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Lecture 7

DiffusionRef:

1. WD Callister Jr. Materials Science and Engineering: An Introduction, John Wiley & Sons.

2. DA Askeland. The Science and Engineering of Materials, Chapman & Hall.

A.K.M.B. RashidDepartment of MME, BUET

Introduction

Mathematical Description of Diffusion

Diffusion in Solids

Factors that Influence Diffusion

Diffusion Mechanisms

Diffusion and Materials Processing

Diffusion in Ionic Compounds and Polymers

Today’s Topics ...

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The phenomenon of transport of mass through material by atomic motion is called diffusion.

What is Diffusion?

Mass transport in gasses and liquids occurs by a combination of convection (fluid motion) and diffusion.

In solids, convection does not occur, and diffusion is the only available mass transport mechanism.

Solid-state diffusion is relatively slower than liquid-state diffusion

Introduction

Important materials processes occur by diffusion:

Case hardening of steel

Doping of semiconductors

Oxidation of metals

Solid-state formation of compounds

Sintering of powders to form dense and strong objects

Diffusion bonding of two solids

Applications of Diffusion

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Steel gear

case hardened to improve hardness and resistance to fatigue

diffusing excess carbon or nitrogen into outer surface layer

Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.)

Result:The presence of C (or, H) atoms makes iron (steel) harder

Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion

Heat-treatment almost always involve atomic diffusion

desired results depends on diffusion rate

Phenomenological approach:

Here we ask:

How can we describe the rate and the amount of mass transport that occurs in terms of parameters we can measure ?

This approach is important to our ability to control such processes as carburising, nitriding, tempering, homogenising of casting, and the like.

Atomic approach:

Here we ask:

What is the atomic mechanism by which atoms move?

This approach is important to our understanding of how diffusion mechanisms affect such processes as precipitation hardening.

Approach in Studying Diffusion

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Mathematical Description of Diffusion

Consider the unidirectional diffusion of carbon from 1wt% C steel rod to a pure iron rod, which is butt welded each other to form a “diffusion couple”and heated to 700 oC to allow diffusion at a significant rate.

After some time at 700 oC, the couple is quenched to room temperature, and analyse the carbon content along the rod.

What is the rate at which carbon atoms move to the right?X

C=

Wt.

Fra

ctio

n C

arb

on

Position, Z

0.01 t = 0

t = t

t =

0

Fe + 1wt% C Pure Fe

Iron-Steel Diffusion Couple

The composition profile might look as shown by the curve labeled t = t.

After t = ∞, the composition will become constant at an average value.

D : diffusion coefficient for diffusing species in solid, distance2

time

minus sign denotes the flux component 1 is towards lower concentrations, i.e. “down the concentration gradient”

: concentration gradient,

C is either mass density or atom density

dCdZ

or, number/volumedistance

mass/volumedistance

J : diffusion flux , or rate of diffusion number of atomsarea-time

massarea-time

or,

J =1

A

dM

dtM = mass/numberA = area

dC1

dZJ1 = - D1

According to the Fick’s First Law, the flux of atoms, J, is proportional to the volume concentration gradient, dC/dZ, i.e.,

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Whenever a concentration gradient is present in metal, a diffusion flux will occur.

Our problem now is:How do we actually determine D ?

One cannot measure J or D directly.

We can only measure composition as a function of Z and t.

Concentration gradient (dC/dZ) varies with both position and time, and so does the flux.

Therefore, we must determine a differential equation for the diffusion process,

by performing a mass balance upon a differential volume element perpendicular to the mass flow direction.

dZ

Jin Jout

1 2

Mass balance during carbon transport,

Mass In – Mass Out = Mass Accumulation

Rate mass in = All mass comes into the volume element through plane 1= Flux at 1 x Area at 1 = (JA)1

Rate mass out = All mass comes out the volume element through plane 2= Rate at 1 + change in rate across the volume element= (JA)1 + [ (JA)/Z ] dZ

Rate accumulation = change in volume concentration in the volumeelement with time = (C . A dZ) / t = A dZ (C/t)

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JA - JA + dZ = A dZ (JA)Z

Ct

Then,

Which reduces to:

JZ

- = Ct

This is known as the Continuity Equation.

Note that, our treatment assumed that, mass transport occurred in only one direction.

The equation holds for all material flow processes when no material is generated within the volume element, for example, flow of heat, neutrons, electrons, etc.

