chapter 13: surfaces - materials science · chapter 13: surfaces ... 13.8 precipitation from a...

Notes on the Thermodynamics of Solids J.W. Morris, Jr.: Fall, 2008

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C h a p t e r 1C h a p t e r 1 33 : S u r f a c e s: S u r f a c e s Chapter 13: Surfaces........................................................................................................210

13.1 Introduction ................................................................................................210 13.2 The Gibbs Construction..............................................................................211 13.3 The Fundamental Equation of the Surface ..................................................212

13.3.1 The shape of the dividing surface .................................................213 13.3.2 Elimination of the curvature terms ................................................214 13.3.3 The fundamental equation.............................................................217

13.4 The Conditions of Internal Equilibrium......................................................218 13.4.1 The general condition of equilibrium............................................218 13.4.2 The conditions of thermal and chemical equilibrium.....................219 13.4.3 Variations that include displacements of the interfaces .................221 13.4.4 The conditions of mechanical equilibrium at an interface..............222 13.4.5 Equilibrium at a junction line on a solid: wetting ..........................224

13.5 The Conditions of Global Equilibrium........................................................224 13.6 The Critical Nucleus ...................................................................................225

13.6.1 The critical nucleus .......................................................................225 13.6.2 The work of formation of a critical nucleus ..................................226 13.6.3 Comments on the work to form a critical nucleus.........................228

13.7 Approximations for the Critical Work ........................................................230 13.7.1 The free energy approximation: one-component system...............230 13.7.2 The Thomson approximation for a one-component system ..........231 13.7.3 The free energy approximation: multi-component system.............232

13.8 Precipitation from a Dilute Solution............................................................234 13.9 Heterogeneous Nucleation..........................................................................237 13.10 The Preservation of a Phase in a Cavity.....................................................239 13.11 A Simple Polygranular Solid.....................................................................240

13.11.1 The equilibrium microstructure...................................................241 13.11.2 Grain coarsening in the polygranular microstructure ..................242 13.11.3 Heterogeneous nucleation in a simple polygranular solid ...........244

13.1 INTRODUCTION The thermodynamics of interfaces in real materials is made difficult by the complexity of the surface structure. The interfaces that separate phases in contact are not strict discontinuities. They are rather thin transition shells across which the materials prop-erties and thermodynamic densities change from the values appropriate to one phase to those appropriate to the other. The reason for the thickness of the interface is relatively straightforward: the two phases perturb one another over a distance that is at least equal to the effective range of the atomic interaction. However, detailed analysis of the surface re-gion is difficult. Even the best modern characterization tools reveal very little about the in-ternal structure of real interfaces. We are hence faced with the problem of writing a ther-


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modynamic analysis of an inhomogeneous material whose internal structure we know very little about. The thermodynamics of surfaces was developed by Gibbs (Equilibrium of Hetero-geneous Substances), who devised a mathematical technique that acknowledges the finite thickness of the interface while avoiding it in a simple formal way. The method uses a ge-ometric representation of the interface, the Gibbs construction, together with the assumption that while the internal state of the interfacial shell may be unknown it is fixed by equi-librium with the bounding phases. In the Gibbs construction the interfacial shell is replaced by a hypothetical dividing surface, which is a two-dimensional surface that is placed so that it is roughly coincident with the physical interface. The bounding phases are imagined to extend homogeneously up to the dividing surface from either side. The hypothetical system is then forced to have a thermodynamic content that duplicates that of the actual transition shell by defining surface excess quantities of energy, entropy and mass that are imputed to the dividing surface itself. When the system is in equilibrium the surface excess quantities are related to one another by a fundamental equation that governs the behavior of the interface. In the following sections we derive the fundamental equation for the surface, estab-lish the conditions of equilibrium for interfaces, and discuss some of the implications of surface thermodynamics for the behavior of multiphase systems. The discussion is re-stricted to the case in which the bounding phases are thermodynamic fluids, and where the interface has fluid properties in the sense that its state does not depend on its orientation. In a crystalline solid the properties of an interface generally depend on its orientation since the detailed configuration of atoms in an interface depend on its specific crystallographic plane. Interfacial models that include the regular atom arrangements within crystalline solids have been developed, but are very restricted in their content and generality and will not be dis-cussed here. 13.2 THE GIBBS CONSTRUCTION Consider a two-phase interface that is, physically, an interfacial shell between two homogeneous phases. Let a mathematical dividing surface be placed so that it lies within the interfacial shell parallel to the physical interface. The interfacial shell can then be re-ferred to the curvilinear coordinates, x1, x2, x3, where x1 and x2 lie in the plane of the sur-face and are everywhere orthogonal, and x3 is normal to the surface. Let two additional imaginary surfaces be drawn so that they are everywhere parallel to the dividing surface and separated from it by a distance, h, that is large enough that these surfaces lie in the homogeneous phases outside the interfacial shell. If the dividing surface is the set of points with the coordinates (x1, x2, 0) then points on the parallel surfaces have the coordinates (x1, x2, ± h). The energy, entropy and chemical content of a system that contains an interface are measurable quantities, as are the thermodynamic contents of the homogeneous phases that


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bound the interface. It follows that the energy, entropy and chemical content of the inter-facial shell, EI, SI, and {NI}, can be found by subtraction. In the Gibbs construction the homogeneous phases, å and ∫, are imagined to ex-tend up to the dividing surface, S, which therefore divides the shell into subvolumes Vå and V∫. Since the thermodynamic content of the hypothetical interface must reproduce that of the real one, surface excess quantities of the energy, ES, entropy, SS, and chemical species, {NS}, are attributed to the dividing surface itself, where ES = EI - Eå

v Vå - E∫v V∫ 13.1

SS = SI - Så

v Vå - S∫v V∫ 13.2

NS

i = NIi - n

åi Vå - n∫

i V∫ 13.3 It is convenient to define surface excess densities of the energy, entropy and chemical con-tent by the equations

ES = ES

A 13.4

SS = SS

A 13.5

©i = NS

iA 13.6

where A is the area of the surface, S. 13.3 THE FUNDAMENTAL EQUATION OF THE SURFACE To find the fundamental equation for the surface we need to identify the set of geo-metric coordinates that determines the surface excess energy. The thermodynamic state of the interfacial shell is fixed by its equilibrium with the bounding phases. Its energy should, then, depend on its entropy, its chemical content, and its geometry, which implies that the surface excess energy is fixed by the surface excess entropy, the surface excess of the chemical content, and the geometry of the dividing surface. The geometry of the dividing surface is given by its placement within the interfacial shell, its area and its shape. (A complete set of variables for the surface energy of a free surface of a crystalline solid would also include the crystallographic orientation of the surface; the set of variables for a grain boundary or interface between crystalline solids would include the crystallographic orienta-tions of both. Fortunately, many of the surface phenomena of interest in crystalline solids can be understood without considering the orientation dependence of the interfacial proper-ties.)


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13.3.1 The shape of the dividing surface The shape of the surface must be quantified before it can be included in the funda-mental equation. In the differential geometry of two-dimensional surfaces an element, dA, is defined by its position in space and its normal vector, n. The direction of the normal vector defines the positive side of the surface. If the surface is closed around a finite vol-ume the direction of n is generally chosen so that it points out of the enclosed volume, and is hence the outward normal vector. If the surface is not closed then either side of it can be regarded as the outward, or positive side; it is only necessary to be clear and consistent. The local shape of the surface at dA is specified by its two principal radii of curvature, R1 and R2, which are the radii of the two circles that match the curvature of the orthogonal sur-face coordinates, x1 and x2, at dA. A radius of curvature is positive if its center of curva-ture, the center of the circle of radius, R, lies on the interior side of the surface, and is negative otherwise. Convex surfaces have positive curvature. Since interfaces are often planar or nearly so, and since the radius of curvature of a plane is infinite, it is usually more convenient to specify the shape of the surface element, dA, by its mean curvature, –K, and Gaussian curvature, K, where

–K = 1

R1 +

1R2

13.7

K = 1

R1R2 13.8

The mean and Gaussian curvatures have the additional advantage that while the values of the radii R1 and R2 depend on the precise choice of the coordinates x1 and x2, –K and K are inherent characteristics of the shape that are the same in every coordinate system. For a sphere,

–K = 2R

K = 1

R2 13.9

where R is the radius. For a cylinder,

–K = 1R

K = 0 13.10 where R is the cylinder radius; for a plane


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–K = K = 0 13.11 [Note that Gibbs characterizes the shape of the surface by the two parameters –K and

