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Nonlinear models for pattern evolution by ion beam sputtering on a material surface present an ongoing opportunity for new numerical simulations. A numerical analysis of the evolution of preexisting patterns is proposed to investigate surface dynamics, based on a 2D anisotropic damped Kuramoto-Sivashinsky equation, with periodic boundary conditions. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method of Manufactured Solutions has been used to verify the proposed scheme. Ripples and hexagonal patterns were obtained from a monomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence of the ion beam with respect to the irradiated surface. Analytical discussions are based on linear and weakly nonlinear analysis.

We present a continuum model of ion-induced surface patterning. The model incorporates the atomic processes of sputtering, re-deposition and surface diffusion, and is shown to display the generic features of the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear and non-linear stability analyses of the evolution equation give estimates of the emerging pattern wavelength and spatial symmetry. The analytical theory is confirmed by numerical simulations of the evolution equation with the Fast Fourier Transform method, where we show the influence of the incident ion angle, flux, and substrate surface temperature. It is shown that large local geometry variations resulting in quadratic non-linearities in the evolution equation dominate pattern selection and stability at long time scales.

Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. We show the existence of a threshold for perturbations of the homogeneous state. Subthreshold perturbations decay exponentially. Superthreshold perturbations induce the emergence of a long-lived structure formed by two back to back fronts that join the two homogeneous states. While in typical excitability the trajectory follows the remnants of a limit cycle, here reinjection is provided by front interaction, such that fronts slowly approach each other until eventually annihilating. This front-mediated mechanism shows that extended systems with no oscillatory regimes can display excitability., We acknowledge individual support (P.P.-R.) and project support (L.G.) by the Research Foundation - Flanders (Fonds Wetenschappelijk Onderzoek, FWO-Vlaanderen), by the IAP, and by the research council of the Vrije Universiteit Brussel (VUB). We also acknowledge financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project ESoTECoS FIS2015-63628-C2-1-R (AEI/FEDER,UE).

Series Editors: da Costa Mattos, Heraldo, Martins Costa, Maria Laura, Laredo dos Reis, João., This paper presents a numerical approach to a model describing the pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model derived from a anisotropic damped Kuramoto-Sivashinsky equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi-implicit splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is verified with time step and grid spacing tests for the pattern evolution. Hexagonal patterns were obtained from a monomodal initial condition for a higher value of the damping coefficient, while spatiotemporal chaos appeared for lower values. The hexagonal ordered character of the structure was shown to be directly proportional to the damping coefficient.

This paper reviews some of the main mechanical properties of carbon nanotubes (CNT), including nonlinear elastic instabilities and buckling, as well as defect generation, dynamics and plasticity. A multiscale modeling approach of CNT dynamics is proposed. It considers CNT as cylindrical shells and describes their dynamics through the corresponding elasticity equations. This description may also incorporate defects such as vacancies, adatoms, dislocations, Stone–Wales defects and the coupling between film elasticity and defect dynamics. Elastic constants, defect formation and interaction energies may be obtained from direct comparison with experimental results or numerical simulations at the atomic level. It is then shown how the apparition and properties of deformation instabilities may be discussed in this framework.

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development of various types of dislocation microstructures (dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc.), depending on the externally applied loading and the intrinsic lattice constraints. The term “dislocation patterning” was introduced over 20 years ago by the third author and a corresponding “gradient dislocation dynamics” framework was suggested to describe such phenomena. In the W–A model proposed at that time by the last two authors, it was shown how coupled nonlinear evolution equations of the reaction-diffusion type for the forest (immobile) and gliding (mobile) dislocation densities can generate dislocation microstructures which correspond to walls perpendicular to the slip direction for Cu-crystals oriented for single slip under cyclic loading conditions. This model is adapted to the multiple slip case here. Weakly nonlinear analysis predicts that dislocation patterns should correspond to domains of walls perpendicular to each slip direction and separated by domain walls in the same orientations. This result is confirmed by numerical analysis and experimental observations. The present model generalizes the original W–A model to the case of multiple slip and considers also explicitly gradient effects by allowing for non-uniform dislocation velocities and internal stress effects.

