Although the microscopic Hamiltonian contains all of the information needed to describe phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct reduced descriptions. Generally a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a useful starting point for theoretical models. The equations of motion of the macrovariables can be derived from the microscopic Hamiltonian, but in practice one often begins with a phenomenological description. The set of macrovariables is chosen to include the order parameter and all other slow variables to which it is coupled. Such slow variables are typically obtained from consideration of the conservation laws and broken symmetries of the system. The remaining degrees of freedom are assumed to vary on a much faster time scale and enter the phenomenological description as random thermal noise. The resulting coupled nonlinear stochastic differential equations for such a chosen “relevant” set of macrovariables are collectively referred to as the Langevin field theory description. In two of the simplest Langevin models, the order parameter ø is the only relevant macrovariable; in model A (introduced in this chapter) it is nonconserved and in model B (described in the next chapter) it is conserved. The labels A, B, etc. have an historical origin from the Langevin models of critical dynamics. The scheme is often referred to as the Hohenberg-Halperin classification scheme (Hohenberg and Halperin, 1977).

Langevin model A

For model A the Langevin description assumes that, on average, the time rate of change of the order parameter is proportional to the thermodynamic force that drives the phase transition.

In the previous chapter generic features of spiral wave dynamics in oscillatory media were described on the basis of the complex Ginzburg–Landau equation. Spiral waves can also exist in complex oscillatory media where the local dynamics can have period-doubled or even chaotic oscillations. In regimes where complexoscillatory behavior is found, the new feature that appears in spiral waves is a line defect across which the phase of the oscillation changes by 2pi;. The presence of line defects leads to spatiotemporal patterns not seen in media with simple local oscillatory dynamics.

Complex periodic or chaotic oscillations do not have simple single-loop trajectories in concentration phase space. For example, a period-n limit cycle is described by a period-n orbit that loops n times in concentration phase space before closing on itself (see Fig. 27.1). In such circumstances no simple single-valued angle variable may be introduced to play the role of the phase. It is often possible to generalize the definition of phase, even for systems whose dynamics is chaotic, and this is related to the phenomenon of phase synchronization (Rosenblum et al., 1997; Pikovsky et al., 2001; Osipov et al., 2003).

A spiral wave is an example of a self-organized structure that is a result of phase synchronization in a medium with complex local dynamics. Reaction–diffusion equation studies (Goryachev and Kapral, 1996a, 1996b; Goryachev et al., 1998, 2000) and experiments (Yoneyama et al., 1995; Park and Lee, 1999, 2002; Guo et al., 2004; Park et al., 2004) have demonstrated that spiral waves with synchronization defect lines exist in spatially distributed systems that undergo period-doubling bifurcations.

The growth of thin solid semiconductor films is at the heart of the development of modern electronic and optical devices. A key element in strategies for nanoscale fabrication is the exploitation of growth and kinetic instabilities to form surface nanostructures and patterns with desirable functionality.

Epitaxy is a term that is commonly used for the growth of a thin solid layer on top of a substrate. Homoepitaxy denotes the growth of crystals of a material on a crystal face of the same material, while the term heteroepitaxy is used if the materials of the substrate and the growing film are different. Molecular beam epitaxy (MBE) is a common experimental technique that is used to grow such solid films. A film that grows without defects is called a coherently grown film. In such a film the constituent atoms arrange themselves on top of the substrate as its natural extension. The film has the same crystal structure as the substrate.

In the epitaxial growth of a crystal film on another crystal, elasticity plays a dominant role and leads to long-range effective interactions between the adatoms on the surface. These interactions are repulsive and compete with the stronger shortrange chemical interactions. The repulsive nature of the long-range interactions can be qualitatively understood as follows. Consider a planar solid surface of a semiinfinite crystal. When an adatom is placed on this surface, its interaction with the atoms in the top layer creates a stress which changes the distance between its nearest neighbors in the top atomic layer of the surface.

Almost all systems we encounter in nature possess some sort of form or structure. It is then natural to ask how such structure arises, and how it changes with time. Structures that arise as a result of the interaction of a system with a template that determines the pattern are easy to understand. Lithographic techniques rely on the existence of a template that is used to produce a material with a given spatial pattern. Such pattern-forming methods are used widely, and soft lithographic techniques are being applied on nanoscales to produce new materials with distinctive properties (Xia and Whitesides, 1998). Less easily understood, and more ubiquitous, are self-organized structures that arise from an initially unstructured state without the action of an agent that predetermines the pattern. Such selforganized structures emerge from cooperative interactions among the molecular constituents of the system and often exhibit properties that are distinct from those of their constituent elements. These pattern formation processes are the subject of this book.