K' = 1

R1 -

1R2

13.12

While there is nothing wrong with this choice the Gaussian curvature is a more convenient parameter, as we shall see.] Consider a segment of the surface that has constant curvature. Its shape is charac-terized by the variables –K and K, so the surface excess energy is given by a relation of the form ES = ¡ES(SS, {NS}, A, –K, K) 13.13 whose differential is

dES = TdSS + ∑k

µkdNSk + ßdA + Cd–K + C'dK 13.14

where T is the surface temperature, µk is the surface chemical potential of the kth specie, ß is the surface tension, and C and C' are the forces conjugate to the curvature variables. Note that the equation 13.14 defines the surface temperature and chemical potentials. We will show later that these are in fact equal to the temperature and chemical potentials of the adjacent phases. 13.3.2 Elimination of the curvature terms Equation 13.13 can be simplified by removing its dependence on the curvature terms. Gibbs used semi-quantitative arguments to justify this step. The argument can be made quantitative when the thermodynamic densities are defined and continuous within the interfacial shell. Assume that the material within the interfacial shell is continuous to a sufficient ap-proximation that an integrable energy density, Ev(r) = Ev(x1,x2,x3) can be defined for every differential volume element within it. The differential volume within a shell centered on the dividing surface can be written dV = ç(x3)dAdx3 13.15 where ç(x3) = 1 - –Kx3 + Kx32 13.16


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In the Gibbs construction the interfacial shell extends over the range - h ≤ x3 ≤ h, and the energy density is Ev(x3) = Eå

v (- h < x3 < 0)

= E∫

v (0 < x3 < h) 13.17 It follows that the surface excess energy is

ES = ⌡⌠S

⌡⌠

-h

h [Ev]ç(x3)dx3 dA 13.18

where the symbol [Ev] represents the discontinuous function [Ev] = Ev - E

åv (-h < x3 < 0)

= Ev - E∫

v (0 < x3 < h) 13.19 Introducing equation 13.16,

ES = ⌡⌠

-h

h [Ev]dx3 - –K⌡⌠

-h

h [Ev]x3dx3 + K⌡⌠

-∞

∞ [Ev](x3)2dx3 13.20

By the mean value theorem the last term on the right-hand side of 13.20 can be re-written

K⌡⌠

-h

h [Ev](x3)2dx3 =

´[Ev]¨K(2h)3

3 13.21

where ´[Ev]¨ is of the order of the average value of [Ev] through the shell. Its magnitude is less than that of the first term by the factor

K(2h)2

3 = 13 ∂

R2 13.22

where ∂ is the thickness of the shell and R = 1/ K is a measure of its radius of curvature. Given that ∂ is of the order of the effective range of atomic interaction, which is micro-scopic, this term is negligible for surfaces whose radii exceed microscopic dimensions. It follows that we may take ç(x3) = 1 - –Kx3 13.23


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Now introducing the volume density of the work function „V = Ev - TSv - ∑

k

µknk 13.24

(which is the negative of the pressure in the homogeneous phases), the surface excess en-ergy density can be written

ES = TSS + ∑k

µk©k + ⌡⌠

-h

h [„V]dx3 - –K⌡⌠

-h

h [„V]x3dx3 13.25

from which it follows that the force conjugate to the mean curvature is

C = ⌡⌠

-h

h [„V]x3dx3

= ⌡⌠

-h

h „Vx3dx3 - ⌡⌠

0

h „∫

V x3dx3 - ⌡⌠

-h

0 „å

V x3dx3

= ⌡⌠

-h

h „Vx3dx3 -

h2

2 „å

V + „∫V 13.26

The value of the last term on the right-hand side of 13.26 is independent of the precise way in which the dividing surface is placed in the transition shell. The value of the first term, however, is dependent on the location of the dividing surface since the kernel of the integral is the product of a function, „V, whose physical value is independent of how the Gibbs construction is made, and a coordinate, x3, that changes sign as it crosses the dividing sur-face. If the dividing surface is displaced normal to itself by the distance ∂xn the value of the coefficient, C, is changed by the amount

∂

⌡⌠

-h

h [„V]x3dx3 = ⌡⌠

-h

h [„V]∂xndx3 = ∂xn⌡⌠

-h

h [„V]dx3 13.27

where we have used the fact that [„V] = 0 at x3 = ± h. The integral in the second form on the right in 13.27 is just the surface tension, which, as we shall show, is necessarily posi-tive. It follows that the sign of ∂C is determined by the sign of ∂xn, and C can be increased or decreased by displacing the dividing surface in the appropriate direction. In particular, the position of the dividing surface can be chosen so that C = 0 13.28


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in which case the surface excess energy is independent of –K. When the dividing surface is placed so that C = 0 it is said to lie at the surface of tension. 13.3.3 The fundamental equation When the dividing surface is located at the surface of tension the fundamental equa-tion of the surface is ES = ESS = ¡ES(SS, {NS}, A) 13.29 which has the differential form dES = TdSS + ∑

k

µkdNSk + ßdA 13.30

where ß is the surface tension, the force conjugate to the surface area. Since 13.29 and 13.30 are independent of the curvature, these equations hold in general; the interface need not have constant curvature. It follows that the surface excess energy is a linear homoge-neous function of its variables, and has the integrated form ES = TSS + ∑

k

µkNSk + ßA 13.31

or ES = TSS + ∑

k

µk©k + ß 13.32

The surface tension, ß, is the surface excess of the work function, „,

ß = „S

A = ⌡⌠

-h

h [„V]dx3 13.33

where the final form holds under the assumption that the thermodynamic densities can be defined within the interfacial region. While various forms of the fundamental equation of an interface can be obtained from equation 13.29 by Legendre transformation, only the work function, „, is particularly meaningful. The reason is that the interface is an equilibrium transition region between bulk phases, and is therefore an open system in contact with reservoirs that fix its thermal and chemical equilibrium. The governing thermodynamic potential is, hence, the work function, „, and the relevant condition of equilibrium for the interface is


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(∂„S) TR{µR}

≥ 0 13.34

where TR and {µR} are the temperature and chemical potentials fixed by the reservoir. To explore this condition further we need to find the relations between the temperature and chemical potentials of the reservoir and those at the interface. These are set, along with the condition of mechanical equilibrium at the interface, by the conditions of local equilibrium. 13.4 THE CONDITIONS OF INTERNAL EQUILIBRIUM 13.4.1 The general condition of equilibrium To find the necessary conditions of equilibrium at the interface we apply the general condition of equilibrium to a multiphase system whose internal interfaces obey the funda-mental equation derived above. For mathematical simplicity we shall neglect external fields. Gibbs derives conditions of equilibrium that include the influence of external fields, but the only affect that is usually important is the variation of the bulk chemical potential and pressure that we have already treated. Consider a closed system that contains two or more phases separated by interfaces. Let each phase be a thermodynamic fluid, and let the different phases be in thermal and chemical communication with one another across the interfaces. All components are, hence, present in all phases. The energy density in the åth phase is Eå

v = ⁄Eåv(Sv, {n}) = TSv + ∑

k

µknk - P 13.35

The density of the surface excess energy is ES = ⁄ES(SS, {©}) = TSS + ∑

k

µk©k + ß 13.36

where T, {µ} and ß are the forces conjugate to SS, {NS} and A, and are determined by the first partial derivatives of the energy function according to equation 13.30. Let the different phases be denoted by the values of the index, å, and let the inter-face between the phases å and ∫ be labeled by the index Ë, where Ë stands for the pair å∫. The total energy of the system is

E = ∑å

⌡⌠V Eå

v dVå + ∑Ë

⌡⌠S EË

s dAË 13.37


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where the first summation is taken over the bulk phases and the second is taken over the interfaces. The general condition of equilibrium is (∂E)S{N}V ≥ 0 13.38 Using the method of Lagrange multipliers this condition can be re-written

∂

E - ŒS - ∑

k

≈kNk ≥ 0 13.39

where œ and {≈} are constant multipliers and the variation is taken at constant total volume. Using equation 13.37 and 13.39 the general condition of equilibrium can be re-written

∂

∑

å

⌡⌠

V

Eå

v - œSåv - ∑

k

≈knåk dVå + ∑

Ë

⌡⌠

S

EË

s - œSËs - ∑

k

≈k©Ëk dAË

≥ 0 13.40 where an arbitrary variation may include changes in the states of the bulk and surface phases together with deformations or displacements of the interfaces. 13.4.2 The conditions of thermal and chemical equilibrium It is simplest to evaluate the conditions of equilibrium in two steps. In the first we test for equilibrium with respect to variations that do not affect the geometry of the inter-faces. In this case the variation in equation 13.40 can be taken inside the integral sign, with the result that

∂[E - œS - ∑k

≈kNk] = ∑å

⌡⌠

V

∂Eå

v - œ∂Såv - ∑

k

≈k∂nåk dVå

+ ∑Ë

⌡⌠

S

∂EË

s - œ∂SËs - ∑

k

≈k∂©Ëk dAË

= ∑å

⌡⌠

V

(Tå - œ)∂Så

v + ∑k

(µåk - ≈k)∂nå

k dVå

+ ∑Ë

⌡⌠

S

(TË - œ)∂SË

s + ∑k

(µËk - ≈k)∂©Ë

k dAË 13.42


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Since the condition of equilibrium must hold for arbitrary variations of the entropy and molar densities at any point in the system, each of the kernels of each of the integrals must vanish independently. The results are the necessary conditions of chemical and thermal equilibrium:

