A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Microfabricated integrated circuits revolutionized computation by vastly reducing the space, labor, and time required for calculations. Microfluidic systems hold similar promise for the large-scale automation of chemistry and biology, suggesting the possibility of numerous experiments performed rapidly and in parallel, while consuming little reagent. While it is too early to tell whether such a vision will be realized, significant progress has been achieved, and various applications of significant scientific and practical interest have been developed. Here a review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. Specifically, this review explores the Reynolds number Re, addressing inertial effects; the Peclet number Pe, which concerns convective and diffusive transport; the capillary number Ca expressing the importance of interfacial tension; the Deborah, Weissenberg, and elasticity numbers De, Wi, and El, describing elastic effects due to deformable microstructural elements like polymers; the Grashof and Rayleigh numbers Gr and Ra, describing density-driven flows; and the Knudsen number, describing the importance of noncontinuum molecular effects. Furthermore, the long-range nature of viscous flows and the small device dimensions inherent in microfluidics mean that the influence of boundaries is typically significant. A variety of strategies have been developed to manipulate fluids by exploiting boundary effects; among these are electrokinetic effects, acoustic streaming, and fluid-structure interactions. The goal is to describe the physics behind the rich variety of fluid phenomena occurring on the nanoliter scale using simple scaling arguments, with the hopes of developing an intuitive sense for this occasionally counterintuitive world.

/pdf/microfluidics-fluid-physics-at-the-nanoliter-scale-4bpahtd8qa.pdf

Microfluidics: Fluid physics at the nanoliter scale

The lithium metal battery is strongly considered to be one of the most promising candidates for high-energy-density energy storage devices in our modern and technology-based society. However, uncontrollable lithium dendrite growth induces poor cycling efficiency and severe safety concerns, dragging lithium metal batteries out of practical applications. This review presents a comprehensive overview of the lithium metal anode and its dendritic lithium growth. First, the working principles and technical challenges of a lithium metal anode are underscored. Specific attention is paid to the mechanistic understandings and quantitative models for solid electrolyte interphase (SEI) formation, lithium dendrite nucleation, and growth. On the basis of previous theoretical understanding and analysis, recently proposed strategies to suppress dendrite growth of lithium metal anode and some other metal anodes are reviewed. A section dedicated to the potential of full-cell lithium metal batteries for practical applicatio...

Toward Safe Lithium Metal Anode in Rechargeable Batteries: A Review.

Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Long-scale evolution of thin liquid films

There is a special reason for reviewing this book at this time: it is the 50th edition of a compendium that is known and used frequently in most chemical and physical laboratories in many parts of the world. Surely, a publication that has been published for 56 years, withstanding the vagaries of science in this century, must have had something to offer. There is another reason: while the book is a standard fixture in most chemical and physical laboratories, including those in medical centers, it is not as frequently seen in the laboratories of physician's offices (those either in solo or group practice). I believe that the Handbook can be useful in those laboratories. One of the reasons, among others, is that the various basic items of information it offers may be helpful in new tests, either physical or chemical, which are continuously being published. The basic information may relate

Handbook of Chemistry and Physics

IN his classic 1952 paper, Turing1 suggested a possible connection between patterns in biological systems and patterns that could form spontaneously in chemical reaction–diffusion systems. Turing's analysis stimulated considerable theoretical research on mathematical models of pattern formation, but Turing-type patterns were not observed in controlled laboratory experiments until 19902,3. Subsequently there has been a renewed interest in chemical pattern formation and in the relationship of chemical patterns to the remarkably similar patterns observed in diverse physical and biological systems4. Numerical simulations of a simple model chemical system have recently revealed spot patterns that undergo a continuous process of 'birth' through replication and 'death' through overcrowding5. Here we report the observation of a similar phenomenon in laboratory experiments on the ferrocyanide–iodate–sulphite reaction. Repeated growth and replication can be observed for a wide range of experimental parameters, and can be reproduced by a simple two-species model, suggesting that replicating spots may occur in many reaction–diffusion systems.

Experimental observation of self-replicating spots in a reaction–diffusion system

Experiments on a bistable chemical reaction in a continuously fed thin gel layer reveal a new type of spatiotemporal pattern, one in which fronts propagate at a constant speed until they reach a critical separation (typically 0.4 millimeter) and stop. The resulting asymptotic state is a highly irregular stationary pattern that contrasts with the regular patterns such as hexagons, squares, and stripes that have been observed in many nonequilibrium systems. The observed patterns are initiated by a finite amplitude perturbation rather than through spontaneous symmetry breaking.

https://science.sciencemag.org/content/sci/261/5118/192.full.pdf

Pattern Formation by Interacting Chemical Fronts

Surface-tension-driven Benard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Benard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths ( 0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

Long-wavelength surface-tension-driven Bénard convection: experiment and theory

Past experiments on the formation of chemical spatial structures1–4 (mainly chemical waves) have been conducted in closed, batch systems such as Petri dishes5–15, and not in open chemostats such as those usually employed in the study of temporal structures. As a consequence, the observed spatial patterns change uncontrollably and ultimately decay as the mixture evolves towards thermody-namic equilibrium. We have overcome this problem in a novel type of open reactor: the reaction occurs in an annular inert gel that is fed at the inner and outer rims. The reaction that we have studied is the Belousov–Zhabotinskii (BZ) reaction, which has become the paradigm for studies of spatial and temporal structures in nonlinear systems. In the ring reactor we observe sustained travelling azimuthal waves for a wide range of conditions.

Sustained chemical waves in an annular gel reactor: a chemical pinwheel

Numerical simulations and laboratory experiments are conducted for thin layers of particles in a vertically oscillated container as a function of the frequency $f$, amplitude $A$, and depth $H$. The same standing wave patterns (stripes, squares, or hexagons oscillating at $f/2$ or $f/4$) and wavelengths are obtained in the simulations and experiments for a wide range of $(f,A)$ and two layer depths. Two model parameters are determined by fits at just two points $(f,A,H)$. Simulation results lead to heuristic arguments for the onset of patterns and the crossover from squares to stripes.

/pdf/patterns-in-3d-vertically-oscillated-granular-layers-1bnsqiwo35.pdf

William D. McCormick

Papers

Experimental observation of self-replicating spots in a reaction–diffusion system

Pattern Formation by Interacting Chemical Fronts

Long-wavelength surface-tension-driven Bénard convection: experiment and theory

Sustained chemical waves in an annular gel reactor: a chemical pinwheel

Patterns in 3D Vertically Oscillated Granular Layers: Simulation and Experiment