The quantitative relations between the density of germ nuclei, growth nuclei, and transformed volume are derived and expressed in terms of a characteristic time scale for any given substance and process. The density of germ nuclei diminishes through activation of some of them to become growth nuclei for grains of the new phase, and ingestion of others by these growing grains. The theory of the kinetics of phase change is developed with the experimentally supported assumptions that the new phase is nucleated by germ nuclei which already exist in the old phase, and whose number can be altered by previous treatment. These equations are coupled to the equation of heat diffuson by the heat production due to the release of latent heat of crystallization. In the case of two parallel confining walls (crystallization in a slab) the result can be transformed into a set of differential equations using a method originated by W. Using Poisson point processes with special intensity measures for the description of the nucleation processes and a deterministic law for the crystal growth we end with a generalized Kolmogoroff equation. We deal here with the influence of confining surfaces and the surface nucleation process occurring on them. One of the biggest advantages of Kolmogoroff's work is, that his results may easily be generalized to more complex situations, which are important in practical situations. Kolmogoroff used stochastic models which we call today Poisson point processes.
Kolmogoroff from 1937 in which the author already gave a formulation of crystallization kinetics in terms of time dependent (bulk) nucleation and crystal growth rate. Almost forgotten was a short paper by A.N. He had derived his results for isothermal conditions and many authors tried to generalize his results to non-isothermal situations without success, often creating new equations with dubious physical meaning. In most publications dealing with crystallization kinetics Avrami's work from 1939 is used. The results obtained for the single-needle crystal show a constant velocity growth, as expected from laboratory experiments. These methods are tested against exact and analytical results available in planar waves, faceted growth, and motion by mean curvature up to extinction time.
The two-dimensional calculations indicate that this efficient method for treating these stiff problems results in very accurate interface determination without interface tracking. The computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field equations that preserve the physically important parameters. By adjusting the parameters, the computations can be varied continuously from single-needle dendritic to faceted crystals. Also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists. The phase field equations can be used to compute a wide range of sharp interface problems including the classical Stefan model, its modification to incorporate surface-tension and/or surface kinetic terms, the Cahn–Allen motion by mean curvature, the Hele–Shaw model, etc.
#Meltys mathmod free
The parameters in the equations are related directly to the physical observables including the interfacial width $\epsilon $, which we can regard as a free parameter in computation.
#Meltys mathmod for free
The phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase (or “order”) parameter and temperature, respectively.
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