P diffusion_1

Luca Nobili - Interfaces in Materials

Quantities involved in diffusion problems

( )rJ i flux of component i (mol m-2 s-1)

It describes the rate at which i flows through a unit area fixed with respect to a specified coordinate system;

Corresponding mass quantities are defined by replacing the number of moles with mass.

iC concentration of component i (mol m-3)

It is the number of moles of component i per unit volume.


Mass transport

vCJN iii +=

Diffusion consists in the transport of chemical species which develops in a system out of equilibrium.

If diffusion occurs in a moving system, the molar flux of i relative to stationary coordinates is made up of two parts:

v = velocity of the bulk motion

A = 1

v

Civ

t = 0 v

Civ t = 1 s


Phenomenological equations

( ) ( ) ( )xLxLxLJ nn ∂∂−−∂∂−∂∂−= µµµ 12121111 ....

A given n-component system at equilibrium can be uniquely determined by specifying T, p, µ1, µ2,...µn-1, φ, where µi is the chemical potential and φ is any relevant scalar potential (e.g. electric potential). If the system is displaced slightly from equilibrium, it can be assumed that the rate of return to equilibrium is proportional to the deviation from equilibrium. Then, the flux of any component is assumed to be proportional to the gradient of each potential; in x direction:

( ) ( ) ( )dxdLdxdpLdxdTL epq φ111 −−−


Simplifying assumptions

( )xLJ ∂∂−= 1111 µ

In an isothermal, isobaric, isopotential system, the flux of any component is proportional to the gradient of the chemical potential of all components.

If we assume that the off-diagonal coefficients Lij are zero and no constraint exists, the phenomenological equation for any flux, say J1, becomes


Mobility

xCMCFMvCJ ∂∂−=== 11111111 µ

Mobility (M) is defined as the ratio between the mean velocity (v) of an atom and the generalized force (F) which acts on the atom: M = v/F The force gives rise to a steady-state velocity, instead of a continuing acceleration, because on the atomic scale atoms are continually changing their direction of motion. Force is considered in its more general sense as the opposite of a potential gradient, then

1111 CML =


Fick’s Law

iii CDJ ∇−=

It is much easier to determine a concentration gradient by experiment than a chemical potential gradient; therefore, the Fick’s law is commonly used:

Di is the intrinsic diffusion coefficient (or diffusivity) Di has dimensions of area divided by time (units m2 s-1) In a lattice with cubic symmetry, D has the same value in all directions (isotropic diffusion)

xCDJ i

ii ∂∂

−=In one-dimensional diffusion


Diffusion equation (1)

The function C(x,y,z,t) can be determined by solving a differential equation, which is obtained by using the Fick’s law and a material balance.

C

x x+dx

x+dx x

J1 J2

One-dimensional diffusion



( )t

CAdxAJJ i

∂∂

⋅=− 21 tCAdxAdx

xJ i

∂∂

⋅=⋅∂∂

−

J1 J2

dx

A

Ci

∂∂

∂∂

=∂∂

xCD

xtC ii Fick’s second equation



2

2

xCD

tC ii

∂∂

=∂∂

tC

rC

rrCD

rCrD

rriiii

∂∂

=

∂∂

+∂∂

=

∂∂

⋅∂∂ 11

2

2

ii J

tC

−∇=∂∂

Rectangular coordinates

Three dimensions, rectangular coordinates

One dimension, constant D:

Cylindrical coordinates

Spherical coordinates t

CrC

rrCD

rCrD

rriiii

∂∂

=

∂∂

+∂∂

=

∂∂

⋅∂∂ 21

2

22

2


Self-Diffusion During self-diffusion, the components diffuse in a chemically homogeneous system. The diffusion can be measured using radioactive tracer isotopes; the tracer concentration (*C1) is measured and its diffusivity (self-diffusivity) is calculated from the evolution of the concentration profile.

x C

C1

C1 *C1+C1

x C

C2

C1+C2 *C1+C1+C2

C2

C1+C2


Thermodynamic factor (1)

xCDxCMJ iiiiii ∂∂−=∂∂−= µ

Flux of component i can be written with chemical potential or concentration appearing as a source:

