Helmholtz free energy configurational

The S-S theory describes the structure of a liquid by a lattice model with cells of the same size and a coordination number of z = 12. The disordered structure of the liquid is modeled by allowing an occupied lattice-site fraction y=y(V,73 of less then 1. The configurational or Helmholtz free energy, F, is expressed in terms of the volume V, temperature 7] and occupied lattice-site fraction y = y(K7), F=F(V,T,y). The value of y is obtained through the pressure equation P = —(9F/9V)r and the minimization condition (dFldy)v,T = 0. The hole fraction is given by the fraction of unoccupied lattice sites (holes or vacancies), which is denoted by h, h(P,T) = —y(P,T). This theory provides an excellent tool for analyzing the volumetric behavior of linear macromolecules but was also applied successfully to nonlinear polymers, copolymers, and blends. Several universal relationships where found which allow an approximate estimation of the fraction of the hole (or excess) free volume h and the total or van der Waals free volume/ [Simha and Carri, 1994 Dlubek and Pionteck, 2008d]. For more details, see Chapters 4, 6, and 14. 

When calculating free energies, one generates, either by molecular dynamics or MC, configuration space samples distributed according to a probability distribution function (e.g., the Boltzmann distribution in the case of the Helmholtz free energy). 

Islands of uniform radius R of condensed phase are assumed to be distributed in a continuous LE phase. Of course, this assumption is no longer valid for > 0.6, because transitions to stripe configurations and further to islands of LE in LC phases occur in such cases,1121 It is convenient to write the Helmholtz free energy as the sum... 

The configurational part (the part involving the intermolecular forces) of the Helmholtz free energy for a system of N molecules in volume V at temperature T is given by... 

Our calculations have shown that the global anharmonicities in the Helmholtz free energies for both pure water clusters [4] and aqueous ionic clusters [6,8,71] fundamental to nucleation are essential to include in the prediction of accurate equilibrium constants and free energies. For example, the size-dependent anharmonic chemical potentials of water clusters are wildly different from the RRHOA results underscoring the importance of configurations far removed from min-... 

V can be varied by varying the number of empty lattice sites in the system. It is easy to show (9-10,19,21-23,27) that the equations that minimize the configurational Helmholtz free energy Ac with respect to the order variables at constant V and T also minimize the configurational Gibbs free energy Gc with respect to the order variables at constant P and T. The most stable state is the state of lowest free energy or lowest chemical potential at constant P and... 

A worthwhile question to ask is why there should be a phase transition from a fluid to a solid in the hard-sphere model. We can formulate an initial perspective by considering the configurational Helmholtz free energy of the hard-sphere system, which is related to the configurational partition function via... 

The extension of the isotropic mixture conformal solution method of Smith (4) to the case of anisotropic molecular systems can be made easily in the following manner. The quantities aijy biiy and Cy are defined by the relations an = 8ykeylcrym. fcy = 8y%Vyr, c j = 8y%v[Pg.134]

This is the negative value of the configurational Helmholtz free energy per molecule of fluid divided by kT. 

To begin, focus on the MC estimation of the Helmholtz free energy A of a system of N classical particles. For simplicity of presentation we restrict ourselves for now to a one-component system of isotropic particles of mass m its configurations can be described by a set of coordinates = (qi, q2,. .., qN), say. Then we have, in the canonical ensemble,... 

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