Overview of processes

The evolution of foams occurs through a series of rapid non-equilibrium processes which can be observed by sparging gas through a glass sinter into a column of water. As the air bubbles ascend, their velocities are principally determined by their sizes, the difference in the viscosities of the liquid and gas phases and the properties of the gas/liquid interface. However, as the bubbles grow in size, they may collide and in cases where only weak foaming agents are present in solution, compaction and coalescence can occur. There are several other processes which play an important role in determining the characteristics of the bubbles and the structure of the foam as the bubbles accumulate at the interface. For example, the drainage process or the downward flow of liquid coupled with liquid flow into the Plateau borders can cause thinning of the liquid films. Also, repulsive interactions across the thin film lamellae resulting from strongly adsorbed chemical surfactants can slow down drainage or even prevent bubble coalescence. During the ascent and mixing of bubbles, another important process known as disproportionation occurs. This involves the diffusion of gas from smaller to larger bubbles, and the driving force for this process is the Laplace pressure (the pressure difference between bubbles of different sizes). Although the term “disproportionation” is commonly used by chemists to describe inter-bubble gas diffusion within foams, it is often referred to as Oswald ripening, which was originally used to define the evaporation–condensation mechanism in two-phase separation of binary alloys. The term “coarsening” is often used but coarsening is also frequently considered to be a combination of inter-bubble gas diffusion and coalescence. This confusion in terminology is due to the fact that researchers engaged in foams come from a variety of disciplines, and each has its own terminology. An overview of some of the processes that occur during sparging are outlined in Fig. 4.1.

Molecular processes such as the adsorption and the mobility of chemical surfactant molecules at the air/water interface and also the depletion of surfactant from solution can occur at high gas flow rates can also influence the stability of the bubbles.

Introduction

Several non-intrusive analytical techniques have been used to provide information on bubble size distributions, gas fraction, texture and general characteristics of foams. Some of these are relatively simple optical methods, but with the application of image and video analysis, the changes in bubble size and texture of 2D foams can be fairly easily studied during ageing. This information can be useful, but since the rate of deterioration of a wet foam depends on the kinetics of coalescence, drainage and gas diffusion, it is sometimes difficult to separate these events and to resolve the main destabilization mechanism. Bisperink and coworkers (1) reviewed how the variation in 2D bubble size distribution in aging foams may be related to drainage, coalescence and the disproportionation process. Although microscopy can be conveniently used to study 2D foams, more complex techniques based on tomography have been applied to probe the interior and to measure bubble size distribution on 3D foams. The processing of the data usually involves three steps: image acquisition, image processing and data extraction. 3D imaging techniques such as nuclear magnetic resonance imaging (NMRI) or X-ray computerized tomography can be used to develop a scale model in a computer memory. Based on these models the relationship between physical properties and the structure of solid foams can be established. UV, NMR and ultrasonic reflection methods have also been used, but overall it is still difficult to characterize bubble size and liquid fraction in real 3D foams, and hence most investigations are carried on quasi-2D foams.

Although wet foams are unstable, the kinetics of breakdown can range from seconds to weeks, and this has resulted in the development of many different types of test methods for measuring foam stability. Overall, stability tests may be broadly classified as (a) dynamic tests, which measure the height or volume of the foam in a state of dynamic equilibrium between formation and decay, and (b) static tests, in which the rate of foam formation is zero (the gas flow into the liquid is eliminated) and the foam is allowed to collapse.

Overview

All foams are thermodynamically unstable due to their high interfacial free energy, the decrease of which causes foam decay. It is well known that there are several different types of mechanisms involved in the stabilization and decay of foams, which has caused a considerable amount of confusion. In the literature there are many conflicting explanations frequently caused by experimental anomalies and the incomplete interpretation of foaming experiments. Another aspect to consider is that the lifetime of a foam can pass through several different stages, and each stage may involve a different type of mechanism. To explain the overall stability in terms of one mechanism is almost impossible, and the interplay of different mechanisms needs to be taken into consideration. During generation, bubbles expand and contract and are subjected to severe vibrations and dynamic disturbances causing distortion of the adsorption layer. During this process, the liquid films separating the bubbles are relatively thick and subject to stretching, and viscous elastic forces play a crucial role. Possibly the most important mechanisms for the survival of a wet foam during this stage involves the surface elasticity theories of Gibbs and Marangoni. Gravitational forces also cause fairly rapid drainage to occur during this preliminary stage, but this can be retarded by a high bulk viscosity. On entering a secondary stage, capillary forces come into play causing suction and thinning of the lamellae, and this occurs at a lower rate. In addition, disproportionation may occur causing the diffusion of gas between bubbles. As all these processes occur under dynamic conditions, the equilibrium adsorption coverage is rarely reached.

