Nonlinear Effects in the Theory of Impurity Diffusion in Solids
Author(s):
Alekcandra Viktorovna Grezina
Lobachevsky State University of Nizhny Novgorod
candidate of physical and mathematical sciences
associate professor
aleksandra-grezina@yandex.ru
Vladimir Semenovith Metrikin
Lobachevsky State University of Nizhny Novgorod
candidate of physical and mathematical sciences
associate professor
v.s.metrikin@mail.ru
Adolf Grigoiyevith Panasenko
Lobachevsky State University of Nizhny Novgorod
candidate of physical and mathematical sciences
associate professor
a.g.panasenko@yandex.ru
Abstract:
We analyze the limit of applicability of the macroscopic
theory of impurity diffusion in solids based on Fick's first law,
with an additional assumption that the diffusion coefficient does not depend
on the impurity concentration. It is shown that this assumption contradicts
the law of conservation of impurity particles and is approximate and
may be used for the studies in the region of small concentration changes.
A non-contradictory new form of the diffusion law is found, which
leads to non-linear non-autonomous differential equations. New results
of numerical and analytical study of the differential equation with
fixed concentration at the boundary and with zero initial conditions
are presented for a one-dimensional problem (semi-infinite region).
An exact finite limit is found for the distance over which diffusing
particles travel in a given time. In classical macroscopic theory the distance
is infinite. An analytical solution for the non-autonomous nonlinear
differential equation with an additional simplification is obtained,
which differs greatly in its form from the classical one, but gives qualitatively
similar results. The presented results of numerical calculations of
impurity concentration which replace the known solutions obtained in
the classical theory make it possible to obtain the area of applicability
and improve the accuracy of calculations according to the classical theory.
Keywords
- diffusion in solid state materials
- mathematical model
- non-linear effects in diffusion
- non-linear non-autonomous differential equations
- numerical/analytical simulation
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