For one-dimensional mass diffusion process, using Fick’s first law:

=Ct

Z

CZ

D

This is known as the Fick’s Second Law of diffusion.

This is a partial differential equation with C as the dependant variable

and Z and t as the two independent variables.

A solution to this equation will give C as a function of Z, t and D.

dC1

dZJ1 = - D1

JZ

- =Ct

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Steady-State Diffusion

Diffusion is a time-dependent process, where the quantity of element transported is a function of time

Steady-state diffusion occurs when the rate of diffusion does not change with time

Diffusion in Solids

gas at pressure PA

gas at pressure PB

direction of diffusion of

gaseous speciesPA > PBand constant

thin metal plate

area, A

Steady-state diffusion across a thin plate

CB

CA

xA xB

Position, x

Con

cent

ratio

n of

di

ffusi

ng s

peci

es, C

Linear concentration profile for stead-state diffusion

J = - DdC

dx

DC

Dx

CB - CA

xB - xA

=dC

dx≅

Example: Steady-state diffusion

A steel plate is exposed to a carburising (C-rich) atmosphere on one side and a decarburising (C deficient) atmosphere on the other side at 700 C.

If a condition of steady state is achieved, calculate the diffusion flux of C through the plate if the concentration of C at positions of 5 and 10 mm beneath the carburisingsurface are 1.2 and 0.8 kg/m3, respectively.

Assume a diffusion coefficient of 3x10–11 m2/s at this temperature.

x1 x2

c1c2

carbonrichgas

carbondeficient

gas

steady state = straight line Given data:C1 = 1.2 kg/m3 X

1 = 5 mmC2 = 0.8 kg/m3 X

2 = 10 mmD = 3x10-11 m2/s

C2 – C1

x2 – x1

J = - D

= 2.4 x 10–9 kg/m2s

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Most practical diffusion situations are non-steady

For non-steady state diffusion

diffusion flux and concentration flux vary with time and at different points of solid

net accumulation or depletion of the diffusing species resulted

Because diffusion in solids is slow, diffusion is almost always transient!

Nonsteady-State Diffusion

Distance

Co

nce

ntr

atio

n o

f d

iffu

sio

n s

pec

ies

t 0

t 1 < t2 < t 3

C

t= D

2C

x2

Non-steady state diffusion is described by the Fick’s second law

Concentration profile, C(x), changes with time

C

t= D

2C

x2

Solution to this differential equation for a semi-infinite solid with constant surface concentration can be done

Cs – C0Cx – C0

C0

Cx

Cs

Distance from interface, x

Co

nce

ntr

atio

n, C

assuming that

1. Initial concentration C0

2. X = 0 at the surface, and increases with distance into the solid

3. At the initial time, t = 0

with the boundary conditions that

1. For t = 0, C = C0 at 0 ≤ x ≤ ∞

2. For t > 0, C = CS (constant surface concentration) at x = 0

3. For t > 0, C = C0 at x = ∞

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General solution to this differential equation

Cx – C0

Cs – C0

= 1 - erfx

2(Dt)

Cx is a function of dimensionless parameter x / (Dt)

erf ( ) : Gaussian error function is defined by

erf (z) = e dy-y22

π

z

0

where x /(Dt) has been replaced by the variable z.

The Error Function

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

Z

erf

(Z

)

z erf (z) z erf (z)

0.00 0.0000 0.70 0.6778

0.01 0.0113 0.75 0.7112

0.02 0.0226 0.80 0.7421

0.03 0.0338 0.85 0.7707

0.04 0.0451 0.90 0.7969

0.05 0.0564 0.95 0.8209

0.10 0.1125 1.00 0.8427

0.15 0.1680 1.10 0.8802

0.20 0.2227 1.20 0.9103

0.25 0.2763 1.30 0.9340

0.30 0.3286 1.40 0.9523

0.35 0.3794 1.50 0.9661

0.40 0.4284 1.60 0.9763

0.45 0.4755 1.70 0.9838

0.50 0.5205 1.80 0.9891

0.55 0.5633 1.90 0.9928

0.60 0.6039 2.00 0.9953

0.65 0.6420

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Example:A steel has a uniform C concentration of 0.25 wt.%. It is carburised at 950 oC using a media having a C content of 1.2 wt.%. How long will it take to achieve a C content of 0.80 wt.% at a position of 0.5 mm below the surface?