The evolution of a monoatomic layer, deposited on a substrate, is described by a dynamical model of the reaction-diffusion type. This model takes into account the possible coexistence of two types of gains with different orientations with respect to the substrate. It combines reaction (adsorption and desorption) and nonlinear diffusion terms, which, close to the critical point of the order-disorder transition of the adsorbed layer, are of the Cahn–Hilliard type. At high temperatures, uniform grain distributions are expected to develop. At low temperatures, uniform grain distributions may become spatially unstable for sufficiently high adatom mobility. Nanoscale spatial patterns may then develop, which correspond to grain distributions, where grains with different orientations coexist. Links with atomistic simulations and experiments are discussed. It is argued that, for Al or Cu layers, deposited on substrates like TiN, Si2, Ta, etc., instability temperatures and critical wavelengths are well in the experimentally accessible range.

It is shown that coverage evolution, during atomic deposition on a substrate, may be described, on mesoscopic scales, by dynamical models of the reaction–diffusion type. The models combine reaction terms representing adsorption–desorption processes and nonlinear diffusion terms of the Cahn–Hilliard type. The combination may lead, below a critical temperature, to the instability of uniform deposited layers. The instability induces to the formation of nanostructures that correspond to regular spatial variations of atomic coverage. Such models also may describe texture evolution during the deposition of polycrystalline films. In this case, grains with different orientations with respect to the substrate may coexist. Grains with lower surface energy usually dominate, except at high temperatures, where grains with faster lateral diffusion may override energetically favored ones. Furthermore, at sufficiently low temperatures, uniform grain distributions may become unstable versus regular spatial variations of grain orientations. The relevance of this approach to the deposition of Al or Cu on TiN or Ta substrates is discussed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

It is shown that coverage evolution during atomic deposition on a substrate may be described, on mesoscopic scales, by dynamic models of the reaction-diffusion type. Such models combine reaction terms representing adsorption-desorption and chemical processes and nonlinear diffusion terms which are of the Cahn-Hilliard type. This combination may lead, below a critical temperature, to the instability of uniform deposited layers. This instability leads to the formation of nanostructures which correspond to regular spatial variations in substrate coverage. Patterns' wavelengths and symmetries are selected by the dynamics and not by variational arguments. For increasing coverage, one should observe, in layers with isotropic atomic diffusion, a succession of structures going from hexagonal arrays of high-coverage dots, to stripes and finally to hexagonal arrays of low-coverage dots. For anisotropic diffusion, stripes perpendicular to the high-mobility direction should be selected. The relevance of this approach to the study of deposited Al, Ti and TiN monolayers is discussed.

Thermal convection in binary fluids with Oldroyd viscoelastic properties is studied close to threshold. The role of retardation time, characteristic of viscoelastic fluids, and Soret effect, characteristic of binary fluids, is emphasized. A weakly nonlinear analysis is performed for stationary convection. The coefficients of the corresponding amplitude equation are determined analytically and the nature of the bifurcation is discussed.

It is shown how that the combination of atomic deposition and nonlinear diffusion may lead, below a critical temperature, to the growth of nonuniform layers on a substrate. The dynamics of such a system is of the Cahn–Hilliard type, supplemented by reaction terms representing adsorption–desorption processes. The instability of growing uniform layers leads to the formation of nanostructures which correspond to regular spatial variations of substrate coverage. Patterns wavelengths and symmetries are selected by the dynamics and not by variational arguments. For temperatures below critical, one should observe hexagonal arrays of high coverage dots on the surface of otherwise uniform growing layers. On decreasing further the temperature, these structures should transform into hexagonal arrays of low coverage domains, within the growing layer.

It is shown how the combination of atomic deposition and nonlinear diffusion may lead, below a critical temperature, to the growth of nonuniform layers on a substrate. The dynamics of such a system is of the Cahn–Hilliard type, supplemented by 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 wavelengths 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.

