Clusters solid-liquid equilibrium

One of the important issues addressed in our simulations is the character of clusters under study. Are these clusters solid or liquid rmder experimental conditions If they arc liquid, then the distribution wc observe in the pick-up and consequently in the photodissociation simulations corresponds to a statistical distribution at a. given temperature. If, however, the cluster is solid then both in the simulations and in the cxj)eriment we observe a quasi-stationary state with a very long lifetime rather than an equilibrium thermodynamical state. This question can be resolved by means of the instantaneous normal modes (INM) density of states (DOS) spectrum. To calculate INM DOS wc construct the Hessian matrix in a mass-weighted atomic Cartesian coordinate basis of N atoms with /r=. r, y, z. The 3N eigenvectors in the form Ci -.Cjj,Cj-,C2, C2/,C2-,.c.vj.,ca/j,c.v de-... 

Not all clusters meet these conditions. Some, such as Lia, pass readily among potential minima, even when they are in their ground states. Others possess potential surfaces which do not allow the clear distinction into two kinds of behavior. Nonetheless the equilibrium behavior of the phases of the cluster tells us that the freezing and melting temperatures are not logically linked, except in the limit of N - oo, and that the first-order solid-liquid phase... 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

The nanocrystals of such type form in various liquid media, such as organic solution [77, 81] or the softened quasi-liquid glass [82, 83], where there are no steric hindrances for the growth of equilibrium crystals without surface defects. At the same time, barriers for aggregation of clusters or atoms to metal nanocrystals in solid system that arises during the cryochemical solid-state synthesis favor the formation of crystals with structural defects,... 

The main feature of the nonequilibrium behavior of solutions dnring cryocrystallization is the appearance of amorphous solids. Generally vitrification of the liquid system depends on the rate of structural relaxation processes, which are substantially determined by the viscosity of the solution. At higher cooling rates and reduced temperatures, the cluster structure of the solution cannot follow the changes, predetermined by the equilibrium behavior of the system, so that even after solidification, the structure of the amorphous solid is very similar to the structure of the solution at low temperatnres. According to modem concepts, the amorphous state can be considered as a kind of snpercooled liqnid with an extremely high viscosity coefficient. 

This is a quite general equation. It does not depend on type (two- or three-dimensional), on state (solid or liquid), or on form (including non-equilibrium forms) or size of the cluster. K the clusters are crystalline, however, there are different ways of obtaining the final arbitrary shape with size S. Each one of these different paths is possible and, depending on the probabilities and Wau > contribute independently to the overall flux rate. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Figure 13. The cluster size dependence of the calculated binding energies per atom for a He) cluster (N = 6.5 x 103 to 1.88 x lO ) of radius R without a bubble (marked as cluster) and for a cluster with a bubble at the equilibrium electron bubble radius Rf, (marked as cluster + bubble). The experimental binding energy per atom in the bulk [232, 248], E /N = —0.616 meV (R, N = cxd), is presented (marked as bulk). Previous computational results for the lower size domain N = 128-728 [51-54, 106, 128, 129] are also included. The calculated data for the large (N = 10 —10 ) clusters (A = 6.5 x 1Q3 to 1.88 x 10 ), as well as the bulk value of Ec/N without a bubble, follow a linear dependence versus 1 /R and are represented by the liquid drop model, with the cluster size equation [Eq. (58)] (solid line). The dashed curve connecting the E /N data with a bubble was drawn to guide the eye. The calculated data for the smaller clusters (N = 128) manifest systematic positive deviations from the liquid drop model, caused by the curvature term, which was neglected.

The liquid-cluster system appears to have equilibrium metastable or stable heterophase states, in which the volume fraction of solid clusters may vary from negligibly small values up to unity. To find these states, the thermodynamics of the liquid-cluster systems will be considered. 

It follows from (6.57, 58) that pmix — /iL can have one or two minima at 0 < x < 1, thus one or two (stable or stable + metastable) states of supercooled liquids can exist. At high temperatures, T > T, and at low temperatures, T < fm, only one equilibrium state exists. If two equilibrium states coexist, they differ by the degree of clusterization. If a clusterized fraction is large enough, the state must be treated as solid one. Indeed, in the system at x > xc 0.16 an infinite (percolated) solid cluster is formed and at (1 — x) > (1 — x)c 0.16 a percolated liquid cluster appears. So, at x > xg 0.84 the mixed state is really a solid with heterophase liquid fluctuations. The temperature at which the stable state with x > xg exists, is the thermodynamic glassing temperature, Tgh. 

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