The temperature is the same everywhere in a multiphase system at equilib-rium:

Tå = TË = œ 13.43

The chemical potential of every species is the same everywhere in a multi-phase system at equilibrium:

µå

k = µËk = ≈k 13.44

An important corollary to the conditions of thermal and chemical equilibrium in a multiphase system is the demonstration that the temperature and chemical potentials that are determined by the fundamental equation of the surface have the same physical meaning as the temperature and chemical potentials in the bulk phases. The temperature of the interface is

T =

∆ ¡ES

∆SS =

∆ ¡ES

∆SS 13.45

while the chemical potential of the kth species at the interface is

µk =

∆ ¡ES

∆NSk

=

∆ ¡ES

∆©k 13.46

The temperature and chemical potentials at the interface are equal to the corresponding val-ues in the bulk phases. The internal conditions of mechanical equilibrium within each of the bulk phases and on each of the interfaces are immediately established from the conditions of thermal and chemical equilibrium. Within each bulk phase, å, Eå

v - TSåv - ∑

k

µknåk = „å

v = - ¡På(T,{µ}) 13.47

while on each interface, Ë, EË

S - TSËS - ∑

k

µk©Ëk = „Ë

S = ßË(T,{µ}) 13.48


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Since T and {µ} are fixed,

the pressure is constant within each homogeneous phase in a multiphase fluid system and the interfacial tension is constant on each interface.

13.4.3 Variations that include displacements of the interfaces Incorporating the conditions of thermal and chemical equilibrium into the general condition of equilibrium, 13.40, yields the simpler condition

∂

∑

å

⌡⌠V „å

V dVå + ∑Ë

⌡⌠S „Ë

S dAË

= ∂

∑

å

⌡⌠V (- På) dVå + ∑

Ë

⌡⌠S ßË dAË ≥ 0 13.49

Since the pressures and interfacial tensions are fixed by T and {µ}, the inequality 13.49 governs equilibrium with respect to changes of state that result from displacements or de-formations of the interfaces. Hence the inequality can be re-written

∑å

⌡⌠V (- På) ∂(dVå) + ∑

Ë

⌡⌠S ßË ∂(dAË) ≥ 0 13.50

To alter the volume of a phase one must make a normal displacement of its bound-ary. If the boundary element, dA, is displaced along its outward normal by the differential distance, ∂xn, it causes the incremental volume change ∂V = ∂xndA 13.51 Hence ⌡⌠

V På ∂(dVå) = ⌡⌠

S På ∂xndA 13.52

The influence of a displacement on the area of a surface is more complicated. An infinitesimal displacement, ∂x, of a differential element, dA, of the surface can be regarded as the sum of a normal displacement, ∂xn, which is positive if it is in the direction of the outward normal, and a tangential displacement, ∂xt. First consider the normal displacement. A normal displacement does not change the area of a plane, but if the surface element, dA, has a non-zero mean curvature, –K, then its area must change when it is displaced while maintaining continuity with the surface ele-ments that are adjacent to it. The result is easy to see when the surface is spherical. If the


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radius of a sphere is increased by the displacement, ∂xn = ∂R, where R is the radius, the fractional change in its area is

∂AA =

2

R ∂R = –K∂xn 13.53

The second form of equation 13.53 is general; if a differential element of surface, dA, with mean curvature, –K, is given a differential displacement, ∂xn, along its outward normal, the fractional change in its surface area is

∂(dA)

dA = –K∂xn 13.54

Second, consider the tangential displacement, ∂xt. A tangential displacement has no effect on the area of a surface unless that displacement occurs at a line along which the sur-face terminates. A displacement normal to the terminal line increases the area of the ele-ment, dA, that is bounded by the line by the amount ∂(dA) = ∂xtdL = (t^∂x)dL 13.55 where t is a unit vector that lies tangent to the surface and normal to the terminal line, ∂x is the vector displacement of the line normal to itself, and dL is the element of length along the terminal line. Using equations 13.54 and 13.55, ⌡⌠

S ß ∂(dA) = ⌡⌠

S ß –K∂xndA + ⌡⌠

L ß(t^∂x)dL 13.56

where the first integral on the right-hand side is taken over the surface and the second is taken over the line on which the surface terminates. 13.4.4 The conditions of mechanical equilibrium at an interface To obtain the conditions of mechanical equilibrium we use the fact that two phases meet at each interface (Ë) and several interfaces meet at each junction line (L). Then, using equations 13.56 and 13.52, the condition of equilibrium, 13.50, becomes

∑Ë

⌡⌠

S [ ]–KßË - ÎPË ∂xndAË + ∑

L

⌡⌠

L

∑

ËL

ßËtË^∂x dL ≥ 0 13.57

The first summation in 13.57 is taken over all interfaces. Since the Ëth interface separates the phases å and ∫, ÎPË = ÎPå∫ = På - P∫ 13.58


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where we have taken the outward normal to the interface to point from ∫ to å. The second summation is taken over the junction lines at which the interfaces meet. The kernel of the integral is a sum over the interfaces that meet at the Lth junction line. The vector tË is a unit vector that lies in the Ëth interface perpendicular to the junction line. The vector ∂x is the displacement of the junction line normal to itself. Since the displacement, ∂xn, is arbitrary, the kernel of each of the integrals in the first term on the left-hand side of 13.57 must vanish everywhere. Hence, the condition of mechanical equilibrium at an interface is ÎP = –Kß 13.59 where ÎP is the pressure difference across the interface (the internal pressure minus the external pressure), –K is its mean curvature, and ß is its interfacial tension. When the (å∫) interface is a spherical shell surrounding phase å,

ÎPå∫ = På - P∫ = 2ßå∫

R 13.60

where R is the radius of the sphere. When the (å∫) interface is plane, På = P∫ 13.61 irrespective of the interfacial tension. The second integral on the left-hand side of 13.57 must vanish for arbitrary direc-tions of the vector displacement of the junction line at every point. Hence the kernel of the integral must vanish everywhere, and the condition of mechanical equilibrium on a junction line is ∑

Ë

ßËtË = 0 13.62

Equation 13.62 has a simple mechanical interpretation. The vector - tË is a unit vector tan-gent to the Ëth interface and perpendicular to the junction line. If the interface is imagined to be a membrane under a surface tension that tends to contract it, then the force the interface exerts on the junction line is - ßËtË, and equation 13.62 is a simple balance of the forces that are exerted on the junction line by the interfaces that meet there. Equation 13.62 also suggests why a junction line is almost invariably a junction of three phases. Whatever the relative values of ßË it is always possible to find a set of three vectors, tË, that will satisfy equation 13.62, which in this case is called the Neumann tri-angle of forces. Since a three-phase junction line in a fluid is mobile it is always possible for it to adjust its position so that the condition of equilibrium is satisfied. A four-phase junction line can also be constructed according to equation 13.62. However, a four-phase


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junction line is always free to decompose into two three-phase lines separated by a short segment of interface. Barring fortuitous degeneracy, this extra degree of freedom always permits the system to find a more favorable configuration. Four-phase junctions are almost never observed in real materials. 13.4.5 Equilibrium at a junction line on a solid: wetting The Neumann triangle, equation 13.62, governs the contact of three fluid phases along a junction line that can be freely displaced in any direction. A slightly modified equation applies in the case of the wetting of a solid surface by a fluid or deposited phase. The solid can usually be assumed rigid. Let its surface be plane, and wet by two fluid phases, ∫ and ©, while the solid phase is å. The junction line at which the (å∫), (å©) and (∫©) interfaces meet can only be displaced in the plane of the solid surface. The Neumann triangle then yields the equilibrium condition ßå© = ßå∫ + ß∫© cos(œ) 13.63 where œ is the contact angle, the asymptotic angle between the (∫©) and (å∫) interfaces. The contact angle is determined by the tensions of the three interfaces by the Young Equa-tion:

cos(œ) = ßå© - ßå∫

ß∫© 13.64

The value of cos(œ) must lie between -1 and 1. However, the value of the right-hand side of equation 13.64 is not so restricted. When the right-hand side is greater than, or equal to 1 the ∫ phase spreads over the å surface as a continuous film, and is said to wet the å surface. When the right-hand side is less than or equal to -1 the ∫ phase will not at-tach to the å surface, and is said to de-wet from å. 13.5 THE CONDITIONS OF GLOBAL EQUILIBRIUM An interface that is in equilibrium is an open system that is in thermal and chemical contact with the phases to either side of it. The global equilibrium of the interface is hence governed by the minima of the work function, „S. The condition of equilibrium is (∂„S)T{µ} = ∂(ßA)T{µ} ≥ 0 13.65 The condition that the work function of the surface be a minimum has three impor-tant corollaries. First, since A is positive definite, „S can be decreased indefinitely by in-creasing the surface area unless ß is also positive. It follows that the surface tension of an equilibrium interface is always positive. We asserted and used this result above. Second, for given surface area, A,


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(∂ß)T{µ} ≥ 0 13.66

The surface tension of an interface has the least value consistent with the given values of T and {µ}.

