Introduction

Following the pioneering experiments on the glass transition in PMMA, which gave a good approximation to monodisperse hard-sphere systems, Pusey and van Megen (1989) and coworkers started a series of experiments with the aim of characterising the transition. Most of the experiments were performed through laser light scattered by density fluctuations characterising the transition from liquid to supercooled liquid and eventually to an amorphous solid. An efficient method was also used to measure the time average in a non-ergodic system using averages over different scattering volumes and wave vectors. The comparison with the predictions of MCT was performed in an extended fashion, showing a relatively good agreement with the experimental findings, to a 20% level of accuracy. The most relevant result of this important set of measurements was the detection of the structural arrest point, a result that is not easy to obtain in normal liquids due to the existence of activated dynamics or hopping effects. The latter are supposedly responsible for the crossing of the barriers that confine the system in a potential well.

MCT was subsequently applied to potentials with an attractive tail following the short-range repulsion, and lead to the behaviour described in the previous chapter on the the theory of supercooled liquids. The most relevant finding was the evidence of the existence of higher-order singularities, which were already defined and studied within MCT, in systems with short-range attractive interactions. Shortly after the predictions of MCT on the consequences of an attractive interaction in hard-sphere systems obtained, many attempts were made to experimentally demonstrate their validity. In particular, first the re-entrant glass line was detected, then the effect of the A3 singularity was shown and finally the higher singularity of type A4 was identified. The experiments on these various aspects of the behaviour of supercooled liquids are illustrated in the following sections.

Introduction

Interactions between particles, both in molecular fluids and colloidal systems, are generally characterised by a strong short-range repulsion, which is responsible for excluded volume effects, followed by an attraction of variable strength. The latter is at the origin of cluster formation, a process that produces many different physical phenomena of great importance in the physics of simple and complex fluids. In atomic and molecular systems the most relevant effect of attraction is the appearance of critical points accompanying phase transitions, while in complex fluids, besides critical effects, peculiar phenomena develop such as aggregation, percolation, glass and sol–gel transitions. Recently the latter have been collectively named arrest phenomena, since their common feature is the pronounced slowing down of the dynamics. We first outline briefly the phenomenology and the approaches based on aggregation and percolation, which describe situations in which the attractive interaction is so strong that the colloidal particles adhere, leading to the formation of macroscopic clusters that eventually invade the physical sample. In the case of reversible aggregation the particles form the so-called physical gels. When the aggregation is irreversible, chemical gels are formed.

Thanks to the possibility of forming reversible or irreversible reactive links, during aggregation clusters of particles tend to coalesce and form larger aggregates. In the case of reversible bond formation a fragmentation process is also present. Aggregation is an ubiquitous process that can be observed in disparate situations at various length and time scales. Examples are polymer chemistry, aerosol systems, cloud physics, clusters of galaxies in astrophysics, etc. Although aggregation in colloidal suspensions has long been studied, it has become a subject of renewed interest in recent years because it is a non-equilibrium phenomenon, the final stage of which may lead, among other things, to the formation of a gel. We briefly summarise various aspects of clustering by introducing the Smoluchowski aggregation formalism, which is used in many different physical approaches to aggregation, and a brief summary of the salient aspects of percolation that are important for the physical phenomena we describe.

Introduction

As we mentioned in previous chapters, structural arrest refers in general to various phenomena which have in common a marked slowing down of the dynamics, including aggregation and cluster formation, percolation and glass transition. In the last few years the study of the glass transition of colloidal systems in the supercooled region revealed a series of new phenomena which were interpreted as the possibility of including in a single framework glass and gel transition in liquids (Sciortino and Tartaglia, 2005).

Summarising what we described at length in previous chapters, the study of the glass transition in supercooled liquids was initiated in hard-sphere systems, where the cage effect is the relevant physical mechanism for the structural arrest. In the case of colloids an interaction potential with a hard core and an attractive tail brings in the usual phase separation with a critical point, but also a new mechanism of structural arrest, predicted theoretically. Besides the repulsive glass transition due to excluded volume effects, a line of attractive liquid-glass appears, the driving mechanism of which is the bonding between the particles. Since the attractive glass line extends to a wide range of volume fractions of the dispersed phase, the colloid could also give rise to a low-density gel, similar to the gelling due to cluster formation and aggregation at very low densities.

In order to clarify the mutual relation between the coexistence curve and the glass lines, a detailed study was performed (Tartaglia, 2007) in a model binary system with hard-core repulsion followed by a short-range square-well attractive potential. The ideal glass transition line has been studied with a simulation of the dynamics of the colloidal system and by evaluating the diffusion coefficient D for long times. The loci of constant D, the iso-diffusivity lines, are used and by extrapolating to the limit D → 0 one gets an indication of the position of the arrest line.

