Curvature dependence of the surface tension and crystal nucleation in liquids

Abstract

To achieve a quantitative agreement of experimental data with theoretical predictions, in classical nucleation theory a curvature‐ or size‐dependence of the surface tension of critical clusters has to be accounted for. For its description, frequently the Tolman equation is chosen. Tolman derived his relation originally in application to droplets or bubbles in one‐component fluids assuming that nucleation is caused by variations of pressure. As shown here his approach and the resulting basic relations are applicable also to the description of crystal nucleation in multi‐component fluids if either pressure or temperature is changed. Estimates of the Tolman parameter in application to crystallization are advanced for both the mentioned cases. The Tolman parameter is shown to depend on the surface tension for a planar interface, the number of components in the liquid, the bulk properties of both the liquid and crystal phases, and the way the metastable state is generated. In addition, we develop a method of improving the precision in the specification of the curvature dependence of the surface tension in melt crystallization going beyond the Tolman equation in its original form. The results are applied successfully to the description of crystal nucleation in silicate glass‐forming melts.

Entropy and the Tolman Parameter in Nucleation Theory

“…In the approximation described by Equation 3, the Tolman parameter, δ ∞ , is, consequently, the main parameter determining the curvature dependence of the surface tension. For this reason, a large number of studies have been devoted to its determination [14][15][16][17][18][19][20][21][22]. It is a quantity that has to be defined for the respective states of the equilibrium coexistence of both phases at a planar interface.…”

“…As is evident from the above-given relations, the Tolman equation is an approximation valid, as a rule, for small deviations from thermodynamic equilibrium, only. For this reason, it may not be sufficiently accurate in order to describe nucleation proceeding with measurable rates only at sufficiently large supersaturations (see also [15]). An overview of the advantages and limitations of the Tolman equation in its application to condensation and boiling can be found in [12,13,[16][17][18][27][28][29].…”

“…As demonstrated there, an appropriate description of the curvature dependence of the surface tension in the analysis of the nucleation of droplets and bubbles requires an expansion of the Tolman equation. Such a generalization can be written in the form (see [15,28,29] and the references cited therein):…”

“…The second of the main aims of the present paper consists of the development of a thermodynamic method of determination of this second parameter, l ∞ , employing, similarly to the specification of δ ∞ , only directly-measurable data on the bulk state parameters of the coexisting phases and its comparison with alternative methods of the specification of its value. In [15], such a procedure was performed in the application to crystal nucleation, and here, we extend the method to condensation and boiling, advancing a new equation for the determination of the surface tension valid in the whole range of metastable initial states between the binodal and spinodal curves of the ambient fluid.…”

“…This is the reason for the success of the application of the Tolman equation in the interpretation of data on the nucleation of drops and bubbles. In detail, this topic was addressed in [15,29]. It is also the origin of the fact that experimental data on crystal nucleation can be explained by introducing the Tolman equation into the description [2,[30][31][32][33][34].…”