0.421 = erf62.5t

Using table, and after interpolation

z = 0.392 = 62.5t

t = 25400 s

C0 = 0.25 % CCs = 1.20 % CCx = 0.80 % Cx = 0.5 mmD = 1.6x10-11 m2/s

Cx – C0

Cs – C0

= 1 - erfx

2(Dt)

Factors that Influence Diffusion

1. Diffusing Species

Magnitude of diffusion coefficient D indicative of the rate at which atoms diffuse

D depends on both the diffusing species as well as the host atomic structure

Depending on this, two categories of diffusion can be recognised:

1. Self-diffusion

2. Inter-diffusion

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Movement of atoms through their own lattice

Occurs in elemental solids, or pure metals

A

B

C

D

B

C

D

Initially After some time

Self-diffusion

Observe the position of labeled atoms after diffusion

A

Atoms of one metal diffuse into another in response to a concentration gradient, resulting a net drift of atoms from higher to lower concentrations (formation of alloy region)

Inter-diffusion

Faster than self-diffusion

Concentration Profile

100 %

0 %

Cu Ni

Initially

Concentration Profile

100 %

0 %

After some time Example: movement of Ni atoms through the

lattice of Cu atoms

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2. Temperature

Has the most profound influence on the coefficients and diffusion rate

Example: Fe in a-Fe

@ 500oC D = 3.0 E(-21) m2/s@ 900oC D = 1.8 E(-15) m2/s approximately six orders higher

Diffusivity increases with T:

D = D0 exp -Qd

RT

D0 = T independent pre-exponentialQd = activation energy [J/mol], [eV/mol]

D = D0 exp -Qd

RT

Dinterstitial >> Dsubstitutional

C in a-Fe Cu in CuC in g-Fe Al in Al

Fe in a-FeFe in g-FeZn in Cu

D has exponential dependence on T

ln D = ln D0 –Qd

R

1

T

log D = log D0 –Qd

2.3R

1

T

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log D = log D0 –Qd

2.3R

1

T

Qd = – 2.3 Rlog D1 – log D2

1/T1 – 1/T2

Thus, knowing two diffusivity data at two different temperatures, the activation energy for the diffusing atom can be calculated.

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3. Role of Microstructure

Diffusivity of atoms depends on the diffusion path.

In general, the diffusivity is greater through the less restricted structural regions

[1] grain boundary[2] dislocation cores[3] external surface

Surface DiffusionActivation energy for diffusion is the lowest, since there are no atoms above the atom of interest; fewer neighbours, fewer bond breaking

Since grain boundaries are relatively more open structure compared to atomic structures inside the grain, the barrier through grain boundary is much less

Grain boundary Diffusion

Pipe DiffusionSince it feels like movement of atoms through a pipe

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Diffusion Mechanisms

Atoms in solids are in constant motion rapidly changing positions.

Diffusion is just the stepwise migration of atoms from one lattice site to other lattice site.

Two conditions for movement:1. There must be an empty adjacent site2. Atom must have sufficient energy to break bonds with neighbour atoms

Atomic vibration Every atom vibrates very rapidly about its lattice position within crystal

At any instant, not all vibrate with same frequency and amplitude

Not all atoms have same energy

Same atom may have different level of energy at different time

Energy increases with temperature

To jump from lattice site to lattice site, atoms need activation energy to break bonds with neighbours, and to cause necessary lattice distortion.

This energy come from the thermal energy of atomic vibration (Qm ~ kT)

The average thermal energy of an atom (kBT = 0.026 eV at RT) is usually much smaller than the activation energy (Qm ~ 1 eV/atom).

So a large fluctuation of energy (when the energy is “pooled together” in a small volume) is needed for a jump.

The probability of such fluctuation, or the frequency of jumps, Rj, depends exponentially on T and defined as

Rj = R0 exp -Qm

kBT

R0 = attempt frequency, proportional to the frequency of atomic vibrationFigure 5.12 A high energy is required to squeeze atoms past one another

during diffusion. This energy is the activation energy Q. Generally more energy is required for a substitutional atom than for an interstitial atom

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Vacancy Diffusion

Involves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy

Necessitates presence of vacancies

Diffusing atoms and vacancies exchange positions

they move in opposite directions

Diffusion rate depends on:[1] no. of vacancies[2] activation energy to exchange

Both self- and inter-diffusion occurs by this mechanism

Interstitial Diffusion

Atoms migrate from an interstitial position to a neighboring one that is empty

Found for inter-diffusion of small impurity atoms, such as hydrogen, carbon, nitrogen, and oxygen, to fit into interstices in host.