( )iiiiii nRTaRT γµµµ lnln 00 +=+=

The general expression of µi in a condensed phase is

µi0 = chemical potential of i in the reference state

ai = activity of i γi = activity coefficient of i ni = atomic fraction of i



( ) ( )[ ]xxnnxRTnMD ii

iiii ∂∂+∂∂∂∂

= γlnln

( ) ( )[ ] xnCDxxnRTCM iiiiii ∂∂=∂∂+∂∂ γlnln

It may be assumed that the total concentration of a solid system is constant; then Ci depends only on ni: Ci = niC

The expressions of Ji become

( ) ( )[ ]iiii nRTMD lnln1 ∂∂+= γ



Di = MiRT in ideal solutions (γi =1)

Di = MiRT in dilute solutions (if γi is constant)

Di < 0 e.g. in a miscibility gap

The deviation of the ratio Di/(MiRT) from unity will depend on the degree of non-ideality; with rising temperature alloys tend to be more ideal, then any deviation of Di/(MiRT) from unity will decrease with rising temperature.


Relationship between Di and *Di

( ) ( )[ ]iiii nRTMD **** lnln1 ∂∂+= γ

RTMD ii** =

In a binary diffusion couple with no concentration gradient, the self-diffusion coefficient (*Di) is given by:

Since the stable and radioactive isotopes are chemically identical, γi will be independent on the *ni/ni ratio and will be constant, then

If it is assumed that *Mi = Mi, a relationship is obtained between intrinsic diffusivity and self-diffusivity:

( ) ( )[ ]iiii nDD lnln1* ∂∂+= γ


Diffusion in a concentration gradient

If the original position of the interface is marked by an insoluble element, it is found that the distance of the marker from the end of the couple changes during the interdiffusion time (Kirkendall effect). This shift occurs because the flux of one component is largely different from that of the other across the same plane (substitutional alloy).

x C

C1

C1+C2 C1+C2

C1

marker


Interdiffusion experiments (1)

xnCDxCDJL ∂∂−=∂∂−= 11111

xnCDJL ∂∂−= 222

Two coordinate systems must be considered: one is fixed relative to the marker (lattice system), the other is fixed relative to the ends of the sample (reference system).

In the lattice system, the flux is given only by diffusion:

In order to visualize the situation, a vacancy mechanism of diffusion may be assumed. In the crystal, the number of lattice sites is fixed and the sum of the fluxes of atoms and vacancies in the lattice coordinate system is zero:

021 =++ vLLL JJJ



( )xnDD

CJJ

CJv

LLv

L

∂∂

−=+

−== 121

21

1221121

1111 nJnJCC

JJJvCJJ LLLL

LLR ⋅−⋅=+

−=+=

( )xCnDnDJR

∂∂

⋅+⋅−= 112211

The flux of vacancies will produce the marker shift, with velocity

where the equation dn1=-dn2 is used.

The flux in the reference system is given by

( )x

CnDnDJR

∂∂

⋅+⋅−= 212212



2

2

xCD

tC ii

∂∂

=∂∂

( ) ( ) ( )[ ]1112*

21* lnln1 nnDnDD ∂∂++= γ

1221 nDnDD ⋅+⋅=

The flux relative to specimen ends can be expressed according to the Fick’ law by defining the interdiffusion coefficient

Consequently, the concentration profile is obtained by solving the diffusion equation

The interdiffusion coefficient can be expressed in term of the self-diffusion coefficients:

From the Gibbs-Duhem equation, ( ) ( ) ( ) ( )2211 lnlnlnln nn ∂∂=∂∂ γγ


Solutions to the diffusion equation constant D, steady-state

02

2

=dx

CdD

02 =

drdCr

drd

( ) ( )LxCCCxC L 00 −+=

Rectangular coordinates, one dimension

Cylindrical coordinates, one dimension

0=

drdCr

drd ( ) ( )