Self-organized structures appear in a variety of different contexts, many of which are familiar from daily experience. Consider a binary solution composed of two partially miscible components. For some values of the temperature, the equilibrium solution will exist as a single homogeneous phase. If the temperature is suddenly changed so that the system now lies in the two-phase region of the equilibrium phase diagram, the system will spontaneously form spatial domains composed of the two immiscible solutions with a characteristic morphology that depends on the conditions under which the temperature quench was carried out.The spatial domains will evolve in time until a final two-phase equilibrium state is reached.

Liquid crystals are ubiquitous. They are in silk, snail slime, and crude oil. They are in mantles of neutron stars, and provide models for cosmic strings. They are in our food (gluten) and drinks (milk). The behavior of hair cells in the inner ear and the function of DNA are affected by them. The insulating coating of the axons of nerve cells is a liquid crystal called myelin. Liquid crystals are very responsive to excitations, which has led to many useful applications, such as liquid crystal displays. A great deal is known and understood about liquid crystalline materials (Chandrasekhar, 1992; de Gennes and Prost, 1993).

Liquid crystalline materials are orientationally ordered soft matter (Palffy-Muhoray, 2007). These materials are composed of large organic molecules, which have a long and rigid core, typically consisting of several linked benzene rings, terminated by a flexible alkyl chain. Such a molecular structure is then often modeled by disk-like or rod-like entities, depending on the cylindrical aspect ratio. Such model molecules have a head–tail symmetry. Thus, at high densities, liquid crystals can naturally create local orientational order. Onsager (1949) showed that hard rods tend to align at volume fractions larger than about four times their breadth-to-length ratio. Many liquid crystal phases can exist, depending on the temperature and solvent concentration. Some of these phases are shown in Fig. 11.1.

An isotropic disordered liquid phase exists at high temperatures. As the temperature is lowered, there is a competition between the positional and orientational entropies: the former favors a random location for a rod and the latter a random orientation.

A Turing pattern forms when a spatially homogeneous steady state, which is stable to small spatially homogeneous perturbations, loses its stability to small spatially inhomogeneous perturbations. The mechanism responsible for such instabilities was first described by Turing (1952), in his paper The chemical basis of morphogenesis, as a model for pattern formation in biology. The appearance ofTuring patterns relies on the interplay between autocatalytic chemical kinetics and diffusion. The basic Turing mechanism can be described in terms of the kinetics of two chemical species termed the activator and the inhibitor. The activator tends to increase the production of chemical species while the inhibitor tends to inhibit such concentration growth. A Turing pattern can form if the diffusion coefficient of the inhibitor is much greater than that of the activator. While there is still controversy over the role of Turing patterns in morphogenesis, these patterns have been unambiguously identified in chemically reacting media.

The formation of a chemicalTuring pattern in a continuously fed unstirred reactor was reported by Castets et al. (1990). The chlorite–iodide–malonic acid system was studied in a thin gel reactor schematically depicted in Fig. 23.1. The top and bottom sides of the thin hydrogel in which the reaction takes place are in contact with reservoirs containing chemical reagents. The chemical species diffuse into the gel, and reaction takes place in a thin layer within the gel shown in the center panel of the figure. Within this reaction zone a stationary inhomogeneous periodic pattern of chemical concentrations develops, as seen in the right panel of the figure.

The existence and dynamics of interfaces played a central role in the description of the domain-coarsening phenomena considered in the previous chapters. In the late stages of domain growth the random forces in the order parameter kinetic equations were suppressed and the interface dynamics was treated deterministically. In this chapter we provide a more detailed treatment of the effects of noise and diffusion on the structure of the interface. One may capture the essential physics of diffusively rough interfaces in a general model often called the Kardar–Parisi–Zhang (KPZ) equation (Kardar et al., 1986).

KPZ equation

Consider a front in a (d + 1)-dimensional system extended along x and moving, on average, in the x1 direction (see Fig. 16.1). The system is assumed to be infinitely extended along x1 and has linear dimension L along x. In contrast to the description in Chapter 7, we neglect the intrinsic thickness of the interface and investigate the effects of diffusion and noise on the dynamics of the interfacial profile. Referring to Fig. 16.1, let h(x, t) be the position of the interface as a function of x at time t, relative to an arbitrarily selected origin. Its mean position at time t is. We assume that the front propagates with velocity v in a direction normal to its interface; noise provides a destabilizing influence on the front while diffusion tends to remove any surface roughness.