In this chapter, we describe the elements of liquid theory. Our intention is not to present all aspects of liquid theory in its complete form, but instead only those that will be useful and sufficient for discussing its different applications for presentations in subsequent chapters. It will be addressed in the context of understanding problems that arise in complex fluids and colloidal science discussed in subsequent chapters.

In Section 2.1 we introduce the concept of the pair correlation function and the structure factor which are fundamental quantities when discussing applications of liquid theories to the analysis of scattering data, including light scattering, X-ray scattering and neutron scattering. We would like to remind the reader here that the pair correlation function and thus the structure factor are intimately connected with the statistical thermodynamics of the system. In this sense the study of the structure factor in the liquid state using scattering techniques is to investigate some aspect of the thermodynamics of the liquid system.

Then we discuss the solution of Ornstein–Zernike equation in its different approximations. In particular, in Section 2.2.3, we illustrate the use of the Baxter method, an elegant analytical method for solving hard-sphere and adhesive hardsphere systems in the Percus–Yevick approximation. These two systems are the model systems for simple liquids as well as for the colloidal solutions. In Section 2.2.4 we present an analytical solution for the case of a narrow-squarewell potential that avoids some aspects of the unphysical features of the Baxter solution of the adhesive hard-sphere system. We shall show in a later chapter the applications of this analytical solution for studying the kinetic glass transition in a micellar system.

We then skip the background introduction to the liquid theories of ionic solutions such as the classical Debye–Hückel theory and the Poisson–Boltzmann theory of ionic solutions, as well as the mean spherical approximation solutions of the so-called primitive model of ionic solution already available in Blum (1980).

The central theme of this book is ‘slow dynamics in supercooled, glassy liquids and dense colloidal systems’ which has been an intense area of current research for some time. Although it can be well described by the mode-coupling theory of dense liquids, controversial viewpoints persist. Thus, the authors have written about the exciting modern aspects of the physics of liquids by selecting only the most interesting contemporary development in this rich field of research in the last decades.

This book presents and summarises a wide variety of recent research on the physics of complex liquids and suggests that the use of established techniques, essentially neutron, X-ray and light scattering together with theoretical and computer molecular dynamic simulation approaches, can be fruitfully applied to solve many new phenomena. These techniques are also central to investigating new interesting findings in liquid water such as liquid–liquid transition and its associated low-temperature critical point.

Although many materials found in nature can be classed as complex fluids, the authors have chosen to focus on water and colloids in this book for the following reasons:

1. • Water is the most important liquid for life on Earth. It covers 71% of the Earth's surface and is probably the most ubiquitous, as well as the most essential, molecule on Earth. It is a vital element controlling not only all aspects of life itself but also the environmental factors that make life enjoyable. Water is a simple molecule yet possesses unique and anomalous properties at low temperatures that have fascinated scientists for many years. Thus in selecting the categories of complex liquids to include in this book, water is the obvious top choice.

2. • Colloids are another class of complex liquids characterised by the slowing down of the dynamics. They are becoming increasingly studied for their potential applications and the availability of degrees of freedom that are relatively simple to vary experimentally through physical and chemical control parameters, giving rise to a much larger variety of phenomena compared to simple liquids.

Introduction

There are several theories and models of glassy behaviour. For example, Binder and Kob discussed these in great details in their book (Binder and Kob, 2011), with extensive references. Theories often cited are, for example, Adam–Gibbs theory (Adam and Gibbs, 1965), free-volume theory (Cohen and Turnbull, 1959; Turnbull and Cohen, 1961, 1970), mode-coupling theory (MCT), random firstorder theory (RFOT), etc. While all these theories and models can certainly enhance our understanding of glass-forming systems, none seems to be flawless. Some are too sophisticated and complex, while others are too macro in their approach and make simple assumptions. Thus there is plenty of room for much more in-depth research and experiments. In this chapter, we shall place more emphasis on recent experimental results, which can help us to understand better the complex landscapes and systems of the glass-forming liquids. Whenever possible, we will relate the results to some of the available theories.

In Chapter 7, we have already discussed the dynamic crossover (often called fragile-to-strong dynamic crossover, FSC) phenomenon in supercooled water found by Chen and coworkers (Faraone et al., 2004). Therefore, in this chapter, we are exploring such phenomena in other glass-forming liquids. It has been found that there is a similar dynamic crossover behaviour in other glass-forming liquids, substantiated not only by mode-coupling theory and MD simulations, but also collaborated with many experimental results (Chen et al., 2009b). We shall expand on this in the following.

8.1.1 Experimental results on transport coefficients

The existing data on transport coefficients of common bulk glass-forming liquids exhibit the same FSC phenomena as water does. For example, in Figure 8.1, for the well-known case of o-terphenyl, the Arrhenius plots of the inverse of the self-diffusion constant (top panel) (Mapes et al., 2006) together with the viscosity (middle panel) and followed by the specific heat (lower panel) are shown (Laughlin and Uhlmann, 1972).

