MSEP 503 Defects, Diffusion and Transformation Slide ... · ,qwhuvwlwldo dwrp...

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Diffusion

Factors that Influence Diffusion Temperature - diffusion rate increases very rapidly with increasing temperature

Diffusion mechanism - interstitial is usually faster than vacancy

Diffusing and host species - Do, Qd is different for every solute, solvent pair

Microstructure - diffusion faster in polycrystalline vs. single crystal materialsbecause of the rapid diffusion along grain boundaries and dislocation cores.

Atomic diffusion is a process whereby the random thermally-activated hopping of atoms in a solid results in the net transport of atoms. For example, helium atoms inside a balloon can diffuse through the wall of the balloon and escape, resulting in the balloon slowly deflating. Other air molecules (e.g. oxygen, nitrogen) have lower mobilities and thus diffuse more slowly through the balloon wall. There is a concentration gradient in the balloon wall, because the balloon was initially filled with helium, and thus there is plenty of helium on the inside, but there is relatively little helium on the outside (helium is not a major component of air).

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General Note.

Diffusion is a flux of matter in which the atoms or molecules of a certain type move differently (rate, amount etc) with respect to the atoms/molecules of other type.

Please note the difference from gas or liquid flow, in this case ALL components move in the same way.

The definition: “Diffusion is the movement of molecules from a high concentration to a low concentration” is wrong because there are cases when diffusion process does just opposite.

Flux of matter can be caused not only by the difference in concentration of the atom that diffuses, but also the difference in concentration of other atoms and or gradient of physical parameters ( temperature, pressure, electric or magnetic field).

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Diffusion

Types of diffusion Diffusion paths:

HRTEM image of an interface between an aluminum (left) and a germanium grain. The black dots correspond to atom columns.

Surface diffusion

Bulk diffusion

Grainbaoundarydiffusion

In general: Dgp >Dsd >Dgb >>Db for hightemperatures and short diffusion times

Diffusion through the gas phase

Self diffusion: Motion of host lattice atoms. The diffusion coefficient for self diffusion depends on the diffusion mechanism:

Vacancy mechanism: Dself = [Cvac] Dvac

Interstitial mechanism: Dself = [Cint] Dint

Inter diffusion, multicomponent diffusion:Motion of host and foreign species. The fluxes and diffusion coefficient are correlated

Diffusion - Mass transport by atomic motion

Mechanisms• Gases & Liquids – random (Brownian) motion• Solids – self, vacancy, interstitial, or inter

diffusion

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Oxidation

Roles of Diffusion

Creep

AgingSintering

Doping Carburizing

Metals

Precipitates

SteelsSemiconductors

Many more…

Many mechanisms

Material Joining Diffusion bonding

Diffusion is relative flow of one material into another Mass flow process by which species change their position relative to their neighbours.

Diffusion of a species occurs from a region of high concentration to low concentration (usually). More accurately, diffusion occurs down the chemical potential (µ) gradient.

To comprehend many materials related phenomenon (as in the figure below) one must understand Diffusion.

The focus of the current chapter is solid state diffusion in crystalline materials. In the current context, diffusion should be differentiated with flow (of usually fluids and

sometime solids).

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Processing Using Diffusion

Case Hardening: Diffuse carbon atoms into

the host iron atoms at the surface.

Example of interstitial diffusion is a case hardened gear.

Result: The presence of C atoms makes iron (steel) harder.

• Doping silicon with phosphorus for n-type semiconductors:

• Process:

3. Result: Dopedsemiconductorregions.

silicon

magnified image of a computer chip

0.5 mm

light regions: Si atoms

light regions: Al atoms

2. Heat it.

1. Deposit P rich layers on surface.

silicon

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Ar H2

Movable piston with an orifice

H2 diffusion direction

Ar diffusion direction

Piston motion

Piston moves in thedirection of the slower

moving species

When a perfume bottle is opened at one end of a room, its smell reaches the other end via the diffusion of the molecules of the perfume.

If we consider an experimental setup as below (with Ar and H2 on different sides of a chamber separated by a movable piston), H2 will diffuse faster towards the left (as compared to Ar). As obvious, this will lead to the motion of movable piston in the direction of the slower moving species.

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A B

Inert Marker is basically a thin rod of a high melting material, which is insoluble in A & B

Kirkendall effect

Let us consider two materials A and B welded together with Inert marker and given

a diffusion anneal (i.e. heated for diffusion to take place).

Usually the lower melting component diffuses faster (say B). This will lead to the

shift in the marker position to the right.

Direction of marker motion

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Mass flow process by which species change their position relative to their neighbours. Diffusion is driven by thermal energy and a ‘gradient’ (usually in chemical potential).

Gradients in other physical quantities can also lead to diffusion (as in the figure below). In this chapter we will essentially restrict ourselves to concentration gradients.

Usually, concentration gradients imply chemical potential gradients; but there are exceptions to this rule. Hence, sometimes diffusion occurs ‘uphill’ in concentration gradients, but downhill in chemical potential gradients.

Thermal energy leads to thermal vibrations of atoms, leading to atomic jumps. In the absence of a gradient, atoms will still randomly ‘jump about’, without any net flow of

matter.