Host or substitutional impurity atoms rarely have interstitial diffusion

Diffusion rate depends on: [1] vacancy concentration[2] jump frequency

Interstitial atoms are smaller and thus more mobile

interstitial diffusion occurs much more rapidly than by vacancy mode

There are more empty interstitial positions than vacancies

interstitial atomic movement have greater probability

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Diffusion FASTER for .... open crystal structures lower density materials lower melting point materials secondary bonded materials smaller diffusing atoms cations

Diffusion SLOWER for .... close-packed structures higher density materials higher melting T materials covalent bonded materials larger diffusing atoms anions

In Summary ....

Diffusion and Materials Processing

Diffusional processes become very important when materials are used or processed at elevated temperatures.

Grain Growth

Materials composed of many grains contains a large number of grain boundaries, which represent a high-energy area because of inefficient packing of atoms.

Grain growth occurs by diffusion of atoms during high-temperature processing and small grains accumulates to form larger fewer grains in order to reduce grain boundary areas.

Since larger grains yield inferior mechanical properties, heat treatment and many other high-temperature processes are carefully controlled to avoid excessive grain growth.

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Figure 5.31 Grain growth occurs as atoms diffuse across the grain boundary from one grain to another

Figure 5.32 Grain growth in alumina ceramics can be seen from the SEM micrographs of alumina ceramics. (a) The left micrograph shows the microstructure of an alumina ceramic sintered at 1350oC for 150 hours. (b) The right micrograph shows a sample sintered at 1350oC for 30 hours. (Courtesy of I. Nettleship and R. McAfee.)

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Diffusion Bonding

It is a method of joining materials in the solid state at elevated temperature.

Two smooth and clean surface are forced together at a high pressure, producing a high atom-to-atom contact. Addition of high temperature causes diffusion of atoms and results atomic bonding. Finally, elimination of voids occurs by a slow volume diffusion process.

Diffusion bonding often used for joining exotic metals (e.g., titanium), dissimilar metals and materials, and ceramics.

Figure 5.33 The steps in diffusion bonding: (a) Initially the contact area is small; (b) application of pressure deforms the surface, increasing the bonded area; (c) grain boundary diffusion permits voids to shrink; and (d) final elimination of the voids requires volume diffusion

Sintering

In the powder process, powders are consolidated using high pressure to form green compact and then heated at high temperature to form the sintered compact. During heating, particles join together and volume of pore between the particles is reduced. Often pressure and temperature are added together (hot isostatic pressing, or HIPing) to form the object.

Most of the high temperature metals (e.g., tungsten carbide cutting tools) and ceramic materials are processed using this technique.

Figure 5.28 Diffusion processes during sintering and powder metallurgy. Atoms diffuse to points of contact, creating bridges and reducing the pore size

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Figure 5.30 The microstructure of BMT ceramics obtained by compaction and sintering of BMT powders. (Courtesy of H. Shirey.)

Figure 5.29 Particles of barium magnesium tantalate (BMT) (Ba(Mg1/3 Ta2/3)O3) powder are shown. This ceramic material is useful in making electronic components known as dielectric resonators that are used for wireless communications. (Courtesy of H. Shirey.)

Diffusion in Ionic Compounds and Polymers

In metals and alloys, atoms can move into any nearby vacancy or interstitial sites. But in other materials, atom movement may be somewhat more restricted.

Ionic Compound

A diffusing ion only enters a site having the same charge.

In order to reach the site, the ion must physically squeeze past adjoining ions, pass by a region of opposite charge, and move a relatively long distance.

Consequently the activation energies are higher and rates of diffusion are lower than for metals

Diffusion of cations are higher than anions (for being smaller in size)

Diffusion of ions also transfer electrical charge. Thus, as temperature increases, diffusion increases and electrical conductivity also increases.

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Polymers

Diffusion of atoms or small molecules occur from one location to another along a long polymer chain. Strong covalent bonds must need to be broken for this to occur.

Diffusion through crystalline polymer is slower than through amorphous polymers, which have no long-range order and consequently have a lower density.

Next Class

Lecture 8

Chemical Kinetics