This important theorem has the consequence, among others, that the chemical and physical reconfigurations that occur spontaneously at an interface always lower its surface tension. For example, it is common to clean a surface before attempting to wet it with a fluid phase or deposit. Cleaning a surface always raises its tension, since it removes adsorbed species or reactants, and promotes wetting according to the Young equation. Third, for given ß, (∂A)T{µ} ≥ 0 13.67 and a fluid interface at equilibrium has a shape that minimizes its area. This theorem ex-plains why liquid droplets are spherical, why cylindrical or plate-shaped bodies tend to break up into spherical particles, and why distributions of small precipitates coarsen into larger ones. For solids with anisotropic tension the quantity minimized is the product ßA, which leads, for example, to the Gibbs-Wulff theorem for the equilibrium shape of a free solid surface (which we shall not discuss here). When a multiphase fluid is in equilibrium its temperature and chemical potentials are constant everywhere, the pressure is constant within each phase and the surface tension is constant on each interface. The pressures of phases in contact, their interfacial tensions, and the curvatures of the interfaces separating them are related by the condition of mechani-cal equilibrium. Since the pressures and tensions are constant, the curvatures are constant as well. The interfaces must either close on themselves or meet at three-phase junction lines that satisfy equation 13.62. These conditions are generally sufficient to specify the state and geometry of a multiphase system. 13.6 THE CRITICAL NUCLEUS Interfacial phenomena are most important when the surface-to-volume ratio is high, that is, when the system is a small particle or when it contains an aggregate of small parti-cles. The prototypic example of a small particle, which was studied by Gibbs, is the critical nucleus that initiates the transformation from a metastable to a stable phase. We shall treat this example first, and then consider aggregates of small particles such as distributions of precipitates or crystal grains. 13.6.1 The critical nucleus Consider a phase transformation that initiates in a metastable phase. Since the initial state is metastable with respect to infinitesimal perturbations the transformation must begin with a finite fluctuation that creates a small volume, or nucleus, of distinguishable material


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that will eventually become the product phase. The material within the microscopic nucleus need not be the same as that of the macroscopic phase that will eventually form, but must be distinctly different from that of the parent phase so that it sets the system on a evolutionary path that leads spontaneously toward the product phase. It must, therefore, be separable from the parent phase by an interface that has a positive tension. Let Ï be the thermodynamic potential that governs the equilibrium of the parent phase. If the nucleus forms through a finite perturbation of the parent phase that is arbi-trarily small in its physical dimensions then Ï must increase since the parent phase is stable. That Ï increases is also evident from the fact that the surface tension is positive and the surface-to-volume ratio increases without bound as the size of the nucleus is made arbi-trarily small. As the nucleus grows to large size, on the other hand, the potential Ï must ultimately decrease since the product phase is preferred. It follows that the potential passes through a maximum as the size of the nucleus increases. The state of the nucleus at the maximum of the potential is called the critical nucleus. The critical nucleus has the property that it grows spontaneously if its size is slightly increased, but shrinks and disappears if its size is decreased. The value of ÎÏ at the maximum is the nucleation barrier that opposes the phase transformation. The preferred state of the critical nucleus is a state of equilibrium with the parent phase. Since the change in Ï on formation of a particle of given size that is in equilibrium with the parent is necessarily less than that for a particle of the same size in a non-equilib-rium state, the least nucleation barrier is associated with an equilibrium state of the critical nucleus. Moreover, if we let the system be the small nucleus itself, the parent phase acts as a reservoir that fixes its temperature, pressure and chemical potentials. The thermodynamic potential that governs the equilibrium of the nucleus is the work function, „, whatever po-tential, Ï, governs the equilibrium of the parent phase. The value of „ for the critical nucleus is the work required to form it. The work of formation of the critical nucleus was obtained originally by Gibbs, who used the derivation given below. Gibbs' result may be the most often misquoted theo-rem in science. It is given incorrectly in virtually every text, treatise and technical paper that touches on the subject of nucleation (Landau and Lifshitz' text is an exception, as usual). 13.6.2 The work of formation of a critical nucleus Let a system contain a large ambient phase, å, that has a temperature, Tå, pressure, På, and chemical potential, µå

i , for each of its n components. Let there be a small, globular inhomogeneity in the interior of å that we shall take to be a spherical nucleus of phase ∫ of radius R. Since we shall use the Gibbs construction it does not matter whether the material within the nucleus is identical to the bulk phase ∫ or is simply something that will become phase ∫ when it grows to reasonable size. If it is reasonable to neglect terms of order (∂/R) where ∂ is the thickness of the interfacial shell, it does not even matter whether the nucleus is spherical in shape.


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The work to form the nucleus is minimum when the nucleus is in equilibrium with the ambient phase, å. The conditions for thermal and chemical equilibrium of the ∫ nucleus are T∫ = Tå 13.68 µ∫

i = µåi 13.69

where (i) is any component. The pressure, P∫, is determined by the fundamental equation for phase ∫ in the Gibbs-Duhem form: P∫ = ¡P∫(T,{µ}) 13.70 The condition for mechanical equilibrium of the spherical nucleus is

ÎP = P∫ - På = 2ßR

13.71

where ß is the tension of the å∫ interface. Let E', S' and {N'} be the total excess of energy, entropy and chemical content in and around the ∫ nucleus, that is, the excess over the amounts of these quantities that would be present if the volume about ∫ were uniformly filled with homogeneous å phase. Since the temperature and chemical potentials are fixed, the local equilibrium of the system is governed by the work function, „. The excess of the work function, „', is just the net work that would have to be done to create the heterogeneity that constitutes the nucleus of phase ∫: W = „' = E' - TS' - ∑

k

µkN'k 13.72

Using the Gibbs construction to define the excess quantities on the å∫ interface,

„' = „S + V∫ „∫V - „å

V 13.73

Given the definition of the surface tension, this relation yields the work of formation of a critical nucleus in the form W = „' = ßAå∫ - (P∫ - På)V∫ 13.74 The work required to form the critical nucleus is a physical quantity, „'. Hence the value of the right-hand side of the equation must be a physical quantity as well, independent of the Gibbs construction. It may not be obvious that this is true, since we do not know the precise state of the material the critical nucleus contains. However, the pressures På and P∫


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are determined by the fundamental equations of the two bulk phases, and the volume, V∫, and surface tension, ßå∫, are related to them and to one another by the condition of mechanical equilibrium, equation 13.71. The condition of mechanical equilibrium can be used to eliminate the spherical radius, R, from equation 13.74 with the result

W = 16πß3

3(ÎP)2 13.75

where ÎP = (P∫-På). Equation 13.75 defines the surface tension of the å∫ interface in terms of the work, W, and the pressure difference, ÎP. Alternatively, ß can be eliminated from the equation to give

W = 12 V∫(ÎP) 13.76

Equation 13.76 defines the volume, V∫, of the critical nucleus. 13.6.3 Comments on the work to form a critical nucleus The derivation of equations 13.74-76 shows that they are tautological relations rather than physical laws; the interfacial tension is defined to be whatever it needs to be to balance the equation. Nonetheless, they are often treated as physical laws in the theory of phase transformations. It is therefore useful to make a few clarifying comments concerning the nature and application of these equations before proceeding further. 1. While equation 13.75 defines the interfacial tension, ß, of the nucleus, it is nonetheless useful to treat it as an equation that determines the work, W. If the critical nu-cleus really is a mass that is approximately in the fluid phase ∫ then the value of ß can be estimated from the surface tension of a bulk ∫ phase in contact with å. It may also be pos-sible to compute the state of the critical nucleus from atomic models that estimate the value of ß directly. Research on model fluid systems suggests, in fact, that the surface tension of the microscopic nucleus is often reasonably close to that of an interface between the macro-scopic phases. 2. In the derivation of equation 13.74 it was never specifically assumed that the heterogeneity actually consists of homogeneous phase ∫, or that it is, physically, a sphere with an atomically sharp interface. The derivation is accurate whatever the internal consti-tution of the nucleus. Moreover, if the phases are thermodynamic fluids and the interface between them is drawn at the surface of tension, then ß is isotropic and is determined by the temperature and chemical potentials in the å phase. It follows that the critical nucleus in a transformation of fluid phases is spherical. In the case of a solid-solid transformation the nucleus may not be spherical, but the analysis is accurate to within a term of order (∂/R) whatever the nucleus shape. 3. The work to form the critical nucleus, W, is the increase in the value of the work function, „', of the subvolume that contains the nucleus. The work function, „, is the


