Chapter 4
Diffusion with a Chemical
Concentration Gradient
Up to this point, we have studied diffusion in mixtures of chemically identical
species and in dilute alloys. We have seen that impurity and host atoms can
have greatly different diffusivities which can depend on the concentration
of the impurity. We have not yet considered the effect of large chemical
concentration gradients. We now consider the general situation, allowing for
the presence of large concentration gradients of chemically dissimilar species.
We see that for ideal or dilute solutions, the diffusive flux is driven by the
entropy increase associated with mixing of the species, while for non-ideal
solutions, the diffusive flux is driven by the enthalpy of mixing as well. We
find that we are able to describe the intermixing of two species using Fick’s
laws with a single diffusion coefficient D̃, which depends on concentration
and the chemical nature of the two species. In order to do so, we examine
the intermixing in a diffusion couple formed by joining two rods of different
composition of two mutually soluble elements. There are two effects which
are important in this situation. The first, known as the Kirkendall effect, is
motion of the atomic planes due to the difference in diffusivities of the two
constituents. The second is the effect of the chemical driving force on the
diffusion.
99
100
4.1
CHAPTER 4. DIFFUSION WITH A CHEMICAL . . .
Kirkendall Effect
For the general case of a diffusion couple of two mutually soluble but chemically different elements, the difference in diffusivities of the two species can
give rise to motion of atomic planes in the interface region where diffusion
is occurring. For example, if in a A-B diffusion couple, the diffusivity of A
is greater than that of B, the flux of A across a given lattice plane will be
greater than the flux of B in the opposite direction. Thus there will be an
accumulation of atoms on one side of the reference plane, resulting in its
motion. In order to examine this phenomena, we define a position coordinate z which is measured from a region in the sample where no diffusion is
occurring. There will be two contributions to the atomic flux relative to this
fixed coordinate system.
• Flux due to diffusive motion of atoms relative to the moving atomic
planes.
• Flux due to the motion of the atomic planes.
The total flux of component A relative to a stationary observer is:
JA = −D̃A
∂cA
+ vcA
∂z (4.1)
where v is the velocity of the atomic planes relative to the fixed reference
frame z , and D̃A is the diffusivity of A in the presence of the concentration
gradient1 . This situation is analogous to the flux of ink in a moving stream.
If we travel in a canoe moving with the water in the stream, then we will
observe a drop of ink spreading in response to a diffusive flux:
−D̃A
∂cA
∂z
where the coordinate z is measured relative to the boat. But since z and z are related by z = z + vt and the composition gradient is independent of the
origin of the coordinate system, the diffusive flux is also given by:
−D̃A
1
∂cA
∂z This diffusivity is sometimes called the intrinsic or chemical diffusion coefficient. We
will see that D̃A reflects the chemical nature of the intermixing of the constituents.
4.1. KIRKENDALL EFFECT
101
An observer sitting on the bank will see this flux as well. However, he will also
see the ink moving by with the velocity of the water. This is an additional
flux vcA due to motion of the water.2
We now wish to find the velocity and see how the diffusion equation is
affected by this extra flux in Eqn. 4.1. We assume that the molar volume is
independent of composition so that:
c = cB + cA = constant
(4.2)
therefore:
∂cA ∂cB
∂c
=
+
=0
∂t
∂t
∂t
To find how the compositions cA and cB change with time, we apply the
conservation equation to both species:
∂JB
∂cB
= − ∂t
∂z
∂cA
∂JA
= − ∂t
∂z
and insert the flux from Eqn. 4.1:
(4.3)
∂cA
∂cB
∂
∂c
=
D̃A − vcA + D̃B − vcB
∂t
∂z
∂z
∂z
∂cA
∂cB
∂
D̃A + D̃B − cv = 0
=
∂z ∂z
∂z
Therefore we have that:
∂cA
∂cB
+ D̃B − vc = constant
(4.4)
∂z
∂z
To find the value of this constant, we recognize that in a region far away
from the interface, where no diffusion occurs, we know that:
D̃A
∂cB
∂cA
=
=0
∂z ∂z And, since there is no diffusion in this region, v = 0 also. Therefore the
constant in Eqn. 4.4 is zero. Solving Eqn: 4.4 for v, we find:
1
∂cA
∂cB
v=
D̃A + D̃B c
∂z
∂z
2
In fact, we have already seen this type of flux in the moving interface problem.
