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National Advisory Committee for Aeronautics, Technical Notes - The Theory of Diffusion in Strained Systems

Because the current theory of solid—state diffusion is limited to
unstrained crystals and cannot be applied readily to strained systems,
Fick‘s first and second laws were generalized to include the effects of
strain on the diffusion rates. The nonhomogeneity introduced into the
atomic jump frequency by strain was found to contribute strain—dependent
terms to the diffusion equations in addition to the terms containing the
concentration gradient.

From a consideration of the effect of strain on the free energy of
activation, it can be shown that for simple strains, such as those re-
sulting from compression, tension, shear, and hydrostatic pressure, the
diffusion coefficient is an exponential function of the lattice parameter.
An examination of the available experimental data for the variation of
diffusion coefficients with pressure confirms this theoretical prediction.

The theory presented herein states that the magnitude of the varia—
tion of the diffusion coefficient with pressure depends on the interatomic
forces as the diffusing atom moves from its equilibrium.position to the
activated position. On the basis of this theory, a parameter depending
upon the interatomic forces can be computed from the experimental data.

In all cases investigated, the magnitudes of this parameter were in agree-
ment with the known characteristics of the interatomic potential-energy
functions of the systems.

The effect of plastic flow on the diffusion rate was also studied
by considering the rate at which vacancies are produced by dislocation
motion and the rate at which vacancies condense at inhomogeneities in
the crystal. The resulting equations predict that for a vacancy mecha-
nism the diffusion coefficient varies linearly with the strain rate.
This conclusion is in agreement with experiment.

The theory of diffusion in solids has been the subject of a great
deal of investigation in recent years and satisfactory theoretical models
have been constructed that adequately describe the basic diffusion proc—
esses in many simple solids. Present theories, however, are limited to
unstrained crystals and are not strictly applicable to strained systems.
Since the diffusion rate is determined by the energy of interaction be—
tween the diffusing atom and the crystal lattice, and since this energy
depends on the interatomic distances, it is to be expected that the dif-
fusion coefficients‘will be altered by a strain superimposed on the crys-
tal.