Kinetics of phase transformation on a Bethe lattice.
A kinetic Ising model is applied to the description of phase transformations on a Bethe lattice. A closed set of kinetic equations for a model with the coordination number q=3 is obtained using a procedure developed in a previous paper. For T close to Tc(T>Tc), where Tc is the phase transition temperature, and zero external field (absence of supersaturation), the rate of phase transformation (RPT) for small deviations from equilibrium is independent of time and tends to zero as (T-Tc). At T=Tc, the RPT depends on time and for large times behaves as t(-1). For T<Tc, we examine the transformation from the initial state with almost all spins "down" to the state with almost all spin "up" after the external field jumped from Bi<0 to Bf>0. The role of different mechanisms responsible for growth (decay), splitting (coagulation), and creation (annihilation) of clusters are examined separately. In all cases there is a critical value Bc of the external field, such that the phase transformation takes place only for Bf>Bc. This result is also obtained from a more simple consideration involving spherical-like clusters on a Bethe lattice. The characteristic time tR at which the polarization becomes larger than zero diverges as (Bf-Bc)(-b) for Bf-->Bc with b=0.47. The RPT has a rapid growth near tR and remains constant for t>tR. The average cluster size (number of spins in a cluster) exhibits a rapid unrestricted growth at a time td approximately tR which indicates the creation of infinite clusters. The only exception to the latter behavior occurs when the kinetics is dominated by cluster growth and decay processes. In this case, the average cluster size remains finite during the transformation process. In contrast to the classical theory, the present approach does not separate the processes of creation of clusters of critical size (nucleation) and of their growth, both being accounted for by the kinetic equations employed.
Berim
Gersh O
GO
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260, USA.
Ruckenstein
Eli
E