Molecular distribution functions and thermodynamics

The book is organized into eight chapters and some appendices. The first three include more or less standard material on molecular distribution functions and their relation to thermodynamic quantities. Chapter 4 is devoted to the Kirkwood-Buff theory of solutions and its inversion which I consider as... 

SCF-MI (Self Consistent Field for Molecular Interactions) and non orthogonal Cl were used to determine a water-water interaction potential, from which BSSE is excluded in an a priori fashion. The new potential has been employed in molecular dynamics simulation of liquid water at 25°C. The simulations were performed using MOTECC suite of programs. The results were compared with experimental data for water in the liquid phase, and good accordance was found, both in radial distribution functions and thermodynamic properties, as well as in geometric parameters. 

U will typically be the sum of potential energies between pairs of particles although, of course, many-body forces or (with minor modification) external forces can be included. Other examples of mechanical quantities would include the molecular distribution functions and the pressure. It would be nice to estimate as well integrals like the configuration integral Q itself since this would give the statistical thermodynamic quantities such as the entropy and free energy. However, this represents a more difficult Monte Carlo (MC) problem, for reasons that will shortly be clear. Some unconventional approaches to this problem are discussed in Chapter 5 of this volume. 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

Relations between thermodynamic quantities and generalized molecular distribution functions... 

The result (4.23) is already a relation between thermodynamic quantities and molecular distribution functions. However, since the derivatives in (4.23) are taken at constant chemical potentials, these relations are of importance mainly in osmotic systems. Here, we are interested in derivatives at constant temperature and pressure. Obtaining these require some simple transformations of the partial derivatives. We first define the elements of the matrix A by... 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on... 

Nevertheless, assuming that the molecular distribution functions are given, then we have a well-established theory that provides thermodynamic information from a combination of molecular information and MDFs. The latter are presumed to be derived either from experiments, from simulations, or from some approximate theories. The main protagonists in this route are the pair correlation functions once these are known, a host of thermodynamic quantities can be calculated. Thus, the less ambitious goal of a molecular theory of solutions has been for a long time route II, rather route I. 

Routes II and III are identical in the sense that they use the same theoretical tools to achieve our goals. There is however one important conceptual difference. Clearly, molecular properties are microscopic properties. Additionally, all that has been learned about MDF has shown that in the liquid phase, and not too close to the critical point, molecular distribution functions have a local character in the sense that they depend upon and provide information on local behavior around a given molecule in the mixture. By local, we mean a few molecular diameters, many orders of magnitude smaller than the macroscopic, or global, dimensions of the thermodynamic system under consideration. We therefore rewrite, once again, route II in different words, but meaning the same as III, namely... 

As a last physical approach we mention, but do not further consider, the scaled-particle-theory (SPT) which was developed about the same time as the Percus-Yevick theory. It gives good results for the thermodynamic properties of hard molecules (spheres or convex molecules). It is not a complete theory (in contrast to the integral equation and perturbation theories) since it does not yield the molecular distribution functions (although they can be obtained for some finite range of intermolecular separations). 

Appendices follow Chapter 6. In Chapter 2, it has been pointed that local entropy may be expressed in terms of same independent variables as if the system were at equilibrium (local equilibrium). The limitations of Gibbs equation have been discussed in Appendix I. At no moment, molecular distribution function of velocities or of relative positions may deviate strongly from their equilibrium form. This is a sufficient condition for the application of thermodynamics method. Some new developments related to alternative theoretical formalism such as extended irreversible are discussed in Appendices II and III. 

There was, however, one important follow-up paper, by Buff and Brout (1955). The reader may have noticed that the Kirkwood-Buff paper concerns exclusively those properties of solutions that can be obtained from the grand potential by differentiation with respect to pressure or particle number. Those such as partial molar energies, entropies, heat capacities, and so forth, are completely ignored. The original KB theory is an isothermal theory. The Buff-Brout paper completes the story by extending the theory to those properties derivable by differentiation with respect to the temperature. Because these functions can involve molecular distribution functions of higher order than the second, they are not as useful as the original KB theory. Yet they do provide a coherent framework for a complete theory of solution thermodynamics and not just the isothermal part. 

After a decade of development and application, a number of original papers on continuous thermodynamics have appeared in the literature. Rtosch and Kehlen [28 30] reviewed the state-of-the-art on systems containing synthetic polymers [28, 29] and those containing petrol fractions and other multicomponent low molecular hydrocarbon systems [30]. Therefore, this overview focuses on systems containing copolymers characterized by multivariate distribution functions and those containing block copolymers. Of source, all important aspects regarding homopolymer systems are automatically included in our discussion. 

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