The atmosphere is a compressible fluid, and the description of such a form of matter is usually unfamiliar to students who are just completing calculus and classical mechanics as part of a standard university physics course. To complicate matters the atmosphere is composed of not just a single ingredient, but several ingredients, including different (mostly nonreactive) gases and particles in suspension (aerosols). Some of the ingredients change phase (primarily water) and there is an accompanying exchange of energy with the environment. The atmosphere also interacts with its lower boundary which acts as a source (and sink) of friction, thermal energy, water vapor, and various chemical species. Electromagnetic radiation enters and leaves the atmosphere and in so doing it warms and cools layers of air, interacting selectively with different constituents in different wavelength bands.

Meteorology is concerned with describing the present state of the atmosphere (temperature, pressure, winds, humidity, precipitation, cloud cover, etc.) and in predicting the evolution of these primary variables over time intervals of a few days. The broader field of atmospheric science is concerned with additional themes such as climate (statistical summaries of weather), air chemistry (its present, future, and history), atmospheric electricity, atmospheric optics (across all wavelengths), aerosols and cloud physics. Both the present state of the bulk atmosphere and its evolution are determined by Newton's laws of mechanics as they apply to such a compressible fluid. Dynamics is concerned primarily with the motion of the atmosphere under the influence of various natural forces. But before one can undertake the study of atmospheric dynamics, one must be able to describe the atmosphere in terms of its primary variables.

The preceding chapters developed the basic principles needed to describe selforganization and self-assembly in a variety of systems in either initially prepared unstable and metastable states or far-from-equilibrium states. The underlying mesoscopic description involved order parameter fields whose evolution was given in terms of either free energy functionals or amplitude equations. The latter approach is used for systems for which the free-energy-based description is not applicable. However, in many applications to physical and biological problems there is no clear-cut distinction between these two approaches. Often physical systems operate far from equilibrium, and the dynamics may involve both a free energy functional component and a component that cannot be expressed in this form. In this and the following chapters we describe several applications that illustrate how the methods developed in the body of the book may be used to construct models that capture the important aspects of the dynamics. We begin with a discussion of laser-induced melting in this chapter. In the following chapter we consider reactive physical and biological systems where phase segregation and reaction–diffusion dynamics are combined. The last chapter considers active materials where the constituent elements undergo driven or self-propelled motion. The analysis involves the combination of liquid crystal free energy formulations with order parameter dynamics to account for the active motion.

Laser-induced melting

When a laser with an appropriate intensity is focused onto a solid semiconductor film it can create a variety of ordered and disordered lamellar patterns of coexisting solid and melt regions (Fig. 29.1). The periodicity of the ordered patterns is commensurate with the wavelength of the incident laser radiation (van Driel et al., 1982).

Growth of order from disorder is a natural phenomenon which is seen in a variety of systems. An important class of such phenomena involves the kinetics of phase ordering and phase separation. The examples of such growth processes that were described in Chapter 2 had common characteristics. Now, we discuss the experimental results that point to common features of the kinetics of phase separation processes. A combination of techniques from nonequilibrium statistical mechanics and nonlinear dynamics is used to study the formation and evolution of spatial structures. Substantial progress in our understanding of the kinetics of domain growth during a first-order phase transition has been made over the past few decades. The knowledge gained in these studies forms the underpinning of the descriptions of many such processes which create order from disorder.

Kinetics of phase ordering and phase separation

Phase separation is usually initiated by a rapid change or quench in a thermodynamic variable (often temperature and sometimes pressure), which places a disordered system in a post-quench initial nonequilibrium state. The system then evolves towards an inhomogeneous ordered state of coexisting phases, which is its final equilibrium state. Depending on the nature of the quench, the post-quench state may be either thermodynamically unstable or metastable (see Fig. 2.1). In the former case, the onset of separation is spontaneous, and the kinetics that follows is known as spinodal decomposition. For the metastable case, nonlinear fluctuations are needed to initiate the separation process. The system is said to undergo phase separation through homogeneous nucleation if the system is pure. Phase separation occurs by heterogeneous nucleation if the system has impurities or surfaces which initiate nucleation events.

The idea for this book arose from the observation that similar-looking patterns occur in widely different systems under a variety of conditions. In many cases the patterns are familiar and have been studied for many years. This is true for phase-segregating mixtures where domains of two phases form and coarsen in time. A large spectrum of liquid crystal phases is known to arise from the organization of rod-like molecules to form spatial patterns. The self-assembly of molecular groups into complex structures is the basis for many of the developments in nanomaterial technology. If systems are studied in far-from-equilibrium conditions, in addition to spatial structures that are similar to those in equilibrium systems, new structures with distinctive properties are seen. Since systems driven out of equilibrium by flows of matter or energy are commonly encountered in nature, the study of these systems takes on added importance. Many biological systems fall into this far-from-equilibrium category.

In an attempt to understand physical phenomena or design materials with new properties, researchers often combine elements from the descriptions of equilibrium and nonequilibrium systems. Typically, pattern formation in equilibrium systems is studied through evolution equations that involve a free energy functional. In far-from-equilibrium conditions such a description is often not possible. However, amplitude equations for the time evolution of the slow modes of the system play the role that free-energy-based equations take in equilibrium systems. Many systems can be modeled by utilizing both equilibrium and nonequilibrium concepts.

Currently, a wide variety of methods is being used to analyze self-organization and self-assembly. In particular, microscopic and mesoscopic approaches are being developed to study complex self-assembly in considerable detail.

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.