Collapse of Transient Nucleation Fluxes in a Cold Ising Ferromagnet

The classical theory of nucleation (1-3) views it as a one-dimensional random walk in the space of nuclei sizes and has an enormous number of applications from con- densation of vapors (4) to crystallization of glass-forming melts and glasses (5,6) and, more recently, to biological systems (7). The corresponding master equation, known as the Becker-Doring equation, has an attractive mathemati- cal structure (8). In steady state, there is an exact solution (2), and the flux, the ''nucleation rate,'' also can be ex- pressed asymptotically as the product of an exponential term determined by thermodynamics (1) and a kinetic preexponential (3) (similarly to the activation flux in the Kramers problem (9)). The preexponential can be refined to account for discrete number of monomers in a nucleus (10), making the asymptotic flux practically indistinguish- able from the exact one. In the time-dependent case, mo- ments of the transient flux can be obtained exactly—see, e.g., Refs. (11,12), and an asymptotic expression for the flux itself also is available (13). Efficient numerical schemes had been developed to solve the Becker-Doring equation under realistic experimental conditions (14). Despite the outstanding role of the classical approach, it is a phenomenological one and there remain fundamental questions with respect to its underlying assumptions. It is thus natural to seek additional insight from models where the description of nucleation is close to ''first principles,'' such as the the Ising model in external field (15-21) with various kinds of spin-flip dynamics. Rather surprisingly, even for this well-studied model, the status of the classical approach remains open, and below we associate this with inherent limitations of the stationary treatment. Transient nucleation was discussed in connection with analysis of high-temperature Monte Carlo simulations (22,23) as- suming the qualitative validity of the Becker-Doring pic- ture for the Ising model, but, otherwise, the thrust of most of the earlier studies had been the stationary nucleation rate Jst or the metastable-state lifetime, usually (depending on precise definition) proportional to 1=Jst. In the present Letter, the nonclassical transient nuclea- tion rates are obtained for the case of Metropolis dynamics which is of nonconservative, Glauber type. Since at low temperatures this can be done without additional strong assumptions (and without using the Monte Carlo methods), connections with the classical phenomenology, includ- ing the limitations of the latter, can be elucidated. The significance of time dependence is increased by the fact that the steady-state treatment allows multiple definitions of a ''nucleus.'' (This flexibility echoes the duality in splitting the nucleation rate into the exponential and pre- exponential factors (19(b)) and, more generally, the un- certainty of classical-type approximations to the rate if both factors are treated as fitting parameters (24).) The time scale associated with postnucleation growth also emerges naturally in the nonstationary approach, allow- ing one to estimate the duration of the entire phase transformation. The method is based on solving a time-dependent ki- netic equation in a multidimensional space of cluster con- figurations assuming noninteracting clusters which evolve due to random gain or loss of a spin. (More complicated events, such as coagulation, can be neglected at small T