Introductory remarks on confined water

Water affects all aspects of life (Ball, 2000; Franks, 2000). Hence, it comes as no surprise that water has tremendous political, cultural and historical significance. The role that water plays in different branches of the natural sciences is varied, but always influential. Understanding the physical characteristics of water has the potential to elucidate how it affects countless systems that are of fundamental importance to biology, chemistry, geology, meteorology, etc. While much more is known about the properties of water than those of most other substances, there are still many open questions and substantial gaps in our understanding. Thus, despite systematic investigation throughout the history of scientific inquiry, the behaviour of water remains an active and exciting area of research.

The properties of water are anomalous. The density maximum of liquid water observed at 4 °C under ambient pressure is probably the most widely known anomaly, but water's thermodynamic response functions and transport coefficients also behave in seemingly counterintuitive ways. On the molecular level, water's ability to form hydrogen bonds that induce an open tetrahedral inter-molecular structure underlies its unique qualities. These hydrogen bonds, which have energy intermediate to van der Waals interactions and covalent bonds, are continuously breaking and reforming while maintaining some average degree of association. At lower temperatures, the hydrogen bonding becomes increasingly prominent and the average degree of association in the fluid increases. The macroscopic properties resulting from this microscopic picture can be conceptualised in terms of a competition between density ordering, associated with the normal liquid state behaviour found in van der Waals' view of the liquid state (Chandler et al., 1983) and bond-orientational ordering associated with the formation of hydrogen bonds (Tanaka, 1998). This antagonism is captured by models of water used in molecular simulations that treat the central oxygen atom with a Lennard–Jones potential and incorporate the preference for bond-orientational ordering through the addition of decentralised hydrogen atoms, virtual sites or three-body interactions.

Introduction

In 1995, while discussing the future of science, P. W. Anderson made the following statement on the unsolved problems of physics:

The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition. This could be the next breakthrough in the coming decade. (Anderson, 1995)

In recent years substantial progress has been made in the understanding of the glass transition, both experimentally and theoretically, but from the point of view of supercooled liquids, and in particular liquids of supramolecular aggregates, more than from the point of view of solid state physics. The general name given to these phenomena, commonly found in soft condensed matter, is dynamically disordered arrested phenomena. Examples belonging to this category are the liquid–glass, the sol–gel and the percolation transitions. The reason for this variety of behaviours is related to the large space and time scales of the aggregates, the various shapes the molecular aggregates can take and the corresponding specific interaction potentials, and the possibility of changing a large number of control parameters both at the level of the inter-particle potential and the state of the system. Moreover the length and time scales involved allow the use of powerful experimental techniques, such as neutron and light scattering and confocal microscopy, the latter allowing tracking of the trajectories of individual particles. Examples of systems where dynamical arrest was observed include thermoreversible physical gels, associating polymers at low concentration, micellar systems, star-polymer mixtures, colloidal gels and block-copolymers. All these colloidal systems show the typical slowing down of the dynamics leading to the arrest, with a mechanism which is in general very sensitive to changes in the external control parameters, and which persists over time scales that might be even longer than the typical observation time of the experiment.

Comparison of inelastic neutron scattering and inelastic X-ray scattering techniques

10.1.1 Complementarity of inelastic neutron scattering and inelastic X-ray scattering spectroscopies

The study of atomic and molecular density fluctuations in condensed systems on the length scale of inter-particle separation is, traditionally, the domain of inelastic neutron scattering spectroscopy. The principal reason for which neutrons are particularly suitable for these studies is the very good matching between the kinematic phase space covered by thermal and cold neutron scatterings and that of phononlike collective excitations in the condensed matter systems and bio-molecular assemblies. In fact, the energy of neutrons with wavelengths of the order of interparticle distances of say 3 Å in liquids and solids is about 9 meV. This energy is comparable to the energies of typical phonons in condensed matter with wavelengths in the nanometre range. As a consequence, one can determine the peak position in the measured dynamic structure factor S (Q, E) without requiring an excessive energy resolution of the inelastic neutron spectrometers used. Thus one is able to utilise the neutron flux from a typical thermal and cold neutron source rather effectively.

In principle, X-rays can also be used to measure the dynamic structure factor S(Q, E) of the systems studied. The inelastic X-ray scattering cross-section, under the prevailing circumstances, has an expression very similar to that for neutrons, and the coupling of X-rays and neutrons to the density fluctuations in the material system is of the same order of magnitude. The X-ray scattering cross-section is derived by considering the interaction between the atomic electrons and the electromagnetic field associated with the X-rays in the Hamiltonian. In the nonrelativistic limit, the interaction Hamiltonian is composed of two terms. They are the Thomson term and the photoelectric absorption term, which is the coupling of the photon field to the electron current.