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appropriate thermodynamic potential since the nucleus is an open system that is in contact with a large body of å phase that acts as a reservoir. Contrary to virtually all elementary texts on the theory of phase transformations, W is not the change in the Gibbs free energy of the nucleus. However, if we focus our attention on the whole system that contains the critical nucleus then W is equal to the change in its governing thermodynamic potential, whatever that potential may be. For example, let the parent å phase be in thermal and mechanical equilibrium with a reservoir across a plane, impermeable wall. The equilibrium states of the system as a whole are then determined by the extrema of its Gibbs free energy. When a critical nucleus of the ∫ phase forms in the interior the change in the Gibbs free energy of the whole system can be evaluated by treating it as a composite system that consists of an open volume with fixed boundaries that surrounds the nucleus and a bulk phase å. The change in the Gibbs free energy is, then, ∂G = ∂ET + ∂Er = ∂ET - T∂ST + På∂VT = E' + ∂Eå - TS' - T∂Så + På∂Vå

= E' - TS' + ∑k

µk∂Nåk = E' - TS' - ∑

k

µkN'k

= „' = W 13.77 The suffix (r) in the first term on the right-hand side designates the reservoir that surrounds the system. The symbols E' and S' and N' are the changes in energy, entropy and chemical content in the fixed volume surrounding the nucleus. The transition from the fourth to the fifth form of the right-hand side uses the fact that the system is surrounded by an im-permeable membrane. Equation 13.77 shows that the increase in the Gibbs free energy of the system is equal to the work done in forming the nucleus. It can be shown as directly that if å is surrounded by an impermeable, diathermal wall then W is the change in its total Helmholtz free energy. More generally, W is equal to the change in the value of whatever thermodynamic potential governs the equilibrium of å. The work W measures the quantitative deviation from equilibrium. 4. When the value of the surface tension, ß, is independent of the radius, R, then the equilibrium of the critical nucleus is always unstable, since it provides the maximum value of „'. This result follows immediately from equation 13.71, which cannot be satis-fied for R > Rc, the radius of the critical nucleus. If ß does vary with the nucleus size then there may be multiple solutions to equation 13.71. Each of these identifies an extremum of W that is a local equilibrium state. Only the extrema that correspond to maxima and minima of „ are of interest. Maxima and minima must alternate as R increases, and the first and last extrema must be maxima since „ is an


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increasing function R near R = 0, and since ß is independent of R for large R. Hence if there are multiple extrema (excluding inflection points) there must be at least three, and at least one of these provides a minimum of „. The local minimum identifies a metastable state for the nucleus; if ß varies with size so that there are multiple extrema it is possible to have small particles of ∫ that do not grow spontaneously. 5. It is common in materials science to treat solid phases as thermodynamic fluids and nucleated phase transformations as if the nucleus were also fluid. A solid phase be-haves as a fluid when it is subject to a hydrostatic pressure. However, the use of a fluid model to describe nucleation in solids may lead to an interfacial tension that has a complex behavior with particle size and has very little numerical relation to the bulk interfacial ten-sion. This is true for two reasons. First, a crystalline solid has an anisotropic surface tension. The ß that is used in the fluid model is an average whose value depends on the shape of the nucleus. A second and often more important consideration is that nuclei within the bulk of a solid phase are almost always coherent, in that their crystal lattices continue the lattice of the matrix in some simple way, and hence introduce an elastic strain whose value depends on the geometric mismatch between the crystal structures of the two phases and on the shape of the nuclear particle. The elastic strain in solid-solid transforma-tions is often the dominant contribution to the effective interfacial tension and has the con-sequence that the value of ß has no relation whatever to its average value on a bulk inter-face. As we shall see later the elastic effect can often be calculated to reasonable accuracy and separated out of the interfacial tension by treating the material as an elastic solid rather than a fluid. When this is not done, however, the elastic interaction is part of the interfacial tension, and one should be extremely cautious in using information obtained from bulk solids to infer the interfacial behavior of embedded solid particles. 13.7 APPROXIMATIONS FOR THE CRITICAL WORK To calculate the work to form a critical nucleus it is not only necessary to have an estimate for the interfacial tension, ß, but also for the pressure difference, ÎP = (P∫ - På), between the two phases at the ambient values of T and {µ}. Since thermodynamic mea-surements are usually made in vacuum or at atmospheric pressure the Gibbs-Duhem func-tion, ¡P(T,{µ}), is rarely known. The quantity that is easily measured is the change in the Gibbs free energy on transformation at constant temperature and pressure, Îg = ¡g∫(T,P,{x∫}) - ¡gå(T,P,{xå}) 13.78 where g is the molar density of the Gibbs free energy. It is usually possible to relate ÎP to this quantity. The relation is particularly simple in a one-component system. 13.7.1 The free energy approximation: one-component system


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In a one-component system g = µ, the chemical potential. The condition of chemi-cal equilibrium for the critical nucleus is µå(På) = µ∫(P∫) 13.79 where På is the ambient pressure and P∫ is the pressure within the nucleus. The free en-ergy change for the transformation from å to ∫ at ambient pressure (a quantity that is more easily measured) is Îµ(På) = µ∫(På) - µå(På) = µ∫(På) - µ∫(P∫)

= - ⌡⌠

På

P∫

∆µ∫

∆P dP = - ⌡⌠

På

P∫

v∫ dP 13.80

where v∫ is the molar volume of the ∫ phase. If ∫ is incompressible, as most condensed phases are to a good approximation, Îµ(På) « - v∫(ÎP) 13.81 which evaluates ÎP in terms of the free energy change on transformation at ambient pres-sure:

ÎP « - Îµ(På)

v∫ = - ÎGv(På) 13.82

where ÎGv(På) is the change in free energy per unit volume when the transformation is carried out at pressure På. Equation 13.82 is the free energy approximation. It provides a partial justification for the relation W = (ÎGv)V∫ + ßS 13.83 that is usually found in textbooks that treat phase transformations. [The Gibbs free energy per unit volume, Gv, is, in any case, a very awkward quantity to use since the Gibbs free energy pertains when the pressure, rather than the volume, is controlled. The appearance of an artificial quantity like ÎGv in eq. 13.83 should give an immediate hint that something is not quite right with this expression.] 13.7.2 The Thomson approximation for a one-component system A slightly different approximation is often useful when one of the phases involved in the reaction is a dilute vapor. Let å be a dilute vapor and ∫ be a condensed phase; the


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å“∫ transformation may, for example, be the condensation of liquid from the vapor. The condition of chemical equilibrium for a nucleus of ∫ in å is µå(På) = µ∫(P∫) 13.84 If P0 is the pressure at which å and ∫ are in equilibrium in bulk form then we also have µå(P0) = µ∫(P0) 13.85 Taking the difference between 13.84 and 13.85 and expressing each side of the equation by an integral yields

⌡⌠

P0

På

vå dP = ⌡⌠

P0

P∫

v∫ dP 13.86

Since å is a dilute gas,

vå « TP 13.87

while, since ∫ is a condensed phase, v∫ is approximately constant. Integration of 13.86 then gives

T ln

På

P0 = v∫(P∫ - P0) 13.88

Condensation from the vapor usually occurs under a relatively small supersaturation. Then P0 « På and

ÎP « Tv∫ ln

På

P0 13.89

The approximation 13.89 is called the Thomson approximation, after William Thomson (who later became Lord Kelvin). It has the significant advantage that it only re-quires measurement of the supersaturation ratio, På/P0. 13.7.3 The free energy approximation: multi-component system If the system contains n chemical components the conditions of chemical equilib-rium for the ∫ nucleus are

µ∫k(T,P∫,{x∫}) = µå

k(T,På,{xå}) 13.90


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While T is the same in the two phases, P and {x} generally are not. If the ∫ phase is nearly incompressible, however, its volume remains constant so long as its composition is fixed, and the pressure difference, ÎP, is related to the Gibbs free energy of ∫ at composition {x∫} by the approximation g∫(P∫,{x∫}) ~ g∫(På,{x∫}) + v∫(P∫ - På) 13.91 Using the relation

g = ∑k=1

n µkxk = µn + ∑

k

–µkxk 13.92

where the second sum is taken over the (n-1) independent components, equation 13.91 can be solved for ÎP as follows:

v∫ÎP = ∑k=1

n µ∫

k(P∫) - µ∫k(På) x∫

k = ∑k=1

n µå

k(På) - µ∫k(På) x∫

k

= ∑k=1

n

µå

kxåk + µå

k x∫

k - xåk - µ∫

kx∫k

= ¡gå(På,{xå}) - ¡g∫(På,{x∫}) + ∑k

–µåk x∫

k - xåk 13.93

where the relative chemical potential, –µå

k , is evaluated at the composition, {xå}, of the am-bient phase, and {x∫} is the composition of the nucleus of phase ∫. If the composition of the critical nucleus is known then equation 13.93 can be used to estimate ÎP. However, the composition of the nucleus is usually unknown and differs from the composition of the bulk phase ∫ that would be in equilibrium with å at the ambi-ent pressure, På. The composition of the ∫ nucleus that minimizes the work, W, is that which maximizes the pressure, ÎP. Differentiating equation 13.93 with respect to the composition of the ∫ phase shows that ÎP is maximized when