Modeling spatio-temporal pattern formation in complex systems far from thermal equilibrium has long been considered as a challenge. Fortunately, during the last decade, a unified framework, which allows the study of generic aspects of pattern formation phenomena, emerged from intensive theoretical and experimental research. It has been successfully applied in several domains, from hydrodynamics to chemistry and nonlinear optics, and, more recently, to materials instabilities. It is shown here, how the methods of nonlinear dynamics may be used to establish conditions for instability in dynamical models for surface processing. Their application to the determination of selected patterns in post-bifurcation regimes is illustrated in a dynamical description of laser-induced thin film deformation.

Strain localization and dislocation microstructure formation are typical features of plastic deformation in metals and alloys. Plastic deformation occurs by the glide of dislocations, and, although dislocation distributions are rather uniform at its onset, they usually become unstable when deformation proceeds and undergo successive transitions towards various types of microstructures such as cells, deformation bands, persistent slip bands, labyrinth structures, etc. This phenomenon is experimentally well documented, but, despite a huge number of theoretical investigations, its modeling is still in its infancy. In the spirit of multiscale modeling, an intermediate step between micro- and macroscopic descriptions, consists in the derivation of mesoscopic rate equations describing the dynamical evolution of dislocation densities. It will be shown on some classical examples why this approach is interesting for the description of mechanical systems undergoing plastic instabilities, and how it may be used to determine the key physical processes for dislocation patterning. It will also be shown that the localization of plastic deformation is a natural consequence of collective behaviors induced by reaction and transport in dislocation populations.

A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into account an angular dependence of the nonlinear couplings between spatial modes with different orientations. Furthermore, the marginal stability curve presents other local minima than the one corresponding to critical modes, as usual in optical systems. Quasipatterns form when one of these secondary minima may be associated with harmonics built on pairs of critical modes. The model is analyzed numerically and in the framework of amplitude equations. The results confirm the importance of harmonics to stabilize quasipatterns and assess the applicability of the model to other systems with similar generic properties. © 2014 American Physical Society., We acknowledge useful discussions with P. Colet, and financial support from FEDER and MINECO (Spain), through Grant No. FIS2012-30634 INTENSE@COSYP, and from Comunitat Autónoma de les Illes Balears.

Irradiation of materials by energetic particles (e.g., electrons, ions and neutrons) is associated with very high internal power dissipation, which can drive the underlying nano- and microstructure far from normal equilibrium conditions. One of the most unusual responses in this connection is the ability of the material's nano- and microstructure to self-assemble in well-organized, two- and three-dimensional periodic arrangements. We review and assess here experimental evidence and theoretical models pertaining to the physical understanding of nano- and microstructure self-organization under irradiation conditions. Experimental observations on the formation of self-organized defect clusters, dislocation loops, voids and bubbles are presented and critically assessed. Implantation of metals with energetic helium results in remarkable self-assembled bubble super-lattices with wavelengths (super-lattice parameters) in the range of 5–8 nm. Ion and neutron irradiation produce a wide variety of self-assembled 3-D defect walls and void lattices, with wavelengths that can be tailored in the range of 10's to 100's of nanometers. Theoretical models aimed at explaining these observations are introduced, and a consistent description of many features is outlined. The primary focus of the most recent modeling efforts, which are based on stability theory and concepts of non-linear dynamics, is to determine criteria for the evolution and spatial symmetry of self-organized microstructures. The correspondence between this theoretical framework and experimental observations is also examined, highlighting areas of agreement and pointing out unresolved questions.

The onset of convection in a binary-viscoelastic Oldroyd-B fluid is investigated. The threshold for oscillatory convection is calculated. It is shown that the critical oscillation frequency may differ by several orders of magnitude on varying separation ratio and Deborah number. The results suggest that binary fluid aspects may not be discarded when studying thermal convection in polymeric solutions.

In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.

Convection in viscoelastic fluids may be induced by oscillatory or stationary instabilities. When the oscillatory instability appears first, the system may be described, in its vicinity by coupled Ginzburg–Landau equations. We study the convective and absolute instabilities of the steady states of these equations. We analyze pattern selection and stability in deterministic and stochastic systems and discuss the possibility of noise-induced phase transitions.

