The following is a survey of some general and useful relations for evaluating chemical potentials and free energy changes. The number of such relations isn't large, but an overview is warranted here. Evaluations of free energy changes are typically the most basic and convincing validations of molecular simulation research. Calculations of free energy changes are typically more specialized undertakings than unspecialized simulations. If the problem at hand has been well-considered and calculations are to be specially directed to evaluate free energy changes, then thermodynamic or coupling parameter integration procedures are likely to be the most efficient possibilities. They are favorably stratified, they can have low bias, it is clear how computational effort can be added effectively as results accumulate, and they can be embarrassingly parallel. Other methods considered here, such as importance sampling and overlap methods, can be incorporated into thermodynamic integration methods, and can improve the results.

Nevertheless, there are cases where the alternatives to thermodynamic integration would be chosen instead. In the first place, there are many cases where the problem hasn't yet been considered fully enough to establish a natural integration path. But in the second place, it would often be argued that nonspecialized calculations are more efficient because they produce ancillary results too. Furthermore, the success of alternative free energy calculations often depends on some physical insight.

Molecular liquids are complicated because the defining characteristics that enliven the interesting cases are precisely molecular-scale details. We argue here that practical molecular theory can be simpler than this first observation suggests. Our argument is based upon the view that an effective tool for developing theoretical models is the potential distribution theorem, a local partition function to be used with generally available ideas for evaluating partition functions. An approach based upon the potential distribution theorem also allows functional theory to ride atop simulation calculations, clearly a prudent attitude in the present age of simulation.

This work is about molecular theory, and emphatically not about how to perform simulations. Molecular simulation is an essential component of modern research on solutions. There are a number of presentations of simulation techniques, but not of the molecular theory that we take up here. We offer this book as complementary theory with simulators in mind.

A goal of this book is, thus, to encourage those performing detailed calculations for molecular solutions to learn some of the theory and some of the sources. The physical insights permitted by those calculations are more likely to become apparent with an understanding of the theory that goes beyond the difficulties of executing molecular simulations. Confronting the enormity and lack of specificity of statistical mechanics usually would not be the practical strategy to achieve that goal.

This book also frequently attempts to persuade the reader that these problems can be simple. Extended discussions are directly physical, i.e., non-technical.

This chapter discusses several statistical mechanical theories that are strongly positioned in the historical sweep of the theory of liquids. They are chosen for inclusion here on the basis of their potential for utility in analyzing simulation calculations, and their directness in connecting to the other fundamental topic discussed in this book, the potential distribution theorem. Therefore tentacles can be understood as tentacles of the potential distribution theorem. From the perspective of the preface discussion, the theories presented here might be useful for discovery of models such as those discussed in Chapter 4. These theories are a significant subset of those referred to in Chapter 1 as “… both difficult and strongly established …” (Friedman and Dale, 1977), but the present chapter does not exhaust the interesting prior academic development of statistical mechanical theories of solutions. Sections 6.2 and 6.3 discuss alternative views of chemical potentials, namely those of density functional theory and fluctuation theory.

The MM and KS expansions

The Mayer–Montroll (Mayer and Montroll, 1941) and Kirkwood–Salsburg (Kirkwood and Salsburg, 1953) expansions are storied parts of basic statistical thermodynamics (Stell, 1985), but have been neglected for practical purposes because of a lack of recognition of how simple and simplifying they can be.

We introduce results with the specific example of a hard-core solute that was previously considered in Section 4.3. The hard-core results give perspective for a direct generalization to more realistic interactions.

We consider a molecular description of solutions of one or more molecular components. An essential feature will be the complication of treating molecular species of practical interest since those chemical features are typically a dominating limitation of current work. Thus, liquids of atomic species only, and the conventional simple liquids, will only be relevant to the extent that they teach about molecular solutions. In this chapter, we will introduce examples of current theoretical, simulation, and experimental interest in order to give a feeling for the scope of the activity to be taken up.

The Potential Distribution Theorem (PDT) (Widom, 1963), sometimes called Widom's particle insertion formula (Valleau and Torrie, 1977), is emerging as a central organizing principle in the theory and realistic modeling of molecular solutions. This point is not broadly recognized, and there are a couple of reasons for that lack of recognition. One reason is that results have accumulated over several decades, and haven't been brought together in a unified presentation that makes that central position clear. Another reason is that the PDT has been primarily considered from the point of view of simulation rather than molecular theory. An initial view was that the PDT does not change simulation problems (Valleau and Torrie, 1977). In a later view, the PDT does assist simulations (Frenkel and Smit, 2002). More importantly though, it does give vital theoretical insight into molecular modeling tackled either with simulation or other computational tools, or for theory generally.

An initial discussion of a quasi-chemical approach appeared in Section 4.6. This chapter gives a fuller development of those ideas. The idea of our initial discussion was to introduce a statistical model capable of a natural description of strong association phenomena in solutions, and the example of ion clustering in electrolyte solutions was considered. But the quasi-chemical approach may be founded on broader concepts, and given a more extensive development. The most primitive idea is to identify an inner-shell region from the rest of the neighborhood of a distinguished solute, and to rely on a painstaking treatment of the inner shell, with full molecular resolution. The remainder of the neighborhood of that distinguished solute – the outer-shell region – can be given an alternative statistical description, and then a proper matching of results for inner and outer shells must be accomplished. The pragmatic approach of using alternative methods for physically distinct spatial regions is important.

Many problems of solution theory cry out for chemical treatment of an obvious inner shell. For example, complexes such as Fe(H2O)63+ naturally present themselves as important solution structures when Fe3+(aq) is considered. But discussion of the thermodynamics of Fe3+(aq) on that basis requires a satisfactory parsing of the thermochemistry associated with the ligand species.

Introduction

Particle monolayers are formed when small colloidal solid particles adsorb at liquid–vapour or liquid–liquid interfaces. Typical examples are latex monolayers at the air–aqueous salt solution and oil–water interfaces. The interaction between particles within the monolayer is dependent on both the properties of the fluids that make up the interface and on the nature of the adsorbed particles. Therefore, a detailed analysis of the interactions in colloidal monolayers is quite complex and distinctions must be made to take into account the different components of the monolayer.

The total interaction between particles in the monolayer determines their stability behaviour. Thus, examples of stable monolayer systems with particles that remain independent for a long time have been reported, in spite of the fact that in a thermodynamic sense, colloidal particles are not stable because of their great surface to volume ratio. Some monolayer systems showed a triangular structure suggesting the existence of long-ranged particle interactions. In other reported systems, however, it was found that particles are unstable and aggregate to form fractal structures or even became organized to form the so-called mesostructures. When fractal structures appear, the particle interaction potential is short ranged and has a minimum at very short distances.