The energetics of prenucleation clusters in lattice solutions

Abstract

According to classical nucleation theory, nucleation from solution involves the formation of small atomic clusters. Most formulations of classical nucleation use continuum "droplet" approximations to describe the properties of these clusters. However, the discrete atomic nature of very small clusters may cause deviations from these approximations. Here, we present a self-consistent framework for describing the nature of these deviations. We use our framework to investigate the formation of "polycube" atomic clusters on a cubic lattice, for which we have used combinatoric data to calculate the thermodynamic properties of clusters with 17 atoms or less. We show that the classical continuum droplet model emerges as a natural approach to describe the free energy of small clusters, but with a size-dependent surface tension. However, this formulation only arises if an appropriate "site-normalized" definition is adopted for the free energy of formation. These results are independently confirmed through the use of Monte Carlo calculations. Our results show that clusters formed from sparingly soluble materials (µM solubility range) tend to adopt compact configurations that minimize the solvent-solute interaction energy. As a consequence, there are distinct minima in the cluster-size-energy landscape that correspond to especially compact configurations. Conversely, highly soluble materials (1M) form clusters with expanded configurations that maximize configurational entropy. The effective surface tension of these clusters tends to smoothly and systematically decrease as the cluster size increases. However, materials with intermediate solubility (1 mM) are found to have a balanced behavior, with cluster energies that follow the classical "droplet" scaling laws remarkably well. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license

Experimental measurement of the diamond nucleation landscape reveals classical and nonclassical features

“…Thus, the nucleation barrier and critical size do not seem to originate from a geometric scaling competition between surface and volumetric energies, but instead appear to arise from either an interfacial energy or surface bonding energy that changes with particle size. These results highlight the importance of revising classical nucleation theory to include complex pathways with metastable molecular precursors that cannot be understood as microscopic clusters with bulk properties (6)(7)(8)(9)(10)34).…”

“…The carbon supersaturation in the plasma drives nucleation, and the nucleation barrier is determined by the plasma-nucleus interfacial energy, which is strongly influenced by the diamond surface termination (38). study of nucleation using model "polycube" clusters shows that size-dependent surface energies may be a key element of nucleation (7). Here, the critical increase in diamond nucleation probability could be explained if the nucleus-plasma interfacial energy abruptly decreases between a cluster size of 22 and 26 carbon atoms.…”

“…Recent theoretical advances (5-7) have extended classical nucleation theory, for example, by including both size and density as order parameters to describe nucleation pathways (6) or including size-dependent interfacial free energies (7). Multistep pathways, additional thermodynamic barriers, and the existence of distinct subcritical and postcritical regimes with differing scaling in nucleation rates and probabilities are key elements of these updated theories.…”

“…1 is only rigorously applicable to nucleation pathways with a single reaction coordinate and where particles grow deterministically upon reaching the critical size. More advanced models have examined how this equation will change under differing assumptions (6,7). By starting with monodisperse monolayers of diamondoid protonuclei, we initialize the system at specific points on the hypothetical nucleation energy landscape in Fig.…”

Supersaturated calcium carbonate solutions are classical

“…We use this reduced model to predict cluster size and shape distributions and the equilibrium constants between the interconversion of species as a function of concentration. Recent work suggests that cluster configurational entropy can significantly influence the size and shape distributions of small nucleation clusters ( 19 ), making efficient statistical sampling of the molecular configurations for concentrated CaCO 3 a necessity. In keeping with the assumption of a dilute limit, we neglect many-body correlations between the ions but retain both two-body correlation effects and the important water-ion correlations that are encoded in the PMFs ( Fig.…”