Diffusion

Chemical potential

ElectricGradient

Magnetic

Stress

First we will consider a continuum picture of diffusion and later consider the atomic basis for the same in crystalline solids. The continuum picture is applicable to heat transfer (i.e., is closely related to mathematical equations of heat transfer). 8

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Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities

Conditions necessary for diffusion An empty adjacent site Enough energy to break bonds and cause lattice distortions during

displacement

Mathematics of diffusion Steady-state diffusion (Fick’s first law) Nonsteady-State Diffusion (Fick’s second law)

Factors that influence diffusion Diffusing species Host solid Temperature Microstructure

Diffusion – How atoms move in solids

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What is diffusion?

Diffusion transport by atomic motion.

Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.

Diffusion

Part1. Constitutional effects

Diffusion is the phenomenon of spontaneous material transport byatomic motion.

Diffusion is classified according to

a) conditions: self-diffusion, diffusion from surface, interdiffusion, fastpath diffusion etc.

b) mechanism: interstitial, substitutional;

Part 2.

Non-constitutional effects. Kirkendall effect, Einstein equation, etc.11 12

Atom migration Vacancy migration

AfterBefore

Diffusion Mechanisms

Vacancy diffusion

To jump from lattice site to lattice site, atoms need energy to break bonds withneighbors, and to cause the necessary lattice distortions during jump.

Therefore, there is an energy barrier.

Energy comes from thermal energy of atomic vibrations (Eav ~ kT) Atom flow is opposite to vacancy flow direction.

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Interstitial atom before diffusion

Interstitial atom after diffusion

Diffusion Mechanisms

Interstitial diffusion

Generally faster than vacancy diffusion because bonding of interstitials tosurrounding atoms is normally weaker and there are more interstitial sites thanvacancy sites to jump to.

Smaller energy barrier Only small impurity atoms (e.g. C, H, O) fit into interstitial sites. The rate of interstitial diffusion is controlled only by the easiness with which a

diffusing atom can move into an interstice. 14

Self-diffusion: In an elemental solid, atoms also migrate.

A

B

C

D

After some time

AB

C

D

Vacancy Diffusion:• atoms exchange with vacancies• applies to substitutional impurities atoms • rate depends on:

-- number of vacancies-- activation energy to exchange.

increasing elapsed time

Probability of an atom jumping over the energy barrier:

𝑃 = 𝑒𝑥𝑝 −𝑄

𝑘𝑇

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Interstitial diffusion – smaller atoms can diffuse between atoms.

Interdiffusion: : In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc.

More rapid than vacancy diffusion

Initially After some time

There is an energy barrier which must be overcome when an atom changes site.

Mechanisms of interdiffusion:

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Before After

(Heat)

Inter-diffusion vs. Self-diffusion

Self-diffusion: One-component material, atoms are of same type.

1 dnJ

A dtFlow direction

Area (A)

Concentration gradient. Concentration can be designated in many ways (e.g. moles per unit volume). Concentration gradient is the difference in concentration between two points (usually close by).

We can use a restricted definition of flux (J) as flow per unit area per unit time: → mass flow / area / time [Atoms / m2 / s].

Steady state. The properties at a single point in the system does not change with time. These properties in the case of fluid flow are pressure, temperature, velocity and mass flow rate. In the context of diffusion, steady state usually implies that, concentration of a given species at a given point in space, does not change with time.

Important terms

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mass atomsJ

area time m s

In diffusion problems, we would typically like to address one of the following problems.(i) What is the composition profile after a contain time (i.e. determine c(x,t))?

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Fick’s 1 law

Assume that only species ‘S’ is moving across an area ‘A’. Concentration gradient for species ‘S’ exists across the plane.

The concentration gradient (dc/dx) drives the flux (J) of atoms. Flux (J) is assumed to be proportional to concentration gradient. The constant of proportionality is the Diffusivity or Diffusion Coefficient (D).

‘D’ is assumed to be independent of the concentration gradient. Diffusivity is a material property. It is a function of the composition of the material and

the temperature. In crystals with cubic symmetry the diffusivity is isotropic (i.e. does not depend on direction).

Even if steady state conditions do not exist (concentration at a point is changing with time, there is accumulation/depletion of matter), Fick’s 1-equation is still valid (but not easy to use).

dx

dcDA

dt

dn

dx

dcJ

dx

dcDJ

dx

dcD

dt

dn

AJ

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Fick’s first law (equation)

As we shall see the ‘law’ is actually an equation

Area

Flow directionThe negative sign implies that diffusion occurs down the

concentration gradient

* Adolf Fick in 1855

A material property

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dx

dcDA

dt

dn

No. of atoms crossing area A

per unit time

Cross-sectional area

Concentration gradient

ve sign implies matter transport is down the concentration gradient

Diffusion coefficient/Diffusivity

AFlow direction

As a first approximation assume D f(t)

Let us emphasize the terms in the equation

Note the strange unit of D: [m2/s]

2 3

1[ ]

dc number numberJ D D

dx m s m m

Let us look at the units of Diffusivity

2

[ ]m

Ds

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Diffusion

How do we quantify the amount or rate of diffusion?

sm

kgor

scm

mol

timearea surface

diffusing mass) (or molesFlux

22J

J slope

dt

dM

A

l

At

MJ

M = mass diffused

time

Measured empirically Make thin film (membrane) of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the

membrane

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Steady-State Diffusion

dx

dCDJ

Fick’s first law of diffusion

D diffusion coefficient

Rate of diffusion independent of time

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12 linear ifxx

CC

x

C

dx

dC

Flux proportional to concentration gradient =

Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn.

If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove?

