The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid

Abstract

This manuscript is concerned with the theory of nucleation and evolution of a polydisperse ensemble of crystals in metastable liquids during the intermediate stage of a phase transformation process. A generalized growth rate of individual crystals is obtained with allowance for the effects of their non-stationary evolution in unsteady temperature (solute concentration) field and the phase transition temperature shift appearing due to the particle curvature (the Gibbs–Thomson effect) and atomic kinetics. A complete system of balance and kinetic equations determining the transient behaviour of the metastability degree and the particle-radius distribution function is analytically solved in a parametric form. The coefficient of mutual Brownian diffusion in the Fokker–Planck equation is considered in a generalized form defined by an Einstein relation. It is shown that the effects under consideration substantially change the desupercooling/desupersaturation dynamics and the transient behaviour of the particle-size distribution function. The asymptotic state of the distribution function (its ‘tail’), which determines the relaxation dynamics of the concluding (Ostwald ripening) stage of a phase transformation process, is derived. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

Ostwald ripening in the presence of simultaneous occurrence of various mass transfer mechanisms: an extension of the Lifshitz–Slyozov theory

“…To find such a 'tail', it is necessary to determine the particle size distribution function at the intermediate stage of the phase transformation, and then, considering its asymptotic behaviour at large times, determine the 'tail'. This 'tail' is found analytically in our previous papers [20][21][22] for supersaturated solutions and supercooled melts. Using the initial condition thus found for the distribution function, we can describe the process of formation of the universal state of a dispersed system based on a previously developed theory for the volume diffusion mechanism [23], as well as for the grain boundary diffusion, diffusion along the dislocations and reaction on the interface surface [24,25].…”

“…Taking this into account, Slezov [19] suggested analysing the 'tail' of the distribution function, which forms during the intermediate stage of a phase transformation and represents the initial condition at the beginning of the Ostwald ripening stage. Note that this 'tail' in the form of power-dependent and exponentially decaying functions is found in a sister paper [22]. Furthermore, we will use these analytical dependences (tails) as initial conditions at the Ostwald ripening stage.…”

Nucleation and growth dynamics of ellipsoidal crystals in metastable liquids

“…This can be done by analogy with the already developed theories of this stage for spherical crystals [70][71][72][73][74][75][76][77][78][79][80][81][82][83]. Third, the theory under consideration makes sense to expand with allowance for the shift in the phase transition temperature due to the Gibbs-Thomson effect and the attachment kinetics of atoms to the interphase boundaries of evolving crystals [84,85].…”

Transport phenomena in complex systems (part 1)

“…A complete analytical solution of an integro-differential model of kinetic and balance equations is found and analysed. These studies are continued by Makoveeva & Alexandrov [23]. Namely, three important effects (the shift in the phase transition temperature induced by the Gibbs-Thomson effects and atomic kinetics, as well as the effect induced by the non-stationary evolution of individual crystals) on the nonlinear dynamics of particulate assemblages in supercooled and supersaturated liquids is analysed.…”

“…The nonlinear and non-local transport phenomena in metastable media occurring as a result of nucleation and the evolution of a polydisperse ensemble of crystals are studied in the next papers [22,23]. When describing the growth of crystal ensembles from metastable solutions or melts, a significant deviation from a spherical shape is often observed, especially in biological systems (the crystallization of insulin and protein crystals).…”