The process of gas diffusion owes its origins to the difference in pressure, surface tension and curvature of the bubbles, but the gas diffusion to the atmosphere also needs to be considered. In addition to diffusive disproportionation theories to explain the changes in size distribution in bubbles, alternate processes have been considered which involve the effect of interfacial rheology on the shrinkage of bubbles.

Introduction and early studies

Surfactants such as soap (sodium and potassium salts of fatty acids) stabilize thin liquid films, which are the basic structural units of foams and act as cell walls encapsulating the gas. The films are stabilized by surfactant monolayers of polar head groups (hydrophilic) solubilized in the water phase and aliphatic hydrocarbon tails (hydrophobic) protruding into the vapor phase. Isolated soap films can be easily produced by dipping a wire loop or rectangular frame into a soap solution and raising the frame slowly, causing the liquid to drain vertically. As the thickness of the film decreases from the top to the bottom of the frame, a pattern of interference fringes rapidly develops. This phenomenon can be spectacular, since each color band flows downward and a swirling motion is frequently observed due to rapid, complex fluid motion. Finally, toward the end of the thinning process, the film reaches a thickness that is less than the wavelength of light, and at this point, black spots appear, which spread rapidly, and eventually the entire film appears black in reflected light.

Foams, bubbles and foam films have been the subject of intense scientific investigations over a period of several hundred years. In the 17th century, both Robert Hooke (1) and later Isaac Newton (2) became captivated by these systems and carried out many fundamental experiments involving film drainage in bubbles in which the brilliant interference colors were carefully observed and reported in the Proceedings of the Royal Society. In Fig. 3.1, a typical spectrum generated by the drainage of thin foam films is illustrated.

It was also noted by scientists such as Newton and Hooke that, as the colored interference bands transformed to thin black films, the thinning ceased and eventually the films collapsed. This can be explained by the formation of a metastable state, resulting from the interaction of intermolecular forces. At first sight, the appearance of black spotswas extremely puzzling. Hooke first thought that these black spots were black holes in the soap bubbles, and Isaac Newton, in his second paper on light and color, described (as illustrated below) the occurrence of several different states of thickness of thin black films on bubble surfaces (as indicated by different shades of black and also the coalescence of the spots).

Antibubbles

Antibubbles, which have sometimes been referred to as negative bubbles or inverted bubbles, have been known for at least 70 years. They were first reported in 1931/2 by Katalinic (1) and Hughes and Hughes (2) and originally considered as a scientific curiosity. Essentially, the structure of an antibubble consists of a thin shell or core of air which is separated by liquids, with the surfactant adsorbed at the inner and outer interfaces of the shell. The polar heads of the chemical surfactant are immersed in the aqueous medium, while the nonpolar tails face and extend into the gas. This is the exact opposite situation to a bubble where the thin shell of liquid separates the air inside and outside the bubble. The shell of air on the antibubble is about 10 nm thick and can generate interference colors. Antibubbles are formed at surfactant concentrations below and above the CMC, and several different types of surfactants have been used in their preparation. Dishwashing detergents were often used in the generation process, but more recently they have been prepared using different types of conventional long-chain nonionic, anion and protein (beer) surfactants (3). It is also interesting to note that antibubbles can be generated and stabilized by partially hydrophobic particles (4). Most studies in antibubbles have been mainly concerned with generation rather than stability. In Fig. 12.1, the structure of a common bubble is compared with chemically stabilized antibubbles and particle-stabilized antibubbles.

In recent years, several interesting applications have been put forward. For example, it has been suggested that they can be used as antifoams or new types of lubricants analogous to ball bearings or filtration systems in which the web of passageways would be permeable to gas molecules (5). Since antibubbles provide twice the surface area of ordinary bubbles of the same size, another potential application could be the control of chemical reactions. It has also been shown that it is possible to replace air in the antibubble inner shell with other liquids or dyes, suggesting that they could be used as a drug delivery system.