102
CHAPTER 4. DIFFUSION WITH A CHEMICAL . . .
Furthermore, from Eqn. 4.2 cA = c − cB so that:
∂cA
∂cB
=− ∂z
∂z
and the velocity is just:
v = D̃A − D̃B
1 ∂cA
c ∂z (4.5)
Thus we see that a difference in diffusivities of the two constituents will result
in a motion of atomic planes in the region where diffusion is occurring.
We can find the diffusion equation for this case by applying the conservation equation to the flux equation, just as we did in deriving Fick’s second
law. We insert the velocity from Eqn. 4.5 into the flux equation (Eqn. 4.1)
and apply the conservation equation (Eqn. 4.3) to find:
∂cA
∂cA
∂
= − −D̃A + vcA
∂t
∂z
∂z
c ∂c
∂cA ∂
A
A
= − −D̃A + D̃A − D̃B
∂z
∂z
c ∂z
∂
= − ∂z
∂
=
∂z −cA D̃A − cB D̃A + cA D̃A − cA D̃B
c
cB D̃A + cA D̃B
c
∂cA
∂z This is just Fick’s Second Law:
∂cA
∂cA
∂
= D̃ ∂t
∂z
∂z
with:
cB D̃A + cA D̃B
c
= xA D̃B + xB D̃A
D̃ =
where xi = ci /c is the atomic fraction i.
∂cA
∂z 4.2. CHEMICAL DRIVING FORCE
4.2
103
Chemical Driving Force
A second effect of a chemical concentration gradient on diffusion is to change
the nature of the driving force. This is because diffusion changes the bonding
in a solid. If, for example, two constituents in a diffusion couple have a
preference for bonding with unlike neighbors, that is, they have a negative
heat of mixing, then the decrease in free energy associated with diffusional
mixing will have an enthalpy contribution as well as the mixing entropy
contribution characteristic of ideal or dilute solutions. This added enthalpy
contribution will act as a driving force to increase intermixing. Conversely,
if the mixing enthalpy is positive, then the diffusional rate will be lower than
that for an ideal or dilute solution. In order to examine this effect we must
generalize Fick’s first law, the flux equation, by realizing that flux occurs as
a result of a system’s drive to approach thermodynamic equilibrium. With
this treatment, generally attributed to Darken, we can describe the mixing of
chemically dissimilar materials with a diffusion coefficient which is a function
of the chemical nature of the solution.
4.2.1
Generalized Flux Equations
Thermodynamic equilibrium is characterized by the absence of spatial or
temporal variations in temperature T , pressure P , external potentials φ, and
chemical potentials of the components µi . This condition does not always
mean the absence of concentration gradients. Hence it is more reasonable to
assert that the rate of return to equilibrium, that is, the flux of atoms, energy,
and defects, is proportional the deviation from equilibrium. Hence, to first
order, the flux will be proportional to gradients in temperature, pressure,
potential, and chemical potential, rather than just to composition gradients.
The flux of the ith component is given by:
Ji = −
Mik ∇µk − MiT ∇T − MiP ∇P − Miφ ∇φ
(4.6)
k
where the Mij ’s are the coupling coefficients between fluxes in i and gradients
in k. These coefficients reflect the strength driving the flux and the mobility
of the species in responding with movement.