Data:

– diffusion coefficient in butyl rubber: D = 110 x10-8 cm2/s

– surface concentrations: C1 = 0.44 g/cm3

C2 = 0.02 g/cm3

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Example: Chemical Protective Clothing (CPC)

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12- xx

CCD

dx

dCDJ

Dtb 6

2

glove

C1

C2

skinpaintremover

x1 x2

Solution – assuming linear conc. gradient

D = 110 x 10-8 cm2/s

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3

x2 – x1 = 0.04 cm

Data:

scm

g 10 x 16.1

cm) 04.0(

)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(

25-

3328-

J

Fick’s 2 law

The equation as below is often referred to as the Fick’s 2 law (though clearly this is an equation and not a law).

This equation is derived using Fick’s 1-equation and mass balance. The concentration of diffusing species is a function of both time and position C =

C(x,t) The equation is a second order PDE requiring one initial condition and two

boundary conditions to solve.

2

2

x

cD

t

c

c cD

t x x

If ‘D’ is not a function of the position, then it can be ‘pulled out’.

Derivation of this equation will taken up next.

Another equation

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Let us consider a 1D diffusion problem. Let us consider a small element of width x in the

body. Let the volume of the element be the control

volume (V) = 1.1. x = x. (Unit height and thickness).

Let the concentration profile of a species ‘S’ be as in the figure.

The slope of the c-x curve is related to the flux via the Fick’s I-equation.

In the figure the flux is decreasing linearly. The flux entering the element is Jx and that leaving

the element is Jx+x. Since the flux at x1 is not equal to flux leaving

that leaving at x2 and since J(x1) > J(x2), there is an accumulation of species ‘S’ in the region x.

The increase in the matter (species ‘S’) in the control volume (V) = (c/t).V = (c/t). x.

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Jx Jx+x

x

xxx JJonAccumulati

xx

JJJonAccumulati xx

xx

JJJx

t

cxx J

sm

Atomsm

sm

Atoms

23.

1

xx

Jx

t

c

x

cD

xt

cFick’s first law

x

cD

xt

c D f(x)2

2

x

cD

t

c

A B

Calculation of units

If Jx is the flux arriving at plane A and Jx+x is the flux leaving plane B. Then the Accumulation of matter is given by: (Jx Jx+x).

cJ

t

In 3D

Arises from mass conservation (hence not valid for vacancies)

2cD c

t

In 3D26

RHS is the curvature of the c vs x curve

x →

c →

x →

c →

+ve curvature c ↑ as t ↑ ve curvature c ↓ as t ↑

LHS is the change is concentration with time

2

2

x

cD

t

c

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Diffusion

Steady stateJ f(x,t)

Non-steady stateJ = f(x,t) D = f(c)

D = f(c)

D f(c)

D f(c)

Steady and non-steady state diffusion

Diffusion can occur under steady state or non-steady state (transient) conditions.

Under steady state conditions, the flux is not a function of the position within the material or time. Under non-steady state conditions this is not true.

This implies that under steady state the concentration profile does not change with time.

In each of these circumstances, diffusivity (D) may or may not be a function of concentration (c). The term concentration can also be replaced with composition.

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0x t

dc J

dt x

Under steady state conditions

0c

Dx x

Substituting for flux from Fick’s first law

2

20

cD

x

If D is constant

Slope of c-x plot is constant under steady state conditions

constantc

Dx

If D is NOT constant

If D increases with concentration then slope (of c-x plot) decreases with ‘c’

If D decreases with ‘c’ then slope increases with ‘c’

x

cD

xt

c cJ

t

In 3DThe general form of the Fick’s 2-equation is:

The equation is a second order differential equation involving time and one spatial dimension. This equation can be simplified for various circumstances and solved, as we will consider one

by one. These include: (i) steady state conditions and (ii) non-steady state conditions.

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Cases:

Steady state

Zero accumulation

Unsteady state

• Flux in ≠ flux out

• Enrichment or depletion

Fick’s second law

Fick’s laws

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The first simplification we make for the non-steady state conditions is that ‘D’ is not a function of the position.

If the diffusion distance is short relative to dimensions of the initial inhomogeneity, we can use the error function (erf) solution with 2 arbitrary constants.

The constants can be solved for from Boundary Condition(s) and Initial Condition(s). (we will

take up examples to clarify this).

Under non-steady state conditions

x

cD

xt

c If D is not a function of position 2

2

x

cD

t

c

2c

D ct

In 3D

Dt

xerfBAtxc

2 ),(

Under other conditions other solutions can be applied. For example, if a fixed amount of material is deposited on the surface of an infinite body and diffusion is allowed to take place, the concentration profile can be determined from the function below.

2

( , ) exp4

M xc x t

DtDt

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0

2exp2

duuErf

Erf () = 1 Erf ( ) = 1 Erf (0) = 0 Erf ( x) = Erf (x)

u →

Exp

(u2 )

→

0

Area

Also For upto x~0.6 Erf(x) ~ x x 2, Erf(x) 1

The error function (erf()) is defined as below. The modulus of the function represents the area under the curve of the exp(u2) function between ‘0’ and (with ‘some’ constant scaling factor). Some properties of the error function are also listed below.

Properties of the error function

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An example where the error function (erf) solution can be used

Let two materials M1 & M2 be joined together and kept at a temperature (T0), where diffusion is appreciable. Let C1 be the concentration of a species in M1 and C2 in M2.