Introduction

Partially hydrophobic small particles have the potential to act as stabilizing agents in many foaming processes, and they behave fairly similar in some ways to chemical surfactant molecules in that they can adsorb (attach) at the bubble interface. However, particles show several distinct differences from chemical surfactants. For example, macro-sized particles are considerably larger than the molecular dimensions exhibited by chemical surfactants. They also behave differently, in that particles cannot aggregate at the interface and are unable to buildup self-assembles and cannot solubilize in the bulk solution. Unlike chemical surfactants, it is difficult to generate bubbles or foams solely without particles, but partially hydrophobic particles can be good foam stabilizers at moderate concentrations (about 1 w%). In fact, if the particles exhibit moderate hydrophobicity, then the foams can be extremely stable (with lifetimes of the order of years). However, generally it is more convenient to add other surface-active components to the particles, such as polymers, dispersants or chemical surfactants, to ensure a higher degree of foamability and foam stability. The different features exhibited by surfactant- and particle-stabilized systems are illustrated in Fig. 8.1.

There are only a few foaming processes operating solely with particles, for example, molten metal foams (where ceramic particles are used) and sometimes hydrophobic particles (such as graphite) in the froth flotation process. Excluding the past two decades, the literature on foams stabilized by particles is sparse, but there has been a revival of attention, which was mostly because of the success achieved with particle-stabilized emulsions (Pickering emulsions). For a considerable period of time it has been well known that particles acted as stabilizers in many established industrial foaming processes, such as froth flotation of mineral particles (1), deinking flotation (2) and food processing (3). However, little effort was made to understand the mechanisms until recent years; today a considerable amount of insight has been achieved in understanding the basics.

Introduction

Foams can exist in the wet, dry or solid state and can be seen almost everywhere, in the home, in the surrounding natural environment and in numerous technological applications. In fact they are prevalent, and it is almost impossible to pass through an entire day without having contact with some type of liquid or solid foam. They have several interesting properties which enable them to fill an extremely wide range of uses; for example, they possess important mechanical, rheological and frictional characteristics which enable them to behave similar to solids, liquids or gases. Under low shear, wet (bubbly) foams exhibit elastic properties similar to solid bodies, but at high shear, they flow and deform in a similar manner to liquids. On the application of pressure or temperature to wet foams, the volume changes proportionately, and this behavior resembles that of gases. Interestingly, it is the elastic and frictional properties of wet foams which lead to their application in personal hygiene products such as body lotions, foaming creams and shaving foams. While shaving, foam is applied to the skin and the layer on the blade travels smoothly over the surface, reducing the possibilities of nicking and scratching. Another example is their use as firefighting foams, where properties such as low density, reasonably good mechanical resistance and heat stability are required in order to be effective in extinguishing gasoline fires. Essentially, they act by covering the flames with a thick semi-rigid foam blanket. The low density allows the water in the foam to float even though it is generally denser than the burning oils. The chemical composition and mechanical properties of these types of foams can be varied to optimize the firefighting utility.

Foams are also found in many food items, either in finished products or incorporated during some stage in food processing. They primarily provide texture to cappuccino, bread, whipped cream, ice-cream topping, bread, cakes, aerated desserts, etc. Surprisingly, several novel types of food foams have been recently produced from cod, mushroom and potatoes, using specially designed whipping siphons powered by pressurized gas with lecithin or gelatin as alternative foaming agents to replace egg and creams (1).

Correlation functions are central quantities in the description of interacting many-body systems, both in the theoretical formulation and in the analysis of experiments. In contrast to single numbers like the total energy, correlation functions reveal far more information about the electrons, how they arrange themselves, and the spectra of their excitations. In contrast to the many-body wavefunctions that contain all the information on the system, correlation functions extract the information most directly relevant to experimentally measurable properties. Dynamic current–current correlation functions are sufficient to determine the electrical and optical properties: one-body Green's functions describe the spectra of excitations when one electron is added to or removed from the system, static and dynamic correlations are measured using scattering techniques, and so forth. In this chapter we present the basic definitions and properties of correlation functions and Green's functions that are the basis for much of the developments in the following chapters.

In general, a correlation function quantifies the correlation between two or more quantities at different points in space r, time t, or spin σ. Very often the correlation function can be specified as a function of the Fourier-transformed variables, momentum (wavevector) k, and frequency ω. It is useful to distinguish between a dynamic correlation function which describes the correlation between events at different times and a static or equal-time correlation function, by which we mean that of a property measured or computed with “snapshots” of the system. Also, the different correlation functions can be classified by the number of particles and/or fields involved.

