sphere on plane

Contact area


203

In the case of line contacts, elliptical contacts and point contacts, these (elastic) stresses can be calculated with the theory of Hertz. This theory will be not discussed in detail here. However, in all the contact situations mentioned, the pressure distribution has an elliptical appearance, with a maximum contact pressure in the middle of the contact. In the case of point contacts, the maximum pressure is 1.5 times as high as the mean contact pressure The contact area will be circular in the case of a point contact. The contact radius is represented by the symbol a.

Figure 7.7: Hertzian pressure distribution

Below the surface, a subsurface stress field will develop. It can be shown that intial plastic failure will take place below the surface, because the highest shear stresses are acting below the surface at a depth of about half the contact radius. The exact location is dependent on the tangential load FT that is acting in the contact. The subsurface stresses are at its maximum at the symmetry axis for the situation that FT=0, as is shown schematically in the following figure. It can be shown that the maximum shear stress in the material is about equal to half the mean contact pressure for a point contact.

p506.0max =τ (7.1) In the case of metals, the maximum shear stress that a material can sustain is half the yield strength in tension according to the Tresca flow criterium. So,

yallowed στ 5.0= (7.2) Therefore, in order that the contact stays elastic, the mean contact pressure has to stay lower than the yield strength of the material, so the mean pressure has to be lower than approximately the yield strength of the softest material:

yp σ≤ (7.3)

pmax

max32 pp =

2a


204

Figure 7.8: Maximum shear stress is below the surface in the center of the contact for the

situation of a sphere contacting a flat Above, the situation of a ball on a flat has been discussed without a tangential force. A similar discussion can be held for elliptical contacts and line contacts with and without tangential force acting. Also for some other contact geometries analytical solutions for the contact pressures and the deformations are known. For more complicated cases, numerical techniques can be used to calculate the contact pressures and the subsurface stress field. For details, the reader is referred to . Contact & Surface roughness In the case of rough surfaces, the contact area will not be a simple point, line or ellipse as they are in respectively point, line and elliptical contacts. In the case of rough surfaces, the load will be carried by a large number of small contacts, the so called microcontacts. The contact situation is schematically shown in the following figure.

Figure 7.9: Nominal contact area and the real contact area. The real contact area is the sum of

the microcontacts. In the figure it can be seen that the load is carried by a large number of microcontacts. The total contact area is composed of many microcontacts with a small contacting area ∆A. The total area in contact is called the real contact area Ar. The contact area over which two

FN

τmax

FT

Nominal contact area An

Real contact area Ar=Σ∆A

∆A ∆A ∆A ∆A FN

An


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surfaces are apparently in contact is called the nominal contact area An. In the case of contacting rough surfaces, the real contact area is typically only a few percent of the nominal contact area. Exceptions are very contacts with very high contact pressures acting, highly elastic surfaces or cases with plastic deformation of the bulk material as is the case in for example deep drawing processes. Here, the real contact area may approach the nominal contact area in some situations.

7.2.2 Friction Strictly spoken friction forces acting at the interface do not have to result in surface damage which is the topic of this chapter. However, in some situations friction is related to damage taking place at the contacting surfaces. An example is a situation where friction forces acting in the contact area will cause the development of heat. If the heat conduction from the surface to the bulk is hindered, for example by insufficient contact area or a thermal conductivity which is too low, a high temperature will arise in the contact area. Then, thermal degradation of the surfaces of the contacting bodies can occur because a critical temperature is exceeded. This critical temperature can be a phase transformation or melting temperature for metals, the glass temperature for semicrystalline polymers or the melting temperature for amorphous polymers. Besides the formation of excessive equilibrium temperature at the interface, also at smaller length scales high temperatures can occur at the interface between contacting bodies. The reason for this is that asperity contacts in a sliding contact are present at a specific location during only a short instant of time, because contact spots are quickly formed and broken again. In such microcontacts, high local temperatures are acting at the asperity tops during only a short instant of time are called flash temperatures. Flash temperatures can also cause (local) thermal degradation if the temperature exceeds a certain critical temperature. An example in thermal decomposition of a lubricant present in a lubricated contact. So although friction is always not causing damage to a surface, the presence of excessive friction forces in the contact area can contribute to the occurrence of thermal surface damage indirectly.

Static versus dynamic friction If we restrict ourselves to sliding friction, two situations can be distinguished The first situation is the case when there is no sliding between two contacting bodies. However, one of the contacting bodies is loaded with a tangential force. The reason that there is no sliding is that the interface ‘is strong enough’, so has a high enough shear strength and a large enough area, to sustain the tangential force. Although there is no macroscopic slip in the contact area in this contact situation, at a smaller scale there will be deformation due to the tangential force. The situation where there is no macrosliding but microscopic deformation is called the static friction regime. The second situation is the case where there is macroscopic sliding. Here, the interface is ‘not strong enough’ to hinder the sliding movement between the contacting bodies. This contact situation is called the dynamic friction regime. Suppose, that starting from a tangential force equal to zero, the tangential load is increased. As explained above, in the beginning there will be no macrosliding in the contact area. Then, increasing the applied tangential force will at a certain moment result in macrosliding. The tangential force at which slding takes place is called the maximum static friction force. The coefficient of static friction is defined as the maximum static friction force divided by the normal forces, so

N

Tstatics F

F max,=µ (7.4)


206

If sliding occurs, then the dynamic (or kinetic) coefficient of friction will be of importance, so then

N

dynamicTk F

F ,=µ

(7.5)

Amontons (1663-1705) defined the so called classical friction laws. This law says that the (kinetic) friction force is (nearly) independent of the nominal contact area An, the velocity v, the temperature T and the roughness of the contacting bodies. Besides this, the friction force is proportional with the normal force FN. If the friction force is proportional to the normal force, then the results in a constant coefficient of friction. In classicial studies often a value of about 0.3 is found. One should remember that these classical friction laws are valid for unlubricated ‘classical’ material combinations like wood and stone. As soon as there is a lubricant present which is able to carry a normal load, the coefficient of friction will not be constant anymore. The same is often true for ‘non-classical’ materials like plastics, where it is often found that temperature and velocity have large influence on the coefficient of friction.

Friction mechanisms Friction forces acting at an interface can be subdivided into two contributions:

• Deformation (ploughing of asperities) • Adhesion

Both effects are schematically shown in the following figure for the case of a sphere contacting a flat.

Figure 7.10: Two friction mechanisms: Ploughing and adhesion

Ploughing For ploughing, the sphere needs to be harder than the flat. The contribution of ploughing to the coefficient of friction is strongly related to the geometry of the ploughing body, represented by the frontal area Af and the normal contact are Ar.

rm

f

N

wp Ap

AFF σ

µ == (7.6)

For a sphere, the fraction Af/Ar is a function of the indentation depth and the radius of the sphere R.

=

Rf

AA

r

f δ (7.7)

µFN

pmAr

R

rAτ

FN

µFN

pmA

σAf

R

δ

ploughing adhesion

FN


207

In theory, the total result for many asperitites with different shapes can be calculated by taking the integral of the forces over all contacting asperities:

∫∫=

rm

fp dAp

dAf

σ (7.8)

However, this is difficult in practice because for rough surface, consisting of many asperities, δ/R is different for different asperities and often unknown, because asperities do not have simple shapes like, for example, spheres. As already told, ploughing is dependent on the roughness of the hardest surface. If hardest surface is not very rough, ploughing is not dominant and µp~0.05. A small hardness difference between the contacting bodies will result in the absence of ploughing. If the hardness difference is smaller than 10%, ploughing effects can be neglected. So, to summarize, the ploughing contribution to friction can be reduced by:

• Hard materials, but hardness difference < 10%. • Low surface roughness of the harder contact partner.

Adhesion & surface energy From experiments it has been observed that attractive forces exist between bodies which are close together. Intermolecular attractive forces are not only present at interfaces between bodies. Intermolecular attractive forces can also be present between a fluid and a solid, and are always present inside solids and fluids and are responsible for ‘keeping the material together’. If intermolecular attractive forces are acting inside a solid, these forces are called cohesive forces while in the second case these forces are called adhesive forces. Real-life examples of strong attractive forces between solids and fluids are the sticking of chewing gum, clay or glue. Adhesive interaction between a fluid and a solid surface is also important for ‘wetting of liquids on solid surfaces’ and therefore for capillary forces, lubrication and coating technology. If attractive forces are present in the interface between two solid bodies, a force will be required to separate the two bodies. The force perpendicular to the surface, required to break the adhesion forces, is called the adhesion force Fa. As already mentioned, the origin of adhesion forces lies in atomic and molecular interactions between the two solids. A distinction can be made between strong bonds and weak bonds between atoms. Within the class of ‘strong bonds’, the following kinds of bonds can be distinguished:

• The ionic bond or electro covalent bond. This bond is caused by attraction forces between ions of opposite charges. The ionic bond is of importance for the cohesion within ionic crystals like NaCl The covalent bond is based on sharing one or more pairs of electrons. An example where covalent bonds are found is a diamond crystal.

• The metallic bond is caused by free electrons moving between positive ions. This force is the basis of cohesion of metals and their alloys both in the solid as well as in the liquid state.

Also within the class of ‘weak bonds’ several types of bonds can be distinguished: • The hydrogen bond. The hydrogen bond is formed by bonding two molecules with an

‘intermediate’ hydrogen atom. The hydrogen atoms, bounded to a molecule, will form a positive pole to the molecule. This positive pole attracts a nearby negative molecule and forms a bond. The hydrogen bond is therefore of electrostatic nature.

• The van der Waals bond. The van der Waals force is caused by electromagnetic interactions caused by continuous fluctuations in the electron distribution of atoms or molecules. If such attractive forces exist between gas molecules, the perfect gas law PV=RT is not valid anymore.

All forces between atoms or molecules mentioned above are of electromagnetic nature, but strong bonds are generally 100 times stronger than weak bonds. Although weak bonds are weak compared to the strong bonds, the working range of weak bonds is generally longer than the working range of strong bonds.


