Here we consider a one-dimensional random walk diffusion of particles through a diffusion field which consists of particle sites, vacancy sites and dead sites. In general, numbers of those sites are independent one another. In our case, a step of diffusion is a movement of a particle from a particle site to a vacancy site and the Kirkendall effect is out of consideration.
Fig. 1 is a schematic diagram of a diffusion system. The notations are as follows: λ (distance between neighboring sites),
n (number of particles in sites at a certain
x position),
p (concentration of vacancies at a certain
x position), τ (mean transfer time when both sides of a particle became vacancy, mean transfer time of vacancy), j (particle flow to the positive direction per unit time), j (particle flow to the negative direction per unit time),
J (net particle flow per unit time).
The probability of the transfer of a particle to each side is 1/2 when both sides of the particle became vacancy. So that following equations are obtained.
j=
n(
p+
Δp)/2τ, j=(
n+
Δn)
p/2(τ+
Δτ),
J=j=-
pΔn/2τ+
nΔp/2τ+
npΔτ/2τ
2From the definitions of a particle concentration and a diffusion coefficient, following relations are obtained.
c=
n/λ,
Ddc/
dx=
pΔn/2τ=
DΔn/λ
2,
D=
pλ
2/2τ
Therefore,
J=-
Ddc/
dx+
Dc1/
pdp/
dx+
Dc1/τ
dτ/
dx (1st diffusion equation)
The 1st diffusion equation is a general form and consists of three terms generally independent one another, the concentration term, the vacancy (or probability) term and the kinetic term. If some conditions are fixed, however, derivatives are obtained as follows: (1) τ=coast.
D=
D0p (
D0…extreme diffusion coefficient or vacancy diffusion coefficient).
J=-
Ddc/
dx+
D0cdp/
dx=-
Ddc/
dx+
cdD/
dx (2nd diffusion eq.) This equation is applied to the analysis of cation interdiffusion in a solid solution with different concentration of cation vacancy, (2)
p=const.
J=-
Ddc/
dx-
cdD/
dx (3rd diffusion eq.) This equation may have some way of application. (3)
p=const. τ=const.
J=-
Ddc/
dx (4th diffusion eq.) This is the simple diffusion equation and applied to the analysis of diffusion with a definite concentration of vacancy. (4)
p=
f(
c), τ=
g(
c)
J=-
Dadc/
dx=-
D{1-
c/
f(
c)
df(
c)/
dc-
c/
g(
c)
dg(
c)/
dc}
dc/
dx (5th diffusion eq.) This equation explains the meaning of the apparent diffusion coefficient
Da. (5)
p=1-
c/
c0, τ=const.
J=-
D0dc/
dx (6th diffusion eq.) This equation is applied to the analysis of the diffusion with a gradient of a vacancy concentration and shows that the apparent diffusion coefficient of such a diffusion is the extreme diffusion coefficient.
C. Wagner obtained an interdiffusion coefficient of cations in a solid solution with different concentration of cation vacancies using the simple diffusion equation but in such a case the 2nd diffusion equation must be applied.
J1=-
D1dc1/
dx+
c1dD1/
dx-
z1c1D1Kdφ/
dx
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