Cluster free energy

Miller, M. A. Reinhardt, W. P., Efficient free energy calculations by variationally optimized metric scaling concepts and applications to the volume dependence of cluster free energies and to solid-solid phase transitions, J. Chem. Phys. 2000,113, 7035-7046... 

Figure 19.3 Solid curves show cluster free energy AC/a vs. cluster size A/, (a)...

The cluster free energy, as a function of number of molecules i within the cluster, is shown for the typical case of ion-induced nucleation in Figure 11.14. The free-energy curves are in fact consistent with the kinetic point of view given above. Thus, for an appropriate value of the supersaturation, the free-energy curve shows a local minimum and a maximum for the case of ion-induced nucleation, corresponding, respectively, to the stable subcritical cluster and the unstable critical cluster. The local minimum disappears for the case of homogeneous nucleation. 

The effect of finite temperature on the shells and supershelis has been analyzed by Genzken for sodium clusters. For this purpose, calculations of the cluster free energy were performed by treating the valence electrons as a canonical ensemble in the heat bath of the ions [23]. (The spherical jellium model is even better at finite temperature.) Finite temperature leads to decreasing amplitudes of shell and supershell oscillations with increasing T. This is particularly important in the region of the first supershell node at N 850, which is smeared out already at a quite moderate temperature of T = 600 K. However, temperature does not shift the positions of the magic numbers. 

The argument used in arriving at the drop model expression for the cluster free energy [Eq. (38)] did not include any mention of rotation and translation. 

In terms of the bra and ket wavefunctions the coupled-cluster free energy functional may be written as Christiansen and Mikkelsen (1999a) and Cammi (2009) ... 

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

In the classic nucleation theory, the free energy of forming a cluster of radius r containing n atoms or molecules is the sum of two terms ... 

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

The simplest approach to understanding the reduced melting point in nanocrystals relies on a simple thennodynamic model which considers the volume and surface as separate components. Wliether solid or melted, a nanocrystal surface contains atoms which are not bound to interior atoms. This raises the net free energy of the system because of the positive surface free energy, but the energetic cost of the surface is higher for a solid cluster than for a liquid cluster. Thus the free-energy difference between the two phases of a nanocrystal becomes smaller as the cluster size... 

Use of a Monte Carlo or a cluster (Hybrid) algorithm to calculate ionization constants of the titratable groups, net average charges, and electrostatic free energies as functions of pH. 

Often the most important effective cluster interaction of the concentration fluctuation expansion of the configurational free energy is the EPI which is defined as... 

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

The path variables of the PPM corresponds to the cluster probabilities of the CVM by which the free energy is minimized to obtain the most probable state. Likewise, under a set of constraints, the PPF is maximized with respect to the path variables for each time step, which yields the optimized set of path variables. Since a set of path variables, + At), relates cluster probabilities t and at time t + At... 

Fig. 3.2. Variation of free energy of a cluster of molecules on a surface as a function of the number of molecules, at a fixed supercooling. N is the number of molecules at which the free energy is a maximum. Any cluster larger than Ns is stable. The dashed curves show the contributions from the increase in surface area and the decrease in bulk free energy. Increasing the supercooling shifts all curves towards the origin and decreases the height of the maximum...

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