208

So, atoms within solids and liquids will experience a force which binds them to the neighboring atoms, and the strength of the binding forces depends on the type of bonds that are present between the atoms. If atoms are bound to neighboring atoms, then the creation of a new surface in a solid or liquid involves the breaking of these cohesive bonds between the atoms or molecules. Besides this, the atoms at the surface will be slightly rearranged because they transform from ‘bulk atoms’ to ‘surface atoms’. The (reversible) amount of energy required (per unit area of the created surface) under isothermal conditions is called the surface energy. The surface energy is represented by the symbol γ and typically expressed in the units [J m-2]. The energy required to generate a surface is related to the energy related to separate two surfaces. More precisely, the energy required to generate a surface, or in other works, the work that has to be performed to separate two bodies, is the surface energy required to generate two surfaces minus the surface energy of the interface between the two contacting bodies, so:

abbaab γγγγ −+=∆=Γ (7.9) In the case of a bulk material is broken into two pieces, the surface energy of the interface is zero and the energy required to generate the two surfaces is equal. Then, the equation gets the following form:

γγ 2=∆=Γab (7.10) The work of adhesion Γab has units [Nm/m2] or [N/m]. The work that has to be performed in order to separate two surfaces is generally called work of adhesion and is represented by the symbol Γ . This work is equal to the energy required to generate the interface, which is represented by ∆γ. For further details, see section 4.4. So, it can be concluded that surfaces influence each other if the distance is smaller than 0.3 nm which is about the atomic distance. In the remainder of this handout we will see that the surface energy is an important quantity for many coating processes, and is, for example, responsible for the type of microstructure of the coating that is formed on the substrate surface. Adhesion forces and friction Like is the case for ploughing, also an expression can be given for the contribution of adhesion to the friction force. Without further comments is will be said that in the case of plastic contact the contact pressure is equal to the hardness H of the material. Then:

HHAA

FA

r

r

N

radh

τττµ === (7.11)

If a tangential load is applied to two adhered surfaces, there are two possibilities, as shown in the following figure. Firstly, shear can take place in the weakest contact partner, which is arbitrarily taken as the sphere in the figure. This will happen is the shear strength of the sphere is smaller than the shear strength of the interface, so:

sphereerface ττ >int (7.12) Then the coefficient of friction is determined by properties of the sphere only, and is given by:

sphere

sphere

r

r

N

radh HHA

AFA τττ

µ ===

(7.13)

Secondly, shear can take place at the interface. Then, the relevant shear strength is the shear strength of the interface τinterface. Shear at the interface will take place if

sphereerface ττ <int (7.14) If the interface is thin and therefore not of influence on the maximum pressures that can be carried by the contact, the relevant hardness still determines the load that can be carried is still the hardness of the sphere. So, the coefficient of friction is determined by the shear strength of


209

the interface and the (bulk) hardness of the softest contacting body. This results in the following expression for the coefficient of friction due to adhesion:

sphere

erface

r

r

N

radh HHA

AFA intτττ

µ ===

(7.15)

Figure 7.11: Shear in the body and shear at the interface Often a thin soft layer is applied in the contact to reduce fricton. A soft layer often has a small shear strength τinterface, Then, the contribution of friction will be low, because the nominator of the equation above is small. However, if the layer is thin enough, the pressures that the contact can carry are still determined by the hardness of the sphere and not by the properties of the thin soft layer. This means that the denominator is still high. A small nominator and a large denominator results in a small contribution of adhesion to friction. In the case of a thick soft layer, the relevant hardness will be the hardness of the thick soft layer, and the contribution of adhesion to friction is increased To reduce adhesion contribution, the following measures can be taken:

• Non-metal / metal or non-metal / non-metal, so non-metallic contacting surfaces • Lubrication, carbonizing, oxidation • High hardness / different hardness (ratio 3 a 5) • Rough surfaces (>0.2 Ra)

Thin layer with low shear strength

7.2.3 Wear Wear mechanisms can be subdivided into single surface wear mechanisms and double surface wear mechanisms.

Single surface wear mechanisms Single surface wear mechanisms are wear mechanisms that cause damage to a surface, and are not related to the contact with a second body. Most of the single surface wear mechanisms are erosion related. The single surface wear mechanisms can be subdivided into four types, were the distinction is made based on the erosion causing medium:

µFN

pmAr

R

rAτ

Shear in body a

Shear in the interface

FN


210

• Cavitation erosion. The damage is caused by the formation if the local pressure in a fluid is reaching values lower than the vapor pressure. Then ‘bubbles’ are formed in the fluid. If these bubbles implode close to a solid surface, pieces of material can be ‘drawn out’ of the surface, hereby causing damage

• Particle erosion. Particle erosion is erosion caused by a stream of solid particles. The particles are often present in another medium, like a gas stream of a fluid stream. A determining importance in particle erosion is the angle at which the stream hits the surface. Polymers and rubbers have advantages at large angles between the stream and the surface, because these materials are sensitive for ploughing effects but not for fatigue. Hard brittle materials have advantages at low angles of incidence because they are not sensitive for ploughing but are sensitive for surface fatigue (fracture) . Metals have properties in between the two extremes

• Droplet erosion. Droplet erosion is erosion by the formation of droplets in, for example, supersaturated steam.

• Fluid or gas erosion. A high speed of gas stream without the presence of particles or droplets can cause damage to a surface.

Double surface wear mechanisms Double surface wear mechanisms are taking place in the contact between to solid bodies, and can be subdivided into the following types:

• Abrasive wear • Corrosive wear • Adhesive wear • Melting wear • Deformation wear • Fretting wear • Surface fatigue

Abrasive wear Abrasive wear can be subdivided into two types:

• Two body abrasive wear • Three body abrasive wear

Two body abrasive wear is abrasion caused by a hard and relatively rough countersurface. Three-body abrasive wear is abrasive wear caused by external particles This can be dust or for example work-hardened and oxidized wear particles. The abrasive wear rate is influenced by the hardness difference between the ploughing body and the ‘victim’ body, the shape of ploughing body and the freedom of movement of the ploughing body. Although abrasive wear can be at a certain level in the beginning at the sliding process, this level may change during the process. This can be caused by wearing off of the sharpest asperities from the rough surface. So, in time asperities may get blunt. Another effect is that in time the roughness valleys can get filled with material from the ‘victim’ countersurface. If the roughness valleys get filled, this will result in less abrasion of the countersurface. This phenomenon is desirable in many plastic-metal contacts. So, in such contacts, some initial abrasive wear is desired in order to be able to form the transfer layer on the metal surface. This also means that a smooth metal countersurface is not the optimal situation for a contact in which the formation of a transfer layer is desirable. It is not always true that the hardest surface will abrade the softest surface. If some cases, wear particles may oxidize and work harden. These harder particles will become embedded into the soft surface and abrade the harder surface. Embedding of wear particles can be advantageous if they ‘sink deep’ into the soft surface, but can also have negative effects because fixed particles can abrade more than free particles. The embedding process is shown schematically in the following figure.


211

Figure 7.12: Effects of embedding of hard particles in a soft matrix Abrasive wear can be reduced by the following measures:

• A hardness difference between the contacting bodies < 10%. • A low roughness of the hardest surface • The formation of a hard surface layer by e.g. nitriding or carbonitriding • Embedding of particles in softest surface • Removal of wear particles (oil filters) • Lubrication. The lubricant film thickness needs to be higher than the size of the

particles. • No external hard particles (sealing).

Corrosive wear Corrosive wear can be considered as a special case of abrasive wear, restricted to a (self healing) oxide layer which is present on the surface. Oxidative wear is in fact a process of formation and removal of chemical reaction products from a (metal) surface. If the oxide layer is strong, oxidative wear results in a mild wear mechanism. If the corrosion rate of the surface is too low, then the absence of a sufficient thick corrosion layer can cause severe adhesive wear. The corrosion rate can, for example, be increased a higher temperature. In many lubricants so called EP additives are used. These additives become active at a certain critical temperature and form reaction layers on the surface which protect the surface. Also the other extreme is possible: If the corrosion rate is too high, then this can result is a high corrosive wear rate, in particular if the oxide layer is weak. A dense and mechanically strong oxide layer will hinder further oxidation. An example is the presence of an Al2O3 layer on a Al surface. The aluminum oxide layer is very hard and dense and protects the aluminum from further oxidation. The formation of an oxide layer on aluminum can be stimulated by anodizing, which is a surface treatment where the Al2O3 oxide layer is formed and thickened in a controlled way.

Lower carried

load Softer surface

Better embedding

Lower wear

Free particles

Rolling, less ploughing


212

Figure 7.13: Corrosive wear of steel and aluminium Corrosive wear is in fact a combination of chemical and mechanical surface degradation mechanisms. Adhesive wear Adhesive wear is caused by welding together of asperities caused by atomic forces. Adhesion has been discussed above, so for details the reader is referred to the friction paragraph. If the interface between two contact partners is very strong, then material transfer at the interface takes place. Adhesive wear can be a initial stage for other wear processes like abrasive wear. This many eventually result in scratching of the surfaces by loose of fixed particles. Adhesive wear is often severe and has to be avoided. Other names for adhesive wear are scoring, scuffing or galling. The measures that can be taken to avoid adhesive wear are the same as the measures that can be taken to reduce adhesion, as discussed above. Material combinations will be more sensitive for adhesive wear if the combination has extensive solid solubility, see section 4.4.2. A reason that metals are sensitive for adhesive wear is the possibility to transfer electrons from one of the surfaces to the other surfaces [1]. If two clean metal surfaces are within each others vicinity at distances less then the diameter of one atom, then electrons from the high-energy metal transfer to the low energy metal. This transfer of electrons results in a strong bond between the two metals, with a strength as strong as the cohesive forces within a body. Electrons will continuously be exchanged between the two bodies. This type of adhesion is not restricted to high temperatures, and the mechanism is related to transfer of electrons. A similar adhesion mechanism, so based on electron exchange, is relevant in the case that a non-metal, like a polymer of a ceramic, contacts a metal. Then, the character of the electron bonding assumes a more chemical character with strong bonding between metals and a reactive non-metal like fluorine or oxygen. An example if a bond between fluorine and metal is the contact between iron and polytetrafluorethylene (PTFE). A second form of adhesion is related to mixing between the contacting bodies based on diffusion. This mechanism is slow and will only be important in the case of high temperatures, because diffusion is a strongly temperature driven phenomenon. Adhesive wear can be minimized by choosing a material with a complex microstructure and a non-metallic character or with non-metallic phases in a metallic matrix. The reason for this is that the probability that two identical microstructural phases touch in the contact is smaller in the case of a complex microstructure.