In the case of a one-dimensional, isothermal, isobaric diffusion with no
external potential gradients of an alloy of two components with a vacancy
104
CHAPTER 4. DIFFUSION WITH A CHEMICAL . . .
mechanism we can write for the flux of the components:
∂µA
∂µB
∂µV
− MAB
− MAV
∂z
∂z
∂z
∂µA
∂µB
∂µV
= −MBA
− MBB
− MBV
∂z
∂z
∂z
∂µA
∂µB
∂µV
= −MV A
− MV B
− MV V
∂z
∂z
∂z
JA = −MAA
JB
JV
(4.7)
Vacancies can only be created or destroyed at sources or sinks such as surfaces
or defects. Hence, throughout most of the crystal, the number of lattice sites
is conserved, so that the fluxes of the three species which can reside on a
lattice site are related by:
JA + J B + J V = 0
If this is to be true for arbitrary gradients, the sum of the coefficients must
be zero, i.e.:
MAA + MBA + MV A = 0
MAB + MBB + MV B = 0
MAV + MBV + MV V = 0
In addition there is a set of reciprocity relations, known as the Onsager
relations, which state that Mij = Mji . Combining these with Eqn. 4.7 we
find:
∂
∂
(µA − µV ) − MAB
(µB − µV )
∂z
∂z
∂
∂
= −MBA (µA − µV ) − MBB
(µB − µV )
∂z
∂z
JA = −MAA
JB
4.2.2
Darken’s Flux Equation
To arrive at Darken’s flux equation, we must make the further assumptions
that the vacancies are in thermal equilibrium everywhere, so that µV = 0,
and that the off diagonal terms are negligible. The flux for a given component
then reduces to:
∂µi
Ji = −Mii
∂z
4.2. CHEMICAL DRIVING FORCE
105
The chemical potential for a given component can be written:
µi = µ0 (T, P ) + kB T ln ai
= µ0 + kB T (ln xi + ln γi )
(4.8)
where ai is the activity of component i, and γi is the activity coefficient of i,
defined as:
ai
γi =
xi
where xi is the atomic fraction of the ith component (xi = ci /c). The term
kB T ln x1 represents the ideal mixing entropy contribution, while the term
kB T ln γi deals with the non-ideality of the solution. For example, in considering the chemical potential of a system of vacancies, impurities, and
vacancy-impurity pairs, we only considered the ideal mixing entropy term
and found:
µideal
= µ0 + kB T ln xi
i
In this treatment, we are interested in deviations from ideality, and so must
use the more general expression for chemical potential Eqn. 4.8.
Our expression for the flux is then:
∂µi
∂z ∂ ln xi ∂ ln γi
= −Mii kB T
+
∂z
∂z
∂ ln xi ∂ ln γi ∂ ln xi
= −Mii kB T
+
∂z
∂ ln xi ∂z
−Mii kB T
∂ ln γi ∂ci
=
1+
ci
∂ ln xi ∂z
Ji = −Mii
(4.9)
(4.10)
Equation 4.10 relates the flux of a component to its concentration gradient,
and as such is a generalization of Fick’s first law. In order to examine the
relationship between Eqn. 4.10 and Fick’s first law, we consider the case of
an ideal solution where ai = xi , or the case of a dilute solution where the
activity follows Henry’s Law, that is ai = γi0 xi where γi0 = constant. In
either case, ∂ ln γi /∂ ln xi = 0 so that flux will be given by:
Ji = −Di
Mii kB T ∂ci
∂ci
=−
∂z
ci
∂z
106
CHAPTER 4. DIFFUSION WITH A CHEMICAL . . .
where Di is the tracer diffusivity of component i, and Di dci /dz is just the
Fick’s Law flux. The tracer diffusivity Di is the diffusion coefficient for the
constituent i which would be measured in a homogeneous alloy where the
only concentration gradients were in the relative concentration of i and a
chemically identical but distinguishable tracer i∗ . Hence, although Di is
affected by the alloying effects discussed in chapter 3, and as such can be
a function of composition, it does not reflect the presence of concentration
gradients of chemically dissimilar materials. In a pure material, the tracer
and self diffusivities are only different by the correlation factor. We see that
the tracer diffusivity is related to the mobility by:
Di =
kB T Mii
ci
We also note that Fick’s first law with the tracer diffusivity results from
considering only the ideal mixing entropy term in the chemical potential.
