Diffusion Flux Quantities

Quantities involved in diffusion problems

()
Ji r 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;

Ci concentration of component i (mol m-3)
It is the number of moles of component i per unit volume.

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

Luca Nobili - Interfaces in Materials

Mass transport
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:
N i = J i + Ci v
v = velocity of the bulk motion

Civ Civ t=1s
A=1

v v
t=0

Phenomenological equations
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:
J1 = − L11 (∂µ1 ∂x ) − L12 (∂µ 2 ∂x ) − .... − L1n (∂µ n ∂x )
− L1q (dT dx ) − L1 p (dp dx ) − L1e (dφ dx )

Simplifying assumptions
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

J1 = − L11 (∂µ1 ∂ x )


Mobility
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
J1 = vC1 = M 1 F1C1 = − M 1C1 ∂µ1 ∂x
L11 = M 1C1


Fick’s Law
It is much easier to determine a concentration
gradient by experiment than a chemical potential
gradient; therefore, the Fick’s law is commonly used:
J i = − Di ∇Ci
∂Ci
In one-dimensional diffusion J i = − Di
∂x
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)

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

One-dimensional
diffusion
x x+dx
J1
J2

x x+dx


J1 J2
Ci
A
dx
∂J ∂Ci
(J1 − J 2 )A = dx ⋅ A ∂Ci − dx ⋅ A = dx ⋅ A
∂t ∂x ∂t

∂Ci ∂  ∂Ci 
= D  Fick’s second equation
∂t ∂x  ∂x 


∂Ci Three dimensions, rectangular
= −∇ J i
∂t coordinates

One dimension, constant D:
∂Ci ∂ 2 Ci Rectangular coordinates
=D 2
∂t ∂x
1 ∂ ∂Ci   ∂ 2Ci 1 ∂Ci  ∂Ci
D⋅r  = D 2 + = Cylindrical coordinates
r ∂r  ∂r   ∂r r ∂r  ∂t

1 ∂ 2 ∂Ci   ∂ 2Ci 2 ∂Ci  ∂Ci Spherical coordinates
D⋅r  = D 2 + =
r ∂r  ∂r   ∂r r ∂r  ∂t
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.
*C1+C1+C2 C1+C2
*C1+C1 C1
C1+C2
C1 C2
C2

C C
x x


Thermodynamic factor (1)
Flux of component i can be written with chemical
potential or concentration appearing as a source:
J i = − M i Ci ∂µi ∂x = − Di ∂Ci ∂x
The general expression of µi in a condensed phase is
µi = µi0 + RT ln ai = µi0 + RT ln(niγ i )
µi0 = chemical potential of i in the reference state
ai = activity of i
γi = activity coefficient of i
ni = atomic fraction of i



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
M i Ci RT [∂ ln (ni ) ∂x + ∂ ln (γ i ) ∂x ] = Di C ∂ni ∂x
∂x
Di = M i RTni [∂ ln(ni ) ∂x + ∂ ln(γ i ) ∂x]
∂ni
Di = M i RT [1 + ∂ ln (γ i ) ∂ ln (ni )]



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
In a binary diffusion couple with no concentration
gradient, the self-diffusion coefficient (*Di) is given by:
*
Di =*M i RT [1 + ∂ ln ( *γ i ) ∂ ln ( * ni )]
Since the stable and radioactive isotopes are chemically
identical, γi will be independent on the *ni/ni ratio and
will be constant, then
*
Di =*M i RT
If it is assumed that *Mi = Mi, a relationship is obtained
between intrinsic diffusivity and self-diffusivity:
Di =*Di [1 + ∂ ln (γ i ) ∂ ln (ni )]


Diffusion in a concentration gradient
C1+C2 C1+C2
C1

C1
C marker
x

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).

Interdiffusion experiments (1)
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:
L
J 1 = − D1 ∂C1 ∂x = − D1C ∂n1 ∂x
L
J 2 = − D2C ∂n2 ∂x
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:
L
J 1 + LJ 2 + LJ v = 0

The flux of vacancies will produce the marker shift, with velocity
Jv L L
J 1 + LJ 2 ∂n1
v= =− = (D1 − D2 )
C C ∂x
where the equation dn1=-dn2 is used.
The flux in the reference system is given by
L
J 1 + LJ 2
R
J 1 = LJ 1 + vC1 = LJ 1 − C1 = LJ 1 ⋅ n2 − LJ 2 ⋅ n1
C
∂C
R
J 1 = −(D1 ⋅ n2 + D2 ⋅ n1 ) 1
∂x
∂C2
R
J 2 = −(D1 ⋅ n2 + D2 ⋅ n1 )
∂x

