Kinetics of phase transformation on a bethe lattice

摘要

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 T_c (T>T_c),where T_c is the phase transformation (RPT) for small deviations from equilibrium is independent of time and tends to zero as (T-T_c).At T=T_c,the RPT depnds on time and for large times behaves as t~(-1).For T0.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 B_c of the external field,such that the phase transformation takes place only for B_f>B_c.This result is also obtained from a more simple consideration involving spherical-like clusters on a Bethe lattice.The characteristic time t_R at which the polarization becomes larger than zero diverges as (B_f-B_c)~(-b) for B_f->B_c with b=0.47.The RPT has a rapid growth near t_R and remains constant for t>t_R.The average cluster size (number of spins in a cluster) exhibits a rapid unrestricted growth at a time t_dapprox=t_R which indicates the creation of infinite clusters.The only exception to the latter behavior occurs when the kinetic 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.