−+=

112

121 ln

ln rr

rrCCCrC

Spherical coordinates, one dimension

( ) ( )12

12121 rr

rrrrCCCrC

−−

⋅−+=


Solutions to the diffusion equation constant D, non-steady-state

2

2

xCD

tC

∂∂

=∂∂

In rectangular coordinates and one-dimension, the equation is

In general, the solutions of this equation fall into two cases: - when the diffusion distance is short relative to the dimensions of the system, the solution C(x,t) can be most simply expressed in terms of error functions (infinite system); - when complete homogenization is approached, C(x,t) can be represented by the first few terms of an infinite trigonometric series. Because of the likeness between the diffusion equation and the heat equation, similar solutions exist for these two equations.


Graphical interpretation

2

2

xCD

tC

∂∂

=∂∂

∂2C/∂x2 > 0 , ∂C/∂t > 0 C increases with time

C

x

C

x

∂2C/∂x2 < 0 , ∂C/∂t < 0 C decreases with time


Infinite system – constant concentration

=

−−

Dtxerf

CCCtxC

2),(

12

1

C2

x

t = 0

L

C1 C1 is constant (e.g. reactions with the atmosphere produces a constant surface concentration)

C(x,0) = C2 C(0,t) = C1 C(∞,t) = C2

C2

C1


The error function

( ) duuzerfz

∫ −=0

2 )exp(2π

( ) 00 =erf

( ) 1=∞erf

( ) 9953.02 =erf

( ) ( )zerfzerf −=−0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

erf(z

)

z


Solution for infinite systems

=

−−

Dtxerf

CCCtxC

2),(

12

1

LDt <<2 ( )LDt <4

Each value of the ratio (C-C1)/(C2-C1) is associated with a particular value of z = x/[2(Dt)1/2]; each composition moves away from the plane of x = 0 at a rate proportional to (Dt)1/2 (except C = C1 which remains at x = 0). The system can be treated as infinite if the diffusion distance is small relative to the length of the system (L):

C2

C1

Dtx 4=


Interdiffusion in semi-infinite solids

=

−−

tDxerf

CCCtxCS

S

2),(

2

( ) 221 CCCS +=

C2

x=0

t = 0

2L

C1

C(x,0) = C2 x > 0 C(x,0) = C1 x < 0

C(-∞,t) = C1

C(∞,t) = C2

C2

C1


Finite system – constant concentration

⋅

⋅−=−−

Lx

LDt

CCCtxC

Si

S

2cos

4exp4),(

2

2 πππ

00

=∂∂

=xxC

05.02 >LDt

C(x,0) = Ci x > 0

C(L,t) = C(-L,t) = Cs Ci

x=0

2L

x=L

x Cs

t = 0

LDt 9.04 >


Finite system – average concentration

05.02 >LDt

⋅−=−−

2

2

2 4exp8)(

LDt

CCCtC

Si

S ππ

( )∫=L

dxtxCL

tC0

,1)(

average concentration

-L L

Cs

Ci

C


Time evolution of the concentration profile

Infinite system erf ( )

-L -L -L L L L

Cs Cs Cs

Ci Ci Ci

Finite system exp( )cos( )

t1

t2 > t1 t3 > t2

Finite system exp( )cos( )


Atomic mechanisms of diffusion in solids

It is assumed that diffusion in solids occurs by the periodic jumping of atoms from one lattice site to another.

The diffusion coefficient can be related to the jump frequency by considering two adjacent lattice planes.

Γ = jump frequency s1, s2 = diffusing atoms per unit area in planes 1 and 2

β

1 2 Assuming that the jump frequency is the same in all orthogonal directions, one-sixth of the atoms will go to the right from plane 1; the net flux from planes 1 to 2 is

( )Γ−= 2161 ssJ


Atomic movement and the diffusion coefficient

The surface atomic density can be related to the concentration: s1 = C1⋅β s2 = C2⋅β

and the net flux becomes J = (1/6)(C1 - C2)βΓ

Usually, C changes slowly with x, then C1 - C2 = -β(∂C/∂x) J = -(1/6)β2Γ(∂C/∂x) This equation is identical to Fick’s law if:

Γ= 2

61 βD

β

1 2

x

This equation applies to the self-diffusion coefficient, because equal jump frequency is assumed in all directions.