The coupling between surface deformation and defect motion may be at the origin of deformation patterns in thin films under laser irradiation. We analyze the dynamics of laser-induced vacancy densities and deformation fields and show how it triggers deformational instabilities, in the case of uniform and focused laser irradiation. Pattern selection analysis is performed, through linear, nonlinear, and numerical methods. In irradiation with extended beams, we show that, according to the relative importance of nonlinearities arising from the defect or from the bending dynamics, square, hexagonal or even quasi-periodic patterns are selected. It appears, furthermore, that one-dimensional gratings are always unstable in isotropic systems. In irradiation with focused laser beams, rose deformation patterns, with petal number increasing with laser intensity, naturally arise in this model, in qualitative agreement with experimental observations. These results claim for more systematic and quantitative experimental investigations of deformational pattern formation under laser irradiation.

The formation of laser-induced deformation patterns on thin films and surfaces may be described by a dynamical model for the coupled evolution of defect densities and deformation fields of the material. Increasing laser intensity induces deformational instability, which may be characterized in the framework of linear stability analysis of undeformed states. However, the selection and stability of deformation patterns are determined by nonlinear effects, and require full nonlinear analysis in the post-bifurcation regime. Analytical and numerical results are presented for uniform laser irradiation of pure metallic thin films, and the conditions for pattern selection are discussed.

We study spatiotemporal pattern formation associated with the polarization degree of freedom of the electric field amplitude in a mean field model describing a Kerr medium in a cavity with flat mirrors and driven by a coherent plane-wave field. We consider linearly as well as elliptically polarized driving fields, and situations of self-focusing and self-defocusing. For the case of self-defocusing and a linearly polarized driving field, there is a stripe pattern orthogonally polarized to the driving field. Such a pattern changes into a hexagonal pattern for an elliptically polarized driving field. The range of driving intensities for which the pattern is formed shrinks to zero with increasing ellipticity. For the case of self-focusing, changing the driving field ellipticity leads from a linearly polarized hexagonal pattern (for linearly polarized driving) to a circularly polarized hexagonal pattern (for circularly polarized driving). Intermediate situations include a modified Hopf bifurcation at a finite wave number, leading to a time dependent pattern of deformed hexagons and a codimension 2 Turing-Hopf instability resulting in an elliptically polarized stationary hexagonal pattern. Our numerical observations of different spatiotemporal structures are described by appropriate model and amplitude equations., This work was supported by QSTRUCT ~Project No. ERB FMRX-CT96-0077!. Financial support from DGICYT ~Spain!, Project No. PB94- 1167, is also acknowledged. M.H. wants to acknowledge fi- nancial support from the FOMEC project 290, Dep. de Fı´sica FCEyN, Universidad Nacional de Mar del Plata, Argentina.

The mechanical behavior of thin films subjected to laser irradiation is described by a dynamical model that is based on coupled evolution equations for the deformation and vacancy density fields. Lattice vacancies are generated in a thin layer as a result of shallow absorption of electromagnetic laser radiation. The strain field associated with lattice dilatation due to vacancies is shown to couple with bending and stretching mechanical deformation fields. The dynamical model developed here is an extension of the work of Emel'yanov in two respects: (1) the coupling between the diffusion and mechanical deformation fields is rigorously developed with additional cross-field contributions; (2) new equations for reduced dynamics are derived from this model, and are used to analyze the physical conditions for the onset of a deformational instability. For a given material, the threshold for this instability is correlated mainly with laser power. We also show that, although the instability threshold and critical wavelengths are given by the linear part of the dynamics, the selection and type of deformation patterns induced by this instability require a nonlinear formulation. Both numerical and analytical analysis are performed here. According to the relative importance of nonlinearities arising from the defect or from the bending dynamics, square or hexagonal planforms are shown to be selected. Furthermore, it appears that one-dimensional gratings are always unstable in isotropic systems. The results for square patterns are consistent with experimental observations, while those for hexagonal and one-dimensional gratings show the importance of anisotropies on their final selection.

