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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).