Thermodynamic and molecular-kinetic considerations of the initial growth of newly born crystals; crystal size distribution; Dissolution of small crystals during Ostwald ripening due to temperature changes

Abstract

This paper aims to present a comprehensive (rather than complete) review of recent studies and efforts to elucidate the initial growth of newly born crystals, their possible dissolution, and ripening due to temperature changes. It is argued that besides describing the birth of crystals, Gibbs’ thermodynamics also predetermines important features of the following crystal growth: the routes of initial crystal growth, dissolution, and ripening of nanocrystals are encoded in the negative branch of the dependence of the Gibbs’ free energy on crystal size. However, the growth and dissolution of crystals are inherently out of thermodynamic equilibria processes and cannot be established thermodynamically; the mechanism and kinetics of the crystallization process are determined by kinetic factors. (But this does not mean that the thermodynamics and the kinetics are opposed concept; rather they supplement each other.)

In this paper, key points of the crystallization theory have been revisited and further elucidated. At first, the initial growth of the just-born crystals has been considered from a thermodynamic point of view; an equation has been derived that quantifies the variation of the Gibbs’ thermodynamic potential with the change in the size of continuously growing crystals. Then, using a molecular-scale kinetic approach, the probabilities for attachment and possible detachment of molecules to/from just-born crystals have been calculated. It is thus shown that the probability of decomposition of super-critically sized crystals down to subcritical dimension is negligibly small already for crystals larger than the critical size by three molecules only.

This paper focuses on crystal ripening because, being the final crystallization stage, it determines the ultimate crystal size distribution - which is of significant interest. It is emphasized that, due to the relatively small driving energy and the diffusion-limited mass transfer, the isothermal Ostwald ripening is an extremely slow process - it proceeds for weeks or even months (therefore, the isothermal ripening does not find technological application). In contrast, with substances having temperature-dependent solubility ripening can be substantially accelerated under the impact of repeated changes in the temperature. The reason is that during the time of increased solubility, that is induced by the temperature change, the smallest crystals, which had been in equilibrium with the solution at the starting temperature, become under-critically sized and can dissolve faster than isothermally. So, to quantify the effect of the temperature changes on Ostwald ripening, the time needed for complete dissolution of small crystals (so small that they obey Gibbs–Thomson rule) is calculated; and since ripening takes place by diffusion of molecules, it has been assumed that the diffusion is the rate-determining step of the crystal dissolution (and growth) processes. This assumption has been supported by comparing the rates of diffusion- and kinetic-controlled crystal growth. Importantly, the equations derived for the time needed to completely dissolve small crystals may be helpful for the practice.

Of course, while the number of the crystals decreases during the ripening, ‘nourished’ from the dissolved solute, the surviving crystals grow larger; and in cases of prolonged processes, all the crystallizable solutes can be integrated in one crystal only (meaning the reachable minimum of crystal surface to volume energies). Therefore, to shed additional light on the ripening process, also the time point when only one big crystal has been grown is determined.