Capillarity theory for the "coexistence" of liquid and solid clusters

The apparent coexistence of liquid and solid clusters of argon, observed in argon jets, is discussed in terms of a simple theory based on capillarity phenomena. A "coexistence" range is easily demonstrated, limited from above by the true melting temperature of the cluster (i.e., where the vapor pressures of solid and liquid clusters are equal) and from below by the temperature of "critical supercooling" for the nucleation of crystals in the melt. However, from the practical point of view the coexistence range is determined by the probabilities of observing either solid clusters (upper limit) or liquid clusters (lower limit). Coexistence is demonstrated to be not a thermodynamic phenomenon but one dependent on the existence of a free-energy barrier that lowers the rate of transition between solid and liquid and vice versa. One sort of cluster is in metastable equilibrium while the other is stable. Definition of the cluster "melting point" as the temperature for which the chances of observing liquid and solid clusters are equal is convenient only when the clusters are prevented from exchanging material with their surroundings, and, indeed, the vapor pressure of the liquid cluster exceeds that of the solid at this temperature (which must be lower than the true depressed melting temperature). Substitution of macroscopic thermodynamic parameters (density, surface tension, and heat of fusion) into the simple theory yields results in fairly good agreement with those obtained from molecular theory and simulation. Even the transition rate is consistent with the simple theory, but "magic numbers" cannot be derived unless the above-mentioned thermodynamic parameters are allowed to depend on both temperature and cluster size.