Steel

Aluminum

porous oxide layer

No oxygen barrier

Oxidation continues

dense oxide layer

oxygen barrier

No continuous oxidation

Anodising 0.02 µm

Hard loose oxides Abrasion

EP additives


213

A ductile surface will promote adhesive wear because in the case of ductile materials tensile adhesive contact between asperities is possible. Examples are high tensile ductile materials like nickel base alloys and stainless steels. Hard materials are less prone to adhesive wear because the microcontacts are not able to become sufficiently deformed spots to be in close contact Silver and gold are relatively chemically inert and because of this reason not sensitive for adhesive wear despite their ductility. Ceramics are not sensitive for adhesive wear because of their limited ductility and their covalent bonds which will minimize electron transfer between the bodies. Grey cast iron is not sensitive for adhesive wear because the graphite in the material is non-metallic and acts as a solid lubricant separating the metal surface from its countersurface. Aluminum and titanium are ductile materials and chemically reactive. This is the reason that they are very sensitive for adhesive wear. Melting wear Frictional heat generation in a tribological contact can, if the temperatures become high enough, cause melting of the material. Because of the severe nature of melting wear, this type is wear is highly not desirable and means failure of the machine. Deformation wear Deformation wear is relevant if shear is taking place in the adhesive contacts. If one body is moving over another body, the velocity difference has to be bridged within a small layer of material close to the interface. This means that a layer of highly plastically deformed material will be formed close to the interface. In time, thin, plate-like wear particles will be formed. Surface fatigue Surface fatigue is in particular important for rolling contraform contacts. Al already mentioned, the maximum shear stress is acting in the subsurface. In the case of a rolling movement, the contact will be intermittent. Although the stresses are below the elastic limit, on a micro scale plastic deformation will take place at inclusion or cracks. The cracks will grow in time and certain moments reach the surface. If the cracks reach the surface, relatively large craters will be formed with typical sizes of 1 mm. Typical for surface fatigue is the incubation period till the cracks reach the surface. The incubation period is inverse proportional with the contact pressures to the 9th power. Surface fatigue is schematically shown in the following diagram.

Figure 7.14: Fatigue wear Fretting wear Fretting wear can be disnguished into three types:

subsurface crack formation

Crack growth to the

surface

Plate-like wear

particles Friction, traction Delamination wear

1000 µm

Incubation period Lifetime


214

• Fretting corrosion (‘passingroest’) • Fretting wear • False brinelling

A characteristic of all fretting wear mechanisms is the small amplitude of the movement. The movement is often resulting from elastic deformation or vibrations. The determining aspect of fretting wear is the amplitude of the movement relative to the contact length. This fraction is smaller than 0.1 for fretting wear. Fretting corrosion The process of fretting corrosion is schematically shown in the following figure.

Figure 7.15: Fretting corrosion It can be seen that failure of the oxide layer in a vibrating contact can cause a combination of adhesion, fatigue and corrosion, finally resulting in loose oxide products which stay as wear particles in the contact, causing 3 body abrasion. It will be clear that fretting corrosion is a combined action of several separate wear mechanisms, like abrasion, adhesion and oxidative wear. Fretting wear Fretting wear is similar to fretting corrosion in the sense that the final result is also 3 body abrasion by particles which stay in the contact. In the case of fretting wear the particles are not formed by an oxidation process, but by a wear process. So fretting wear can also take place in the case that there is no oxidation. The mechanism is schematically shown in the following figure.

Figure 7.16: Fretting wear False brinelling

Small amplitude Breaking through oxide layer

Adhesion Corrosion

3-body abrasion oxide products Wear particles

Fatigue

Small amplitude Wear particles stay in contact

3-body abrasion


215

False brinelling is a special form of a fretting process, because it takes place in equipment which is in principle not operating, like roller bearings which are not rotating but subject to vibrations. Examples are the rolling bearings in emergency compressors. If roller bearings are stationary, pits may be formed in the ring by the high momentary local contact pressures resulting in plastic deformation

Figure 7.17: False brinelling is caused during standstill If in an emergency situation a reserve system is switched on, vibrations will be present when the system is running because of the pits that are formed in the ring in the stationary situation. The high loads caused by these vibrations will cause failure of the system. So, brinelling is a complex sequence of deformation and failure mechanisms, finally resulting in failure of the system.

Figure 7.18: False brinelling is causing failure at running The following measures can be taken to reduce fretting:

• Take away movement (vibrations), so take away the loads on the system • Enlarge amplitude, so give the possibility for the wear particles to leave the system • Solid lubricants of a soft coating that can embed wear particles and allow slip in the

interface. • Damping (rubber), resulting in less vibrations in the contact area • The use of hard and wear resistant surfaces, which will reduce wear.

Fretting is a generally difficult wear mechanism to solve.

Vibrations Pitting in ring

Plastic deformation Pitting in ring

False brinelling

Brinelling

Stand still

Vibrations generated by pits

failure

Movements


216

7.2.4 Wear rate After a so called running-period, many wear processes tend to stabilize a constant volume worn away per sliding length if the conditions under which the tribological system is operating, so the load, velocity and temperature, stay constant. If a stable regime is reached after the running-in period, then the volume worn away is proportional with the sliding distance. This proportionality can be seen in Archard’s law:

sFVkskFVN

N∆

=⇒=∆ (7.16)

In this expression, k is called the specific wear rate or the specific wear coefficient and is mostly expressed in the dimensions [mm3 N-1 m-1] or [m2 N-1]. Although Archard’s equation suggests that k is a constant, it is also sometimes found to be a function of FN and s. Further, wear is a system characteristic, so the k value will generally change if for example the countersurface or the environment is changed. A material combination, or more accurate a tribological system, is generally considered wear resistant if k<10-15 m2 N-1 = 10-6 mm3 N-1 m-1. Wear rates, can also be expressed in classes as: Table 7.1: k values and wear classes

k [10-6 mm3/Nm] class

0.0001-0.001 0

0.001-0.01 1

0.01-0.1 2

0.1-1 3

1-10 4

10-100 5

100-1000 6

1000-10000 7

Some examples:

• Polymers like POM and PA against steel under unlubricated conditions operate in class 5.

• Polymers like POM and PA against steel + Teflon under water lubricated conditions operate in class 3.

• A grinding process, were in fact a high wear rate is desired, is operating higher than class 7.

• Adhesive wear often results in wear rates around class 6 or 7. Remember than the classes express wear at a logarithmic scale. So the difference between class 3 and class 5 is a factor 100 in wear rate, and if Archard’s law is considered, also a factor 100 in lifetime expressed in sliding distance s. The most interesting regime in practice is often around class 3-4. The expression in classes is not used extensively and the reason for this is that the scale is not discriminating enough. For example, if the lifetime of a tool or a machine component is increased by a factor of 2, which might be very interesting for the industrial practice, then this cannot be expressed as a different class.


217

7.3 (Electro)chemical damage: Aqueous corrosion In the previous section, attention has been paid to damage caused to surfaces by mechanical processes. In many practical cases, surface damage is of chemical nature. Metals, which are commonly used in engineering applications, are not the chemically stable form in an oxide- hand humidity rich environment. This means that energetically it is favorable for metals to form oxides on the surface. The most important mechanisms which are chemical in nature are summarized in the following figure.

Figure 7.19: High temperature corrosion and aqueous corrosion As is clear from the figure above, there a three basic mechanisms causing chemical damage to surfaces:

• Aqueous corrosion. This chemical damage mechanism is taking place in an aqueous environment by salts dissolved in water or acids.

• High temperature corrosion. This chemical damage mechanism takes place at high temperatures

• Chemical degradation of biological or biochemical nature. Examples of chemical attack of biological nature are bacteria which may use some solids as food, or which excrements cause oxidation of the material on which they are deposited

Corrosion can be defined as undesirable attack of a material by electrochemical reactions with components in the environment. Here, a clear difference is seen with wear, which is undesirable attack of a material when it is in contact with a counterbody.

Inorganic aqueous chemical reactions

High temperature reactions

Rusting of metals

Softening of ceramics and

cermets

Corrosion activated by

bacteria

Oxidation, sulphidization,

Chloridization of metals and non-oxide ceramics

Solvation damage by liquid metals

and sals

Slow oxidation Catastrophic oxidation

Biological and biochemical processes

Destruction of wood and artificial

materials by organisms


218

7.3.1 Aqueous corrosion This section described corrosion in aqueous environments. Most of this section has been taken from [5].The most well known example of aqueous oxidation is rusting of steel in a aqueous medium. In the case of iron oxidation, iron is converted into iron hydroxide. The reaction shown below shows the chemical corrosion reaction reaction of iron (Fe) with oxygen:

43223 OFeOFe ⇒+ (7.17) Iron can also corrode chemically by water:

2432 443 HOFeOHFe +⇒+ (7.18) Or by sulfur:

FeSSFe 23 2 ⇒+ (7.19) Contrary to these chemical corrosion reactions, in aqueous solutions electrochemical corrosion reactions take place. Aqueous solutions are watery solutions with dissolved ions. The ions in these solutions can also cause corrosion of metals. In order to understand some basic aqueous corrosion mechanisms, it is important to explain the basics of electrochemistry. Some electrochemical corrosion reactions are: By acids:

222 HFeHFe +⇒+ ++ (7.20)

Or by oxygen: −+ +⇒++ OHFeOOHFe 4222 2

22 (7.21) Or by nobler ions:

CuFeCuFe +=⇒+ ++ 22 (7.22) . The figure below shows an electrochemical cell, which is in fact a system composed of two dissimilar metals in a vessel containing a weak acidic solution. The metals are called the electrodes and the acidic solution is called the electrolyte. The electrodes are connected with a wire. If the switch is closed, the following phenomena will happen:

• The zinc electrode will slowly dissolve. • Electrons will go from the zinc electrode to the copper electrode. • Gas bubbles, more precise hydrogen bubbles. are emitted from the copper electrode.

As [5] points out, prevention of corrosion is in fact the recognition of hidden electrochemical cells in engineering structures. Possible corrosion mechanisms in engineering structures will be discussed later more extensively. In corrosion problems, the rate at which the electrochemical reaction progresses is crucial because this factor determines the lifetime of the system. The faster the electrochemical reaction progresses, the shorter the lifetime of the system. So slowing down the electrochemical reaction rate is important to increase the lifetime. Electrochemical reactions are prevented by dense, stable and adherent oxide layers. A famous example is aluminum, which is protected with a dense Al2O3 layer, blocking the electrochemical circuit in the cell. If the electrochemical cell is again considered, the formation of such a oxide layer will ‘open the switch’. The same is true for chromium, which forms a stable Cr2O3 layer on the surface.


219

Figure 7.20: Electrochemical cell consisting of azinc electrode, a copper electrode and an electrolyte In an electrochemical cell, two electrochemical reactions are taking place at the surfaces of the two electrodes. At the zinc electrode in the electrochemical cell shown above, the following reaction occurs:

−+ +⇒ eZnZn 22 (7.23) So zinc atoms are dissolved into the solution as zinc ions. After dissolving into the solution, the zinc ions are almost immediately surrounded by water molecules in the solution. If a Zn atom is dissolved into the solution, in fact two electrons are freed from the Zn atom. The ‘extra’ electrons at the zinc electrode will flow, when the circuit is closed by the switch, to the copper electrode. At the copper electrode, the following reaction takes place:

OHHOHe 223 222 +⇒+ +− (7.24) The copper is not involved in electrochemical reactions but is just the site at which the hydrogen will be formed. The zinc electrode is called the anode and the copper electrode is called the cathode.

