Page 174 - Materials Science and Engineering An Introduction

146 • Chapter 5 / Diffusion

Solutions to this expression (concentration in terms of both position and time)
are possible when physically meaningful boundary conditions are specified.
Comprehensive collections of these are given by Crank, and Carslaw and Jaeger (see
References).
3
One practically important solution is for a semi-infinite solid in which the surface
concentration is held constant. Frequently, the source of the diffusing species is a gas
Tutorial Video:
Steady-State and phase, the partial pressure of which is maintained at a constant value. Furthermore, the
following assumptions are made:
Nonsteady-State
Diffusion 1. Before diffusion, any of the diffusing solute atoms in the solid are uniformly
What are the distributed with concentration of C 0 .
Differences between 2. The value of x at the surface is zero and increases with distance into the
Steady-State and solid.
Nonsteady-State
Diffusion? 3. The time is taken to be zero the instant before the diffusion process begins.
These conditions are simply stated as follows:

Initial condition
For t = 0, C = C 0 at 0 x
Boundary conditions

For t > 0, C C s (the constant surface concentration) at x 0
For t 0, C C 0 at x
Application of these conditions to Equation 5.4b yields the solution
Solution to Fick’s
second law for the
condition of constant C x - C 0 = 1 - erf a x b (5.5)
surface concentration C s - C 0 21Dt
(for a semi-infinite
solid) where C x represents the concentration at depth x after time t. The expression erf(x/ 21Dt)
4
is the Gaussian error function, values of which are given in mathematical tables for
various x/21Dt values; a partial listing is given in Table 5.1. The concentration pa-
rameters that appear in Equation 5.5 are noted in Figure 5.6, a concentration profile
taken at a specific time. Equation 5.5 thus demonstrates the relationship between con-
centration, position, and time—namely, that C x , being a function of the dimensionless
parameter x/1Dt, may be determined at any time and position if the parameters C 0 ,
C s , and D are known.
Suppose that it is desired to achieve some specific concentration of solute, C 1 , in an
alloy; the left-hand side of Equation 5.5 now becomes

C 1 - C 0
= constant
C s - C 0

3 A bar of solid is considered to be semi-infinite if none of the diffusing atoms reaches the bar end during the time
over which diffusion takes place. A bar of length l is considered to be semi-infinite when l 7 101Dt.
4 This Gaussian error function is defined by

z
2 2
erf (z) = 3 e -y dy
1p 0
where x/21Dt has been replaced by the variable z.