This is a 1D diffusion problem (i.e. the species diffuses along x-direction only). The initial concentration profile (at t = 0, c(x,0)) of a species is like a step function (blue

line). If M1 and M2 are pure materials, then C1 would be zero. We can define an average composition of the species as: (C1 + C2)/2.

M2 M1

x →

Con

cent

rati

on →

Cavg

C1

C2

C(+x, 0) = C1

C(x, 0) = C2

The initial conditions (at t = 0) can be written as:

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M2 M1

x →

Con

cent

rati

on →

Cavg

↑ t

t1 > 0 | c(x, t1)t2 > t1 | c(x, t1) t = 0 | c(x,0)

Flux

f(x)|t

f(t)|x

Non-steadystate

If D = f(c) c(+x,t) c(x,t)i.e. asymmetry about y-axis

C(+x, 0) = C1

C(x, 0) = C2

C1

C2

A = (C1 + C2)/2 B = (C2 – C1)/2

Dt

xerfBAtxc

2 ),(

AB = C1

A+B = C2

1 2 2 1( , ) 2 2 2

C C C C xc x t erf

Dt

With increasing time the species ‘S’ diffuses into M1 leading to a depletion of S in the region close to the interface on the M2-side and enrichment on the M1-side.

This implies that we are dealing with non-steady state (transient) diffusion. From the initial conditions the arbitrary constants A & B can be determined and the

concentration profile as a function of time (t) and position (x) can be determined. Such a profile for two specific times (t1 and t2) are shown below.

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Examples of Solutions:

1. A fixed quantity of solute (B) is plated onto a semi-infinite bar

0

),(and0)0,( BdxtxCxCBoundary conditions:

Solution:

Dtx

Dt

BtxC

4exp),(

2

This case is realized if a thin film of diffusant is deposited on a surface.

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2. Interdiffusion of ONE component and diffusion from constant source.

0)0,(and),0( CtxCCtxC s Boundary conditions:

Solution:

Dtx

erfCCCtxC ss 4)(),(

2

0

x

duuxerf0

2 )exp(:Reminder

2BA

s

CCC

Notice that the surface concentration remains fixed.

In the case of interdiffusion of TWO components concentration profiles may be very different!

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In ideal case the point of constant concentration propagates with a rate of (4Dt)-½

If there is a way to trace a point of constant concentration then diffusion coefficient can be determined explicitly.

x2

t

The slope is 4D

0 2 4 6 8 10

0.5

1.0

Con

cent

ratio

n

Depth

x2/(4Dt)= 1 2 4 8 16 32 64

x

This method can be used to measure diffusion coefficient by measuring experimentally :

Dt

x 2

41

constDt

xconsttxC

4),(

2

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Temperature dependence of diffusivity

Arrhenius type

Diffusion is an activated process and hence the Diffusivity depends exponentially on temperature (as in the Arrhenius type equation below).

‘Q’ is the activation energy for diffusion. ‘Q’ depends on the kind of atomic processes (i.e. mechanism) involved in diffusion (e.g. substitutional diffusion, interstitial diffusion, grain boundary diffusion, etc.).

This dependence has important consequences with regard to material behaviour at elevated temperatures. Processes like precipitate coarsening, oxidation, creep etc. occur at very high rates at elevated temperatures.

Diffusion coefficient increases with increasing T.

= pre-exponential [m2/s]

= diffusion coefficient [m2/s]

= activation energy [J/mol or eV/atom]

= gas constant [8.314 J/mol-K]

= absolute temperature [K]

D

Do

Q

R

T

𝐷 = 𝐷 𝑒𝑥𝑝 −𝑄

𝑅𝑇

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Non-Steady State Solution of Diffusion - Superposition Principle

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Non-Steady State Solution of Diffusion - Superposition Principle

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Non-Steady State Solution of Diffusion – Application of Superposition Principle

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Non-Steady State Solution of Diffusion – Leak Test & Error Function

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Non-Steady State Solution of Diffusion – Semi-Infinite Source

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Determination of Diffusivity – Grube method

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Determination of Diffusivity – Boltzmann-Matano

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Non-Steady State Solution of Diffusion – Separation of Variable

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60

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Diffusion along High Diffusion Path – Grain Boundary Diffusion Model

dx

dCD

tL

mJ L

LL

2

dx

dCLDm LL

2

dx

dCD

Lt

mJ gb

gbgb

2

dx

dCLDm gbgb 2

LD

D

dx

dCLD

dx

dCLD

m

m

L

gb

L

gb

L

gb 22

2

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Phenomenological description does not give dependence of the diffusion coefficient on any physical parameters.

Consider two adjacent planes in the crystal one can get thatN sites with n1 atoms

N sites with n2 atoms

1 2

Q

Energy profile

a v is the number of jumps per second

Q is the energy barrier separating two sites

N is the number of atoms per plane

Microscopic Mechanisms of Diffusion

In ideal case diffusion coefficient exponentially depends on temperature and written as:

𝐷 = 𝑣

6𝑁𝑒𝑥𝑝 −

𝑄

𝑘𝑇

𝐷 = 𝐷 𝑒𝑥𝑝 −𝑄

𝑅𝑇62

Diffusion: A thermally activated process I

Energy of red atom= ER

Minimum energy for jump = EA

Probability that an atom has an energy >EA:

PEN EAexp

EA

kT

Diffusion coefficient

D D0 exp EA

kT

D0: Preexponential factor, a constant which is a function of jump frequency, jump distance and coordination number of vacancies

Numberof atoms

EnergyEA ER

Boltzmann distribution

T2T1

T1 < T2

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Diffusion: A thermally activated process II

The preexponential factor and the activation energy for a diffusion process can bedetermined from diffuson experiments done at different temperatures. The result are presented in an Arrhenius diagram.

ln D0

lnD

1/T

EA

k

ln D ln D0 EA

k

1

T

D D0 expEA

kT

In the Arrhenius diagram the slope is proportional to the activation energy and the intercept gives the preexponential factor.