Estimate of the jump frequency

It is expected that β is approximately the interatomic distance in a lattice, then the order of 0.1 nm;

near their melting points, most fcc and hcp metals have a self-diffusion coefficient close to 10-12 m2 s-1;

by taking β = 10-10 m, the order of magnitude of Γ results to be 108 s-1.

The vibrational frequency (Debye frequency) of the atoms is 1013 to 1014 s-1, so the atoms only changes position on one oscillation in 105.


Vacancy mechanism

A substitutional atom diffuses by a vacancy mechanism when it jumps into an adjacent vacant site.

In close-packed structures, the displacement of the diffusing atom requires a local dilatation of the lattice.

The vacancy mechanism is usually the dominant diffusion mode in pure metals and substitutional alloys; it also is found in ionic compounds and oxides.


Interstitial sites

Interstitial sites are a set of atomic positions distinct from the lattice sites.

Interstitial sites in an fcc lattice


Interstitial mechanism

An atom diffuses by an interstitial mechanism when it passes from one interstitial site to one of its nearest-neighbour interstitial sites.

The movement of the interstitial atom implies a local distortion of the matrix lattice.

The interstitial mechanism mainly operates in alloys with interstitial solutes (e.g. C in Fe).


Self-diffusion coefficient The average number of jumps per second for each tracer atom (Γ) will be proportional to the number of nearest-neighbour sites (z), to the probability that any adjacent site is vacant (pv) and to the probability per unit time that the tracer will jump into a particular vacant site (w): Γ = z·pv·w

The probability pv will be equal to the fraction of vacant sites nv;

the self-diffusion coefficient will be given by the expression D = zl⋅a2·pv·w

where a is the lattice parameter and zl is a constant dependent on the number of nearest neighbours in an adjacent plane and on the ratio β/a (β is the distance between planes).


Diffusion coefficient of interstitial solutes

In very dilute solutions, w is independent of composition and the fraction of vacant interstitial sites is essentially unity; then D for the interstitial element is D = zi·a2·w

where zi is a geometric constant depending on the lattice features.

The interstitial atoms always has many vacant sites in the nearest-neighbour shell and this is the reason why their diffusivity is typically much larger than that of substitutional atoms.


Atomic movement and diffusion coefficient

An atom which jumps in a vacant site moves through a midway configuration, which is treated as an activated state.

Ene

rgy

a c

b

a b c


Frequency of vacancy occupation The frequency of vacancy occupation w can be evaluated by calculating the fraction of activated complexes, i.e. sites containing an atom midway between two equilibrium sites (saddle point). The change in Gibbs free energy for the activated state is given by the work done reversibly to move an atom from its initial site to the saddle point: ∆Gm = ∆Hm - T∆Sm

The equilibrium fraction of atoms in the saddle point (nm) can be calculated by the same procedure used to obtain the equilibrium fractions of vacancies: nm = exp(∆Sm/R)exp(-∆Hm/RT)


Calculation of the diffusion coefficient

The frequency w can be expressed by the equation w = ν nm

where ν is the mean vibrational frequency of an atom about its equilibrium site; it is usually taken equal to the Debye frequency.

Empirically it is found that the diffusion coefficient can be described by the equation D = D0 exp(-Q/RT) D0 and Q will depend on composition but are independent of temperature, as long as the same mechanism is dominant.


Self-diffusion by a vacancy mechanism

∆+∆−

∆+∆

=RT

HHR

SSazD mvmvl expexp2ν

∆+∆

=R

SSazD mvl exp2

0 ν

For diffusion in a pure metal, equations previously obtained give the following expression for D: D = zl·a2·nv·ν nm

mv HHQ ∆+∆=


Interstitial diffusion

∆−

∆=

RTH

RSazD mm

i expexp2ν

∆=

RSazD m

i exp20 ν

In the case of interstitial diffusion, the expression for D is D = zi·a2ν nm

mHQ ∆=


Empirical rules for Q and D0

Brown and Ashby examined data for a wide variety of solids and proposed these correlations: - the diffusion coefficient at the melting temperature, D(Tm), is a constant; - the ratio of activation energy to RTm is a constant.

