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

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6.3 Steady-State Diffusion • 165

hence, the probability of interstitial atomic movement is greater than for vacancy

diffusion.

6.3 STEADY-STATE DIFFUSION

Diffusion is a time-dependent process—that is, in a macroscopic sense, the quantity

of an element that is transported within another is a function of time. Often it is

necessary to know how fast diffusion occurs, or the rate of mass transfer. This rate is

diffusion flux frequently expressed as adiffusion flux(J), defined as the mass (or, equivalently, the

number of atoms)Mdiffusing through and perpendicular to a unit cross-sectional

area of solid per unit of time. In mathematical form, this may be represented as

J=

M

At

Definition of (6.1a)

diffusion flux

whereAdenotes the area across which diffusion is occurring andtis the elapsed

diffusion time. In differential form, this expression becomes

J=

1

A

dM

dt

(6.1b)

The units forJare kilograms or atoms per meter squared per second (kg/m^2 -s or

atoms/m^2 -s).

If the diffusion flux does not change with time, a steady-state condition exists. One

steady-state diffusion common example ofsteady-state diffusionis the diffusion of atoms of a gas through

a plate of metal for which the concentrations (or pressures) of the diffusing species

on both surfaces of the plate are held constant. This is represented schematically in

Figure 6.4a.

When concentrationCis plotted versus position (or distance) within the solidx,

concentration profile the resulting curve is termed theconcentration profile;the slope at a particular point

xA xB

Position, x

Concentration of diffusing species,

C

(b)

(a)

CA

CB

Thin metal plate

Area, A

Direction of

diffusion of

gaseous species

Gas at

pressure PB

Gas at

pressure PA

PA > PB

and constant

Figure 6.4 (a)

Steady-state

diffusion across a

thin plate. (b)A

linear concentration

profile for the

diffusion situation

in (a).