Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.4, pp.321--352

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2000, Volume 3, Number 4, pp.321--352

Kinetic Approach to the Nucleation-and-Growth Phase Transition in Complex Systems.
Adam Gadomski

This study deals mostly with kinetic anomalies occurring in nucleation-and-growth phenomena in complex systems, e.g. polycrystals, partly ordered alloys, quasicrystalline assemblies or mesomorphs. The main kinetic phenomenological approach utilized in a versatile way is in fact an anomalous random walk approximation, though the process is not thought to be realized in a position space (a most expected case) but in the space of grains (clusters) sizes. Two effective descriptions of the processes are discussed. The first, in which a supreme role of capillary forces as the leading kinetic mechanism is proposed. The second, for which the Fick's law is fulfilled, and the effective area of the clusters, like in the diffusion-limited cluster cluster aggregation, is of prior importance. A novel contribution to the kinetic problem mentioned is offered which seems to be very suitable for revealing kinetic anomalies in such systems. It relies on assuming that the systems under study are not only statistically self-similar when looking at their distribution over the available physical space, but that the processes proceed also in a self-similar manner when carefully inspecting their time behaviour. Therefore, the basic kinetic coefficients characterizing the system's behaviour are often assumed to be inverse power law time-dependent, which is by the way the main assumption of the so-called dispersive or long-tailed kinetics frequently applied to, e.g. reactive and fluctuating soft-matter systems, like model biomembranes or polymers. The presented description, which unfortunately offers no explicit microscopic insight, is compared to the standard approaches of theoretical analysis of heterogeneous phase transformations as the Avrami-Kolmogorov (Mehl-Johnson) concept, utilized mostly in metallic polycrystals, or to some extent, the Mullins-Sekerka-like instability mechanism applied to biopolymers. The study is completed by a brief consideration of the propagation of mechanical stresses in a polycrystal along crystalline bundaries and some order-disorder effects manifested, for example, in lipid mesomorphs. A critical comparison with other available kinetic approaches has been done as well.
Key words: phase transitions; nucleation-and-growth, anomalous kinetics, complex systems, power laws

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