7.3.2 Thermodynamics: The Nernst equation Many processes spontaneously go to a lower energy content. A simple example is a falling body, where the potential energy in minimized to the lowest possible value when the body is lying in the ground. Another example is burning of fuel. By the formation of strong chemical bonds with oxygen , energy is released. If a process takes place at constant temperature, it is more correct to speak about enthalpy than about energy. Then processes go in the direction of minimum enthalpy H, which is related to the energy content U by:

pVUH += (7.25) The minimization of enthalpy H (or energy) can be seen as the driving force of the process. However, in the practice of electrochemical damage, the difference between enthalpy and energy is often neglected. Besides that processes tends to develop in the direction of a lower entropy H, some processes also tend to develop in the direction of increasing internal energy. An example is melting of a solid or evaporation of a fluid. In the case of melting of a solid, atoms leave their equilibrium

Copper electrode

Zinc electrode

Surface damage on zinc electrode

switch

Gas bubbles

after closure switch

Weak acid solution in water


220

crystal position, which means that the attractive forces between the atoms are overcome in the melting process. This requires energy. This phenomenon is related to a second phenomenon in nature, which is the tendency to develop in the direction of maximum chaos. In the case of a liquid, the chaos is increased compared to a crystalline solid. The same is true for a vapor compared to a liquid. This tendency to develop in the direction of increasing chaos gains importance at higher temperatures. The ‘chaos content’ of the system is expressed with the entropy S. Here, the increase of entropy is the driving force of the processes In reality, the driving force is a combination of increasing entropy and decreasing enthalpy for many processes, as is the case for chemical processes. This combination is expressed by the Gibbs free energy or free enthalpy. The Gibbs free energy is given by:

TSHG −= (7.26) Now, the change in free enthalpy under the condition of constant pressure and constant temperature can be expressed as

VpSTHG ∆+∆+∆=∆ (7.27) The last term is usually neglected. The process will develop spontaneously if ∆G<0 and will be in an equilibrium situation if ∆G=0. At equilibrium, the free enthalpy content of the system is as small as possible. If T or ∆S are small, then the second term in the Gibbs equation is small, and the system will develop in the direction of minimal enthalpy. The system will develop in the direction of chaos if T or ∆S are large. In the chemical domain, a substance be said to have a free enthalpy or a thermodynamic or chemical potential. The thermodynamic or chemical potential per mole is often expressed by µ. The chemical potential (per mole) is by definition equal to

Tsh −=µ (7.28) With h the molar enthalpy and s the molar entropy. The molar enthalpy is independent of pressure and almost independent of temperature. However, the molar entropy is dependent of temperature. For a gas X, which may be part of a mixture of gases with partial pressure p, the molar entropy is given by:

( ) ( ) ( )XpRTXsTXs gas ln,, 0 −= (7.29) The standard molar entropy s0 is the entropy of 1 mole gas at 1 atmosphere pressure and a temperature T. This equation can be interpreted as that at a lower pressure, there are less molecules in a certain volume. This means that at a lower pressure the chaos, and thus the entropy is higher. Similarly in liquid or solid mixtures, the following equations can be written for the molar entropy

( ) ( ) ( )XcRTXsTXs micture ln,, 0 −= (7.30) With c(X) the concentration of X in the mixture in [mol l-1]. Summarzing, this results in following expressions for molar entropy s, molar enthalpy h and chemical potential µ:

Table 7.2: expressions for molar entropy s, molar enthalpy h and chemical potential µ

Substance Molar enthalpy h Molar entropy s Chemical potential µ Solid substance ( )Xh 1 ( )X0µ Gas ( )Xh ( ) ( )XpRTXs ln,0 − ( ) ( )( )xpX ln0 +µ Solution ( )Xh ( ) ( )XcRTXs ln,0 − ( ) ( )( )xcX ln0 +µ Let us now consider the following reaction equation as an example:

ba BAbBaA 22 =+ (7.31) Then the driving force is the change is free enthalpy per mole ∆g, which is given by:

( ) ( ) ( )22 BbAaBAg ba µµµ −−=∆ (7.32) An example of such a reaction could be the dissolution of iron, given by:


221

( ) ( )gHFeHsFe 222 +⇔+ ++ (7.33)

Then the change of free enthalpy is given by: ( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

+∆=

++++

−−−=∆

+

+

++

++

22

20

22022

0

00

)()().(

ln

lnln

ln22

HcHpFecRTG

HpRTHFecRTFeHcRTHFeg

µµ

µµ

(7.34)

Again, if ∆g<0: a spontaneous reaction will tend to progress. If ∆g=0, the system is in an equilibrium situation. The term behind the ln-term is called the equilibrium constant. For a general reaction A+B→C+D, the equilibrium constant f(p,c) can be expressed as:

( ) ( ) ( )( ) ( )BcAc

DcCccpfDCBA =⇒+→+ , (7.35)

With c is equal to 1 for pure materials, c equal to the partial pressure [bar] for gases and c equal to the concentration for solutions [mol l-1]. So, a general expression for the driving force is:

( )( )

( ) ( ) ( )( ) ( )BcAc

DcCccpfK

cpfRTGg

==

+∆=∆

,

,ln0

(7.36)

And in an equilibrium situation: ( )( )

( ) ( ) ( )( ) ( )BcAc

DcCccpfK

cpfRTGg

==

−=∆⇒=∆

,

,ln0 0

(7.37)

Before we continue, a small discussion about the pH value has to be done: In every solution, water molecules will in a small amount dissociate into H+ and OH- ions. This phenomenon is called self-ionization.

−+ +⇒ OHHOH 2 (7.38) It is known that at room temperature:

( ) ( ) 1410−−+ =OHcHc (7.39) This means that in pure water

( ) ( ) 710−−+ == OHcHc (7.40) In acidic solutions

( ) 710−+ >Hc (7.41) And in alkalic solutions:

( ) 710−+ <Hc (7.42) In order to avoid working with powers, the pH is defined as:

( )+= HcpH log (7.43) So for a neutral solution pH=7, for an acid pH<7 and for an alkalic solution pH>7 Take as an example the following equation, where Ni is oxidized by gaseous oxygen:

NiOONi ⇔+ 221 (7.44)

Then the equilibrium constant is given by:

RT)Gexp(p1

c(Ni).pc(NiO)K 0

21

2O

21

2O

∆−=== (7.45)


222

So this results in an equilibrium pressure for O2 gas for this reaction:

∆−=

RTGp

mequilibriuO

0

2exp2

(7.46)

If the actual pressure of the oxygen p(O2) is higher than p(O2(equilibrium), then spontaneous oxidation will take place. If the actual pressure of oxygen p(O2) is smaller than p(O2)(equilibrium) then dissociation of oxide will occur. If values for the equilibrium pressures are calculated and if the partial gas pressure of the environment is known, then the possibility of corrosion can be evaluated based on the thermodynamic consideration discussed above. Similar expressions for the free enthalpy can be written for electrochemical reactions. As an example it is known that if a steel part is immersed in a solution containing Cu2+ ions, that a copper layer is deposited on the Fe part. The reaction that is taking place is the following:

CuFeCuFe +⇒+ ++ 22 (7.47) This means that Fe is oxidized from the iron part and that Cu is reduced from the solution. Before we continue, it is important to define the terminology, as is done in the following table: Table 7.3: Oxidation and reduction

Oxidation Reduction Production of electrons Consumption of electrons

Takes place at anode Takes place at kathode

Apparently there is a driving force for this reaction denoted above. Calculation of the the driving force, so the free enthalpy, results in:

( ) ( ) ( )( ) ( ) ( )( )( )

+∆=+

+++−−=∆

+

+

++++

)()(ln

lnln

2

20

0

220

2200

CucFecRTGCu

FecRTFeCucRTCuFeg

µ

µµµ

(7.48)

So, the equilibrium constant is dependent of the concentrations of the Fe and Cu ions in the solution. Although the reaction is denoted as above, it cal also be thought as composed of two reactions:

CuFeCuFeCueCueFeFe

+⇒+⇒

⇒+

+⇒ ++−+

−+22

2

2

22

(7.49)

This two partial reactions can be separated by the following electrochemical cell:


223

Figure 7.21: Electrochemical cell consisiting of two electrodes Here, a Fe electrode is put in a solution contaning Fe2+ ions and the Cu electrode is put in the solution containing Cu2+ ions. The membrane in the middle will avoid mixing of the solutions, but will allow electrons to pass, so will allow a current to flow. Considering the reaction equation (7.47), when the switch is closed, electrons will pass from the Fe electrode to the Cu electrode caused by Fe atoms going into solution and Cu2+ ions being deposited. If the electrons flow from ‘left’ to ‘right’, the current flows by definition from ‘right’ to ‘left’, because the direction of the current is in the direction of the positive charges. If a Voltmeter would be mounted in the connecting wire, a potential would be measured between the Fe electrode and the Cu electrode. This potential is called the potential of the electrochemical cell. In electrochemical reactions, the free enthalpy change ∆g is related to potential of electrochemical cell E according to:

nFEg −=∆ (7.50) With n the number of electrons involved in the reaction, F the Faraday constant, which is equal to the electrical charge of a mole of electrons and is equal to 96500 C. The potential of the electrochemical cell is represented by E. This equation can be interpreted as follows: ∆g is the energy change for a mole reaction. But the term nFE is in fact the charge of the total number of electrons involved times the potential. We now from electricity that P=IV, so the power is the current times the voltage. This means that the energy is given by the current over a certain time, so the transported electrical charge, times the potential. If the thermodynamic investigation above is substituted in equation (7.50), this leads to the Nernst equation below:

00

2

202

20

)()(ln)(

)(ln

gE

CucFec

nFRTEE

nFEgCucFecRTgg

∆=

−=⇒

−=∆

+∆=∆

+

++

+

(7.51)

Here, E is the electrochemical potential, E0 called the standard electrochemical potential. R is the gas constant, equal to 8.31 [J mol-1 K-1], n is the number of electrons involved and F is the Faraday constant, equal to 96500 [C]. This very important equation learns us that electrochemical potentials are not absolute constants but are dependent on temperature T and the concentration c of the electrolyte solution.