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Diffusion and TemperatureD has exponential dependence on T

Dinterstitial >> Dsubstitutional

C in a-FeC in -Fe

Al in AlFe in a-FeFe in -Fe

1000K/T

D (m2/s)

0.5 1.0 1.510-20

10-14

10-8

T(C)

1500

1000

600

300

Tracer diffusion coefficients of 18O determined by SIMS profiling for various micro-and nanocrystalline oxides: coarse grained titania c-TiO2

(- - - -), nanocrystalline titania n-TiO2 (- - - -), microcrystalline zirconia m-ZrO2 (– – –), zirconia doped with yttrium or calcium (YSZ —· · —, CSZ — · —), bulk diffusion DV ( ) and interface diffusion DB (♦) in nanocrystalline ZrO2 (——), after Brossmann et al. 1999.

Diffusion coefficients I

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Diffusion coefficients II

Self diffusion coefficient for cations and oxygen in corundum, hematite and eskolaite. Despite having the same structure, the diffusion coefficient differ by several orders of magnitude.

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ExampleAt 300 ºC the diffusion coefficient and activation energy for Cu in Si are D(300 ºC) = 7.8 x 10-11 m2/s and Qd = 41.5 kJ/mol. What is the diffusion coefficient at 350ºC?

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202

1lnln and

1lnln

TR

QDD

TR

QDD dd

121

212

11lnlnln

TTR

Q

D

DDD d

transform dataD

Temp = T

ln D

1/T

1212

11exp

TTR

QDD d

T1 = 273 + 300 = 573 K

K 573

1

K 623

1

K-J/mol 314.8

J/mol 500,41exp /s)m 10 x 8.7( 211

2D

T2 = 273 + 350 = 623 K

D2 = 15.7 x 10-11 m2/s

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Non-steady State Diffusion - Example

B.C. at t = 0, C = Co for 0 x

at t > 0, C = CS for x = 0 (constant surface conc.)

C = Co for x =

• Copper diffuses into a bar of aluminum.

pre-existing conc., Co of copper atoms

Surface conc., C of Cu atoms bar

s

Cs

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Solution:

C(x,t) = Conc. at point x at time t

erf (z) = error function

erf(z) values are given in Tables

CS

Co

C(x,t)

Dt

x

CC

Ct,xC

os

o

2 erf1

dye yz 2

0

2

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• Sample Problem: An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.

• Solution: use

Dt

x

CC

CtxC

os

o

2erf1

),(

Non-steady State Diffusion - Example

– t = 49.5 h x = 4 x 10-3 m

– Cx = 0.35 wt% Cs = 1.0 wt%

– Co = 0.20 wt%

)(erf12

erf120.00.1

20.035.0),(z

Dt

x

CC

CtxC

os

o

erf(z) = 0.812572

Solution (cont.):

We must now determine from Table 5.1 the value of z for which the error function is 0.8125. An interpolation is necessary as follows

z erf(z)

0.90 0.7970z 0.81250.95 0.8209

7970.08209.0

7970.08125.0

90.095.0

90.0

z

z 0.93

Now solve for D

Dt

xz

2

tz

xD

2

2

4

/sm 10 x 6.2s 3600

h 1

h) 5.49()93.0()4(

m)10 x 4(

4

2112

23

2

2

tz

xD

69 70

71 72

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For diffusion of C in FCC Fe

Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

73

• To solve for the temperature at which Dhas the above value, we use a rearranged form of Equation (5.9a); )lnln( DDR

QT

o

d

/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(

J/mol 000,14821125

T

Solution (cont.):

T = 1300 K = 1027ºC

74

Example: Chemical Protective Clothing (CPC) Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it

also may be absorbed through skin. When using this paint remover, protective gloves should be worn.

If butyl rubber gloves (0.04 cm thick) are used, what is the breakthrough time (tb), i.e., how long could the gloves be used before methylene chloride reaches the hand?

Data

– diffusion coefficient in butyl rubber:

D = 110 x10-8 cm2/s

glove

C1

C2

skinpaintremover

x1 x2

Solution – assuming linear conc. gradient

Dtb 6

2

cm 0.04 12 xx

D = 110 x 10-8 cm2/s

Breakthrough time = tb

Time required for breakthrough ca. 4 min

min 4 s 240/s)cm 10 x 110)(6(

cm) 04.0(28-

2bt

Assumptions:

• These are 2 different metals in ratio 1:1

• They are joined by welding

• They are not completely miscible with each other

Let’s consider a chemical diffusion which occurs in presence of a contact between two metals.

Metal A Metal B

Diffusion in Multiphase Binary System

75

Diffusion Coefficient – Inter Diffusion

76

73 74

75 76

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Diffusion Coefficient – Inter Diffusion

77

Diffusion Coefficient – Self/Tracer Diffusion

78

Diffusion Coefficient – Intrinsic Diffusion Coefficient

79

Diffusion Coefficient – Inter Diffusion Coefficient

80

77 78

79 80

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21

B

BBAAB Nd

dDNDND

ln

ln1)(

~ **

Inter-diffusion Coefficient in a binary alloy linked to intrinsic diffusion by the Darken’s relation

Intrinsic diffusion Coefficient composed of mobility term (Tracer Diffusion) and

thermodynamic factor

B

BBB Nd

dDD

ln

ln1*

Tracer diffusion Coefficient – as a function of composition & temp.