Evidence of noise-sustained patterns in nonlinear optical systems is given. They are found in passive optical cavities, filled by Kerr type nonlinear media, when the angle of incidence of the pump beam is not zero, in a regime of convective instability. These patterns arise as a macroscopic manifestation of dynamically amplified noise, with amplification factors of up to 105. We characterize the difference between noise-sustained and deterministic patterns in terms of statistical properties of the field spectral intensity., This work is supported by QSTRUCT (Project No. ERB FMRX-CT96-0077). Financial support from DGICYT (Spain) Project No. PB94-1167 is also acknowledged.

The coupling between surface deformation and defect dynamics may be at the origin of deformation patterns in thin films under laser irradiation. We analyze a simple model describing the dynamics of such systems in the case of focused laser irradiation. We show, through linear, nonlinear, and numerical analysis, how rose deformation patterns, with the petal number increasing with laser intensity, naturally arise in this model, in agreement with experimental observations.

A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Benard convection and results are obtained for very large system sizes and for very long times on small computational platform.

Recent computer simulations on dislocation patterning have provided remarkable results in accordance with empirical laws. Moreover, several analytical models on dislocation dynamics have provided qualitative insight on dislocation patterning. However, a model, based on partial differential equations, which gives a dynamical evolution of dislocation patterns in function of measurable variables still missing. Here, we give a re-formulation of a model proposed some years ago. From this formulation, we obtained that the onset of a dislocation instability is related to the applied stress. The analytical and numerical results reported are partial and studies on this direction are under development.

The effect of mean flows on pattern stability in systems where oscillatory instabilities of the Hopf type interact with stationary ones is investigated. In particular, it is shown that pattern selection may be strongly modified when the absolute instability threshold of the trivial uniform steady state is rejected beyond the stationary instability. The effect of spatially distributed noise including the competition between noise sustained and dynamically sustained structures is also discussed.

When thin films are grown on a substrate by chemical vapor deposition, the evolution of the first deposited layers may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type. For monatomic layers, such models describe the evolution of atomic coverage due to the combined effect of reaction terms representing adsorption-desorption and chemical processes and nonlinear diffusion terms that are of the Cahn-Hilliard type. This combination may lead, below a critical temperature, to the instability of uniform deposited layers. This instability triggers the formation of nanostructures corresponding to regular spatial variations of substrate coverage. Patterns wavelengths and symmetries are selected by dynamical variables and not by variational arguments. According to the balance between reaction- and diffusion-induced nonlinearities, a succession of nanostructures including hexagonal arrays of dots, stripes, and localized structures of various types may be obtained. These structures may initiate different growth mechanisms, including Volmer-Weber and Frank-Van der Merwe types of growth. The relevance of this approach to the study of deposited layers of different species is discussed. © 2013 American Physical Society.

We investigate a pattern-forming system close to a Hopf bifurcation with broken translational symmetry. In one-dimensional geometries, its evolution is governed by two coupled complex Ginzburg-Landau equations which describe the amplitude of the counterpropagating traveling waves that develop beyond the instability. The convective and absolute instabilities of the possible steady states are analyzed. In the regime of strong cross coupling, where traveling waves are favored by the dynamics, the results of previous analysis are recovered. In the weak cross-coupling regime, where standing waves are favored by the dynamics, traveling waves nevertheless appear, in the absence of noise, between the uniform steady state and the standing-wave patterns. In this regime, standing waves are sustained by spatially distributed external noise for all values of the bifurcation parameter beyond the Hopf bifurcation. Hence, the noise is not only able to sustain spatiotemporal patterns, but also to modify pattern selection processes in regimes of convective instability. In this weak coupling regime we also give a quantitative statistical characterization of the transition between deterministic and noise-sustained standing waves when varying the bifurcation parameter. We show that this transition occurs at a noise-shifted point and it is identified by an apparent divergence of a correlation time and the saturation of a correlation length to a value given by the system size., We acknowledge financial support from the DGICYT grant PB94-1167

Pattern selection and stability in viscoelastic convection are studied in the framework of amplitude equations derived in the vicinity of stationary and oscillatory instabilities. The oscillatory instability corresponds to a Hopf bifurcation with broken translational symmetry. When this instability is the first to appear with increasing Rayleigh number, such systems may be described by coupled one-dimensional complex Ginzburg-Landau equations for counterpropagating waves. The coefficients of these equations, as computed from the underlying Navier-Stokes equations, are such that the selected pattern corresponds to standing waves. The phase dynamics of these waves is derived and leads to coupled Kuramoto-Sivashinsky equations. Their stability range is also determined for different typical fluid parameters. \textcopyright{} 1996 The American Physical Society.