Material D(Tm) (m2/s) Q/RTm

Fcc metals 5.5·10-13 18.4

Bcc trans. metals 2.9·10-12 17.8 Hcp (Mg,Cd,Zn) 1.6·10-12 17.3 Alkali halides 3.2·10-13 22.7

A.M. Brown, M.F. Ashby, Acta Met., 28 (1980) 1085.


Correlation effects

So far, the directions of successive jumps of the diffusing atom have been assumed to be independent of one another; this is not true and correlation between successive jumps has to be considered.

Correlation effects will be examined only in self-diffusion of tracer isotopes in pure metals.

The correlation factors in dilute alloys are evaluated with similar reasoning; they can be quite marked when an impurity atom is strongly attracted to a vacancy, because the vacancy-impurity exchange rate becomes much greater than the vacancy-solvent exchange rate.


Correlation in self-diffusion

After any jump of the tracer by a vacancy mechanism, the most probable next jump direction for the tracer is just back to the site that is now vacant.

A good approximation of the correlation factor (f) can be obtained as f = 1- 2/z (error of ~4% for an fcc lattice); this approximation is equivalent to say that two successive jumps having probability 1/z (z = coordination number) produce no net movement if the atom returns to its original position.

Then *D =f⋅ zl·a2·nv·ν nm


Self-diffusion in dilute alloys

In a binary substitutional solution, there are relationships between self-diffusion coefficients of components 1 and 2. It has been observed that if D1 decreases with the atomic fraction of solute 2 (n2), then D2 also decreases with n2. These relationships are often expressed by the equations: D1(n2) = D1(0)[1+b1n2] D2(n2) = D2(0)[1+B1n2]

thus b1 and B1 have the same sign.

The effect of the solute can be thought to be the addition of regions of a different jump frequency (a higher frequency if b is positive), probably because the effect of the solute is to modify the vacancy concentration.


Calculation of interdiffusion coefficient

The interdiffusion coefficient can be estimated using the expressions of the self-diffusion coefficient and the thermodynamic factor.

In order to remove the assumption that *Mi = Mi, it must be considered that in presence of a concentration gradient vacancies will more frequently approach any given atom from one side than from the other. This vacancy flux (vacancy wind) increases the apparent D for the fastest moving component and decreases that for the slower one. Then, the interdiffusion coefficient has to be corrected by a proper correlation factor.


Diffusion with traps (hydrogen)

Trapping at defects can have a large effect on diffusion in solids of solute with a low equilibrium solubility. Hydrogen diffusion is a typical example since it diffuses so easily that even shallow traps will produce a measurable effect on D (DH > 1010⋅DC at 300 K in Fe).

The observed solubility of H in Fe at room temperature can be much greater than the lattice solubility (≈0.5 at. ppm), the exact value depending on the density of low energy sites represented by dislocations, matrix-precipitate interfaces, grain boundaries, microvoids, etc.; these low energy sites serve as traps which inhibit the diffusion of hydrogen.


Effective diffusion coefficient

LLLt CD

tC

tC

tC 2∇=

∂∂

+∂∂

=∂∂

LLLL

L

t CDt

Ct

CdCdC 2∇=

∂∂

+∂∂

Lt

Le dCdC

DD+

=1

Hydrogen is either in traps (Ct) or perfect lattice sites (CL), then the mass balance equation becomes

If we assume that equilibrium exists between H atoms on trap sites and lattice sites (e.g. H2 trapped in internal voids), an equilibrium relationship will exist between (Ct) and (CL), then

The effective diffusion coefficient can be defined as

LLt

LL CdCdC

Dt

C 2

1∇

+=

∂∂


Molecular hydrogen in internal voids

Lt

L

gL

Le CC

DKC

DD2121 +

=+

=

Hydrogen dissolves in metals in atomic form and the equilibrium solubility in the lattice is proportional to the square root of the hydrogen pressure in the gas bubbles, then Ct = (CL)2Kg

where Kg is a quantity dependent on temperature. The effective diffusion coefficient becomes

Voids can be significant as hydrogen traps in cold worked two-phase alloys, where deformation creates holes or cracks at the interface between hard particles and the matrix.