–µåk({xå}) - –µ∫

k({x∫}) =

∆v∫

∆x∫k

ÎP =

∆v∫

∆x∫k

2ßR 13.94

for k = 1,...,n-1, where the relative chemical potentials are evaluated at the ambient pressure, På. The set of equations 13.94 simplifies considerably in two common cases. First, let the molar volume of phase ∫ be independent of its composition. For example, let ∫ be a


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nearly pure component or let the phase separation occur on a fixed crystal lattice. Then the composition that maximizes ÎP is the solution to the relatively simple set of equations –µå

k({xå}) = –µ∫k({x∫}) 13.95

Since the relative chemical potential of a stable phase is an increasing function of its com-position, and since –µå

k must be greater than the value at which the å and ∫ phases are in equilibrium at pressure På (otherwise the ∫ phase would not nucleate), equation 13.95 has the consequence that the ∫ nucleus is richer in solute than the equilibrium ∫ phase. Second, let the molar volumes of the species that make up phase ∫ have fixed val-ues, which is a reasonably accurate assumption for almost all common condensed phases. Then

v∫ = ∑k=1

n vkx∫

k = ∑k

–vkx∫k + vn 13.96

where –vk is the relative molar volume of the kth specie, –vk = (vk - vn), and the composition of the ∫ nucleus is determined by the condition

–µåk({xå}) - –µ∫

k({x∫}) = 2ß–vk

R 13.97

Using the composition determined by equation 13.95 as a reference, the nucleus is rela-tively lean in solute when the volume of the solute is greater than that of the solvent, and is relatively rich in solute when the solute volume is less. A possible exception to equation 13.93 occurs when the molar free energy of the ∫ phase is less than that of å at the same pressure and composition: ¡g∫(På,{xå}) < ¡gå(På,{xå}) 13.98 It is then possible for the ∫ phase to form congruently, that is, without a change in com-position. While the work to form a congruent nucleus is higher than that for a nucleus whose state satisfies equation 13.93, the congruent nucleus requires no diffusion and is, hence, often preferred on kinetic grounds. 13.8 PRECIPITATION FROM A DILUTE SOLUTION Let å be a dilute binary solution of composition (x) and let ∫ be a solute-rich phase that precipitates out of å when it is quenched into the two-phase region of the binary phase diagram. If å and ∫ behave as fluids then the å∫ interface has an isotropic surface tension. The ∫ particle has the form of a sphere (or spherical cap) and an internal pressure


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P∫ = På + 2ßå∫

R 13.99

where R is its radius. The differential of the molar Gibbs free energy of a solution with respect to its pres-sure is

∆g

∆P T{x} = ∑

k

xk

∆µk

∆P T{x}

= ∑

k

vkxk = v 13.100

where v is the molar volume and vk is the partial molar volume of the kth component. Since the chemical potentials are the same in the two phases, µå

k(På,{xå}) = µ∫k(P∫,{x∫}) = µ∫

k(På,{x∫}) + vkÎP

= µ∫k(På,{x∫}) +

2vkßR 13.101

where we have assumed that the molar volume of the kth component does not depend on the pressure. If component (1) is dilute in phase å and concentrated in phase ∫ then the equilib-rium concentration of å when phase ∫ has the form of a sphere of radius R is determined by the equation

µd1(På) + T ln(xå) = µ0

1(På) + T ln(x∫) + 2v1ß

R 13.102

When the two phases are in equilibrium across a plane interface at pressure På the equilib-rium concentrations are determined by µd

1(På) + T ln(xåe ) = µ0

1(På) + T ln(x∫e ) 13.103

Equations 13.102 and 13.103 show that the concentration of (1) in a dilute solution that is in equilibrium with a particle of radius R is related to the equilibrium concentration by


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xå

xåe =

x∫

x∫e exp

2v1ß

RT ~ exp

2v1ß

RT 13.104

where we have assumed x∫ « x∫

e « 1. Equation 13.104 has three important applications. First, it can be used to find the critical radius, Rc, of a nucleus of ∫ that forms out of a supersaturated solution of concen-tration xå. Since Rc is the radius of a ∫ particle that is in equilibrium with the solution

Rc = 2v1ß

T ln(xå/xåe )

13.105

Since the concentration of a macroscopic phase, å, is not sensibly changed by the growth of a small, isolated particle of ∫, the equilibrium between ∫ and å is lost as soon as R > Rc. Equation 13.102 shows that µ∫

1 < µå1 for all larger radii. The critical nucleus is unstable

and the ∫ phase continues to grow if it is perturbed to a size R > Rc. However, since the formation of a single ∫ particle of size, R, does not change the concentration within a macroscopic solution it is possible (and is the usual case) that many such particles form simultaneously. Their growth depletes the matrix of solute, and their volume fraction changes with their radii as they grow in equilibrium with an increasingly lean matrix. The second application of equation 13.104 is to estimate the volume fraction of precipitate particles as a function of their mean size. Assuming that the precipitates have a uniform size, R, the equilibrium matrix concentration in equilibrium with them is xå(R), and the equilibrium mole fraction of precipitate particles of size R is

f∫ ~ N∫

N0 = xå - xå(R) = xå

1 - xå(R)

xå

= xå

1 - exp

2v1ß

T

1

R - 1Rc

13.106

where N∫ is the aggregate number of atoms in the ∫ precipitates, N0 is the number in the whole system, and Rc is the radius of a ∫ particle that is equilibrium with the parent å solu-tion of composition xå. In most cases equation 13.106 shows that the volume fraction of ∫ approaches its equilibrium value at a very early stage in precipitate growth. The third application of equation 13.104 is to determine the rate of coarsening of a distribution of precipitates of unlike size. If the precipitate particles were uniform in size and in equilibrium with the surrounding matrix then the particles would cease to grow. However, the equilibrium of each of the particles is unstable; since xå(R) is a decreasing function of R a small perturbation that increases the size of any one of the particles causes it to grow further. In fact, in all real cases the precipitate particles have a distribution of sizes that reflects their different nucleation times and growth histories. If the particles can grow


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easily, that is, if there is no significant kinetic barrier to growth or dissolution, then each of the particles is in local equilibrium with a matrix of composition xå(R), where R is its radius. Since xå(R) is a decreasing function of R a distribution of equilibrium precipitates of different sizes creates concentration gradients in the matrix in which solute diffuses from the vicinity of particles of small size to particles of large size. This causes the larger particles to grow at the expense of the smaller ones, and leads to a general coarsening of the precipitate distribution. To compute the kinetics of coarsening it is necessary to incorporate the kinetics of diffusion, which we shall defer to a later point. 13.9 HETEROGENEOUS NUCLEATION The nuclei that initiate phase transformations in condensed phase often appear pref-erentially at free surfaces, internal boundaries and foreign particles. It is obvious that nu-cleation should usually be catalyzed by internal surfaces. Since an interface necessarily has a positive tension, the formation of a nucleus at an interface eliminates a portion of it, and hence regains some of the work that is expended in forming the critical nucleus. Let a nucleus of phase ∫ form out of an ambient å phase on a rigid surface of phase ©. The å ambient fixes the temperature, T, and chemical potentials, {µ}. Hence the pres-sure within the ∫ nucleus is P∫ = ¡P∫(T,{µ}) 13.107 If the å and ∫ phases are fluids with interfacial tension, ßå∫, the å∫ interface must be spherical with radius

R = 2ßå∫

ÎP 13.108

At the three-phase junction line the å, ∫ and © interfaces form the equilibrium contact angle that is determined by the Young equation:

cos(œ) = ßå© - ß∫©

ßå∫ 13.109

Equations 13.107-109, together with the geometry of the © surface on which the contact line must lie, determine the equilibrium shape of the nucleus. The work to form the critical nucleus is W = Î„ = - (ÎP)V∫ + (ß∫© - ßå©)S∫© + ßå∫Så∫ = - (ÎP)V∫ + ßå∫[Så∫ - cos(œ)S∫©] 13.110


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Whatever the shape of the solid surface, V∫ varies as R3 and Så∫ and S∫© vary as R2. Whenever the geometry of the solid surface is uniform we can, therefore, write

V∫ = 43 πR3v©(œ) 13.111

Så∫ - cos(œ)S∫© = 4πR2s©(œ) 13.112 where the functions v©(œ) and s©(œ) depend on the contact angle and the precise geometry of the © interface. Using equation 13.108 we then have

