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

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6.4 Nonsteady-State Diffusion • 169

Distance from interface, x

Concentration,

C

Cx – C 0

C 0

Cx

Cs

Cs – C 0

Figure 6.6 Concentration profile for

nonsteady-state diffusion; concentration

parameters relate to Equation 6.5.

relationship between concentration, position, and time—namely, thatCx, being a

function of the dimensionless parameterx/

√

Dt, may be determined at any time and

position if the parametersC 0 ,Cs, andDare known.

Suppose that it is desired to achieve some specific concentration of solute,C 1 ,in

an alloy; the left-hand side of Equation 6.5 now becomes

C 1 −C 0

Cs−C 0

=constant

This being the case, the right-hand side of this same expression is also a constant,

and subsequently

x

2

√

Dt

=constant (6.6a)

or

x^2

Dt

=constant (6.6b)

Some diffusion computations are thus facilitated on the basis of this relationship,

as demonstrated in Example Problem 6.3.

EXAMPLE PROBLEM 6.2

Nonsteady-State Diffusion Time Computation I

For some applications, it is necessary to harden the surface of a steel (or iron-

carbon alloy) above the hardness of its interior. One way this may be accom-

plished is by increasing the surface concentration of carbon in a process termed

carburizing carburizing;the steel piece is exposed, at an elevated temperature, to an atmo-

sphere rich in a hydrocarbon gas, such as methane (CH 4 ).

Consider one such alloy that initially has a uniform carbon concentration of

0.25 wt% and is to be treated at 950◦C (1750◦F). If the concentration of carbon

at the surface is suddenly brought to and maintained at 1.20 wt%, how long

will it take to achieve a carbon content of 0.80 wt% at a position 0.5 mm below

the surface? The diffusion coefficient for carbon in iron at this temperature is

1.6× 10 −^11 m^2 /s; assume that the steel piece is semi-infinite.