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2014 | OriginalPaper | Buchkapitel

20. Entropy Production and Morphological Selection in Crystal Growth

verfasst von : Leonid M. Martyushev

Erschienen in: Beyond the Second Law

Verlag: Springer Berlin Heidelberg

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Abstract

This chapter discusses morphological transitions during non-equilibrium crystallization and coexistence of crystals of different shapes from the viewpoint of the maximum entropy production principle (MEPP).

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Binodal

(binodal curve or coexistence curve) denotes the condition at which two distinct equilibrium or non-equilibrium phases may coexist. Beyond the binodal, the perturbations (or fluctuations) of phase will lead to phase transition. Before the binodal, the phase will be stable with respect to any perturbations (or fluctuations). For equilibrium phase transition, the binodal is defined by the condition at which the chemical potential is equal in each equilibrium phase. There is hypothesis that for non-equilibrium phase transition, the binodal is defined by the condition at which the entropy production is equal in each non-equilibrium phase.

Spinodal

(spinodal curve) denotes the boundary of absolute instability of equilibrium or non-equilibrium phases. Beyond the spinodal, infinitesimally small perturbations (or fluctuations) of phase will lead to phase transition. Before the spinodal, the phase will be at least stable or metastable with respect to perturbations (or fluctuations).

It is only in this approximation that the analytical solutions can be advanced sufficiently far.

Superscript “Max” refers the fact that this is the largest possible critical size; the actual critical size will be smaller under finite-amplitude perturbation.

Above this size the crystal itself growth and below this size it shrinks.

In line with the terminology accepted in the theory of equilibrium phase transitions, it is reasonable if the calculated minimum critical size R _C ^min is called, by analogy, a binodal, and the stability size R _C ^max , which was observed when the perturbation amplitude was almost zero, is termed a spinodal of a non-equilibrium transition.

This principle can be most generally formulated as follows [29‐31]: at each level of description, with preset external constraints, the relationship between the cause and the response of a non-equilibrium system is established such as to maximize the entropy production.

In line with the terminology used in the theory of equilibrium phase transitions, this curve is named as binodal (it separates the region, in which the phase is stable, from the region, in which it is metastable and unstable).

For example, one morphological phase implies the initial (spherical or cylindrical) form of growth, and the other phase means the initial form with some added harmonic.

These radii were rendered dimensionless to the critical radius of nucleation.

The results were similar for a spherical crystal [28].

Martyushev L. M. has also published under the alternate (French) spelling Martiouchev L.M.

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Titel: Entropy Production and Morphological Selection in Crystal Growth
verfasst von: Leonid M. Martyushev
Verlag: Springer Berlin Heidelberg
Buch: Beyond the Second Law
Print ISBN: 978-3-642-40153-4

Electronic ISBN: 978-3-642-40154-1

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-642-40154-1_20