)0(* BB ND

RTNQB

oBBB

BBeNDTND /)(* )(),(

: tracer impurity diffusion coefficient

: self-diffusion of A in the given structure)0(* BA ND

Diffusion Coefficient – Modeling

selfABB DND ** )0(

81

Diffusion Coefficient – Modeling

assuming composition independent D o

21

2

221

1

1

221

21

21

1

**/

221

21

21

1* nn

n

Nnn

n

N

RTQnn

nQ

nn

n

oBB DDeDN

nn

nN

nn

nD

Linear composition dependence of QB in a composition range N1 ~ N2

221

21

21

12

21

21

21

1 )()( Qnn

nQ

nn

nN

nn

nN

nn

nQNQ

Tracer diffusion coefficient at an intermediate composition is a geometrical mean of those at both ends – from experiments

the same for the D o term

RTNQNQRTNQNDBB

BBBoBBBB

oB eeeeTND /)()(/)()(ln* ),(

Both Q o & Q are modeled as a linear function of composition82

A hypothetical phase diagram A-B

A diffusion couple made by welding together pure A and pure B will result in layered structure containing α, β and γ.

83

Annealing at temperature T1 will produce a phase distribution and composition profile like that:

where:a, b, c and d – are solubility limits of the phases at T1.

84

81 82

83 84

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Concentration profile across the α/β interface and its associated movement assuming diffusion

control

})(~

)(~

{1

x

bBC

Dx

aBC

DaBCb

BCdt

dxv

a

dxaBCb

BC )(

dtx

aBC

Dx

bBC

D )})(~

())(~

{(

a

temerature absolute T constant gas R

energy activationQ~

constantoD

tcoefficien sioninterdiffu D~

)/~

exp(~

:where

RTQoDD

85

Optical micrograph of ion-nitrided ironshowing the multiplayer structure. The sample was ion nitrided at 605 °C for 2.5 h

Nitrogen concentration profile of ion-nitrided iron. The profile was measured

by electron probe microanalysis

Example

86

Nitrogen concentration profile of ion-nitrided iron. The profile was

measured by electron probe microanalysis

Example

87

Atomic Models of Diffusion

1) Interstitial Diffusion

Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that most of the interstitial sites are vacant. Hence, if an interstitial species (like carbon) wants to jump, this is ‘most likely’ possible as the the neighbouring site will be vacant.

Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very important (especially at low temperatures).

The diffusion of two important types of species needs to be distinguished: (i) species in a interstitial void (interstitial diffusion)

(ii) species ‘sitting’ in a lattice site (substitutional diffusion).

1 2

1 2

Hm

At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier Hm (enthalpy of motion).

→ frequency of vibrations, ’ → number of successful jumps / time.

kT

H m

e ' 88

85 86

87 88

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2) Substitutional diffusion via Vacancy Mechanism

For an atom in a lattice site (or a large atom associated with the motif), a jump to a neighbouring site can take place only if it is vacant. Hence, vacancy concentration plays an important role in the diffusion of species at lattice sites via the vacancy mechanism.

Vacancy clusters and defect complexes can alter this simple picture of diffusion involving vacancies.

Probability for an atomic jump (probability that the site is vacant) (probability that the atom has sufficient energy)

'f m

H

kTH

kT e e

kT

HH mf

e '

Hm → enthalpy of motion of atom across the barrier.

’ → frequency of successful jumps.

89

kT

Hm

eD 2

For Substitutional Diffusion

kT

HH mf

eD 2

D (C in FCC Fe at 1000ºC) = 3 1011 m2/s

D (Ni in FCC Fe at 1000ºC) = 2 1016 m2/s

0 f mH H

kTD D e

0 mH

kTD D e

which is of the form

A comparison of the value of diffusivity for interstitial diffusion and substitutional diffusion is given below. The comparison is made for C in -Fe and Ni in -Fe (both at 1000C).

It is seen that Dinterstitial is orders of magnitude greater than Dsubstitutional.

This is because the “barrier” (in the exponent) for substitutional diffusion has two ‘opposing’ terms: Hf and Hm (as compared to interstitial diffusion, which has only one term).

For Substitutional Diffusion

which is of the form

2D

Hence, ’ is of the form:

( )EnthalpykT e

If is the jump distance then the diffusivity can be written as:

( )

2Enthalpy

kTD e

90

Important During self-diffusion there is no change of chemical potential. Realization of each of the mechanisms depends on Type of intrinsic defects that prevails in the solid Activation energy for each of the mechanisms, if more than one

may be realized. Presence of other defects (vacancies).

Realization of vacancy or kick-out diffusion is possible only at the temperatures with sufficient concentration of vacancies. Therefore, prevailing mechanism may change with temperature.

In general, EVERY component in solid undergoes self-diffusion, however, if a solid contains more than one component, the ratio between self-diffusion coefficient depends on the type of bonding: Solids with covalent bonding typically have very low self-

diffusion coefficients. Solids with ionic bonding may have very different self-diffusion

coefficients for anion and cation. Metals and metal alloys usually show fast self-diffusion. 91

Diffusion Paths with Lesser Resistance

Qsurface < Qgrain boundary < Qpipe < QlatticeExperimentally determined activation energies for diffusion

The diffusion considered so far (both substitutional and interstitial) is ‘through’ the lattice.