A sketch of a portion of the interface profile h(x, t) as a function of x at time t is shown in Fig. 16.2.

Self-organization and self-assembly take different forms, and their description involves the consideration of different principles when the elements comprising the system undergo active motion. The active motion may arise either because the elements are self-propelled or because external forces or fluxes are applied to the system to induce motion. Biology provides many examples of such active media. Microorganisms such as Escherichia coli move by using molecular motors to drive flagella that propel the organism. The amoeba Dictyostelium discoideum moves by using pseudopods that change shape through actin polymerization and depolymerization processes. Large numbers of such active agents often display collective behavior: for example, Dictyostelium discoideum colonies form streaming patterns, and rippling patterns are seen in myxobacteria such as Myxococcus xanthus. Myxobacteria patterns have been modeled using reaction–diffusion-like descriptions (Börner et al., 2002; Igoshin and Oster, 2004). Flocking behavior is also exhibited by birds, fish, mammals, and a variety of microorganisms (Reynolds, 1987). The spatial patterns seen in these systems span a large range of length scales, from kilometers for herds of wildebeest to micrometers for colonies of the amoeba Dictyostelium discoideum.

Often particle-based models are employed to describe the nonequilibrium dynamics of active media. Studies based on such models show that collective motion arises as a result of the emergence of orientational long-range order in a system with many degrees of freedom. Simple discrete models are able to capture many of the essential features of the collective behavior (Vicsek et al., 1995; Grégoire et al., 2003). Figure 31.1 shows configurations depicting cohesive flocks obtained from simulations of a discrete model.

Excitable media are spatially distributed systems with a stable state that responds to perturbations in a distinctive way. If the normal resting state of the medium is perturbed sufficiently strongly, the perturbation is amplified before the system returns to the resting state. Such excitable media are commonly found in nature, and self-organized wave patterns in these systems control the behavior of many physical and biological systems (Zykov, 1987; Mikhailov, 1994; Kapral and Showalter, 1995). Surface catalytic oxidation reactions often proceed through the propagation of excitable waves of oxidation that sweep across the surface of the catalyst. The oxidation of CO on Pt surfaces has been especially well studied in this context (Ertl, 2000). In biological systems waves of this type occur in the aggregation stage of the slime mould Dictyostelium discoideum, where the chemical signaling is through periodic waves of cAMP; also, the Ca+2 waves in systems such as Xenopus laevis oocytes and pancreatic β cells fall into this category (Goldbeter, 1996). Electrochemical waves in cardiac and nerve tissue also depend on the excitability of the medium, and the appearance and/or breakup of spiral wave patterns (Fig. 24.1) are believed to be responsible for various types of arrhythmia in the heart (Winfree, 1987; Fenton et al., 2002; Clayton and Holden, 2004). Excitable waves have been extensively studied (Belmonte et al., 1997) for the BZ reaction, one of the first systems in which such waves were observed (Zaikin and Zhabotinsky, 1970; Winfree, 1972). Chemical waves in excitable media often take the form of spirals, and Fig. 24.2 shows spiral waves in the Belousov–Zhabotinsky reaction under conditions where this chemical medium is excitable.

Phase separation in systems with competing interactions involves two dynamic phenomena: segregation into two phases, and the creation of supercrystal (modulated phase) ordering. These two processes occur on very different time scales. The early and intermediate-time regimes were discussed in Chapter 14. In these regimes, all important information about the system may be obtained from the scalar order parameter ψ. During the intermediate-time regime, the domain size reaches its saturation value and the time evolution is ultimately governed by this time independent length scale. Systems with a scalar order parameter form domains of the ordered phase separated by domain walls, the relevant topological defect, and evolve so as to decrease the domain-wall energy.

In the presence of long-range repulsive interactions, the late stage of phase ordering involves the evolution from a disordered liquid of minority phase droplets towards the crystalline (hexagonal) ground state through the gain of orientational and positional order. As discussed in Chapter 9, systems with continuous order parameters have point, line, and other more complex defect structures. The late stages of the phase separation processes are dominated by the motion of these defects and, as time evolves, both their density and energy decrease. This is in contrast to model B in the absence of long-range repulsive interactions, where the late-stage kinetics is curvature driven and the conservation law plays an important role.As long-range repulsive interactions become important, qualitatively different late-stage effects emerge, since dipolar forces compete with forces arising from line tension.