The flux relative to specimen ends can be expressed according
to the Fick’ law by defining the interdiffusion coefficient
D = D1 ⋅ n2 + D2 ⋅ n1
Consequently, the concentration profile is obtained by solving
the diffusion equation
∂Ci ∂ 2 Ci
=D 2
∂t ∂x
The interdiffusion coefficient can be expressed in term of the
self-diffusion coefficients:
D = ( * D1n2 + *D2 n1 )[1 + ∂ ln (γ 1 ) ∂ ln (n1 )]

From the Gibbs-Duhem equation, ∂ ln (γ 1 ) ∂ ln (n1 ) = ∂ ln (γ 2 ) ∂ ln (n2 )


Solutions to the diffusion equation
constant D, steady-state
Rectangular coordinates, one dimension
d 2C x
D 2 =0 C ( x ) = C0 + (C L − C0 )
dx L

Cylindrical coordinates, one dimension
d  dC  C2 − C1  r 
r =0 C (r ) = C1 + ln 
dr  dr  ln (r2 r1 )  r1 
 

Spherical coordinates, one dimension
d  2 dC  r2 r − r1
r =0 C (r ) = C1 + (C2 − C1 ) ⋅
dr  dr  r r2 − r1


Solutions to the diffusion equation
constant D, non-steady-state
In rectangular coordinates and one-dimension, the equation is
∂C ∂ 2C
=D 2
∂t ∂x
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
∂C ∂ 2C
=D 2
∂t ∂x
C C

x x

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


Infinite system – constant concentration
L
C1 is constant
C1 t=0
(e.g. reactions with the atmosphere
produces a constant surface
C2 concentration)
x

C(x,0) = C2 C(0,t) = C1 C(∞,t) = C2

C1
C ( x, t ) − C1  x 
= erf  
C2 − C1  2 Dt 
C2


The error function

2
erf ( z ) =
z

∫ exp(−u 2 )du
π 0

1.2
erf (0 ) = 0 1.0

erf (∞ ) = 1
0.8

erf(z)
0.6

erf (2 ) = 0.9953 0.4

erf (− z ) = −erf ( z )
0.2
0.0
0 1 2 3
z


Solution for infinite systems
C1

C ( x, t ) − C1  x 
= erf  
C2 − C1  2 Dt  C2

x = 4 Dt

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):
2 Dt << L (4 Dt < L )

Interdiffusion in semi-infinite solids
2L C(x,0) = C2 x>0
C1 t=0 C(x,0) = C1 x<0
C(-∞,t) = C1
C2
C(∞,t) = C2
x=0

C ( x, t ) − C S  x 
C1

= erf 
 

C2 − C S  2 Dt 
CS = (C1 + C2 ) 2
C2


Finite system – constant concentration
2L

Ci C(x,0) = Ci x>0

t=0 C(L,t) = C(-L,t) = Cs
 
Cs ∂C
=0
x ∂x x =0

x=0 x=L

C ( x, t ) − C S 4  π 2 Dt   π x  Dt
= exp − ⋅ 2  cos ⋅  > 0.05
Ci − C S π  4 L   2 L
2
L
4 Dt > 0.9 L


Finite system – average concentration

Ci
1L
C C (t ) = ∫ C ( x, t )dx
L0
Cs average concentration

-L L

C (t ) − CS 8  π 2 Dt  Dt
= 2 exp − ⋅ 2  > 0.05
Ci − C S π  4 L  L2


Time evolution of the concentration profile
t1 t2 > t 1 t3 > t 2
Ci Ci Ci

Cs Cs Cs

-L L -L L -L L

Infinite Finite Finite
system system system
erf ( ) exp( )cos( ) 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.
Assuming that the jump frequency is the 1 2
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
1
J = (s1 − s2 )Γ
6 β
Γ = jump frequency
s1, s2 = diffusing atoms per unit
area in planes 1 and 2

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 1 2
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: x
1 2 β
D= β Γ
6
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.

a b c

b
Energy

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

 2  ∆S v + ∆S m   ∆H v + ∆H m 
D =  zl a ν exp  exp − 
  R   RT 
 ∆S + ∆S m 
D0 = zl a 2ν exp v 
 R 
Q = ∆H v + ∆H m


Interstitial diffusion

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

 2  ∆S m   ∆H m 
D =  zi a ν exp  exp − 
  R   RT 
 ∆S m 
D0 = zi a ν exp
2

 R 
Q = ∆H m


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
Hydrogen is either in traps (Ct) or perfect lattice sites (CL), then
the mass balance equation becomes
∂C ∂Ct ∂C L
= + = DL ∇ 2C L
∂t ∂t ∂t
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
dCt ∂C L ∂C L ∂C L DL
+ = DL ∇ C L
2
= ∇ 2C L
dC L ∂t ∂t ∂t 1 + dCt dC L
The effective diffusion coefficient can be defined as
DL
De =
1 + dCt dC L

