Entropy Production and Morphological Selection in Crystal Growth
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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Entropy Production and Morphological Selection in Crystal Growth Leonid M. Martyushev
Abstract This chapter discusses morphological transitions during nonequilibrium crystallization and coexistence of crystals of different shapes from the viewpoint of the maximum entropy production principle (MEPP). Notation Symbol Meaning Roman Symbols D Diffusion coefficient V Local velocity of the crystal k, l Wave-numbers of perturbing modes u Solute concentration u1 Equilibrium solute concentrations near the crystal surface u2 Density of the crystal R Size of a crystal R* Critical radii of nucleation of a crystal Rcrit Critical size for instability of a crystal Rmax Maximal size for instability of a crystal (spinodal) Rmin Minimal size for instability of a crystal (binodal) Rmin Minimum critical size, which was calculated using entropy production EP Greek Symbols a Dimensionless parameters characterizing the growth regime b Coefficient of attachment kinetics
L. M. Martyushev (&) Institute of Industrial Ecology, Russian Academy of Sciences, 20A Sophy Kovalevskaya Street, Ekaterinburg, Russia 620219, e-mail: [email protected] L. M. Martyushev Ural Federal University, 19 Mira Street, Ekaterinburg, Russia 620002,
R. C. Dewar et al. (eds.), Beyond the Second Law, Understanding Complex Systems, DOI: 10.1007/978-3-642-40154-1_20, Ó Springer-Verlag Berlin Heidelberg 2014
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d R X
L. M. Martyushev
Initial amplitude of perturbation Local entropy production The equation of the crystal surface
Subscripts C Cylindrical S Spherical
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). We advance the following hypothesis: a necessary condition for a morphological transition is a larger entropy production in the final non-equilibrium phase, and equality of the entropy productions of two non-equilibrium phases is a condition for their coexistence. The basic results are as follows. (1) We explain the experimentally observed phenomenon of coexistence of morphological phases during non-equilibrium crystallization by metastability of crystallization growth regimes. (2) We prove by analytical and numerical methods that metastable regions limited by a minimum size (binodal) and a maximum size (spinodal) exist for simple morphological transitions during non-equilibrium crystallization. (3) Considering an agreement between analytical and numerical calculations, we infer that the maximum entropy production principle allows finding the binodal of a morphological transition during non-equilibrium crystallization.
20.1 Morphological Stability and the Coexistence of Morphological Phases In the non-equilibrium phenomenon of crystal growth from a supersaturated/ supercooled solution/vapor/melt, a regular-shaped nucleus starts distorting and transforming into, for example, dendrite structures upon reaching a certain size. The formation of snowflakes (ice crystals) from supersaturated
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