Entropy and the Tolman Parameter in Nucleation Theory

Abstract

Thermodynamic aspects of the theory of nucleation are commonly considered employing Gibbs’ theory of interfacial phenomena and its generalizations. Utilizing Gibbs’ theory, the bulk parameters of the critical clusters governing nucleation can be uniquely determined for any metastable state of the ambient phase. As a rule, they turn out in such treatment to be widely similar to the properties of the newly-evolving macroscopic phases. Consequently, the major tool to resolve problems concerning the accuracy of theoretical predictions of nucleation rates and related characteristics of the nucleation process consists of an approach with the introduction of the size or curvature dependence of the surface tension. In the description of crystallization, this quantity has been expressed frequently via changes of entropy (or enthalpy) in crystallization, i.e., via the latent heat of melting or crystallization. Such a correlation between the capillarity phenomena and entropy changes was originally advanced by Stefan considering condensation and evaporation. It is known in the application to crystal nucleation as the Skapski–Turnbull relation. This relation, by mentioned reasons more correctly denoted as the Stefan–Skapski–Turnbull rule, was expanded by some of us quite recently to the description of the surface tension not only for phase equilibrium at planar interfaces, but to the description of the surface tension of critical clusters and its size or curvature dependence. This dependence is frequently expressed by a relation derived by Tolman. As shown by us, the Tolman equation can be employed for the description of the surface tension not only for condensation and boiling in one-component systems caused by variations of pressure (analyzed by Gibbs and Tolman), but generally also for phase formation caused by variations of temperature. Beyond this particular application, it can be utilized for multi-component systems provided the composition of the ambient phase is kept constant and variations of either pressure or temperature do not result in variations of the composition of the critical clusters. The latter requirement is one of the basic assumptions of classical nucleation theory. For this reason, it is only natural to use it also for the specification of the size dependence of the surface tension. Our method, relying on the Stefan–Skapski–Turnbull rule, allows one to determine the dependence of the surface tension on pressure and temperature or, alternatively, the Tolman parameter in his equation. In the present paper, we expand this approach and compare it with alternative methods of the description of the size-dependence of the surface tension and, as far as it is possible to use the Tolman equation, of the specification of the Tolman parameter. Applying these ideas to condensation and boiling, we derive a relation for the curvature dependence of the surface tension covering the whole range of metastable initial states from the binodal curve to the spinodal curve.

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

“…In CNT, the thermodynamic driving force of crystallization, , is expressed commonly in a first approximation via the difference of the bulk contributions to the Gibbs free energy between liquid and crystal per unit volume of the crystal phase [ 3 , 4 , 19 , 30 ] both taken at the pressure, p , and temperature, T , of the liquid phase, i.e., with …”

“…By this reasoning, this general relation is transformed into expressions allowing one to compute directly by knowing the parameters ( ) of some particular macroscopic equilibrium state liquid-crystal and the respective deviations of temperature, , and pressure, , from it. Following such ideas, the dependence of thermodynamic driving force of crystallization on temperature and pressure can be described in a good approximation in the form (for details see [ 4 , 19 , 30 , 58 ]) …”

“…Clusters of the newly evolving phase are assumed to be of spherical shape and surface contributions to the thermodynamic quantities are described by the surface tension. The theoretical foundation of the possibility to treat critical crystal nucleation (critical crystals may have, in general, quite different shapes and the relations describing the surface contributions to the thermodynamic potentials are, in general, much more complex) in terms of such simplified model is described in detail in [ 18 , 19 ]. This model we will use in the present study.…”

“…It has the same value for both nucleation and growth processes, i.e., the possibility that critical clusters have properties different from the properties of the evolving crystalline phase is, as a rule, not accounted for. Corrections in the theoretical description of crystal nucleation rates are introduced in CNT into the description exclusively via curvature corrections to the surface tension [ 18 , 19 ]. In contrast, employing the generalized Gibbs approach [ 4 , 21 , 22 , 23 ], the bulk properties of the critical clusters are also shown to deviate, as a rule, significantly from the properties of the newly evolving macroscopic phases and can be appropriately accounted for.…”

“…Once the expressions for the thermodynamic driving force are considered to be widely correct, the only method which can be used in CNT in order to reconcile experimental data with theoretical predictions consists in the introduction of a size dependence of the surface tension [ 16 , 21 , 53 , 54 , 55 , 56 ] of the critical clusters (or, equivalently, a dependence of the surface tension on pressure and temperature determining the degree of metastability of the melt, respectively, the thermodynamic driving force). Based on the Stefan–Skapski–Turnbull rule, in recent papers [ 18 , 19 , 30 , 57 , 58 ] expressions for the dependence of the surface tension on both external control parameters have been derived. These relations are utilized in the present analysis to arrive at expressions for the derivatives of the work of critical cluster formation with respect to the thermodynamic driving force.…”