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Entropy Production and Morphological Selection in Crystal Growth

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Beyond the Second Law

Part of the book series: Understanding Complex Systems ((UCS))

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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Notes

  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

    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.

  6. 6.

    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).

  7. 7.

    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.

  8. 8.

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

  9. 9.

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

  10. 10.

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

Abbreviations

D :

Diffusion coefficient

V :

Local velocity of the crystal

k, l :

Wave-numbers of perturbing modes

u :

Solute concentration

u 1 :

Equilibrium solute concentrations near the crystal surface

u 2 :

Density of the crystal

R :

Size of a crystal

R * :

Critical radii of nucleation of a crystal

R crit :

Critical size for instability of a crystal

R max :

Maximal size for instability of a crystal (spinodal)

R min :

Minimal size for instability of a crystal (binodal)

R minEP :

Minimum critical size, which was calculated using entropy production

α :

Dimensionless parameters characterizing the growth regime

β :

Coefficient of attachment kinetics

δ :

Initial amplitude of perturbation

Σ :

Local entropy production

Ω:

The equation of the crystal surface

C :

Cylindrical

S :

Spherical

Reference

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

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Author information

Authors and Affiliations

  1. Institute of Industrial Ecology, Russian Academy of Sciences, 20A Sophy Kovalevskaya Street, Ekaterinburg, Russia, 620219

    Leonid M. Martyushev

  2. Ural Federal University, 19 Mira Street, Ekaterinburg, Russia, 620002

    Leonid M. Martyushev

Authors
  1. Leonid M. Martyushev
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Corresponding author

Correspondence to Leonid M. Martyushev .

Editor information

Editors and Affiliations

  1. Australian National University, Research School of Biology, Canberra, Aust Capital Terr, Australia

    Roderick C. Dewar

  2. Australian National University, Research School of Astronomy and Astrophysics and the Research School of Earth Sciences, Canberra, Aust Capital Terr, Australia

    Charles H. Lineweaver

  3. School of Engineering and Information Technology, University of New South Wales at ADFA, Canberra, Aust Capital Terr, Australia

    Robert K. Niven

  4. School of Earth and Environment, University of Western Australia, CSIRO Earth Science and Resource Engineering, Crawley, West Australia, Australia

    Klaus Regenauer-Lieb

Glossary

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).

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Martyushev, L.M. (2014). Entropy Production and Morphological Selection in Crystal Growth. In: Dewar, R., Lineweaver, C., Niven, R., Regenauer-Lieb, K. (eds) Beyond the Second Law. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40154-1_20

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  • DOI: https://doi.org/10.1007/978-3-642-40154-1_20

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