In this chapter we discuss a theory for the freezing of an isotropic liquid into a crystalline solid state with long-range order. The transformation is a first-order phase transition with finite latent heat absorbed in the process. We focus on a first-principles orderparameter theory of freezing that originated from the pioneering work of Ramakrishnan and Yussouff (1979). The theory approaches the problem from the liquid side and views the crystal as a liquid with grossly inhomogeneous density characterized by a lower symmetry of the corresponding lattice. This is in contrast to description of the crystal in terms of phonons. The crucial quantity characterizing the physical state of the system in this non-phonon-based model is the average one-particle density function n(x). The thermodynamic description of either phase involves a corresponding extremum principle for a relevant potential. The latter, obtained as a functional of the one-particle density n(x) and the stable thermodynamic state of the system, is identified by the corresponding density required for invoking the extremum principle. This approach, which is generally referred to as the density-functional theory (DFT) of freezing (Haymet, 1987; Baus, 1987, 1990; Singh, 1991; Löwen, 1994; Ashcroft 1996), has been improved over the years and successfully applied for the study of liquid-to-crystal transitions in various simple liquids, the solid–liquid interface, two-dimensional systems, metastable glassy states, etc. For applications of density-functional methods in statistical mechanics there exist general reviews (Evans, 1979; Henderson, 1992).

The fluctuating-hydrodynamics approach discussed earlier takes into account only the transport properties at the level of completely uncorrelated motion of the fluid particles. The corresponding dissipative processes are expressed in terms of bare transport coefficients of the fluid. The strongly correlated motion of the fluid particles which occurs at high density is not take into consideration here. This is reflected through the Markov approximation of the transport coefficients and the short correlation of the corresponding noise representing the fast degrees of freedom in the system. The Markovian equations for the collective modes involving frequency-independent transport coefficients constitute a model for the dynamics of fluids with exponential relaxation of the fluctuations. The corresponding equations of motion for the collective modes are linear. However, exceptions occur in certain situations in which the description of the dynamics cannot be reduced to a set of linearly coupled fluctuating equations with frequency-independent transport coefficients. In this chapter we will consider the nonlinear dynamics of the hydrodynamic modes for studying the strongly correlated motion of the particles in a dense fluid.

Nonlinear Langevin equations

We present in this section the formulation of a set of nonlinear stochastic equations for the dynamics of the many-particle system. We first discuss the physical motivation for extension of the fluctuating-hydrodynamics approach to include nonlinear coupling of the slow modes.

We have discussed the construction of the nonlinear Langevin equations for the slow modes in a number of different systems in the previous chapter. Next, we analyze how the nonlinear coupling of the hydrodynamic modes in these equations of motion affects the liquid dynamics. In particular, we focus here on the case of a compressible liquid in the supercooled region. In this book we will primarily follow an approach in which the effects of the nonlinearities are systematically obtained using graphical methods of quantum field theory. Such diagrammatic methods have conveniently been used for studying the slow dynamics near the critical point (Kawasaki, 1970; Kadanoff and Swift, 1968) or turbulence (Kraichnan 1959a, 1961a; Edwards, 1964). The present approach, which is now standard, was first described by Martin, Siggia, and Rose (1973). The Martin–Siggia–Rose (MSR) field theory, as this technique is named in the literature, is in fact a general scheme applied to compute the statistical dynamics of classical systems.

The field-theoretic method presented here is an alternative to the so-called memoryfunction approach. The latter in fact involves studying the dynamics in terms of non-Markovian linearized Langevin equations (see, for example, eqn. (6.1.1) which are obtained in a formally exact manner with the use of so called Mori–Zwanzig projection operators. This projection-operator scheme is described in Appendix A7.4). The generalized transport coefficients or the so-called memory functions in this case are frequency-dependent and can be expressed in terms of Green–Kubo forms of integrals of time correlation functions.

In Chapter 4 we introduced the Kauzmann temperature TK as a possible limiting temperature for the existence of the supercooled liquid phase. The original hypothesis due to Kauzmann proposes eventual crystallization in the supercooled liquid at very low temperatures as a possible way out of the paradoxical situation in which the entropy of the disordered state becomes less than that of the crystal. Another possible explanation of the Kauzmann paradox could be that the simple extrapolation of the high-temperature result to very low temperature is not correct and the entropy difference between supercooled liquid and crystal remains finite down to very low temperature (Donev et al., 2006; Langer, 2006a, 2006b, 2007), finally going to zero only near T = 0. Either of these resolutions, however, leaves us with no understanding of the dramatic slowing down and associated phenomenology of the supercooled region above Tg. The difference of the entropy of the supercooled liquid from that of the solid having only vibrational motion around a frozen structure represents the entropy due to large-scale motion and is identified with the configurational entropy Sc of the liquid. The rapid disappearance of the configurational entropy of the disordered liquid or the so-called “entropy crisis” poses an important question that is essential for our understanding of the physics of the glass-transition phenomena and the divergence of the relaxation time at Tg. Apart from having a characteristic large viscosity, the supercooled liquid shows a discontinuity in specific heat cp at Tg due to freezing of the translational degrees of freedom in the liquid.