Hydrogen embrittlement

Diffusion of hydrogen in metals may produce embrittling effects through different mechanisms, which include - interaction of hydrogen atoms with dislocations - pressure increase in internal voids and cracks - reduction of surface energy in crack growth - formation of brittle hydrides (TiH2, ZrH2, etc.) - formation of gaseous species (H2O, CH4, etc.)

Dislocations are typical saturable traps, because only a limited number of low-energy sites exist around a dislocation.


Saturable traps (1)

The effective diffusion coefficient in presence of saturable traps can be obtained under these simplifying assumptions: - only one type of traps exists; - each trap site can only hold one hydrogen atom; - the enthalpy difference between trap sites and lattice sites is ∆Hb; - equilibrium exists between H atoms on trap sites and lattice sites;

The following fractions are defined: θt = fraction of occupied trap sites θL = fraction of occupied lattice sites


Saturable traps (2)

LL

Ltt KCN

CKNC+

=

Under assumptions similar to those used to derive the McLean isotherm, the equilibrium condition is

where the assumption θL « 1 is made and the entropy change Sb is neglected. If the material contains NL lattice sites and Nt trap sites per unit volume, the following expressions can be written: Ct = Ntθt CL = NLθL The equilibrium condition becomes

KRTH

Lb

Lt

t θθθ

θ=

∆−=

−exp

1


Saturable traps (3)

( )ttL

LLe CC

CDDθ−+

=1

Calculation of the derivative dCt/dCL and substitution in the general expression of De give the following expression:

Two limiting cases exist → θt ≈ 1 De ≈ DL traps are saturated and give no significant contribution to diffusion

→ θt « 1 De=DLCL(CL+Ct)-1 traps are receptive to H atoms and reduce the effective diffusivity; the extent of the reduction depends on the ratio Ct/CL


Saturable traps (4)

De

DL

0,00

0,05

0,10

0,15

0,20

θt

1/T →

ln D

→


Kinetics of hydrogen release (1)

( )RTHCAdtdC att ∆−⋅−= exp

If the sample is heated in a vacuum after hydrogen adsorption, the release is thermally activated and its rate is taken to be proportional to the concentration in the traps Ct

where ∆Ha=∆Hb+∆Hm and ∆Hm is the activation energy for lattice diffusion. When ∆Hb » RT, traps are said “irreversible”.

∆Hb

∆Hm

Distance

Ent

halp

y


Kinetics of hydrogen release (2)

If the sample is continuously heated at a rate α and all traps have the same value of ∆Hb, the evolution rate will attain a peak at the temperature Tp, which can be obtained by setting the differential of dCt/dt equal to zero:

∆Ha can be calculated from the experimental value of Tp. If there are traps with different binding energies, hydrogen will be released independently from each type of traps, then peaks in the evolution rate will appear at different temperatures during heating.

( )

∆−=

∆

p

a

p

aRTHA

TRH exp2α


Internal traps

Values of ∆Ha for hydrogen traps in α-Fe are*

∆Ha = 18.5 kJ/mol at Fe/Fe3C interface ∆Ha = 26.9 kJ/mol at dislocations ∆Ha = 86.9 kJ/mol at Fe/TiC interface

Other examples of diffusion with irreversible trapping are - internal oxidation, where oxygen is the rapidly diffusing element (like hydrogen) which combines with a reactive solute, such as aluminium in silver, to form fine oxide particles; - diffusion of carbon in multicomponent alloys with carbide precipitation.

* H.G. Lee, J.Y. Lee, Acta Met., 32 (1984) 131-6.


