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Spatio-temporal pattern formation in physico-chemical systems far from thermal equilibrium has long been a puzzling phenomenon. Until the last decade, understanding pattern selection and stability mechanisms was considered as a challenge. Fortunately, thanks to intensive theoretical and experimental research, a unified framework is now available to study pattern formation phenomena. It has been successfully applied to several systems, in different fields, such as hydrodynamics, chemistry, and nonlinear optics. They are now being applied to various types of materials instabilities, and will hopefully lead to a better understanding of phenomena such as the formation and evolution of defect microstructures in plastically deformed or irradiated materials, the formation and symmetries of regular deformation patterns in surfaces and thin films under laser irradiation, the role and the control of instabilities in surface modification technologies, etc. In this context, defect microstructures appear as the result of defect motion and nonlinear interactions, which naturally destabilize uniform distributions. The applicability of the methods of nonlinear dynamics to materials instabilities is analyzed, and an appropriate methodology is proposed. The importance of nonlinear analysis beyond instability thresholds in the determination of pattern selection and stability is emphasized. Several examples are discussed, with references to relevant reviews and technical publications.

Coverage evolution, during atomic deposition on a substrate may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type, which combine reaction terms representing chemical processes such as adsorption-desorption and nonlinear diffusion terms. Below a critical temperature, uniform deposited layers are unstable, which leads to the formation of nanostructures corresponding to regular spatial variations of substrate coverage. For increasing mean coverage close to one-half, the dynamics is of the Cahn-Hilliard type, and one should observe a succession of structures going from hexagonal arrays of high coverage dots, to stripes and finally to hexagonal arrays of low coverage dots. For mean coverage close to one, the nanostructures are highly nonlinear, and have to be obtained numerically. Structures obtained in the case of Al layers deposited on TiN substrates are presented. Comparisons between weakly and highly nonlinear structures are performed.

The deposition of monoatomic layers on a substrate is described by a reactiondiffusion model which combines atomic adsorption, desorption and nonlinear diffusion. Close to the adsorbed layer instability, this dynamics is of the Cahn-Hilliard type, supplemented by linear reaction terms representing adsorption-desorption processes. The instability of uniform layers leads to the formation of nanostructures which correspond to regular spatial variations of substrate coverage. Patterns wavelengthes and symmetries are selected by the dynamics and not by variational arguments. For increasing coverage, one should observe a succession of structures going from hexagonal arrays of high coverage dots, to stripes and finally to hexagonal arrays of low coverage dots. At low deposition rates instability may occur for certain time intervals only, and induce transient structures which may provide templates for further structure development.

The onset of oscillatory convection in Rayleigh-Benard experiments with viscoelastic binary fluids is explicitly determined as a function of separation ratio and rheological parameters. In particular, it is shown that critical oscillation frequency may differ in order of magnitude on varying separation ratios and Deborah number. The results suggest that binary fluid aspects may not be discarded when studying thermal convection in polymeric solutions.

Pattern forming instabilities in spatially extended dissipative systems driven away from equilibrium have been the focus of a large activity for many years. The goal of this chapter is to present some theoretical concepts that have been developed to understand and describe these dissipative structures [1] from a macroscopic point of view. Although these methods present generic features we shall only be concerned with chemical patterning and shall not discuss here instabilities in hydrodynamics, liquid crystals and nonlinear optics that all present similar types of organization because the latter have been the subject of recent well-documented reviews [2–5]. Moreover, we essentially consider the self-organization of structures discarding the spatial patterning resulting from boundary conditions.

Understanding the origin of spatio-temporal order in open systems far from thermal equilibrium and the selection mechanisms of spatial struc tures and their symmetries is a major theme of present day research into the structures of continuous matter. The development of methods for pro ducing spatially ordered microstructures in solids by non-equilibrium methods opens the door to many technological applications. It is also be lieved that the key to laminar/turbulence transitions in fluids lies in the achievement of spatio-temporal order. Let us also emphasize the fact that the idea of self-organization in it self is at the origin of a reconceptualisation of science. Indeed, the appear ance of order which usually has been associated with equilibrium phase transitions appears to be characteristic of systems far from thermal equi librium. This phenomenon which was considered exceptional at first now the rule in driven systems. The chemical oscillations obtained appears to be in the Belousov-Zhabotinskii reaction were initially considered to be ther modynamically impossible and were rejected by a large number of chemists. Now these oscillations and related phenomena (waves, chaos, etc. ) are the subject of intensive research and new classes of chemical oscil lators have been recently discovered. Even living organisms have long been considered as the result of chance rather than necessity. Such points of view are now abandoned under the overwhelming influence of spatio-tem poral organization phenomena in various domains ranging from physics to biology via chemistry, nonlinear optics, and materials science ."

One of the most fascinating and intriguing aspects of natural phenomena is that complex systems may undergo symmetry-breaking instabilities leading to pattern formation or coherent behavior over macroscopic space and time scales (1).

We want to present in this note some topics in the study of stationary twodimensional nonequilibrium structures. Two experiments which have stimulated a reniewed interest in the theory of spatial patterns are described. The problem of pattern selection both in isotropic and anisotropic systems is reviewed. The dynamics of the phase is also presented.

When driven away from equilibrium by the effect of an external constraint, many nonlinear systems present spatio-temporal order beyond a threshold value for the control parameter [1]. At this point, the homogeneous steady state becomes unstable versus inhomogeneous or oscillatory perturbations and various spatial or temporal patterns may nucleate spontaneously in the medium. This has long been a puzzling phenomenon from experimental and theoretical point of views and numerous questions related to the pattern formation, selection and stability mechanisms in physical, chemical and biological systems remain still unanswered. Due to the complexity of the dynamics a complete analytical solution of the kinetic equations governing the behavior of the system is usually impossible. However near threshold great simplifications occur as a consequence of the separation between the time and space scales of the stable and unstable modes [2], The evolution of the system may effectively be reduced to its slow mode dynamics which leads to amplitude equations for the spatio-temporal patterns. This important simplification which reduces the complete set of nonlinear kinetic equations to the kinetics of the unstable modes only is valid and may be justified by multiple scale analysis in the vicinity of the instability [3].