Atomic diffusion theory challenging the Cahn-Hilliard method
Résumé
Our development of the self-consistent mean-field (SCMF) kinetic theory for nonuniform alloys leads to
the statement that kinetic correlations induced by the vacancy diffusion mechanism have a dramatic effect
on nanoscale diffusion phenomena, leading to nonlinear features of the interdiffusion coefficients. Lattice rate
equations of alloys including nonuniform gradients of chemical potential are derived within the Bragg-Williams
statistical approximation and the third shell kinetic approximation of the SCMF theory. General driving forces
including deviations of the free energy from a local equilibrium thermodynamic formulation are introduced. These
deviations are related to the variation of vacancy motion due to the spatial variation of the alloy composition.
During the characteristic time of atomic diffusion, multiple exchanges of the vacancy with the same atoms may
happen, inducing atomic kinetic correlations that depend as well on the spatial variation of the alloy composition.
As long as the diffusion driving forces are uniform, the rate equations are shown to obey in this form the
Onsager formalism of thermodynamics of irreversible processes (TIP) and the TIP-based Cahn-Hilliard diffusion
equation. If now the chemical potential gradients are not uniform, the continuous limit of the present SCMF
kinetic equations does not coincide with the Cahn-Hilliard (CH) equation. In particular, the composition gradient
and higher derivative terms depending on kinetic parameters add to the CH thermodynamic-based composition
gradient term. Indeed, a diffusion equation written as a mobility multiplied by a thermodynamic formulation
of the driving forces is shown to be inadequate. In the reciprocal space, the thermodynamic driving force has
to be multiplied by a nonlinear function of the wave vector accounting for the variation of kinetic correlations
with composition inhomogeneities. Analytical expressions of the effective interdiffusion coefficient are given
for two limit behaviors of the vacancy, the latter treated as either a conservative species (fixed concentration) or
a nonconservative species (time-dependent equilibrium concentration). Relying on the same vacancy diffusion
model, we perform kinetic Monte Carlo simulations starting from a sinusoidal composition modulation in binary
model alloys, with no interaction or nearest-neighbor interactions leading to clustering or ordering tendencies,
along the [100] crystallographic direction of a body centered cubic (bcc) lattice. The resulting temporal variation
of the modulation amplitude is compared to the corresponding SCMF equations. Qualitative and satisfying
quantitative agreements systematically strengthen our theoretical conclusions. The model alloys are shown to be
representative enough of some real alloys, so that one may expect these new heterogeneous correlation effects to
be non-negligible in these alloys.