Simulative determination of kinetic coefficients for nucleation rates

Abstract

Nucleation kinetics can be formulated generally and rigorously as a set of master equations that govern the time evolution of the cluster distribution that underlies the observable rate process. However, this general formulation is only useful if the magnitudes of the coefficients that describe the loss and gain (evaporation and condensation) of molecules by a cluster are quantitatively known. Moreover, these coefficients can refer to multiple losses and gains of molecules (several molecules in a single step). In order to measure these coefficients accurately and efficiently, we have devised a molecular dynamics (MD) simulation that follows the development and equilibration of a single cluster in a small container (volume) that involves only a small number of molecules (in our case 216). There is evidence that such a system can provide a reliable picture of the behavior of a cluster in a larger system. This approach has been applied to supersaturated argon vapor at 85 K. In particular, we have been able to study the fluctuation in the size of the “equilibrium” cluster that develops in the small volume and, from these observations, to determine the evaporation and condensation coefficients. Besides yielding the values of these coefficients, the study has allowed us to establish several points, including the validity of detailed balance within the simulation, the importance of multimolecular losses and gains of molecules, and the intrinsic nature (nonimportance of the surrounding vapor) of the evaporation coefficients. Also, it is shown that the clusters disappear by a first order decay law, thus establishing the relevance of the linear form of the set of master equations that can be used to describe the nucleation process. It is also established, by our first estimates of the condensation coefficients, that they are an order of magnitude larger than those predicted by the simple molecular kinetic theory used in classical nucleation theory (CNT), suggesting the effects of the diffuse outer layers of the actual physical cluster and the role of the cluster’s attractive potential. In addition, we have performed an analysis, involving the statistics of correlation, that strongly supports the idea that multimolecule losses and gains experienced by a cluster are chiefly due to the departure and arrival of smaller “clusters.” Finally, we have modeled the nucleation process in the small system, using CNT, and have found that in many respects CNT provides a good account of the phenomena observed by means of MD. Because of the “intrinsic nature” of the evaporation coefficient, it is possible to perform the simulations at quite high levels of supersaturation, thereby accelerating the approach to equilibrium, and requiring less computer capacity. The evaporation coefficient of the “equilibrium cluster” that forms the object of our measurement is insensitive to the level of supersaturation of the surrounding medium. The condensation coefficient can then be determined by an application of the principle of detailed balance, once the equilibrium distribution of clusters in a particular nucleating system is known. Thus apart from our focus on evaporation and condensation coefficients, the small system appears to be useful in the modeling of nucleation phenomena in general.