Electrochemical cell

Cu- electrode Kathode

Fe-electrode

switch

Fe2+ Cu2+

SO42-

SO42-


224

Now let us consider the following electrochemical cell:

Figure 7.22: When two electrodes are connected in an electrochemical cell, both electrodes will not be at equilibrium Here, the left electrode is a platinum electrode, with hydrogen gas bubbling along the electrode. The Standard hydrogen electrode is chosen in such a way that pH2= 1 atm and C(H+)= 1 [mol l-1], so at a pH value of 1. Then, the net reaction that is taking place in the cell is given by:

222 2 HFeHFe +=+ +++ (7.52)

Now the potential is measured of the Fe electrode. For the electrochemical cell above, the free enthalpy change is given by:

+∆=∆

+

+

)()(ln 2

20

2Fep

HcRTggH

(7.53)

Which results in a cell potential:

[V] )(

)(ln 2

20

2

+=

+

+

FecpHc

nFRTEE

H

(7.54)

If pH2= 1 atm and C(H+)= 1 [mole l-1] is substituted as well as T=25oC, then this results in the following expression for the equilibrium potential of the Fe electrode

( )[V] log2059.0 20 ++== FecEE (7.55)

If now c(Fe2+) is chosen as 1 [mole l-1], then: E=E0. This value is called the standard electrode potential. The following table gives an overview of standard electrode potentials of metals. Table 7.4: : Summary of standard electrode potentials

Reaction E0 [V]

Fe + 2 H+ → Fe2+ + H2.

Fe2+

SO42

- SO4

2

-

V

Pt

Fe

H+

H2


225

Mg2+ + 2 e- = Mg - 2.34 Al3+ + 3 e- = Al - 1.67 Zn2+ + 2 e- = Zn - 0.76 Cr3+ + 3 e- = Cr - 0.71 Fe2+ + 2 e- = Fe - 0.44 Cd2+ + 2 e- = Cd - 0.40 Ni2+ + 2 e- = Ni - 0.25 Sn2+ + 2 e- = Sn - 0.14 Pb2+ + 2 e- = Pb - 0.13 2 H+ + 2 e- = H2 0.000 Cu2+ + 2e- = Cu 0.34 Ag+ + e- = Ag 0.80 Pt2+ + 2e- = Pt 1.2 Au+ + e- = Au 1.68

If such an electrode reaction as in the table above is combined with a hydrogen electrode, then the net reaction is given by:

MnHneMH n +=++ +−+22

1 (7.56)

If for a 1 Molar solution (1 mole per liter), the value for E0 is negative, then ∆g0=-nFE will be positive. This means that the reaction equation above will tend to the ‘left’ and the metal will oxidize which implies that corrosion takes place. So, if the value of the standard electrode potential is negative, the material will corrode in a 1M solution with pH=1 at room temperature. However, the standard electrode potential can also be for other electrode combinations. Up to now we will still consider standard conditions, as in the electrochemical cell above. If iron-zinc reaction is considered, the net reaction is given by:

FeZnZnFe +=+ ++ 22 (7.57) Then:

VFeZnFeEeZnZn

EFeeFe

EEEFeZnFe

Zn

Fe

FeZn

323.0ZnV 763.02

V 440.02

Zn

2202

02

0022

−=+=+⇒

−=→+=

−=→=+

−=⇒+=+

++

−+

−+

++

(7.58)

The result is that E<0. This means that ∆g>0 and no corrosion of Fe will take place under the conditions mentioned. For non-standard conditions, the Nernst equation can be used. In the equation below still a temperature of T=25oC is used. However, this can also be changed, resulting in other pre-log factors.

( ) ( )++

++

−−+

=−=⇒+=+2020

22

log03.0log03.0

Zn

FecEZncE

EEEFeZnFe

FeZn

FeZn (7.59)

Now the value of E, so the corrosion tendency, is dependent on concentrations of ions (also pH for hydrogen reaction) and T. If the logarithmic term is negative, so for the case that the concentration of Fe ions is higher than the concentration of Zn ions, then an increased temperature will raise the potential and the reaction will be more energetically favorable. If the logarithmic term is positive, so for the case that the concentration of Fe ions is lower than the concentration of Zn ions, the electrochemical potential will be reduced at increasing temperature. Then, at a high enough temperature the electrochemical potential can become negative which means that iron can under such conditions become anodic to zinc. So in the case of extreme concentration differences the electrochemical potential can be reversed, in particular at high temperatures. Also in practice it is found that zinc coatings do not protect iron or steel when they are immersed in hot water.


226

Up to now, metal dissolution reactions and the hydrogen reaction have been discussed. However also other reactions are possible in corrosion environments. An example is a reaction with oxygen:

( )( )( )

( )14log06.0log015.0401.0

log06.0log015.0401.0

ln4

EE

[V]401.0;442

2

2

2

40

022

−−+

=−+=

⇒+=

==++

−

−

−−

pHpOHcpE

OHc

pF

RT

EOHeOHO

O

O

O

(7.60)

If pH=7 and the partial pressure pO2=0.2 bar as under atmospheric conditions, then: ( ) 804.014log06.0log015.0401.0

2=−−+= pHpE O (7.61)

The standard potentials of many metals, even Cu and Ag are lower than this value. So many metals tend to oxidize in a neutral solution containing oxygen. Removing oxygen from a closed system (like a heating or cooling system) by, for example, chemical binding, is therefore an effective method to reduce corrosion problems. The main conclusion is that metal with lowest electrochemical potential tends to corrode when in an electrochemical cell with another metal. However, only a thermodynamic consideration is not sufficient for a complete analysis of corrosion problems. The thermodynamic situation only described if the material tends to corrode. There is no information about the corrosion speed. The corrosion reaction ma be a very slow reaction, resulting a very small amount of corrosion although it is thermodynamically spoken taking place. Besides this, a complicating factor is the possible presence of dense corrosion layers on surfaces. An important example is the presence of dense Al2O3 oxide layers on Al substrates, reducing corrosion by isolating the Al electrode from the environment. The phenomenon of passivation, as this is called, will be discussed later.

7.3.3 Kinetics: The Tafel equation An important question is always the lifetime of a component with respect to corrosion. This is related to the rate of corrosion, and is often represented in terms of the corrosion current, which is the current that passes between the anode and the cathode of an electrochemical cell. Let us consider a Cu electrode, where the following reaction is taking place:

CueCuanodic

cathodic

I

I

← →

+ −+ 22 (7.62)

If the electrode is isolated and placed in a solution containing Cu2+ ions, a continuous exchange between the electrolyte and the copper electrode will take place. Under equilibrium conditions Ianodic+Icathodic=0, so there is no net current between the electrode and the solution. If there is no equilibrium two situations can take place:

• |Ianodic|>|Icathodic|: Current flows out of electrode and dissolution of Cu occurs • |Ianodic|<|Icathodic|: Current flows in electrode and plating of Cu occurs

In both cases a non-equilibrium is existing and a net chemical reaction is taking place at the electrode. It is also possible that there is no external current, but there is still a net chemical reaction taking place at the electrode. This is possible for the case that the electrode is a combined anode and cathode. This will be discussed later. When a current flows in or out an electrochemical cell, then the potential differs from equilibrium potential according to Nernst equation. This effect is called polarization. The effect of polarization is the formation of an overpotential E*, defined by:

mequilibriuEEE −=* (7.63)


227

With E* the overpotential, E the potential of the electrochemical cell and Eequilibrium the equilibrium potential calculated with the Nernst equation. The overpotential is caused by rate limiting steps in the total current cyclus. There are two main causes:

• Diffusion polarization. Here, diffusion of reactants to or from the surface where the corrosion reaction takes place

• Charge transfer polarization. Here, transfer of charge between atoms, ions and molecules in the solution is the rate limiting step.

By definition the overpotential E* ∙ I>0. So: o Net anodic reaction, so a net e- out the electrode, so a positive I results in an

overpotential E*>0 o Similarly a net cathodic reaction results in net e- in the electrode, so negative

I and an overpotential E*<0 Let us now consider the following electrochemical cell with a variable resistance. If the resistance is very high, there is no current I and the measured potential of the electrochemical cell corresponds to the electrochemical potential according to the Nernst equation. If the resistance is reduced, the potential difference between the electrodes will decrease and the current will increase. In the limiting case when the resistance is zero, the potential of both electrodes will be the same. In the setup shown below it is not possible to measure the electrode potential of the individual electrodes. However, these could be measured by two additional standard hydrogen electrodes and volt meters, one for each electrode of the electrochemical cell.

Figure 7.23: Electrochemical cell with a variable resistance If these measurements are performed, the following graphs can be the result: In this graph, the absolute current is shown. This diagram is called the Evans diagram. It is clear from the figure that, at increasing current, the Zn potential increases and the potential of the hydrogen electrode decreases. If the resistance is zero, the potentials of both electrodes are the same, and this situation is represented by the crosspoint of the two lines in the graph below, from [5]

Zn- electrode Cathode

H+ Zn2+

SO42-

SO42-

e-

I

No equilibrium situation

A V

Pt

H2


228

Figure 7.24: Graphical representation of the Tafel equation

In the figure above, the potential-current curves were simple straight lines. This is not always the case. The actual shape of the curve is determined by the rate limiting steps in the electrochemical cell. This means that the ‘slowest’ process will determine ‘how much’ current will flow between the anode and the cathode. There are two main effects:

• Charge transfer polarization. This means transfer of charge between atoms, ions and molecules in the solution or between solution and electrode

• Diffusion polarization. Diffusion of reactants to or from the surface where the a reaction takes place

Below, these two effects will be further discussed. Charge transfer polarization As discussed above, in the case of two connected electrodes, the electrochemical electrode is not developing its own equilibrium potential with the solution according to the Nernst equation, but is acting on a potential in equilibrium with the external electrode. If under the influence of the other electrode the potential of the first electrode is more negative than the equilibrium potential, then electrons are forced into the electrode. The more negative the potential, the higher the current resulting from this electrode. Also in the case of an applied positive potential, the current will increase with increasing potential because the electrode is made more receptive for electrons at a higher positive potential. The case of an applied positive potential and the related current as well as the case of an applied negative potential and the related current is shown in the following figure. If follows from the figure that if the non-linear region close to the origin, so at low overpotentials, is neglected, then both logarithm of the current in the anodic case, the anodic current, as well as the logarithm of the current in the cathodic case, the cathodic current, depend linearly on the potential.