In the previous chapters we have discussed the transition of the liquid from a disordered fluid state to an ordered crystalline state through a first-order phase transition at the melting or freezing point Tm. In the present chapter we consider the behavior of the liquid supercooled below Tm and the associated phenomenon of the liquid–glass transition.

The liquid–glass transition

Almost all liquids can, under suitable conditions, be supercooled below the freezing point Tm while avoiding crystallization. The undercooled liquid continues to remain in the disordered state and is characterized by very rapidly increasing viscosity with decreasing temperature. The characteristic relaxation time τ of the liquid grows with increasing supercooling. Eventually, at low enough temperature, the supercooled liquid becomes so viscous that it can hold shear stress and behaves like a solid. At this stage the supercooled liquid is said to have transformed into a glass. The latter is an amorphous solid without long-range order. It is in fact in a nonequilibrium state on the time scale of the experiment. The relaxation time τ required for the supercooled liquid to equilibrate is longer than the typical time scale τexp of an experiment. Apart from the viscosity, other dynamic quantities such as the diffusion coefficient, dielectric response function, and conductivity change strongly with increasing supercooling. In contrast, thermodynamic properties such as the specific heat, enthalpy, compressibility, and static structure factor do not show any strong change with supercooling.

If the liquid is cooled beyond the corresponding freezing point Tm at which the liquid and crystalline phases coexist in equilibrium, a thermodynamic driving force builds up towards forming the crystal. In this chapter we will discuss how the liquid transforms into a crystal, focusing on how the changes in the liquid are initiated and on the nature of the crystalline region that is formed. This process is referred to as nucleation. The thermodynamic force favoring the formation of the crystal seed in the supercooled liquid competes with the process of forming an interface between the solid and the liquid. The cost of the interfacial free energy therefore presents a barrier to the formation of the new phase. Only when the driving force is made large enough by moving deep into the supercooled state does crystallization occur on laboratory time scales. Thus pure water can be cooled to -20 °C or below without freezing. Our focus here will be mainly on the process of crystallization of solid from the melt. The condensation of vapor into liquid is a very thoroughly studied process that has been discussed in various reviews (Stanley, 1971; Evans, 1979; ten Wolde et al., 1998). For condensation from a low-density gas or crystallization from dilute solution, it is easier to identify the nucleating bubbles since they differ widely in composition from the surrounding phase.

This book is aimed at teaching the important concepts of the various theories of statistical physics of dense liquids, freezing, and the liquid–glass transition. Both thermodynamic and time-dependent phenomena relating to transport properties are discussed. The standard tools of statistical physics of the dense liquid state and the associated technicalities needed to learn them are included in the presentation. Details of some of the calculations have been included, whenever needed, in the appendices at the ends of chapters. I hope this will make the book more accessible to beginners in this very active field of research. The book is expected to be useful for graduate students and researchers working in the area of soft-condensed-matter physics, chemical physics, and the material sciences as well as for chemical engineers.

We now give a brief description of what is in the book. The first chapter reviews the basics of statistical mechanics necessary for studying the physics of the liquid state. Key concepts of equilibrium and nonequilibrium statistical mechanics are presented. The topics covered here have been chosen keeping in mind the theories and concepts covered in the subsequent chapters of the book. Following this introductory chapter, we focus on the physics of liquids near freezing. In Chapter 2, we demonstrate how the disordered liquid state as well as the crystalline state of matter with long-range order can be understood in a unified manner using thermodynamic extremum principles.

Theoretical developments on the dynamics of a dense liquid using a statistical-mechanical approach primarily involve a small set of slow collective densities termed hydrodynamic modes. The time scales of relaxation of these modes are much longer than those for the microscopic modes of the system. The basic approach adopted here is the analysis of the time correlation functions (introduced earlier in Chapter 1) of the slow modes. In the present chapter and the next two chapters we discuss microscopic methods for calculating the correlation functions involving the fluctuation or hydrodynamic approach. We focus primarily on the simplest type of correlation functions involving fluctuations at two different spatial and time coordinates. Owing to time translation invariance, equilibrium two-point correlation functions of hydrodynamic modes at the same time over different spatial points are time-independent and provide us with information on the thermodynamic behavior of the system. On the other hand, the dynamic behavior of the system is linked to the correlation of physically observable quantities at two different times. The time correlation function of density fluctuations is particularly important for our discussion of the slow dynamics in a liquid. In the simplest of the theoretical models, the decay of the correlation with time is exponential. We discuss here how such exponential relaxation behavior can be understood using linear dynamics of the fluctuations. The formalism developed in the later parts of this chapter allows in a natural way the extension of the macroscopic hydrodynamics to intermediate length and time scales, and is referred to as generalized hydrodynamics.