Diffusion in ionic solids

Diffusion in solids with ionic bonds is more complicated than in metals because site defects are generally electrically charged. Electric neutrality requires that neutral complexes of charged defect exist in the solid. Therefore, diffusion involves more than one charged species.

The Kröger-Vink notation will be used to represent the defects: the subscript indicates the type of site the species occupies and the superscript indicates the excess charge associated to the species in that site; (•) superscript → positive unit charge (′) superscript → negative unit charge (equal to the electron charge) (×) superscript → zero charge


Intrinsic self-diffusion

An example of intrinsic self-diffusion in an ionic material is given by pure stoichiometric KCl. As in many alkali halides, the main point defects are cation and anion vacancy complexes (Schottky defects) and then self-diffusion takes place by a vacancy mechanism. For stoichiometric KCl, the anion and cation vacancies are created in equal numbers because of the electroneutrality condition; these vacancies can be created by moving K+ and Cl- ions from the bulk to an interface, a dislocation or a surface ledge.

K+ Cl-

VK′ VCl

•


Self-diffusivity in alkali halides (1)

[ ] [ ] ( )RTGKVV SeqClK ∆−==⋅ • exp'

•+= ClK VVnull '

The defect creation can be written as a reaction:

The equilibrium constant of this reaction (Keq) is related to the molar free energy of formation (∆GS) of the Schottky pair; for small concentrations of the vacancies, activities can be taken equal to the site fractions:

the square brackets indicate a site fraction.


Self-diffusivity in alkali halides (2)

[ ] [ ]•= ClK VV '

[ ] [ ] ( )RTGVV SClK 2exp' ∆−== •

Electrical neutrality requires that

Then

The self-diffusivity of K is given by:

( )[ ] ( )[ ]RTHHRSSfazD CVmS

CVmSlK ∆+∆−∆+∆= 2exp2exp2* ν

A similar expression applies to Cl self-diffusion on the anion sublattice.

CVmS HHQ ∆+∆= 2

The activation energy for self-diffusion is then


Self-diffusion with interstitial cations (1)

Self diffusion of Ag cations in the silver halides involves Frenkel defects, consisting in an equal number of vacancies and interstitials. Both vacancies and interstitials may contribute to the diffusion; however, experimental data for AgBr indicate that cation diffusion by the interstitialcy mechanism (exchange between interstitial cation and lattice cation) is dominant.

Ag+ Br - VAg′

Agi•


Self-diffusion with interstitial cations (2)

[ ] [ ] ( )[ ]RTGVAg FAgi 2exp' ∆−==•

[ ] [ ] ( )RTGKVAg FeqAgi ∆−==⋅• exp'

'AgiAg VAgAg += •×

The reaction of formation of cation Frenkel pairs is:

The activity of the lattice cation is unity, then

The electrical neutrality requires that fractions of vacancies and interstitials are equal:

The activation energy for self-diffusivity of the Ag cations is ImF HHQ ∆+∆= 2


Extrinsic diffusion (1)

Charged point defects can be induced to form in an ionic solid by the addition of substitutional cations or anions with charges that differ from those in the host lattice.

Electrical neutrality demands that each addition results in the formation of defects of opposite charge (extrinsic defects) that can contribute to the diffusivity or electronic conductivity.

For example, extrinsic cation vacancies can be created in KCl by the adding Ca++ ions, that is by doping KCl with CaCl2; electrical neutrality requires that each substitutional divalent cation in KCl be balanced by the formation of a cation vacancy.



[ ] [ ] [ ]'KClK VVCa =+ ••

[ ] [ ] [ ]( ) ( ) [ ]2''' exp pureKSKKK VRTGCaVV =∆−=−⋅ •

Fractions of anionic and cationic vacancies are related to the content of the extrinsic Ca++ impurity:

By inserting this relationship in the expression of the equilibrium constant, the following equation is obtained:

Solution to this equation is

[ ] [ ] [ ][ ]

++=

•

•21

2

2'' 4

112

K

pureKKK

Ca

VCaV



[ ] [ ]•>> KpureK CaV '

[ ] [ ]•<< KpureK CaV ' [ ] [ ]•= KK CaV '

There are two limiting cases for the behaviour of the cation vacancy fraction:

The intrinsic case applies at small doping levels or at high temperatures; the activation energy for cation self-diffusion is the same as in the pure material.