This volume is devoted to the presentation of the main contributions to the workshop ‘‘From oscillations to excitability: A case study in spatially extended systems,’’ organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self‐oscillatory or excitable behavior. It tries to give a better understanding of the transition from the oscillatory to the excitable regime and of its effect on the properties of spiral waves, and to fill the gap between the theories and concepts used to describe both regimes in the so‐called ‘‘active media.’’

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/ annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro-structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef-Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction-diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model, the first one comprising linear diffusion of the density of mobile dislocations and the second one, with nonlinear diffusion of said variable. Subsequently, the paper focus on a finite difference, second order in time Cranck-Nicholson semi-implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer 0, 35 , 87, 2002) for solving the model evolution equations in two dimensions. The discussion on the WA model and on the numerical scheme was already presented on a conference paper by the authors (Pontes et al., AIP Conference Proceedings, Vol. 1301 pp. 511-519, 2010). The first results of four simulations, one with linear diffusion of the mobile dislocations and three with nonlinear diffusion are presented. Several phenomena were observed in the numerical simulations, like the increase of the fundamental wavelength of the structure, the increase of the walls height and the decrease of its thickness.

Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two-dimensional geometries, patterns of different symmetries may be simultaneously stable. We study here the phase stability of hexagonal planforms in the framework of amplitude equations and reduced dynamical models close to the instability points.

The formation and evolution of defect microstructures in irradiated materials is analysed in the framework of a dynamical model for the evolution of the two fundamental defects of irradiated microstructures, namely vacancy and interstitial clusters. The effects of irradiation on materials is described by dynamical equations for two mobile atomic size species (vacancies and interstitial atoms), and two basic immobile elements of the microstructure (vacancy and interstitial clusters). It is shown that uniform vacancy and interstitial loop distributions may become unstable during irradiation and that they will form large-scale spatially organized distributions, in a specific range of irradiation and material conditions. The selection and stability of the resulting microstructures are studied in the quasi-static approximation and in the weakly non-linear regime around the bifurcation point. It is shown that, after transients corresponding to three-dimensional BCC patterns, the final pattern should correspond to planar wall structures in agreement with experimental observations.

The effect of temporal modulations on wave patterns induced by spatiotemporal Hopf bifurcations is discussed in the framework of amplitude equations of the Ginzburg-Landau type. The approach is well adapted to the study of pattern formation in liquid crystals which, on the other hand, provide a class of easily forced systems. A few examples, related to experimental realizations, are presented. In particular, it is shown how pure temporal modulations may stabilize standing waves or two-dimensional wave patterns in regimes where they are otherwise unstable. The properties of the defects which are associated to these structures are also discussed.

This chapter examines the morphological instabilities of interfaces between the liquid and solid phases as the phase transformation process is completed. It first presents the problem of melting and solidification, in which the classical ‘Stefan problem’ is introduced as a fundamental model upon which melting and solidification studies are based. It then discusses some general aspects of phase change in pure, single-species systems. The attractiveness of the Stefan problem stems from the fact that the model is amenable to analytical solutions, and as such, it furnishes the physical understanding of more complex analysis of coupled heat and mass transport in a binary system. It also leads to the analysis of morphological surface instabilities and dendrite formation. The emergence of interfacial morphological instabilities in electro-chemical deposition is studied, exposing the rich variety of interfacial structures that arise in this case.

The irradiation of materials with energetic particles may induce drastic changes in their microstructure, and hence may induce important variations in their physicochemical properties. This chapter discusses the phase stability of metals and alloys under irradiation, self-organization of defect microstructure, the rate theory of microstructure self-organization, nonlinear evolution dynamics, dislocation and void dynamics, and spatial instability.