229

Overpotential versus current density, from [5] overspanning versus log|current density| (Tafel-diagram), from [5]

Figure 7.25: Polarisation curves

In this case, the corrosion current can also be calculated with the Tafel equation. The Tafel equation gives a relation between the electrochemical potential and the corrosion current and predicts a linear relation between the logarithm of the corrosion current and the overpotential:

=−=

exexternal i

iKEEE log* (7.64)

Here, Eexternal is the externally applied voltage, E is the electrochemical potential according to the Nernst equation, i the current density passing through the electrode [A cm-2], iex is the exchange current density [A cm-2] and K is the Tafel constant [V]. The above also shows a minimum non-zero value for the electrode current, when extrapolated to the current axis, so at zero overpotential. At this point, there is no net flow of electrons from one electrode to the other electrode. At this point, the ions that are deposited on the electrode are exactly balanced by the ions that go into solution at the electrode. This value of the current is called the exchange current and is represented by iex. The exchange current is very important for the rate of corrosion as is the electrochemical potential. The electrochemical potential and the exchange current are independent properties and do not correlate with each other. Up to now, we have mostly considered the behavior at a single electrode, but we have not yet discussed the potential at which the complete electrochemical cell settles. This potential, as shown above, determines the overpotential of the single electrode. About this topic, the following aspects have to be mentioned:

• In a complete electrochemical cell two electrodes together determine the resulting electrochemical behavior. In an electrochemical cell, most of the resistance in the system is the electrolyte and in the surface layer of the electrodes. Therefore both electrodes will work at closely the same voltage if they are connected with a connecting wire with a low resistance.

• The current that is running through the electrodes is also the same for both electrodes, because there is no other electrical circuit through which the electrons can flow, except from one electrode to the other electrode.

The total electrochemical behavior can (in principle) be determined by finding the intersection of two Tafel diagrams of the two separate electrodes. The current of both electrodes should be the same, so the corrosion current is given by the intersection of both Tafel diagrams. The potential at which the two Tafel diagrams cross is called the corrosion potential, and this potential is the potential at which the electrochemical cell settles. Remember that the current density is plotted in a Tafel diagram instead of the current, and the current of both the anode and the cathode are the same. So if the area of the anode and the cathode is different, this factor needs to be taken into account.

η

i0

ic

ia

i

ic i0ia

η

log|i|

i ex


230

If the corrosion current is known, it is simple to calculate the mass loss from the reaction equations. This will not be discussed here, but is discussed by Faraday’s law when discussing electrochemical deposition processes. Diffusion polarization Sometimes the rate determining step is the diffusion of one of the reactants to or from the metal surface. The First law of Fick, which will be discussed in later chapters about diffusion surface treatments, describes the diffusion rate and says that this rate is proportional with the concentration gradient. The maximum diffusion rate will occur is the concentration at the surface of the electrode is exactly zero, Then, the maximum flux will diffuse to the boundary layer. This current density in this situation is called the limiting current density iL and is given by:

dnFDci

dcnFDi

dccnFDi

dxdCD

dtd

l00

lim0 =−=⇒

−−=⇒=

Φ (7.65)

With c0 the concentration in the electrolyte. The concentration in the boundary layer is schematically shown in the following figure, from [5]

Figure 7.26: Concentration gradient in the boundary layer around theh electrode. The diffusion through the boundary layer limits the current density to the limiting current density value If diffusion polarization takes place, the polarization curve will deviate from the Tafel diagram at high current densities close to the limiting current density iL [5]:

Distance to metal

concentration

c0

δ

Limiting current density il = n.F.D. c0 / d


231

Figure 7.27: The ffect of diffusion polarization of the polarisation curve

From the equation for the limiting current it can be concluded that: • A higher concentration in solution c0 increases the concentration gradient • An increasing flow decreases thickness d

It is also known that diffusion is a strongly temperature dependent process, which follows from the fact that increasing T increases diffusivity D. In all cases methioned, this results in a higher limiting current density iL . Passivity A next important phenomenon in the kinetics of corrosion is passivity. Passivity is the formation of dense and adhering oxide layer on the surface. This stops the corrosion process and increases the potential of the electrochemical cell. Some materials which show this phenomenon are aluminium and chromium as well as steel alloys containing Cr of Al and Cu alloys with 10% Al, also called aluminium bronze. A polarization curve including passivity is shown below [5]:

Figure 7.28: The effect of passivity on the polarization curve


232

It can be seen that at a low potential dissolution of metal ions takes place. This stage is often characterized by charge transfer polarization. At higher potentials a critical passivation current density icr is reached. If i is higher than icr, then passivity with very low current density takes place.

7.3.4 Polarisation curves and corrosion In the paragraphs above, polarization curves and the effects of diffusion polarization and passivity on the polarization curves have been shown. Corrosion problems can be analyzed by studying the polarization curves of the anodic reaction and the cathodic reaction. The anodic reaction is the reaction by which metal goes into solution. This reaction almost always shows charge transfer polarization. The polarization curve for the cathodic reaction is often the hydrogen generation reaction, so

−+ +→ eHH 22 2 (7.66) For this reaction, also diffusion polarisation possible. However, in corrosion problems the cathodic reaction can also be a reaction with oxygen or another metallic reaction. Charge conservation means that the total current ‘consumed’ at the cathodic reactions has to equal the total current ‘generated’ at the anodic reaction. This means that the crossing of two polarization curves gives the equilibrium situation in which the electrochemical cell settles. The current at this crosspoint is called the corrosion current density and the potential is called the corrosion potential. It should be remembered that in many corrosion processes the anodic reaction and cathodic reaction can take place at same metal surface [5], resulting in a combined electrode.

Charge transfer polarisation, from [5] Diffusion polarisation, from [5] Figure 7.29: The effect of charge transfer polarization and diffusion polarization on the corrosion current density and the corrosion potential In the figures above it has been assumed that the areas of the anode and the cathode are equal. This can be done in the situation that the anode and the cathode are the same surface. The following curve shows the polarization diagram for corrosion of metal in an acid solution [5]. Here, the anodic reaction is M → Mn+ + n e- and is characterized by charge transfer polarization. The cathodic reaction is the hydrogen reaction H+ → ½ H2 + e-. Here, it can be seen that at pH<4, charge transfer polarization takes place while at pH>4 diffusion polarization takes place due to lack of H+ ions at the surface. From the diagram it follows that the corrosion speed decreases with increasing pH, from [5]

log | i |

E

log igc

log i0a

log i0c

log is = log igc

Eev,c

Es

Eev,a

log | i |

E

log is

log i0c

log i0a

Eev,a

Eev,c

Es

Corrosion potential

Corrosion current density


233

Figure 7.30: Increasing corrosion speed with increasing pH

The following diagram shows the polarisation diagram for contact corrosion with hydrogen reduction at the ‘noble’ electrode as the determining reaction, from [5]

Figure 7.31: Polarisation diagram for contact corrosion with hydrogen reduction at the ‘noble’

electrode as the determining reaction Contact corrosion is the phenomenon by which bringing two unequal metals in contact results in increased corrosion of the least noble material. In this diagram above N is the noble metal and M is the less noble metal. In this diagram, not the equilibrium potential of the noble metal itself, but the hydrogen formation reaction is responsible for the accelerated corrosion. The hydrogen reaction is very fast at noble metal due to high exchange current density. This is also a reason why the presence of impurities causes more corrosion of metals. It can also be seen from the diagram that the area of the anode and the cathode are very important. When the area of anode and cathode are equal, then the situation can be described by the cross point of the polarization curves. If the area of the cathode is 10 times the area of the anode, then the current density at the cathode should be a tenth of the current density at the anode. From the diagram it follows that an area difference results in a higher potential and a high current density. This also means a higher corrosion rate. The following diagram shows three situations including passivity.

log |i|

Eis5 is4

is0

is1is2

is3

i0a

i0c

0

12

34

5

pH

-6 -5 -4 -3 -2log | i |

E

icorr (Ac=10Aa)

2H++2e-→H2

M→M2++2e-

op M op N

icorr (Ac=Aa)icorr

(alleen M)


234

At high potentials, the lines will cross in the passivation region, as shown in the first figure. At lower overpotentials, there are two possibilities: The cathodic line may pass ‘before the nose’ or may cross at three points. In the first case, passivity occurs. In the case of three cross points, only two are electrically stable. If the system starts in the passivated situation and there is no distortion, the layer will stay passive. However, if the passivation layer is mechanically damaged, then the active spot will not be repassivated and corrode according to point 3 will take place. The situation is then an active-passive element with a small anode and a large cathode. This can result in severe localized corrosion, from [5].

Figure 7.32: Polarisation curves and passivation The critical passivation current density icr can be influenced by factors like temperature, fluid flow velocity, the solution and the meta. The following graph shows the polarization curves as a function of chromium content. It is clear that at 12% Cr content (so called stainless steel), spontaneous passivation takes place. Sometimes metals are alloyed with specific elements, like Mo to increase the corrosion resistance in Cl containing environments, from [5].

Figure 7.33:The effect of the chromium content on passivation. At 12 % Cr, spontaneous passivation takes place Up to now it was assumed that no potential difference was present over the electrolyte. However, if the electrolyte has a certain resistance, then this can result in a lower corrosion current. The resistance of the solution can be increased by removing salts. So reduction of salts will reduce corrosion due to the higher resistance of the solution and therefore a smaller corrosion current.

log | i |

E

Ep

icorr=ip

i0c

log | i |

E

Ep

1 2

3

i0c2

i0c1

log | i |

E

Ep i0cicorr

0

1

2

3

4

5

-7 -6 -5 -4 -3 -2 -1 0

log |i| (A/cm2)

zuurstofreductie

0% Cr6% Cr12% Cr

-6

-5

-4

-3

-2

-1

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Cr-gehalte (gew%)

log(

icp)

(A/c

m2)


235

7.3.5 Corrosion rate If the corrosion current in known from the polarization diagrams, the rate of electrochemical corrosion can be calculated. One should be aware that the corrosion current only gives an accurate description for the lifetime in the case of non-localized corrosion mechanisms. The mass that is lost in the electrochemical reaction can be calculated using Farays law which says that the charge required to reduce moles of metal is equal to

mnFeamnNQ =−= (7.67)

The total mass that is lost can be calculated from:

( )∫== dttInFM

nFQMM ww (7.68)

Sometimes, like in the tables at the end of this chapter, the material constants MW, n and F are expressed in the electrochemical equivalent e. So it can be seem that the amount of mass lost per unit time is proportional to the corrosion current with proportionality constant e

eIt

M⋅=

∆∆ (7.69)

7.3.6 Corrosion prevention For the prevention of corrosion, there are several options:

• Change of materials (Materials selection) • Change the environment (Composition) • Change of operational conditions (temperature, fluid flow) • Electrochemical measures • Change of design, construction or finishing methods • Separation of material and environment by coatings

The best option of obtain sufficient corrosion resistance is to have an inert surface. The second option is to have a solid reaction product being formed at the surface. This is called passivation. The worst case for corrosion is the production of soluble ions of metal. Also temperature is of importance for corrosion.

Electrochemical corrosion in practice

In many machines and structures, many different materials are in close contact, which means that many potential electrochemical cells are present in a typical application. Besides this, pure metals are not so often used in applications but mostly alloys or other composite materials. The application of composite materials of alloys means that there are many electrochemical cells possible at a micro scale. Another effect is that machines and other structures are increasingly exposed to more hostile environments like higher temperatures of increasing concentrations of acidic of alkalic solutions. Most information in this section is taken from [1] and [5].