The extrinsic case applies at large doping levels or at low temperatures; the fraction of cation vacancies is equal to the impurity fraction and is therefore temperature independent; the activation energy consists only of the migration term.

Intrinsic

Extrinsic

[ ] [ ]pureKK VV '' =



The expected Arrhenius plot for cation self diffusion in KCl doped with Ca++ shows a two-part curve which reflects the intrinsic and extrinsic behaviour

Slope = –(∆Hm+∆HS/2)/R

Slope = –∆Hm/R

ln (*

DK)

1/T

Intrinsic range

Extrinsic range


Diffusion in nonstoichiometric materials Transition metals have different valence states with small energy difference between them; then, many compounds of these metals are non-stoichiometric. Nonstoichiometry of semiconductor oxides can be induced by the material’s environment. For example, oxides such as FeO, NiO and CoO can be made metal-deficient (oxygen-rich) in oxidizing environments and TiO2 and ZrO2 can be made oxygen-deficient under reducing conditions. These stoichiometric variations cause large changes in point-defect concentrations and affect diffusivity and electrical conductivity. In pure FeO, the point defects are primarily Schottky defects that satisfy the equilibrium relationships.


Equilibrium with the environment (1) When FeO is oxidized through the reaction FeO + x/2 O2 = FeO1+x

each O atom takes two electrons from two Fe++ ions according to the reactions 2 Fe2+ = 2 Fe3+ + 2 e-

1/2 O2 + 2 e- = O2-

e

O2- O2- O2-

O2- O2- O

e Fe2+

O2-

O2- O2- O2-

O2- O2-

Fe3+

Fe3+


Equilibrium with the environment (2)

''2 2212 FeOFeFe VOFeOFe ++=+ ×•×

''2 221 FeOFe VOhO ++= ×•

The overall reaction is 2 Fe2+ + 1/2 O2 = 2 Fe3+ + O2-

A cation vacancy must be created for every O atom added to ensure electrical neutrality:

This reaction can be written in terms of holes h in the valence band created by the loss of an electron from an Fe2+ ion producing an Fe3+ ion

×•• −≡ FeFeFe FeFeh


Equilibrium with the environment (3)

[ ] [ ]( ) ( )RTGKp

hVeq

O

FeFe ∆−==⋅ •

exp21

2''

2

''2 221 FeOFe VOhO ++= ×•

[ ] [ ]''2 FeFe Vh =•

Reaction

Equilibrium Constant

Neutrality condition

The equilibrium fraction of cation vacancies is

[ ] ( ) ( )[ ]RTGpV OFe 3exp41 61

31''

2∆−

=

The cation self-diffusivity due to the vacancy mechanism varies as the one-sixth power of the oxygen partial pressure


Arrhenius plot

In the extrinsic range, the cation self-diffusivity is controlled by the impurity content (e.g. Cr3+)

Slope = –(∆Hm+∆H/3)/R

Slope = –∆Hm/R ln

(*D

Fe)

1/T

Extrinsic range

( ) 61*

2OFe pD ∝


Phenomena related to defects in oxides

- Oxidation of metals parabolic kinetics is controlled by diffusion in the oxide layer

- Solid-state electrolytes e.g. electrolytes in Solid Oxide Fuel Cells

- Gas-sensing e.g. doped zirconia for oxygen sensors

- High-temperature superconductivity e.g. YBa2Cu3O7-x x ~ 0.07 for optimum superconducting properties

- etc.


References

These slides are based on the textbooks:

- Shewmon P., Diffusion in solids, The Mineral, Metals & Materials Society, 1989.

- Balluffi R.W., Allen S.M., Carter W.C., Kinetics of materials, Wiley, 2005.

Date post:	10-Jun-2015
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