Molecular hydrogen in internal voids
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
DL DL
De = =
1 + 2C L K g 1 + 2 Ct C L

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)
Under assumptions similar to those used to derive the McLean
isotherm, the equilibrium condition is
θt  ∆H b 
= θ L exp −  = θLK
1 − θt  RT 
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
KN t C L
Ct =
N L + KC L


Saturable traps (3)
Calculation of the derivative dCt/dCL and substitution in the
general expression of De give the following expression:
DLC L
De =
C L + Ct (1 − θ t )
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)
0,20

0,15

θt
0,10

0,05

0,00
ln D →

DL

De

1/T →


Kinetics of hydrogen release (1)
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
dCt dt = − A ⋅ Ct exp(− ∆H a RT )
where ∆Ha=∆Hb+∆Hm and ∆Hm is the activation energy for
lattice diffusion. When ∆Hb » RT, traps are said “irreversible”.
Enthalpy

∆Hm

∆Hb

Distance

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:
∆H a  ∆H a 
α = A exp − 
R (T p )2  
 RT p 
∆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.


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.

VK′
VCl•

K+ Cl-

Self-diffusivity in alkali halides (1)
The defect creation can be written as a reaction:
null = VK' + VCl
•

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:
[V ]⋅ [V ] = K
K
' •
Cl eq
= exp(− ∆GS RT )
the square brackets indicate a site fraction.


Self-diffusivity in alkali halides (2)
Electrical neutrality requires that [VK' ] = [VCl ]
•

Then [V ] = [V ] = exp(− ∆G
K
' •
Cl S
2 RT )

The self-diffusivity of K is given by:
*
DK = zl a 2 fν exp[(∆S S 2 + ∆S m ) R ]exp[− (∆H S 2 + ∆H m ) RT ]
CV CV

The activation energy for self-diffusion is then
Q = ∆H S 2 + ∆H m CV

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


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.

VAg′
Ag+ Br -
Agi•


Self-diffusion with interstitial cations (2)
The reaction of formation of cation Frenkel pairs is:
Ag × = Ag i• + VAg
Ag
'

The activity of the lattice cation is unity, then
[Ag ]⋅ [V ] = K
•
i
'
Ag eq = exp(− ∆GF RT )

The electrical neutrality requires that fractions of vacancies
and interstitials are equal:
[Ag ] = [V ] = exp[− (∆G 2) RT ]
•
i
'
Ag F

The activation energy for self-diffusivity of the Ag cations is
Q = ∆H F 2 + ∆H m
I


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.


Fractions of anionic and cationic vacancies are related to
the content of the extrinsic Ca++ impurity:
[Ca ]+ [V ] = [V ]
•
K
•
Cl
'
K

By inserting this relationship in the expression of the
equilibrium constant, the following equation is obtained:
[V ]⋅ ([V ]− [Ca ]) = exp(− ∆G
'
K
'
K
•
K S RT ) = V [ ]' 2
K pure

Solution to this equation is

[Ca ]  
[ ] 
12
4 VK pure  
' 2

[V ] = 2
•
' K 1 + 1 + 
K
 

  [ ] • 2  
Ca K 




There are two limiting cases for the behaviour of the cation
vacancy fraction:
Intrinsic [V ]
'
K pure [ ]•
>> Ca K [V ] = [V ]
'
K
'
K pure

Extrinsic [V ]
'
K pure << [Ca ]
•
K
[V ] = [Ca ]
'
K
•
K

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.


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

ln (*DK)
Slope = –(∆Hm+∆HS/2)/R

Intrinsic range
Slope = –∆Hm/R

Extrinsic range
1/T


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 Fe3+
O
O2- O2- O2- O2- O2-

O2- O2- O2- O2- O2- O2-

e Fe3+
Fe2+

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:
2 FeFe + 1 2O2 = 2 FeFe + OO + VFe
× • × ''

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
1 2O2 = 2hFe + OO + VFe
• × ''

hFe ≡ FeFe − FeFe
• • ×


Reaction 1 2O2 = 2hFe + OO + VFe
• × ''

Equilibrium [V ]⋅ [h ] = K
'' • 2

= exp(− ∆G RT )
(p )
Fe Fe
Constant 12 eq
O2

Neutrality
condition
[h ] = 2[V ]
•
Fe
''
Fe

The equilibrium fraction of cation vacancies is
13

[VFe'' ] =  1  ( pO )1 6 exp[− ∆G (3RT )]
 
4 2

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


Arrhenius plot

ln (*DFe)
Slope = –(∆Hm+∆H/3)/R

DFe ∝ ( pO )
* 16
2
Slope = –∆Hm/R

Extrinsic range
1/T

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


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.


Diffusion Flux Quantities

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