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-

/pdf/local-and-global-instabilities-in-spatially-developing-flows-ja4x14nrk6.pdf

Local and global instabilities in spatially developing flows

The atomic force microscope (AFM) can be used to image the surface of both conductors and nonconductors even if they are covered with water or aqueous solutions. An AFM was used that combines microfabricated cantilevers with a previously described optical lever system to monitor deflection. Images of mica demonstrate that atomic resolution is possible on rigid materials, thus opening the possibility of atomic-scale corrosion experiments on nonconductors. Images of polyalanine, an amino acid polymer, show the potential of the AFM for revealing the structure of molecules important in biology and medicine. Finally, a series of ten images of the polymerization of fibrin, the basic component of blood clots, illustrate the potential of the AFM for revealing subtle details of biological processes as they occur in real time.

https://science.sciencemag.org/content/sci/243/4898/1586.full.pdf

Imaging crystals, polymers, and processes in water with the atomic force microscope.

Measurement of the fractal dimension, $D$, of colloidal aggregates of small silica particles is reported. We observe power-law decay of the structure factor $[S(k)\ensuremath{\sim}{k}^{\ensuremath{-}D}]$ by both light and x-ray scattering showing that the aggregates are fractal. $D$ is found to be 2.12\ifmmode\pm\else\textpm\fi{}0.05, which is intermediate between recent numerical results for the kinetic models of diffusion-limited aggregation ($D=2.5$) and cluster aggregation ($D=1.75$), but is rather close to the value for lattice animals ($D=2.0$), which are equilibrium structures.

Fractal geometry of colloidal aggregates

We present experimental results showing that colloidal silica spheres can be induced to aggregate either slowly or very rapidly. The slow process always yields clusters with a fractal dimensionality ${\mathrm{d}}_{\mathrm{f}}$=2.08\ifmmode\pm\else\textpm\fi{}0.05, but the rapid process can produce clusters with either ${\mathrm{d}}_{\mathrm{f}}$=1.75\ifmmode\pm\else\textpm\fi{}0.05 or ${\mathrm{d}}_{\mathrm{f}}$=2.08\ifmmode\pm\else\textpm\fi{}0.05. However, clusters with ${\mathrm{d}}_{\mathrm{f}}$=1.75 are always observed to restructure so as to yield ${\mathrm{d}}_{\mathrm{f}}$=2.08\ifmmode\pm\else\textpm\fi{}0.05.

Restructuring of colloidal silica aggregates.

We report experiments on convection patterns in a cylindrical cell with a large aspect ratio. The fluid had a Prandtl number [sigma][approx]1. We observed a chaotic pattern consisting of many rotating spirals and other defects in the parameter range where theory predicts that steady straight rolls should be stable. The correlation length of the pattern decreased rapidly with increasing control parameter so that the size of a correlated area became much smaller than the area of the cell. This suggests that the chaotic behavior is intrinsic to large aspect ratio geometries.

David S. Cannell

Papers

Imaging crystals, polymers, and processes in water with the atomic force microscope.

Fractal geometry of colloidal aggregates

Restructuring of colloidal silica aggregates.

Spiral defect chaos in large aspect ratio Rayleigh-Bénard convection.

Traveling waves and spatial variation in the convection of a binary mixture