W = 16π(ßå∫)3

3(ÎP)2 [ ]3s©(œ) - 2v©(œ) = WHf©(œ) 13.113

where

WH = 16π(ßå∫)3

3(ÎP)2 13.114

is the work to form a homogeneous nucleus separated from the boundary, and f©(œ) = 3s©(œ) - 2v©(œ) 13.115 is the catalytic factor for the © interface. The catalytic factor, f©(œ) ≤ 1, since the nucleus is always free to form away from the boundary. It depends on both the equilibrium contact angle and the geometry of the © interface, and measures the fractional reduction in the work to form an equilibrium nucleus when heterogeneous nucleation is possible. When the solid substrate, ©, has a flat, rigid interface

f©(œ) = 14[ ]2 - 3 cos(œ) + cos3(œ) 13.116

The work, W, then increases monotonically from the value 0 at œ = 0, which corresponds to complete wetting, to the value WH at œ = π, which corresponds to complete de-wetting. It follows that the interface virtually always catalyzes nucleation. The only exception is when the substrate is completely de-wet by the ∫ phase in which case the substrate has no effect on nucleation. The case of complete wetting (œ = 0) requires further comment. While cos(œ) is confined to the range |cos(œ)| ≤ 1, the right-hand side of equation 13.109 can have any value. It follows that W may be less than zero in the case of complete wetting, and is when ßå∫ + ß∫© < ßå© 13.117


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In this case the system prefers a ∫ film on the å surface, even when the å phase is stable. It is difficult to predict the range of conditions over which the ∫ film will continue to exist, since this depends on the thickness of the film and the change in the effective surface ten-sion f the film as the film thickness decreases. However, it necessarily follows that a film of ∫ can form under conditions where the å phase is stable in the bulk. 13.10 THE PRESERVATION OF A PHASE IN A CAVITY When a phase forms a film at an interface and equation 13.117 is satisfied the phase can be preserved under conditions that lie outside its normal equilibrium range. More gen-erally, almost any phase can be preserved outside its equilibrium range in suitable cavities in a solid surface. As an illustrative example, let a phase, ∫, that is equilibrium with an ambient phase, å, in a one-component system have the form of a droplet on a rigid, solid surface that contains a cylindrical cavity. Let ∫ be the low-temperature phase, and let it be preferred at T < T0. First assume T < T0. At equilibrium the droplet of phase ∫ has a positive radius that is given by

R = 2ßå∫

ÎP = - 2ßå∫

ÎGv 13.118

where we have used the free energy approximation on the assumption that ∫ is incompress-ible. The droplet has contact angle œ along its periphery, where we assume 0 < œ < π. Let the ∫ droplet just fill a cylindrical cavity of radius, r, on the solid surface. When r < R, the equilibrium radius of the droplet, it is always possible to satisfy the conditions of equilibrium at the top of the cavity since a spherical cap of radius R > r can always cover the top of the cavity and since any contact angle can be established at the lip of the cavity. Now let the temperature be increased gradually from T < T0. As the temperature increases the radius of curvature of the å∫ interface increases; the interface is flat at T = T0. When T > T0 the interface has negative curvature. The interface bows into the cavity and its center of curvature is in the å phase. However, barring finite fluctuations the droplet remains stable for some range of T > T0 since it is pinned by the cavity lip. It cannot pene-trate the cavity until it can achieve an equilibrium configuration that permits motion of the line of contact down the cylindrical cavity wall, that is, until it takes on a sufficiently nega-tive curvature that the equilibrium contact angle can be established on the inner wall of the cavity. Given a cylindrical cavity of radius, r, equilibrium of the junction line on the inner cavity wall requires that the radius of the drop, R, and the contact angle, œ, satisfy the rela-tion R cos(œ) = - r 13.119


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where the negative sign is due to the fact that R is negative (center of curvature in the å phase). From equation 13.118, this equation is satisfied for an incompressible phase, ∫, when

ÎGv = 2ßå∫cos(œ)

r 13.120

The solution to this equation requires a positive value of ÎGv, that is, the å phase is pre-ferred to ∫. If the latent heat of the å “ ∫ transformation is approximately constant, ÎGv ~ ÎSv(T0 - T) = - |ÎSv|(T0 - T) 13.121 since ÎSv is negative for a transformation from a low-temperature phase to a high-tempera-ture one. The temperature, T, at which the å phase penetrates the cavity and eliminates the residual ∫ phase is

T = T0 + 2ßå∫cos(œ)

|ÎSv|r 13.122

If the cavity is small the second term on the right may have an appreciable magnitude; ∫ is preserved to temperatures well above the equilibrium transformation temperature. The behavior described above is one likely source of "memory" in phase transfor-mations: the common observation that a transformation å “ ∫ becomes easier to accom-plish if the system is first cycled through the transformation å “ ∫ “ å. If phase ∫ can be trapped within cavities in the walls of the system or in inclusions within the system the transformation is easily reversed since the ∫ phase is already present and need not be nu-cleated; the temperature need only be lowered into the ∫ field far enough to liberate ∫ phase from the cavities. Consistent with this picture it is often found that the "memory" of a cy-cled phase transformation fades. Memory is lost if the system is held in the å field for a sufficiently long time, which increases the probability that the residual ∫ will be eliminated by fluctuations or reaction with the substrate. Memory is also lost if the å phase is heated to a temperature well above T0, which eliminates the ∫ phase within small pores according to equation 13.122. 13.11 A SIMPLE POLYGRANULAR SOLID Most crystalline solids are polygranular; they consist of individual crystal grains that are separated by grain boundaries that are interfaces between gains that differ in the ori-entation of the crystal axes. While the tension of a grain boundary depends on its orienta-tion, most of the interesting features of the behavior of polygranular solids can be studied in a simpler model in which the solid is treated as a thermodynamic fluid. In this case the solid is under hydrostatic pressure and its surfaces have isotropic tension.


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13.11.1 The equilibrium microstructure A simple solid does not have a microstructure in its most stable state. Introducing grain boundaries into a simple solid that is otherwise in equilibrium adds a net positive work Î„ = ßbS ≥ 0 13.123 where ßb is the tension of the boundary and S is the total boundary area. However, it is possible to design a microstructure that satisfies the local conditions of equilibrium every-where. This equilibrium microstructure is sometimes useful in understanding the behavior of real polygranular solids. Since the grains of a polygranular solid are in thermal and chemical equilibrium with one another the pressure is constant throughout (ignoring external fields). The condi-tion of mechanical equilibrium at the grain boundaries then requires that the mean curvature be zero everywhere: –K = 0 13.124 Moreover, where three grain boundaries meet at a junction the Neumann triangle of forces requires that ∑

k

ßbtk = 0 13.125

Since ßb is constant this equation has the consequence that the three surfaces make angles of 120º with one another. Equation 13.125 is satisfied by a hypothetical solid whose microstructure has only plane grain boundaries. Unfortunately, there is no way of filling space with plane-surfaces grains (polyhedra) whose interfaces always meet at 120º at junction lines. However, as first noticed by William Thomson (later Lord Kelvin) the conditions of equilibrium are ap-proximately satisfied when the grains have the shape of fourteen-sided polyhedra known as "Thomson tetrakaidecahedra." The Thomson tetrakaidecahedron can be most easily visual-ized by recognizing that it is the Wigner-Seitz cell of the BCC structure (or, equivalently, the first Brillouin zone of the FCC structure). To construct it imagine a body-centered cubic lattice. Selecting a central lattice point, draw planes that bisect the lines that connect the central point with all of its neighbors. The inner envelope of these planes is the four-teen-sided Thomson tetrakaidecahedron. The surfaces of the Thomson tetrakaidecahedron do not all meet at exactly 120º, but it is not necessary that a surface be plane for its mean curvature to vanish. The surface can also be slightly puckered into a saddle shape so that its two radii of curvature are equal and opposite and –K = 0. It has been shown experimentally with soap films (though, to my