In a microstructure there are many features, which can provide ‘easier’ paths for diffusion. These paths have a lower activation barrier for atomic jumps.

The ‘features’ to be considered include grain boundaries, surfaces, dislocation cores, etc. Residual stress can also play a major role in diffusion.

The order for activation energies (Q) for various paths is as listed below. A lower activation energy implies a higher diffusivity.

However, the flux of matter will be determined not only by the diffusivity, but also by the cross-section available for the path.

The diffusion through the core of a dislocation (especially so for edge dislocations) is termed as Pipe Diffusion.

92

89 90

91 92

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Comparison of Diffusivity for self-diffusion of Ag → single crystal vs polycrystal

1/T →

Log

(D

) →

Schematic

Polycrystal

Single crystal

← Increasing Temperature

Qgrain boundary = 110 kJ /mole

QLattice = 192 kJ /mole

If the ‘true’ effect of the high diffusivity of a low cross-section path is to be observed, then we need to go to low temperatures. At low temperatures, the high activation energy (low diffusivity) path is practically frozen and the effect of low activation energy path can be observed.

93

Applications based on Fick’s 2 law Carburization of steelCarburization of steel

Surface is often the most important part of the component, which is prone to degradation.

Surface hardening of steel components like gears is done by carburizing or nitriding.

Carburizing is done in the -phase field, where in the solubility of carbon is higher that that in the a phase. The high temperature enhances the kinetics as well.

In pack carburizing, a solid carbon powder used as C source.

In gas carburizing Methane gas is used a carbon source using the following reaction.CH4 (g) → 2H2 (g) + C (the carbon relased diffuses into steel).

It is usually assumed that the carbon concentration on the surface (CS) is constant (i.e. the carburizing medium imposes a constant concentration on the surface). An uniform homogeneous carbon concentration (C0) is assumed in the material before the carburization. Transient diffusion conditions exist and C diffuses into the steel.

C(+x, 0) = C0

C(0, t) = CS

A = CS

B = CS – C0

( , ) 2

xC x t A B erf

Dt

94

Approximate formula for depth of penetration

12

x Dt

0

0

( , )1

2S

c x t C xerf

C C Dt

Let the distance at which [(C(x,t)C0)/(CSC0)] = ½ be called x1/2

(which is an ‘effective penetration depth’)

1 211

2 2

xerf

Dt

1 2 1

22

xerf

Dt

1 1

2 2erf

1 2 1

22

x

Dt

penetrationx Dt

The depth at which C(x) is nearly C0 is (i.e. the distance beyond which is ‘un’-penetrated):

0 12

xerf

Dt

Erf(u) ~ 1 when u ~ 2 22

x

Dt

4x Dt

0

( , )=

2S

S

C x t C xerf

C C Dt

Often we would like to work with approximate formulae, which tell us the ‘effective’ depth of penetration and the depth which remains un-penetrated.

12

4x x 95

Another solution to the Fick’s 2 law

A thin film of material (fixed quantity of mass M) is deposited on the surface of another material (e.g. dopant on the surface of a semi-conductor). The system is heated to allow diffusion of the film material into the substrate.

For these boundary conditions we can use an exponential solution.

2

( , ) exp4

M xc x t

DtDt

Boundary and Initial conditions

C(+x, 0) = 0

0cdx M

Initially the species is only on the surface

The total mass of the species remains

constatant

The exponential solution

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93 94

95 96

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Ionic materials are not close packed (like CCP or HCP metals).

Ionic crystals may contain connected void pathways for rapid diffusion.

These pathways could include ions in a sublattice (which could get disordered) and hence the transport is very selective alumina compounds show cationic conduction Fluorite like oxides are anionic conductors.

Due to high diffusivity of ions in these materials they are called superionic conductors. They are characterized by: High value of D along with small temperature dependence of D Small values of D0.

Order disorder transition in conducting sublattice has been cited as one of the mechanisms for this behaviour.

Diffusion in Ionic Materials

97

There are materials where structural properties allow ultra-fast ion movement: superionics. In these materials ( for example AgRb3I4 one of the ions is much smaller than the available sites and there are far more available sites than ions.

Diffusion in polymers and glasses can be described by “randomly opening path” theory. Temperature dependence of diffusion coefficient in these materials is very complicated and time to time activation energy may become negative=> Diffusion coefficient may decrease with temperature.

Diffusion coefficient in anisotropic solids is a strong function of direction. Example: diffusion coefficient of Li and other alkaline metals in graphite along and across the layers may differ by 4 orders of magnitude.

Diffusion in Other Materials

98

Element Hf Hm Hf + Hm Q

Au 97 80 177 174

Ag 95 79 174 184

Calculated and experimental activation energies for vacancy Diffusion

99

SolvedExample

A 0.2% carbon steel needs to be surface carburized such that the concentration of carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface concentration of carbon of 1.4% and the process is carried out at 900C (where, Fe is in FCC form).

