Insights into phase transitions from phase changes of clusters

Abstract

The phases and phase transitions of bulk matter differ in several important ways from the phases or phase-like forms of small systems, notably atomic and molecular clusters. However, understanding those differences gives insights into the nature of bulk transitions, as well as into understanding the behaviour of the small systems. Small systems exhibit dynamic phase equilibria, large fluctuations and size-dependent behaviour in ways one cannot see with macroscopic systems. The Gibbs phase rule loses its applicability, the distinction between first-order and second-order transitions reveals a size dependence, and one can see phases in equilibrium (as minority populations) that are never the most stable thermodynamically. Introduction: phases of clusters and of macroscopic systemsThis review addresses the relations between phases and phase changes exhibited by bulk, macroscopic systems and their counterparts in small systems, atomic and molecular clusters and nanoscale particles. The differences between the phase behaviour of bulk matter and of clusters can, for the most part, be attributed to two factors. The more important is the enormous free energy differences between phases of bulk matter, which make only the most favoured phase observable, compared with the small free energy differences for clusters, which make less favoured phases nearly as observable as more favoured or most favoured phases [1]. The other is the relative contribution of the surface in a macroscopic and a small system. In a d-dimensional system containing N atoms, approximately N (d−1)/d of atoms will be on the surface; for large N the relative fraction N −1/d is very small and the surface effects on the bulk properties of the system are commonly neglected. In contrast, a high proportion of atoms or molecules are in surface layers of clusters which makes the surface as important for many cluster properties as the interior part is. In the spherical approximation used above to estimate the surface contribution one finds that about one half of the particles are on the surface of a cluster containing 500 particles. Because the ratio of surface area to bulk is large for small

Thermodynamic Stability of Transition States in Nanosystems

“…In Ref. [1,2] a fraction of dynamically coexisting phases was added as the additional coordinate; in the present research we find that system size serves better as the additional coordinate of a phase diagram. Summarizing the calculations presented above, we can construct a size-volume phase diagram of a closed system capable of undergoing a solid/liquid phase transition and stabilizing the TS, see Fig.…”

“…OP is a low-dimensional characteristic of a particular transformation in a multi-dimensional space. 2 The transformation is fully characterized by the coarse-grained free energy, which may be significantly simplified by taking into account all the symmetries of the system. Introduction of the OP allows one to define a phase as a locally stable homogeneous in the OP state of the system.…”

“…This means that an equilibrium state of an open system (P = const) remains at equilibrium if the system is closed, (2). The local stability of the state, however, may change drastically as a result of the change of the boundary conditions (open to closed system).…”

“…[1,2], "one cannot represent phase equilibrium of (nanostructures) with conventional, two-dimensional phase diagrams. One needs another dimension."…”

“…icosahedral [7,8]. Distinction between first-order and second-order transitions in clusters is blurred [1,2]. Researchers have found that a trajectory of a cluster through the phase space {r t ,ṙ t } has a high probability of absorbing entropy instead of producing it-fluctuation theorems [9,10].…”