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knowledge, it has never been shown analytically) that the surfaces of a microstructure of Thomson tetrakaidecahedra can distort slightly so that they meet at 120º while maintaining zero mean curvature. The distorted tetrakaidecahedral microstructure is, therefore, an example of an equi-librium microstructure, and is the only one that has ever been identified. Its theoretical value is limited since it is unstable with respect to perturbations (such as the disappearance of a grain) and since it is not the obvious end point to the evolution of any plausible real microstructure. However, it does provide a well-defined model system that has been widely used in studies of microstructural evolution. The geometry of the equilibrium mi-crostructure is responsible for the fact that those who study microstructure theoretically of-ten assume that the grains are arranged in a BCC pattern. 13.11.2 Grain coarsening in the polygranular microstructure The conditions of equilibrium for the polygranular microstructure in a fluid-like solid require homogeneous temperature and chemical potential, which ensures a uniform pressure. The mean curvature is zero on every grain boundary and the boundaries meet at 120º at every junction line. If the microstructure is not the equilibrium microstructure de-fined above then it is not possible to meet all of these conditions simultaneously. The mi-crostructure will necessarily adjust itself until the conditions of equilibrium are satisfied. There are only two possible ending points for the evolution of the microstructure: the equi-librium microstructure and the single crystal. Both intuition and experiment suggest that the equilibrium microstructure is an unlikely result (to take a familiar example, consider the evolution of the foam in a glass of beer, which has the same equilibrium conditions). The normal evolution of the microstructure is toward a single grain, and the mechanism of evolution is the growth of the more favored grains with respect to the less favored. A grain grows through the motion of its boundaries. We can gain some insight into the direction of coarsening by setting up a logical hierarchy in the kinetic order in which the various equilibrium conditions are satisfied. The kinetic order in the usual case begins with mechanical equilibrium, which can be established by small local displacements of the atoms and hence has a rate of the order of the speed of sound. The second equilibrium is thermal, which is governed by the rate of heat conduction. The third equilibrium is chemical, which is achieved at a rate governed by the atomic diffusivity. If we include interfaces and junction lines the problem is more subtle. If an inter-face separates two grains of the same phase that are not in equilibrium with one another then the state of the interface is not, strictly, determined by the fundamental equation for the interface that was developed in the first part of this chapter. The problem is particularly clear in the case of a one-component system whose grains are separated by a curved inter-face. Mechanical equilibrium requires that the pressures be different on the two sides of the interface, but have uniform values in each of the grains. But if the material within each gain is in local equilibrium then the Gibbs-Duhem equation requires that dµ = vdP 13.126


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It follows that the chemical potential is constant within each grain, but is discontinuous across any curved interface. The chemical potential on the interface is, hence, undefined in the Gibbs model. The fundamental equation for the surface ß = ¡ß(T,µ) 13.127 is either not obeyed, or is obeyed with a chemical potential that is, a priori, unknown. In either case the surface tension, ß, is an undetermined quantity. If the kinetic hierarchy discussed above is correct then the appropriate thermody-namic model is one in which each of the grains is internally in equilibrium, but is out of chemical communication with its neighbors. The conditions of thermal and mechanical equilibrium are satisfied at the boundary, but the condition of chemical equilibrium is not. The condition of mechanical equilibrium at the interface ÎP = –Kß 13.128 is obeyed with ß whatever value it must be to satisfy the equation. To see the meaning of this consider a membrane such as a soap film or balloon sur-rounding a gas. As the internal pressure is changed the membrane stretches or contracts, changing its tension, to satisfy the conditions of mechanical equilibrium. If the interfacial shell has constant chemistry it can achieve this result by stretching or contracting the inter-atomic bonds in the interfacial layer. (This possibility has led to a distinction between the interfacial "stress" and the interfacial "tension". The distinction is meaningful, but intro-duces a whole series of subtleties, even for a continuous interface in a simple fluid. Since the situation considered here involves non-equilibrium interfaces and the interfacial tension is an equilibrium property, the distinction is not meaningful.) If we assume mechanical equilibrium and return to the derivation of the general conditions of equilibrium for a multiphase system, the only one of the first-order conditions of equilibrium that is violated is the condition of chemical equilibrium for changes that oc-cur with the interfaces fixed in position. Consider two grains, labeled 1 and 2, that are separated by the 12 interface. The variation of the energy at constant entropy, volume and chemical content is (∂E)S,V,N = ∂(E - TS - µN) = ∂„T,µ = µ1∂N1 + µ2∂N2 + µ12∂N12 - µ(∂N1 + ∂N2 + ∂N12) = µ1∂N1 + µ2∂N2 + µ12∂N12 13.129 Neglecting the material adsorbed on the interface and assuming incompressibility, ∂„ = ∂N1(µ1 - µ2) = ∂N1[v(P1 - P2)]


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= dN1(v–Kß) 13.130 Equation 13.130 is the driving force for grain coarsening. It asserts that material will be transferred from the grain with the higher pressure to that with the lower. While the value of the interfacial tension, ß, is not necessarily the value that is in equilibrium with the state of either grain, it is necessarily positive, and has the consequence that the interface will de displaced toward its center of curvature. Thus, grains grow at concave interfaces an shrink at convex interfaces. If we make the further assumption that ß deviates only slightly from its equilibrium value then mechanical equilibrium requires that the interfaces meet at 120º at the three-phase junction lines, since equilibrium at the junction line involves only the interfacial tension (this equilibrium need only involve the differential elements of surface at the junction line). The Thomson equilibrium microstructure shows that it is possible for a grain to have zero mean curvature on all of its interfaces, consistent with equilibrium on the junction lines, when the grain is fourteen-sided. It follows that a grain with less than fourteen facets will tend to have convex interfaces while one with more than fourteen facets will tend to have interfaces that are concave. Hence grains with more than fourteen facets tend to grow at the expense of those with less than fourteen facets. These conditions can be made precise in two dimensions. If a two-dimensional grain has constant surface tension and 120º junction lines then its boundaries must be con-vex (concave) if the number of sides is less than (greater than) six. Hence a two-dimen-sional grain shrinks (grows) if it has less than (more than) six sides. 13.11.3 Heterogeneous nucleation in a simple polygranular solid A polygranular microstructure offers three kinds of heterogeneous sites on which heterogeneous nuclei may form: grain boundaries, three-grain junction lines, and nodal points at which four junction lines meet. When phase ∫ forms at any of these sites it still must satisfy the conditions of mechanical equilibrium; hence its surfaces are spherical caps that meet in a configuration that satisfies the Neumann triangle. When the ∫ nucleus forms on a grain boundary, for example, it has the form of a double lens whose periphery has the contact angle ßåå = 2ßå∫cos(œ) 13.131 where ßåå is the grain boundary tension and œ is the asymptotic angle between the å∫ in-terface and the plane of the grain boundary. The work to form critical nuclei in each of the possible configurations in the poly-granular solid, grain boundaries, junction lines and nodal points, was worked out by Clemm and Fisher (Acta. Met., 3, 70, 1955). The geometric relations are rather compli-cated. We shall discuss only the results. In all cases the work to form the critical nucleus can be written


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Wi = WHfi(œ) 13.132 where Wi is the work to form a nucleus on a heterogeneous site of type (i), WH is the work to form a homogeneous nucleus, and fi(œ) is the catalytic factor for sites of type (i). The various heterogeneous sites can be labeled by their dimension, i = 2 for grain boundaries, 1 for junction lines and 0 for nodal points. For given œ the function fi(œ) decreases in value with (i), so the new phase always forms most easily on nodes, then junction lines, then grain boundaries. In each case there is a critical value of cos(œ), or, equivalently, of the ratio between the grain boundary and interphase tensions, ßåå/ßå∫, beyond which Wi = 0 and the ∫ phase can exist at or beyond its equilibrium range. These ratios can be found by simple geometry. For phase ∫ to be stable on a grain boundary, a film of ∫ on the grain boundary must decrease the net interfacial tension. This is only possible when

ßå∫ = ßåå

2 13.133

For a film of ∫ to form along a three-phase junction line it must be preferred when it has the form of a ribbon of triangular cross-section along the junction line. This is true when œ ≤ π/3, or ßå∫ ≤ 3 ßåå 13.134 For ∫ to be stable as an isolated particle at a four-grain node it must be preferred when it has the shape of a tetrahedron with plane sides that is centered on the node. Hence

ßå∫ ≤ 223 ßåå 13.135

The conditions for retention of a phase beyond its normal region of preference are most easily satisfied at the four-grain node, equation 10.135. A low-temperature phase, ∫, will be preserved at temperatures above its transformation temperature whenever equation 10.135 is obeyed, and, if the left-hand side of 10.135 is much less than the right-hand side, may be preserved well into the å stability field, leading to a strong memory effect in the re-transformation of å to ∫. The surface tension of a high-entropy phase is usually lower than that of a phase of the same substance with lower entropy; the low-entropy phase is more highly ordered, and is hence less able to adjust its physical state at the interface to accommodate the physical discontinuity there. Hence the phases found along the interfaces in polygranular micro-structures are usually high-temperature phases or liquids.


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A particularly important case of wetting within a polygranular microstructure occurs when equation 10.133 is satisfied for a liquid phase. If the liquid forms a continuous film along the grain boundaries the solid loses its integrity and falls apart in an intergranular mode. A classic example of this behavior is the tendency of many solids to rupture under load at temperatures close to the melting point because of premature melting along the grain boundaries. Another example is the phenomenon of liquid metal embrittlement, in which a low-melting metal wets the grain boundaries of a solid so that it falls apart. Liquid metal embrittlement is perhaps most striking in the catastrophic effect of gallium and mercury on aluminum. Both cause a dramatic loss of cohesion in Al due to the formation of liquid phases that wet the grain boundaries.

chapter 13: surfaces - materials science · chapter 13: surfaces ... 13.8 precipitation from a...

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