Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

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168 • Chapter 6 / Diffusion

Table 6.1 Tabulation of Error Function Values

z erf(z) z erf(z) z erf(z)

0 0 0.55 0.5633 1.3 0.9340

0.025 0.0282 0.60 0.6039 1.4 0.9523

0.05 0.0564 0.65 0.6420 1.5 0.9661

0.10 0.1125 0.70 0.6778 1.6 0.9763

0.15 0.1680 0.75 0.7112 1.7 0.9838

0.20 0.2227 0.80 0.7421 1.8 0.9891

0.25 0.2763 0.85 0.7707 1.9 0.9928

0.30 0.3286 0.90 0.7970 2.0 0.9953

0.35 0.3794 0.95 0.8209 2.2 0.9981

0.40 0.4284 1.0 0.8427 2.4 0.9993

0.45 0.4755 1.1 0.8802 2.6 0.9998

0.50 0.5205 1.2 0.9103 2.8 0.9999

One practically important solution is for a semi-infinite solid^2 in which the surface

concentration is held constant. Frequently, the source of the diffusing species is a gas

phase, the partial pressure of which is maintained at a constant value. Furthermore,

the following assumptions are made:

1.Before diffusion, any of the diffusing solute atoms in the solid are uniformly

distributed with concentration ofC 0.

2.The value ofxat the surface is zero and increases with distance into the solid.

3.The time is taken to be zero the instant before the diffusion process begins.

These boundary conditions are simply stated as

Fort= 0 ,C=C 0 at 0≤x≤∞

Fort> 0 ,C=Cs(the constant surface concentration) atx= 0

C=C 0 atx=∞

Application of these boundary conditions to Equation 6.4b yields the solution

Cx−C 0

Cs−C 0

= 1 −erf

(

x

2

√

Dt

)

(6.5)

Solution to Fick’s

second law for the

condition of constant

surface

concentration (for a

semi-infinite solid) whereCxrepresents the concentration at depthxafter timet. The expression

erf(x/ 2

√

Dt) is the Gaussian error function,^3 values of which are given in mathe-

matical tables for variousx/ 2

√

Dtvalues; a partial listing is given in Table 6.1. The

concentration parameters that appear in Equation 6.5 are noted in Figure 6.6, a

concentration profile taken at a specific time. Equation 6.5 thus demonstrates the

(^2) 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 lengthlis considered to
be semi-infinite whenl> 10
√
Dt.
(^3) This Gaussian error function is defined by
erf(z)=
2
√
π
∫z
0
e−y
2
dy
wherex/ 2
√
Dthas been replaced by the variablez.