Data: 4 20D (C in -Fe) 0.7 10 m / s 157 /Q kJ mole

Given: T = 900° C, C0 = C(x, 0) = C(, t) = 0.2 % C,

Cf = C(0.2 mm, t1) = 1% C (at x = 0.2 mm), Cs = C(0, t) = 1.4% C

The solution to the Fick’ second law: ( , ) 2

xC x t A B erf

Dt

The constants A & B are determined from boundary and initial conditions:

(0, ) 0.014SC t A C , 0( , ) 0.002C t A B C or 0( ,0) 0.002C x A B C

S 0B C C 0.012 , ( , ) 0.014 0.012 2

xC x t erf

Dt

-44

1

1

2 10(2 10 , ) 0.01 0.014 0.012

2C m t erf

Dt

S S 0( , ) C (C C ) 2

xC x t erf

Dt

0

( , )=

2S

S

C x t C xerf

C C Dt

(2)

(1)

4

1

1 2 10

3 2erf

Dt

100

97 98

99 100

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x (in mm from surface)

% C

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

t =0

t = t1 = 14580s

t = 1000st = 7000s

t

0.4 0.6 0.8 1.0 1.2 1.4

x (in mm from surface)

% C

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

t =0

t = t1 = 14580s

t = 1000st = 7000s

t

0.4 0.6 0.8 1.0 1.2 1.4

The following points are to be noted:

The mechanism of C diffusion is interstitial diffusion

The diffusivity ‘D’ has to be evaluated at 900C using: 0 e x pQ

D DR T

0 expQ

D DRT

34 157 10

(0.7 10 )exp8.314 1173

2127.14 10

m

s

-4

1

121

2 10(0.3333) 0.309

2 7.14 10erf

t

24

1 12

1 1014580

0.337.14 10t s

From equation (2)-4

1

1 2 10

3 2erf

Dt

101

1 2

Vacant site

c = atoms / volume c = 1 / 3

concentration gradient dc/dx = (1 / 3)/ = 1 / 4

Flux = No of atoms / area / time = ’ / area = ’ / 2

242

''

)/(

dxdc

JD

kT

H m

eD 2

20 D

kT

Q

eDD 0

On comparisonwith

102

3. Interstitialcy Mechanism

Exchange of interstitial atom with a regular host atom (ejected from its regular site and occupies an interstitial site)

Requires comparatively low activation energies and can provide a pathway for fast diffusion

Interstitial halogen centres in alkali halides and silver interstitials in silver halides

D = f(c)

D f(c)C1

C2

Steady state diffusion

x →

Con

cent

rati

on →

103

Diffusion of impurities.

a) Interstitial b) Vacancy b) Kick-out

Important. The diffusion mechanism of an impurity depends on many factors:

type of the solution: interstitial or substitutional; size of the diffusant and size of the host sites; temperature; presence of other impurities; electronic structure of the host: metal, dielectric or

semiconductor.104

101 102

103 104

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Ionic and charged in impurities in solids can drift in electric field. As a first approximation one can assume that the flux of ions is proportional to electric field and concentration, i.e., one can use a concept of mobility. In this idealized case the flux of ions is given:

ECgJ iii

Diffusion in the Presence of Electric Field: Electromigration

where gi is mobility of ions, Ci is the concentration of ions and E is electric field.

E

Electromigration Diffusion

Diffusion coefficient and mobility are linked

Thus mobility is kT

Dezgi

(Nernst) Einstein equation

105

Limitations

Electrical mobility and diffusion coefficient are linked to eachother.

Nernst-Einstein equation is valid whenever the followingconditions are met:

The system is not far from equilibrium, i.e., gradient ofpotential and concentration are small

The diffusion species follow Boltzmann statistics, i.e., theydo not interact with the host and with each other.

Important: Nernst-Einstein equation is applicable to electrons in somesemiconductors.

Nernst-Einstein equation is not valid for systems with stronginteractions.

106

Nernst-Einstein equation is a low electric field approximation! Itimplies that the energy acquired by ion during one jump is mushsmaller than the activation energy. =>

Systems with very low activation energy do not obey Nernst –Einstein equation.

Application of a sufficiently high electric field maysignificantly increase mobility. This electric field is, in fact,comparable with crystal field, the electric field between ionsin crystal.

Materials with fast-path diffusion may have different electricfields for each path at which non-linear dependence betweenmobility and diffusion coefficient becomes noticeable.

Comments

107

If a homogeneous alloy is placed in a temperature gradient, one of the elements willdiffuse under the influence of the temperature difference. This is known as the Soréteffect, and again shows an example of diffusion occurring without a compositiongradient. In the presence of temperature gradients we cannot use Gibbs freeenergies to define equilibrium conditions, so chemical potential arguments can notbe used.

Thermal Diffusion

T1 < T21 ≠ 2

In practice thermal diffusion (also calledthermomigration) can occur both down andup the temperature gradient.

Carbon in austenite thermomigrates up a temperature gradient, because theactivation energy in this case is required mainly for preparing the destination site.As the carbon moves, two Fe atoms have to separate to create room for the Catom. This occurs more easily at a higher temperature, so the carbon movespreferentially to the hotter region.

Thermal diffusion in ionic solids with onlyone atom mobile leads to thermo-electricvoltage, similar to Seebeck effect inelectronic conductors.

108

105 106

107 108

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The presence of strain in the material can have a significant effect on thechemical potential of a solute. For example, in the case of an interstitialsolute such as carbon in iron, a tensile strain will increase the space availablefor the interstitial and so reduce the chemical potential.

Strain Field Induced Diffusion

The impurities that expand the lattice drift toward dilated regions andimpurities that cause contraction of the lattice drift towards compressedregions.

109

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