4.2.3
Relationship Between Tracer and Intrinsic Diffusivities
If we now return to the more general case of a nonideal, nondilute solution,
we can write:
∂ci
Mii kB T
Ji = −D̃i
=−
∂z
ci
From this we can find that:
D̃i
Mii kB T
=
ci
∂ ln γi
1+
∂ ln xi
∂ ln γi
1+
∂ ln xi
∂ ln γi
= Di 1 +
∂ ln xi
∂ci
∂z
(4.11)
This gives us a relationship between the tracer diffusivity, Di which is measured in dilute solution or by tracer diffusion in a otherwise homogeneous
alloy, and the intrinsic diffusivity, D̃i , which takes into account the effects of
a concentration gradient and nonideality of the solution.
If we write our expression for the flux in terms of the tracer diffusivities:
Ji = −Di
∂ci
∂ ln γi ∂ci
− Di
∂z
∂ ln xi ∂z
4.2. CHEMICAL DRIVING FORCE
107
we see that the first term in this expression comes from the concentration
driving force arising from the ideal entropy of mixing, and the second term
arises from the non-ideality of the solution.
4.2.4
Chemical Diffusion Coefficient
By combining the effect of the thermodynamic biasing with the results we
found by examining the Kirkendall effect of the moving atomic planes we can
find the chemical interdiffusion coefficient D̃ with which we can describe the
intermixing of two constituents which form a nonideal solution. Recall our
expression for D̃:
D̃ = xA D̃B + xB D̃A
where D̃A and D̃B are the intrinsic diffusivites of the two components. We can
find a relationship between the thermodynamic biasing of these diffusivities
by using the Gibbs-Duhem relation:
xA dµA + xB dµB = 0
(4.12)
Looking at our expression for µ:
µA = µ0 + kB T ln aA = µ0 + kB T (ln xA + ln γA )
we find:
xA dµA = kB T (dxA + xA d ln γA )
∂ ln γA
= kB T 1 +
dxA
∂ ln xA
Plugging this into the Gibbs-Duhem relation (Eqn 4.12) we find:
∂ ln γA
kB T 1 +
∂ ln xA
∂ ln γB
dxA + kB T 1 +
∂ ln xB
And since dxA = −dxB we can find that:
∂ ln γB
∂ ln γA
=
∂ ln xA
∂ ln xB
dxB = 0
108
CHAPTER 4. DIFFUSION WITH A CHEMICAL . . .
Plugging Eqn. 4.11 into our expression for the chemical diffusivity, D̃, and
using the above relation, we find:
D̃ = D̃A xB + D̃B xA
∂ ln γA
∂ ln γB
= DA x B 1 +
+ DB xA 1 +
∂ ln xA
∂ ln xB
∂ ln γA
= (DA xB + DB xA ) 1 +
∂ ln xA
This is our final expression relating the chemical diffusivity, D̃, which is a
measure of how a diffusion couple intermixes and is defined by Fick’s laws,
and the tracer diffusivities, Di , which measure the interdiffusion of dilute
or ideal solutions, and the non-ideality of the solution represented by the
activity coefficient, γi .
4.2.5
Regular Solution Example
As an example of the thermodynamic driving force for diffusion, we consider
a regular solution of N atoms, where the entropy of mixing is given by the
ideal solution mixing entropy:
∆Smix = −kB N (xA ln xA + xB ln xB )
and the non-ideality of the solution is represented by the enthalpy of mixing,
which in the quasichemical approximation is given by:
∆Hmix = xA xB nΩRS
where ΩRS is a measure of the strength of unlike bonds, and is given by:
Ω
RS
!
1
= z HAB − (HAA + HBB )
2
where z is the number of nearest neighbors, n is the total number of moles
of atoms, and Hij is the bond enthalpy per mole for i − j bonds. Here the
bonding enthalpy is negative for a stable bond, so the enthalpy of mixing
∆Hmix is negative for systems where the A-B bond is stable relative to A-A
and B-B bonds.