The methods that we introduce in the next four chapters are quite different from those in Parts II and III: stochastic or quantum Monte Carlo methods. In stochastic methods, instead of solving deterministically for properties of the quantum many-body system, one sets up a random walk that samples for the properties. Historically the most important role of QMC for the electronic structure field has been to provide input into the other methods, most notably the QMC calculation of the HEG [109], used for the exchange– correlation functional in DFT. A second important role has been as benchmarks for other methods such asGW. There are systems for which QMC is uniquely suited, for example the Wigner transition in the low-density electron gas, see Sec. 3.1. In this chapter we introduce general properties of simulations, in particular Markov chains, and error estimates. In the following chapters we will apply this theory to three general classes of quantum Monte Carlo algorithms, namely variational (Ch. 23), projector (Ch. 24), and path-integral Monte Carlo (Ch. 25); Ch. 18 already introduced the QMC calculation of the impurity Green's function used in the dynamical mean-field method.We note that there are a variety of other QMC methods not covered in this book.

Simulations

First let us define what we mean by a simulation, since the word has other meanings in applied science. The dimensionality of phase space (i.e., the Hilbert space for a quantum system) is large or infinite. Even a classical system requires the positions and momenta of all particles, and, hence, the phase space for N classical particles has dimensionality 6N.

The previous chapters discuss examples of experimental observations where effects of interactions can be appreciated with only basic knowledge of physics and chemistry. The purpose of this chapter is to give a concise discussion of models that illustrate major characteristics of interacting electrons. These are prototypes that bring out features that occur in real problems, such as the examples in the previous chapter. They are also pedagogical examples for the theoretical methods developed later, with references to specific sections.

The Wigner transition and the homogeneous electron system

The simplest model of interacting electrons in condensed matter is the homogeneous electron system, also called homogeneous electron gas (HEG), an infinite system of electrons with a uniform compensating positive charge background. It was originally introduced as a model for alkali metals. Now the HEG is a standard model system for the development of density functionals and a widely used test system for the many-body perturbation methods in Chs. 10–15. It is also an important model for quantum Monte Carlo calculations, described in Chs. 23–25.

To define the model, we take the hamiltonian in Eq. (1.1) and replace the nuclei by a rigid uniform positive charge with density equal to the electron charge density n.

Deterministic quantum methods have difficulties. For example, the Hartree–Fock method assumes the wavefunction is a single Slater determinant, neglecting correlation. If one expands as a sum of determinants, it is very difficult to have the results size-consistent since the number of determinants needed will grow exponentially with the system size. As we have seen, the DMFT method introduced in Ch. 16 assumes locality. In Ch. 6 we discussed general properties of many-body wavefunctions. Using Monte Carlo methods, we can directly incorporate correlation into a wavefunction, without having to make any further approximations other than the form of the correlation factors. In many cases the energy and other properties are very close to the exact results. Some of the usual restrictions on the form of the many-body wavefunction are not an issue in variational Monte Carlo. The most important generalization of the HF wavefunction is to put correlation directly into the wavefunction via the “Jastrow” factor. At next order, one can use the “backflow” wavefunction, in which correlation is also built into the determinant.

The variational Monte Carlo method (VMC) was first used by McMillan [44] to calculate the ground-state properties of superfluid 4He. One of the key problems at that time was whether the observed superfluid properties were a consequence of Bose condensation.

The title of this book is Interacting Electrons. Of course, there are no non-interacting electrons: in any system with more than one electron, the electron–electron interaction affects the energy and leads to correlation between the electrons. All first-principles theories deal with the electron–electron interaction in some way, but often they treat the electrons as independent fermions in a static mean-field potential that contains interaction effects approximately. As described in Ch. 4, the Hartree–Fock method is a variational approximation with a wavefunction for fermions that are uncorrelated, except for the requirement of antisymmetry. The Kohn–Sham approach to DFT defines an auxiliary system of independent fermions that is chosen to reproduce the ground-state density. It is exact in principle and remarkably successful in practice. However, many properties such as excitation energies are not supposed to be taken directly from the Kohn–Sham equations, even in principle. Various other methods attempt to incorporate some effect of correlation in the choice of the potential.

This chapter is designed to highlight a few examples of experimentally observed phenomena that demonstrate qualitative consequences of electron–electron interactions beyond independent-particle approximations. Some examples illustrate effects that cannot be accounted for in any theory where electrons are considered as independent particles. Others are direct experimental measurements of correlation functions that would vanish if the electrons were independent. In yet other cases, a phenomenon can be explained in terms of independent particles in some effective potential, but it is deeply unsatisfying if one has to invent a different potential for every case, even for different properties in the same material. A satisfactory theory ultimately requires us to confront the problem of interacting, correlated electrons.