Rusting or atmospherical corrosion

Rusting is the aqueous corrosion of zinc, iron, copper, aluminum or other metals. The metal is covered by a corrosion product or in severe cases turned into a corrosion product at a large scale. Corrosion products are often oxides or hydroxides, but also carbonates and sulfates are possible. Because the environment is of large influence for the corrosion behavior of the component, an accurate prediction of rusting is very difficult. The most well known example of atmospherical corrosion is rusting of steel in a humid environment, where iron is converted in a brown layer of hydrated iron oxides and ironhydroxides. During the rusting process the anodic and cathodic spots move over the surface. The result is a more or less uniform degradation of the surface


236

In humid conditions, a surface will be covered with typically a 1 m thick water film. In sea environments, there will be salt crystals in the air, which makes coverage of a surface with a water layer possible at lower values of the humidity. In a polluted atmosphere, sulfates and nitrates are added to the solution. So, in particular at horizontal surfaces in the environments mentioned, small electrochemical cells are formed. Light rain will increase the amount of water available for corrosion, but heavy rain will wash the electrolytes away and will delay rusting. If iron is immersed in water, then rusting will be less severe because oxygen cannot reach the surface of the iron. The mechanism of rusting of iron in the open air is schematically shown in the following figure [1].

Figure 7.34: Mechanism of rusting of iron in the open air

Initially, iron is dissolves inside an anodic pool formed around for example a salt crystal, so:

−+ +⇒ eFeFe 22 (7.70) Then, hydroxide ions will be formed in the surrounding metal, forming a cathode.

−− ⇒++ OHeOHO 442 22 (7.71) A combination of hydroxide ions and ferrous ions inside a layer of rust will stimulate the formation of more rust:

+

+

+

⇒++

OHFeOOHOOHFe

3

222

44104

(7.72)

The most important components in the atmosphere that are responsible for atmospheric corrosion are oxygen (20%), water (maximum 2.3% at 20oC) and carbondioxide (0.03%). Other active components are for example SO2 (vulcanos and sulfur containing fuels), H2S (rotting processes), nitrogenoxides (combustion engines and chemical industry). The composition of the athmosphere has a large influence on the corrosion behaviour. Besides the environment, also the composition of steel is of large influence on the atmospherical corrosion rate. Small addition of Cu, Cr and Ni often improve the corrosion behaviour of steel. Weathering steels are steels wich form a closed, compace ad well adhering oxide layer on the surface. This means that they can be used without much other corrosion protecting measures like painting. Examples of applications for this material are buildings, bridges and railway applications. Atmospheric corrosion can be reduced by lowering the relative humidity. This can be done by ventilation of hollow structures or totally closing hollow structures in order to avoid the entrance of humidity. The rate of atmospheric corrosion is strongly dependent on


237

environment, so whether the atmosphere can be characterized as a city, maritime, industrial atmosphere.

Pitting corrosion

Corrosion typically does not proceed in a uniform way over a large area, but is more severe at some parts than at other parts. In some cases deep and narrow pits are formed at the surface. In the case of active materials some external conditions like localized humidity of mechanical damage of protective layers can cause localized corrosion. In these porous corrosion products formed at the surface, humidity and other aggressive components can collect at that location, and cause more severe corrosion at that specific spot. In the case of passive materials like stainless steel and aluminum, a more important and more dangerous form of pitting corrosion can occur. Pitting corrosion for this type of materials typically occurs in specific environments like chloride containing environments. If a material is covered with a passive layer, localized mechanical damage to this layer has to occur before pitting can occur. Then, a small active anode is present at the spot where the passive layer is broken, as well as a large cathodic spot which is passive. If no repassivation of the active spot occurs, then a high local corrosion rate occurs. So, pitting is in fact the localized failure of passivating film on the surface. This means that a small area is exposed to unrestrained corrosion, while a large area is still protected by a passivating layer. In time, the pit deepens and subsurface corrosion occurs. Pitting is determined by the electrochemical potential at the surface. There is a minimum potential known as the pitting potential required for pitting to occur. The metallurgy plays an important role in pitting, like

• Passivating surface films reduce pitting because they provide resistance to nucleation. • Manganese sulfide inclusions promote pitting. • Chromium steels combined with heat treatments may result in chromium depleted

areas which are sensitive to pitting. • Stainless steels are more sensitive for pitting than plain steel. • Alloying steel with Molybdenum inhibits pitting. Titanium can also be used, but is

often too expensive. Pitting corrosion can be reduced by a higher fluid flow, a higher starting pH value or by avoiding Cl- ions.

Crevice corrosion

In many typical structures there are hidden interior surfaces, like enclosed spaces between a nut and a bolt. The reason for this type of corrosion is the difference in oxygen content in the enclosed space. The different environment is also clear from the fact that crevice corrosion typically occurs in narrow gaps or under dirt which is attached to the surface. Such closed spaces easily become anodic sites and are therefore sensitive for corrosion. This also means that much corrosion can take place at spots which cannot be found by superficial inspection. The best method to avoid crevice corrosion is to avoid crevices by closing small gaps, or by widening gaps so that the solution in the gap can be refreshed.

Selective dissolution

If an alloy is formed by two components with a large difference in electrochemical potential then selective damage to the alloy can occur. The least noble material will go into solution, while the noble component will stay as a porous and mechanically weak mass. This type of corrosion often occurs below dirt or in narrow gaps.


238

Intercrystalline corrosion

Many materials used in practice are not single crystals or amorphous, but polycrystalline materials. So, the material is composed of many crystallites with crystal boundaries in between the crystallites. If the crystal boundaries are damaged, then the structure of the material will be damaged. So, although only very limited mass loss can be found, it will cause large damage to the strength of the structure. The most important example of intercrystalline corrosion is intercrystalline corrosion of stainless steel. For this material, chromium carbides can be formed at the grain boundaries if the material is heated in a certain temperature range, like is the case for welding or heat treatments, according to:

623623 CCrCCr ⇒+ (7.73) These compounds will preferably be formed at the grain boundaries, and the reaction goes particularly fast between 600 and 750 oC. Formation of chromiumcarbides results in depeting of chromium close to the grain boundaries. The reason is that diffusion of chromium to the grian boundary is a very slow process. Intercrystalline corrosion is a dangerous form of corrosion because little mass loss results is a large damage to the material. The most important measures than can be taken to avoid intercrystalline corrosion are the application of stainless steel with a low carbon content below 0.03%. It is also possible to use a carbon-binding element of the alloy, like titanium of niobium.

Stress corrosion and corrosion fatigue

Stress corrosion and corrosion fatigue are two corrosion mechanisms that are related to stresses in the material due to mechanical loading or internal stresses in the material. In the case of cyclic loads corrosion fatigue can occur. Tensile stresses combined with corrosion effects will lead to the formation of cracks which can go through crystallites (transcrystalline cracks) or go along the grain boundaries (intercrystalline cracks). For almost all alloys, this type of corrosion only occurs in specific environments. Examples are corrosion of alloys in combination with ammonia or sulfurdioxide (intercrystalline) or low alloy steels in nitride solutions and concentrated caustic solutions (intercrystalline). Also stainless steels in hot chloride solutions (trans crystalline) or in chloride containing steam (both transcrystalline and intercrystalline)m aluminum alloys in seawater (intercrystalline) are sensitive for stress corrosion [5]. Corrosion fatigue is a special case of stress corrosion, which is the growing of intercrystalline or transcrystalline cracks by the combined action of corrosion and residual stresses in the material. The most important measures to avoid stress corrosion and corrosion fatigue are:

• Avoiding high tensile stresses and in particular stress concentrations. • Avoiding aggressive components in the environment. • Application of cathodic protection, inhibitors or coatings

Erosion corrosion

Acceleration of corrosion by a fluid flow is called erosion-corrosion. The principle is that liquid flow removes diffusion limitations of reagents of electrochemical corrosion. Besides this, a passivating film which may be present at the surface will be exposed to a mechanical load. Because passivating films are often mechanically weak and can easily become detached from the metal substrate, this can result in repeated removal and formation passivating films at the surface. Surfaces become unprotected in high velocity flows and due to the high velocities also impingement at the surfaces occurs then, in some cases, a very fast localized perforation of a metal component can occur.


239

Hydrogen embrittlement and hydrogen blistering

A phenomenon related to corrosion is hydrogen blistering. Hydrogen blistering is found in steel components with sites of cathodic reactions. Hydrogen blistering is an accumulation of gaseous hydrogen inside the metal. This gaseous hydrogen may reach sufficient pressure to form a blister at the metal surface. Hydrogen embrittlement is a related phenomenon and is caused by hydrogen ions penetrating the metal and rapidly diffusing into the material. Diffusion is relatively simple for hydrogen because hydrogen is a small ion. These ions reach voids and forms gaseous hydrogen. Gaseous hydrogen is a larger molecule and can less easily leave the void. This results in a high gas pressure in the void followd by crack growth to release the high pressure.

Corrosion inhibitors

Corrosion inhibitors are solid or liquid substances that reduce corrosion. Most inhibitors used are organic or inorganic chemicals which react with the surface to form a passive film or adsorbed layer acting as barrier film. There are four types of corrosion inhibitors:

• Passivators. Passivity of a surface is a phenomenon where a reactive surface turns out to be resistant against corrosion. A reason for this can be that a passivating surface layer is formed. A passivating layer can be mechanically weak, despite the fact that it provides enough resistance against electrochemical corrosion.

• Inhibitors are additives that slow down the corrosion rate by adsorption to the surface which is subject to corrosion. Cathodic passivators inhibit reduction of hydrogen and by this reduce the corrosion rate. Such materials are often toxic to humans (e.g. arsenicum) and can cause hydrogen blistering. Barrier inhibitors work by the formation of a protective layer of the surface by e.g. dense chemical reaction film. In this way the surface is effectively separated from the environment. Three types of inhibitors that can be absorbed at the surface are Organic adsorption inhibitors which are absorbed by either physical adsorption or chemisorption. Inorganic precipitation inhibitors, which work by reactions with environment and metal surface that cover surface (carbonate and phosphate films from carbonate and phosphate salts. phosphating as a surface treatment is an example of an inorganic inhibitor. Vapor phase inhibitors (e.g. gas) which are injected in critically precision instruments and absorb to the surface

• Scavengers work because they react with oxygen and remove it from (closed) system. For example: Sulfides or phosphites become sulfates and phosphates.

• Neutralizers reduce acidity of environment, which is determined by the concentration of hydrogen ions.

