Comparison of thermodynamic and kinetic models of single‐layer crystal–mother‐phase interface

Abstract

Simple analytical kinetic model of single-layer crystal-mother-phase interface is proposed, which provides results that can be compared directly with thermodynamic Jackson model. Both models are based on zeroth order approximation known from lattice-gas models, in which solid and fluid growth units are treated as mixed randomly in the interface layer. It is shown that the kinetic and thermodynamic approaches can lead to very similar predictions about growth mechanism. The parameters characterising growth conditions, obtained from these models, are significantly different from those obtained from Monte Carlo simulations applied to study stable states of single-layer interface. Monte Carlo calculations describe crystal growth in detail and their results can predict characteristic parameters for real experiment. Observed differences seem to be strongly influenced by the use of zeroth order approximation.

Kinetic block model of crystal–mother‐phase interface with preferential clustering ‐ single layer case

“…In the paper the authors proposed a kinetic model of crystal–mother‐phase interface in which the probability of attaching or detaching a solid block (creation or annihilation) was determined with the use of the Bernoulli formula. The obtained equation described changes of the earlier defined fraction of solid blocks x .…”

“…For ideal solutions, this parameter ‐ related to kinetic roughening ‐ can be directly connected with the relative supersaturation σ of the solution by the following expression: When supersaturation is small, this formula can be simplified: Unfortunately, values of both the parameters differ substantially when one compares the conditions of stable crystal growth yielded by thermodynamic models and Monte Carlo simulations. One way of solving that inconsistency was pursued in the paper , where the authors proposed a kinetic equation which could be solved analytically, eliminating the need for Monte Carlo simulations. That equation emerged from joining the fundamental assumptions proposed yet in the Jackson model with a simple probabilistic model.…”

“…The equation described changes of the quantity x with time, where x denoted the amount of solid blocks ( N S ) expressed as a fraction of the total number of blocks ( N ): The parameter x also appears in the thermodynamic models. In the approach proposed in the paper , we were only interested in solutions obtained after a very long time for different values of the α and β parameters. This let us determine which values of the parameters lead to stable crystal growth.…”

“…They were also compared with Monte Carlo simulations, where the results were significantly different. However, the model proposed in the paper involved many simplifications, similarly to other thermodynamic models. One of them is known as Bragg‐Williams or the zeroth‐order mean field approximation.…”

“…The new model proposed in this work also covers the case of crystal face growth from vapour, which is characterized by asymmetry of the changes in free energy of the system during attaching and detaching of a solid block to/from the crystal surface. This case is not included in the thermodynamic models known from the literature or the zeroth‐order kinetic model considered in our previous paper .…”