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.

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Pattern formation outside of equilibrium

Solid lipid nanoparticles: Production, characterization and applications

Single-molecule force spectroscopy has emerged as a powerful tool to investigate the forces and motions associated with biological molecules and enzymatic activity. The most common force spectroscopy techniques are optical tweezers, magnetic tweezers and atomic force microscopy. Here we describe these techniques and illustrate them with examples highlighting current capabilities and limitations.

Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

The goal of this survey is to review recent developments in the hydro­ dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. We wish to dem­ onstrate how these notions can be used effectively to obtain a qualitative and quantitative description of the spatio-temporal dynamics of open shear flows, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, etc. In this review, we only consider open flows where fluid particles do not remain within the physical domain of interest but are advected through downstream flow boundaries. Thus, for the most part, flows in "boxes" (Rayleigh-Benard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc.) are not discussed. Further­ more, the implications of local/global and absolute/convective instability concepts for geophysical flows are only alluded to briefly. In many of the flows of interest here, the mean-velocity profile is non-

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Local and global instabilities in spatially developing flows

Convection in a binary mixture

We report an experimental study of the dynamics of thermal fluctuations in a 4.86 mm thick layer of ${\mathrm{CS}}_{2}$ heated from above. Stabilizing gradients ranged from 10.3 to $61.7\text{ }\text{ }\mathrm{K}/\mathrm{cm}$. Power spectral measurements were made over the wave vector range $9\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}1}lql100\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}1}$, using a shadowgraph apparatus. Over this range, the spectra change from having a well-resolved oscillatory mode at low $q$ to monotonic decay at higher $q$. For the smallest gradient, the results for the power spectra compare very well with theory, but slight deviations are seen for larger gradients.

Dynamics of long-wavelength fluctuations in a fluid layer heated from above.

A method and apparatus for analyzing a fluid containing light scattering components. The apparatus includes a laser adapted to direct abeam of light into the fluid such as a liquid containing particles, and at least two detectors to receive light scattered by the fluid, which received light is used to calculate a physical property of the fluid. A focusing lens may be used to focus the light beam to a focal waist in the fluid to enhance the degree of spatial coherence of the scattered light. The two detectors are aligned onto the focal waist and are adapted to send a signal to a data processor which correlates the signal received from said two detectors. The two detectors are spaced a substantially equal distance from the focal waist and are oriented at substantially the same angle with respect to the direction of propagation of the incident beam. The data processor is adapted to cross-correlate the signals received from the two detectors. A polarizer can be positioned between the focal waist and the two detectors to aid in positioning the detectors and to enhance the efficiency and speed of the measurement.

Cross-correlation method and apparatus for suppressing the effects of multiple scattering

The pure conduction state of a horizontal layer of fluid heated from below becomes unstable with respect to a convecting state when the temperature difference exceeds a critical value. We examine the question of how real, physical systems evolve from conduction to convection. Most experimental cells contain geometric or thermal inhomogeneities which render the bifurcation to convection imperfect. In that case the pure conduction state never exists and the convecting state evolves continuously and smoothly as the temperature difference is raised. When a sufficiently perfect experimental cell is constructed to eliminate this route to convection, then dynamic imperfections will usually prevail. When the temperature difference across the cell is raised, the vertical gradients in the sidewalls evolve at a rate which differs from that in the fluid. The resultingtransient horizontal thermal gradients initiate the convective flow. This phenomenon can be eliminated by providing sidewalls which have the same thermal diffusivity as that of the fluid. When that is done, the convective flow is started by random noise which exists in any experimental system. Analysis of experiments shows that the noise source is considerably stronger than thermal noise, but its origin is unclear at this time.

David S. Cannell

Papers

Convection in a binary mixture

Dynamics of long-wavelength fluctuations in a fluid layer heated from above.

Cross-correlation method and apparatus for suppressing the effects of multiple scattering

Deterministic and stochastic effects near the convective onset

Analysis of sound propagation in xenon near the critical point