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.

New features appear in the kinetics of phase ordering and phase separation in systems where long-range repulsive interactions (LRRI) compete with the shortrange attractive interactions considered earlier. Competing interactions can lead to the emergence of modulated phases, where a particular symmetry, wavelength, and amplitude are selected (Seul and Andelman, 1995). Both in equilibrium and nonequilibrium systems such modulated phases have domain structures with various shapes, patterns, and morphologies. Figure 13.1 shows some domain structures seen in systems displaying modulated phases. Modulated phases in materials are important in technological applications (Park et al., 1997; Black et al., 2000). An understanding of such phases is crucial in order to be able to design materials with specific properties and control their morphology.

Many systems in nature can be modeled through the inclusion of long-range interactions. Examples of such systems are uniaxial ferromagnetic films, ferromagnetic surface layers, ferrofluid films, ferroelectrics, Langmuir (lipid) monolayers, block copolymers, and cholesteric liquid crystals. A uniaxial ferromagnetic film in the presence of an external magnetic field can be modeled by augmenting the standard scalar order parameter model A with an additional long-range interaction arising from the parallel orientation of magnetic dipoles (Roland and Desai, 1990). This repulsive interaction competes with the attractive domain wall energy. An external magnetic field makes the film's magnetization a nonconserved quantity so that a description based on model A is appropriate. Block copolymers and Langmuir monolayers are examples of conserved order parameter (model B) systems where the connectivity between the covalently bonded blocks of the polymer chains results in an effective LRRI (Sagui and Desai, 1994).

The late stages of model B dynamics for asymmetric quenches, where the initial condition places the post-quench system just inside and quite near to the coexistence curve, exhibit characteristic features. For an asymmetric system, ψ0 is a measure of the extent of off-criticality of the system. For ψ0 > 0 the majority phase equilibrates at ψ+ = +1 and the minority phase at ψ–= –1. At late times, the minority-phase clusters have a characteristic radius R(τ) which is much larger than the interface width ξ. An important coupling exists between the interface and the majority phase through the surface tension σ. The conservation law dictates that the minority phase will occupy a much smaller “volume” fraction than the majority phase in the final equilibrium state. The dynamics is governed by interactions between the different domains of the minority phase. At late times, these domains have spherical and circular shapes for three- and twodimensional systems, respectively. Late-stage coarsening is referred to as Ostwald ripening.

The late-stage dynamics may be mapped onto a diffusion equation with sources and sinks (domains) whose boundaries are time dependent. The classic papers by Lifshitz and Slyozov (1961) and Wagner (1961) form the theoretical cornerstone for the description of domain coarsening dynamics for model B. The Lifshitz– Slyozov–Wagner (LSW) theory of coarsening is based on the assumption that each interface between a minority phase domain and the majority phase background is infinitely sharp. It describes the diffusive interactions between the domains through a mean-field treatment with precisely defined boundary conditions at each of the interfaces.