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.

Introduction

Froth flotation has a long history (over 100 years) of development and widespread applications. Essentially, the process involves the attachment of finely dispersed hydrophobic particles to air bubbles to produce so-called “three-phase froths” on the surface of the flotation cell with the hydrophilic particles remaining dispersed in the suspension. In this way, the particles are separated, based on their differences in surface wettability. Although froth flotation includes several major elementary sub-processes, one of the most important operations involves the interaction of the selected suspended particles with a chemical reagent (a flotation collector) in order to make the surfaces sufficiently hydrophobic and become “targets” for bubbles generated in the cell. The “gangue” particles remain hydrophilic and do not interact with the collector reagent but remain dispersed in the suspension.

The following unit processes are also important:

1. (i) Generation and dispersion of gas bubbles in the presence of a surfactant (frother) in the pulp and the formation of the froth layer.

2. (ii) Collision of hydrophobic particles with gas bubbles.

3. (iii) Adhesion of hydrophobic particles to gas bubbles and the formation of particle–bubble aggregates.

4. (iv) Ascension of particle–bubble aggregates from the pulp into the three-phase froth.

Both the fundamental and practical aspects of froth flotation have been well studied and developed but the process still undergoes modification and advancement.

Solid particles of colloidal dimensions (nm—μm) adsorb at fluid interfaces, either liquid—vapour or liquid—liquid, in many products and processes. Examples include fat crystals around air bubbles in certain foods, particles of sand or clay partially coating water drops in crude oil and the selective attachment of mineral particles to bubbles in froth flotation. The properties of these systems are due in part to the irreversible nature of particle adsorption, and such particles behave in many ways like surfactant molecules. The pioneering work in the area of particle-stabilised foams and emulsions was conducted by Ramsden and Pickering, respectively, early in the 20th century. During the last 10 years or so, there has been a revival of interest in this field, and in the behaviour of particles at planar liquid interfaces, and we felt that it was time to prepare the first book encompassing most of this activity. It is anticipated that this will be the start of a new series in this rapidly evolving field.

Following an introductory chapter to the whole area by the editors, the book is divided into two parts. The first part, dealing with particles at planar interfaces, contains chapters describing simulation and theoretical approaches to the structure, and dynamics of particle monolayers and how particles can assist with the wetting of oils on water.

Introduction

Particulate additives are found in the formulations of a great many high-surface area products in the form of emulsions and foams. Their presence is normally desirable for the purposes of stability. In the case of ice cream foams, tiny fat globules can attach themselves to the surfaces of the air pockets and hinder the process of coarsening by Ostwald ripening. In this case, the particles are a natural ingredient. In other instances, colloidal particles are deliberately added and Pickering emulsions are an important example. The occurrence of particles leading to stabilization can also be unwelcome, as in the case of emulsions formed when seawater and crude oils vigorously mix. This environmental problem can lead to very stable emulsions as a result of particles formed by asphaltenes or clay collecting at the oil–water interface.

The presence of particles at a fluid–fluid interface leads to numerous, profound consequences. Since a very large amount of energy is normally required to remove a particle from an interface (see Chapter 1 for a detailed explanation of this point), particles in these monolayers are normally irreversibly attached. Furthermore, as described in Chapter 2, these systems can be modified by exquisite tuning of interparticle forces, particle chemistry and particle size to create a wide range of morphologies of these “2-D suspensions”.