Page 171 - Materials Science and Engineering An Introduction

5.3 Fick’s First Law • 143

Motion of a host or Figure 5.3 Schematic representations of
substitutional atom (a) vacancy diffusion and (b) interstitial
diffusion.

Vacancy

Vacancy

(a)
Position of interstitial Position of interstitial
atom before diffusion atom after diffusion

(b)

such as hydrogen, carbon, nitrogen, and oxygen, which have atoms that are small enough
to fit into the interstitial positions. Host or substitutional impurity atoms rarely form inter-
stitials and do not normally diffuse via this mechanism. This phenomenon is appropriately
interstitial diffusion termed interstitial diffusion (Figure 5.3b).
In most metal alloys, interstitial diffusion occurs much more rapidly than diffusion
by the vacancy mode, because the interstitial atoms are smaller and thus more mobile.
Furthermore, there are more empty interstitial positions than vacancies; hence, the prob-
ability of interstitial atomic movement is greater than for vacancy diffusion.

5.3 FICK’S FIRST LAW
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 frequently
diffusion flux expressed as a diffusion flux (J), defined as the mass (or, equivalently, the number of
atoms) M diffusing through and perpendicular to a unit cross-sectional area of solid per
unit of time. In mathematical form, this may be represented as

Definition of M (5.1)
diffusion flux J = At

where A denotes the area across which diffusion is occurring and t is the elapsed diffusion
2
time. The units for J are kilograms or atoms per meter squared per second (kg/m # s or
2
atoms/m # s).
The mathematics of steady-state diffusion in a single (x) direction is relatively simple,
dC
Fick’s first law— in that the flux is proportional to the concentration gradient, through the expression
diffusion flux for dx
steady-state diffusion dC
(in one direction) J = -D (5.2)
dx