7.3.7 Data This section summarizes the properties of metals and its oxides, relevant for electrochemical corrosion. Table 7.5: Properties of metals and their oxides [5]

Metal Name Atomic mass / Molar mass [g mol-1]

Density [g cm-3]

oxides Density oxides [g cm-3]

Ag Silver 107.88 10.5 Ag2O 7.143 Al Aluminium 26.98 2.702 Al2O3 3.97 Cr Chromium 52.01 7.20 Cr2O3 5.21


240

Co Cobalt 58.94 8.9 CoO Co3O4

5.6-6.7 6.07

Cu Copper 63.54 8.92 Cu2O CuO

6.0 6.40

Fe Iron 55.85 7.86 FeO Fe3O4 Fe2O3

5.7 5.18 5.24

Mg Magnesium 24.32 1.74 MgO 3.58 Na Natrium 22.99 0.97 Na2O 2.27 Ni Nickel 58.71 8.90 NiO 7.45 Pb Lead 207.20 11.34 PbO

PbO2 9.53 9.38

Sn Tin 118.7 7.28 SnO SnO2

6.45 6.95

Ti Titanium 47.90 4.50 TiO2 4.26 Zn Zink 65.30 7.14 ZnO 5.47 Zr Zirconium 91.22 6.4 ZrO2 5.6 Table 7.6: Thermodynamic properties of oxides [5]

Oxides ∆g0 [kJ mol-1]

∆h0 [kJ mol-1]

∆s0 [kJ mol-1]

Ag2O -11.2 -31.1 -66.5 Al2O3 -1582.4 -1675.7 -312.9 Cr2O3 -1058.1 -1139.7 -273.6 CuO Cu2O

-146.0 -129.7

-168.6 -157.3

-75.8 -92.6

FeO Fe3O4 Fe2O3

-245.1 -1015.5 -742.2

-266.3 -1118.4 -824.2

-70.9 -345.2 -275.0

NiO -211.7 -239.7 -94.0 TiO2 -852.7 -912.1 -199.3 ZnO -318.3 -348.3 -100.5 ZrO2 -1022.6 -1080.3 -193.7 H2O(l) -237.2 -285.8 -163.2 H2O(g) -228.6 -241.8 -44.4 CO(g) -138.2 -112.0 +88.0 CO2(g) -394.3 394.0 -0.84

Data according to equation

nmOMeOnmMe =+ 22 (7.74)

For temperatures close to T=298 K we can write ( ) ( ) ( ) ( ) ( ) ( )298298298298298 00000 sThsThTg ∆−−≈∆−∆≈∆ (7.75)

Table 7.7: Electrochemical equivalents [5]

Metal Reaction

Electrochemical equivalent [kg C-1]

Ag Ag+ + e- = Ag

1.118 10-6

Al Al3+ + 3 e- = Al 0.0932 10-6 Au Au3+ + 3e- = Au

Au+ + e- = Au 0.681 10-6 2.044 10-6


241

Br2 Br2+2e- = 2Br- 0.8282 10-6 Cd Cd2+ + 2 e- = Cd 0.5824 10-6 Cl2 Cl2+2e- = 2Cl- 0.3674 10-6 Co Co2+ + 2 e- = Co 0.3054 10-6 Cr Cr3+ + 3 e- = Cr

Cr6+ + 6 e- = Cr 0.1797 10-6

0.0898 10-6 Cu Cu2+ + 2e- = Cu 0.3294 10-6 F2 F2

+ 2e- = 2F- 0.1969 10-6 Fe Fe2+ + 2 e- = Fe

Fe3+ + 3 e- = Fe 0.2894 10-6 0.1929 10-6

H2 2H+ + 2e- = H2 0.01045 10-6 I2 I2 + 2e- = 2I- 1.315 10-6 Mg Mg2+ + 2 e- = Mg 0.1260 10-6 Mn Mn2+ + 2 e- = Mn 0.2846 10-6 Ni Ni2+ + 2 e- = Ni 0.3041 10-6 O2 O2+2H2O+4e-=4OH-

O2+2H++2e-=H2O2 0.08291 10-6 0.1658 10-6

Pb Pb2+ + 2 e- = Pb Pb4+ + 4 e- = Pb

1.074 10-6 0.5368 10-6

Sn Sn2+ + 2 e- = Sn Sn4+ + 4 e- = Sn

0.615 10-6 0.3075 10-6

Zn Zn2+ + 2 e- = Zn 1.118 10-6 Table 7.8: Standard potentials of redox reactions [5]

Electrode reaction E0 [V] +−+ =+ 23 CreCr -0.41

222 HeH =+ −+ 0.000 +−+ =+ CueCu 2 0.153

( ) ( ) −−− =+ 46

36 CuFeeCuFe 0.356

−− =++ OHeOHO 442 22 0.401 −− =+ IeI 222 0.53

222 22 OHeHO =++ −+ 0.69 +−+ =+ 23 FeeFe 0.771 −− =+ BreBr 222 1.06

OHeHO 22 244 =++ −+ 1.230 −− =+ CleCl 222 1.360

OHeHOH 222 222 =++ −+ 1.77

FeFe 222 =+ − 2.85

7.3.8 References [1] A.W. Batchelor, L.N. Lam, M. Chandrasekaran, Materials degradation and its control

by surface engineering, Imperial College Press, London. [2] K.L. Johnson, Contact Mechanics [3] K. Holmberg and A. Matthews, Coatings Tribology: Properties, Techniques and

Applications in Surface Engineering [4] Karl J. Hemmerich, Polymer Materials Selection for Radiation-Sterilized Products,

http://www.devicelink.com/mddi/archive/00/02/006.html

http://www.devicelink.com/mddi/archive/00/02/006.html


242

[5] P. Gellings, Inleiding to corrosie en corrosiebestrijding, ISBN 9036509793


243

Chapter 8 Appendix C: Explicit equations for stresses beneath a sliding contact In this appendix equations are given for stresses under the surface of a spherical Hertzian (so elastically deforming) contact. Both equations for stresses originating from a normal load FN as stresses originating from a tangential load FT are given. The equations are taken from [18].

8.1 Caused by an applied normal load FN

( ) ( ) ( ) ( )

−+−

−+−

−−−

−++=

SzaMxyxNMzaaANNSNz

rxy

rz

aFN

x

22232

2

22

23 22321111

23

ννν

νφνπ

σ (8.1)

( ) ( ) ( ) ( )

−+−

−+−

−−−

−++=

SzaMyxyNMzaaANNSNz

ryx

rz

aFN

y

22232

2

22

23 22321111

23

ννν

νφνπ

σ (8.2)

+−=

SazMN

aFN

z 323π

σ (8.3)

( ) ( ) ( ){ }

+−−+++−++−−

= aMzNS

aMrrxyzaaMzNzASNNr

rxy

aFN

xy

2

43

32

322

43 22123 νπ

τ (8.4)

+−−= 2232

3HG

yzHS

yNza

FNyz π

τ (8.5)

+−−= 2232

3HG

xzHS

xNza

FNzx π

τ (8.6)

With


244

=

−=

+=

+=

−+=

+=

−

Ma

ASN

ASM

zaAS

azrAyxr

1

222

222

222

tan

2

2

4

φ

zMaMMNHaNzMNMG

++=−+−=

2

22

(8.7)

If r=0 (so, at the axis through the center of the contact), the equations need to be rewritten to

( ) ( )

+

+

−

+== −

22

31

3 2tan1

23

zaaa

zaz

aFN

yx νπ

σσ (8.8)

+

−= 22

3

323

zaa

aFN

z πσ

(8.9)

0===yyzzxxy τττ (8.10)


245

8.2 Caused by a tangential load FT

( )

( ) ( ) ( )

( )

−

−

+

−−+

+−−−−−

−

+

+−+++−

−+

+−

=

ν

ννν

νν

ννφν

πσ

21223

34

47

43

2121

321

62

23

24

722231

4

23

2

2

4

3

222222

2

2

4

22222

22

2

4

3

rx

rxza

rrSxaazAS

rx

rxzN

rxrSzxzAS

rx

raxMx

aFT

x

(8.11)

( )[ ]

( ) ( )

( )

−

−

+

−−+

−−−−−−

−

+

++++−

−+−

=

ν

ννν

ννφν

πσ

21221

34

443

23

221

321

62

21

4322

21

43

23

2

2

4

3

22

222

2

2

2

4

222

222

2

4

3

ry

rxza

rrSyaazAS

rx

rxzN

rSzyzrAS

ry

raxMx

aFT

y

(8.12)

++

−=S

azr

r

zxN

a

FTz

222

231

22

3

πσ

(8.13)


246

( )

( )

( )

−

−

+

−−++

−−−

+−

−

+

−++

+−

−−++

−

=

ν

νν

νφν

πτ

21221

34

32422

3236

12221

221

2422

2121

22

23

2

2

4

3

22222222

2

2

4

2

22

222

2

222

4

3

rx

ryza

Sxy

SxarrazAS

rx

ryzN

rxzrrx

rxAS

Szx

rayMy

aFT

xy

(8.14)

( )

+++−+

−−+= 222

222

43 513222

32

121

223 rza

SzNraz

SaM

rxyz

aFT

yz πτ

(8.15)

( )

−+−−++−+

−++= 2

22222

2

2

2

2

23

221

241

432

431

23

23

rxzrazAS

rN

Sx

rx

razMz

aFT

zxφ

πτ

(8.16)

At the axis r=0 all stresses are non-existent except τzx

( )

+

−

+−= −

22

21

3 2tan

23

23

zaaz

zaza

aFT

zx πτ

(8.17)

At the surface z=0. There, the stresses outside the contact, so for x>a, are given by

+−+

−−+

+−= 22

2

2

2204

03 2

472

23

41

23 rxr

rxM

raxMx

aFN

x νν

νφν

πσ

(8.18)

+

−+−= 2

2

22

40

3 432

21

43

23 r

rya

raxMx

aFN

y ννφνπ

σ (8.19)


247

++−

−−+

−= 222

2

2204

03 2

14122

211

2223 rrx

rxM

rayMy

aFN

xy νννφν

πτ

(8.20)

0

20

===−=

zxyzz

aarMττσ

(8.21)

At the surface z=0. There, the stresses inside the contact, so for x<a, are given by

+−= 1

4223

3

νππ

σx

aFT

x (8.22)

−=

83

23

3

xa

FTy

πνπ

σ (8.23)

−−= 1

2423

3

νππ

τy

aFT

xy (8.24)

[ ]2232

3 raa

Fz Tzx −−=

πτ

(8.25)

0== yzz τσ (8.26)

8.3 References [18] Hamilton, G.M., 1983, Explicit equations for the stresses beneath a sliding spherical contact, Proceedings of the institution of mechanical

engineers, Vol 197 C, p. 53-5

handout%20solids%20&%20surfaces%202009-2010

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