In materials, critical phenomena such as phase transitions, plastic deformation and fracture are intimately related to self-organization. Understanding the origin of spatio-temporal order in systems far from thermal equilibrium and the selection mechanisms of spatial structures and their symmetries is a major theme of present day research on the structure of continuous matter. Furthermore, the development of methods for producing spatially-ordered and self-assembled microstructure in solids by non-equilibrium methods opens the door to many technological applications. In order to describe and understand the behaviour of such materials, dynamical concepts related to non-equilibrium phenomena, irreversible thermodynamics, nonlinear dynamics, and bifurcation theory, are required. The generic presence of defects and their crucial influence on pattern formation and critical phenomena in extended systems is now well-established. Similar to observations in hydrodynamical, liquid crystal, and laser systems, defects in materials have a profound effect. This book is divided into two volumes. The first volume is devoted to the most basic concepts of the physics, mechanics, and mathematical theory utilized in the analysis of non-equilibrium materials. The book presents a background on material deformation, defect theory, transport processes, and the statistical mechanics and thermodynamics of phase transitions. Mathematical concepts of non-linear dynamics, such as bifurcation and instability theory, the dynamics of complex systems near pattern forming instabilities, the generic aspects of pattern formation, selection and stability are presented. Stochastic and numerical methods used in this field are also introduced. The methods and techniques developed in the first volume are applied in the second volume to specific problems in various advanced technologies.

This chapter begins with a discussion of phase transformations and the design of materials. It then presents a case study on the art of steel-making. This is followed by a discussion of precipitate evolution and structure, phase instabilities, stress effects on spinodal decomposition, and martensitic phase transformations.

This chapter begins with a discussion of reduced dynamics and amplitude equations. It then discusses the generic aspects of pattern selection and stability, the effect of external fields; group velocity, convective, and absolute instabilities; and pattern formation and front propagation.

This chapter begins with a discussion of the causes of the roughening and grooving of solid surfaces. It then discusses Asaro-Tiller surface instability, phase field models, surface patterns caused by energetic photons and ions, surface stress during rapid heating, stress corrosion cracking, and numerical methods.

This chapter presents theoretical developments in the treatment of atomic clustering. The theory is applicable to atomic species, which broadly include lattice or surface atoms, molecules, impurities or point defects. The theory is applied in two general areas: on studies of nucleation and growth of precipitates during solidification of metallic alloys, and on atomic clustering on surfaces during deposition of vapour or energetic atoms.

This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial differential equations. It begins by presenting the basic ideas of stability analysis in systems described by ordinary differential equations, introducing Lyapunov functions and their utilization in stability analysis. The stability of systems described by partial differential equations is discussed, emphasizing some of the basic ideas that allow quantitative descriptions of patterns. Specific models illustrating the concepts behind Hopf and Turing instabilities are also discussed.

This chapter aims to develop the fundamental equations of thermodynamics which govern equilibrium between reacting species or phases. It begins with a review of the basic thermodynamic state variables of a homogeneous phase, followed by a discussion of the basic concepts of phase equilibrium in pure species as well as in multi-component systems. The ideal and non-ideal behaviour of mixtures, which facilitates the development of free energy formalisms associated with the mixture, is discussed in the subsequent sections. An introduction to phase diagram calculation techniques for binary and briefly for multi-component systems is presented. Finally, the concept of metastability and equilibrium suppression is introduced, and the significance and applications of metastable phase diagrams are outlined.

Computational modelling of materials behaviour is becoming a reliable tool of scientific investigation, complementary to traditional theory and experimentation. The Multiscale Materials Modelling (MMM) approach reflects the realization that continuum and atomistic analysis methods are complementary. This chapter describes the most popular numerical techniques of each component that make up the MMM paradigm for modelling nano- and micro-systems: Quantum Mechanics (QM), Molecular Dynamics (MD), Monte Carlo (MC), Dislocation Dynamics (DD), Statistical Mechanics (SM), and Continuum Mechanics (CM).