Solutions to the
Diffusion Equation
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Solutions to Fick’s Laws
• Fick’s second law, isotropic one-dimensional
diffusion, D independent of concentration
"c
"2 c
=D 2
"t
"x
Figure removed due to copyright restrictions.
See Figure 4.1 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.
Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.
Linear PDE; solution requires one initial
!
condition and two boundary conditions.
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Steady-State Diffusion
• When the concentration field is independent of
time and D is independent of c, Fick’s second
law is reduced to Laplace’s equation, "2 c = 0
For simple geometries, such as permeation
!
through a thin membrane, Laplace’s
equation can
be solved by integration.
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• (a)
Examples of steady-state profiles
Diffusion through a flat plate
Figure removed due to copyright restrictions.
See Figure 5.1 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.
Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.
(b)
Diffusion through a cylindrical shell
Figure removed due to copyright restrictions.
See Figure 5.2 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.
Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.
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Error function solution…
• Interdiffusion in two semi-infinite bodies
Figure removed due to copyright restrictions.
See Figure 4.2 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.
Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.
Solution can be obtained by a “scaling” method
that involves a single variable, " # x 4Dt
# x &
)c # x &
c(") = c%
( = c + erf %
(
$ 4Dt '
$ 4Dt '
! 2
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• erf (x) is known as the error function and is
2 x $% 2
defined by
erf x "
e d%
( )
• &
# 0
An example:
c( x < 0, 0!
) = 0;c( x > 0, 0) = 1;c("#,t ) = 0; c(#,t ) = 1; D = 10"16
t = 102, 103, 104, 105
!
• Application to
problems with
fixed c at surface
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• Movie showing time dependence of erf solution…
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Superposition of solutions
• When the diffusion equation is linear, sums of
solutions are also solutions. Here is an example
that uses superposition of error-function solutions:
Figure removed due to copyright restrictions.
See Figure 4.4 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.
Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.
Two step functions, properly positioned, can be
summed to give a solution for finite layer placed
between two semi-infinite bodies.
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• Superposed error functions, cont’d
The two step functions (moved left/right by Δx/2):
# x + "x 2 &,
c0 )
cleft = +1+ erf %
(.
$ 4Dt '2*
# x / "x 2 &,
c0 )
cright = / +1+ erf %
(.
$ 4Dt '2*
and their sum
!
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!
clayer
# x ) "x 2 &c0 * # x + "x 2 &
= ,erf %
( ) erf %
(/
$ 4 Dt '.
2 + $ 4Dt '
9
• Superposed error functions, cont’d
An example:
c("#,t ) = 0; c(#,t ) = 0; c( x $ "%x / 2, 0) = 0;c("%x / 2 < x < %x / 2) = 1;
c( x & %x / 2, 0) = 0;%x = 2 '10"6 ; D = 10"16
t = 101, 103, 104, 105
• Application to
problems with
zero-flux plane
at surface x = 0
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• Movie showing time dependence of superimposed
erf solutions…
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The “thin-film” solution
• The “thin-film” solution can be obtained from the
previous example by looking at the case where Δx
is very small compared to the diffusion distance,
x, and the thin film is initially located at x = 0:
N
# x 2 ( 4 Dt )
c( x,t ) =
e
4 "Dt
where N is the number of “source” atoms per unit
!
area initially
placed at x = 0.
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Diffusion in finite geometries
• Time-dependent diffusion in finite bodies can
soften be solved using the separation of variables
technique, which in cartesian coordinates leads to
trigonometric-series solutions.
A solution of the form
c( x, y, z,t ) = X ( x) "Y ( y) " Z ( z) " T (t )
is sought.
!
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• Substitution into Fick’s second law gives two
ordinary-differential equations for onedimensional diffusion:
dT
= "#DT
dt
d2X
= "#X
2
dx
where λ is a constant determined from the
boundary conditions.
!
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• Example: degassing a thin plate in a vacuum
c(0 < x < L, 0) = c0 ; c(0,t ) = c( L,t ) = 0
The function X(x) turns out to be the Fourier series
representation of the initial condition—in this
!
case, it is a Fourier sine-series representation of a
constant, c0:
)
&
#
x
Xn ( x) = * an sin%n" (
$ L'
n=1
with
!
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2c0
an =
L
L
$ #'
* sin&%n" L )( d#
0
15
• degassing a thin plate, cont’d
The function T(t) must have the form:
2 2
'
$
n
#
°
Tn (t ) = Tn exp&" 2 Dt )
% L
(
and thus the solution is given by KoM Eq. 5.47:
4c0
!
c( x,t ) =
"
2
* 1
#
x & # (2 j +1) " 2 &1,, 2 j +1 sin%$(2 j +1)" L (' exp%) L2 Dt (//
$
'.
j=0+
0
!
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• degassing a thin plate, cont’d
Example:
c(0,t ) = 0; c( L,t ) = 0; c(0 < x < L, 0) = 1
L = 10"5 ; D = 10"16
!
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Other useful solution methods
• • • • Estimation of diffusion distance from x " 4Dt
Superposition of point-source solutions to get
solutions for arbitrary initial conditions c(x,0)
!
Method of Laplace transforms
Useful for constant-flux boundary conditions,
time-dependent boundary conditions
Numerical methods
Useful for complex geometries, D = D(c), timedependent boundary conditions, etc.
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