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

Optical tweezers use the forces exerted by a strongly focused beam of light to trap and move objects ranging in size from tens of nanometres to tens of micrometres. Since their introduction in 1986, the optical tweezer has become an important tool for research in the fields of biology, physical chemistry and soft condensed matter physics. Recent advances promise to take optical tweezers out of the laboratory and into the mainstream of manufacturing and diagnostics; they may even become consumer products. The next generation of single-beam optical traps offers revolutionary new opportunities for fundamental and applied research.

http://physics.nyu.edu/grierlab/nreview2c/nreview2c.pdf

A revolution in optical manipulation

With few exceptions biological tissues strongly scatter light, making high-resolution deep imaging impossible for traditional⎯including confocal⎯fluorescence microscopy. Nonlinear optical microscopy, in particular two photon–excited fluorescence microscopy, has overcome this limitation, providing large depth penetration mainly because even multiply scattered signal photons can be assigned to their origin as the result of localized nonlinear signal generation. Two-photon microscopy thus allows cellular imaging several hundred microns deep in various organs of living animals. Here we review fundamental concepts of nonlinear microscopy and discuss conditions relevant for achieving large imaging depths in intact tissue.

/pdf/deep-tissue-two-photon-microscopy-2aqu6d6n65.pdf

Deep tissue two-photon microscopy

Recent theory has predicted a superlens that is capable of producing sub–diffraction-limited images. This superlens would allow the recovery of evanescent waves in an image via the excitation of surface plasmons. Using silver as a natural optical superlens, we demonstrated sub–diffraction-limited imaging with 60-nanometer half-pitch resolution, or one-sixth of the illumination wavelength. By proper design of the working wavelength and the thickness of silver that allows access to a broad spectrum of subwavelength features, we also showed that arbitrary nanostructures can be imaged with good fidelity. The optical superlens promises exciting avenues to nanoscale optical imaging and ultrasmall optoelectronic devices.

http://science.sciencemag.org/content/sci/308/5721/534.full.pdf

Sub-Diffraction-Limited Optical Imaging with a Silver Superlens

The widespread use of controlled molecular-level motion in key natural processes suggests that great rewards could come from bridging the gap between the present generation of synthetic molecular systems, which by and large rely upon electronic and chemical effects to carry out their functions, and the machines of the macroscopic world, which utilize the synchronized movements of smaller parts to perform specific tasks. This is a scientific area of great contemporary interest and extraordinary recent growth, yet the notion of molecular-level machines dates back to a time when the ideas surrounding the statistical nature of matter and the laws of thermodynamics were first being formulated. Here we outline the exciting successes in taming molecular-level movement thus far, the underlying principles that all experimental designs must follow, and the early progress made towards utilizing synthetic molecular structures to perform tasks using mechanical motion. We also highlight some of the issues and challenges that still need to be overcome.

Synthetic molecular motors and mechanical machines.

Double emulsions are highly structured fluids consisting of emulsion drops that contain smaller droplets inside. Although double emulsions are potentially of commercial value, traditional fabrication by means of two emulsification steps leads to very ill-controlled structuring. Using a microcapillary device, we fabricated double emulsions that contained a single internal droplet in a core-shell geometry. We show that the droplet size can be quantitatively predicted from the flow profiles of the fluids. The double emulsions were used to generate encapsulation structures by manipulating the properties of the fluid that makes up the shell. The high degree of control afforded by this method and the completely separate fluid streams make this a flexible and promising technique.

/pdf/monodisperse-double-emulsions-generated-from-a-2mkl984v4i.pdf

Monodisperse Double Emulsions Generated from a Microcapillary Device

The maximal clique problem has been solved by means of molecular biology techniques. A pool of DNA molecules corresponding to the total ensemble of six-vertex cliques was built, followed by a series of selection processes. The algorithm is highly parallel and has satisfactory fidelity. This work represents further evidence for the ability of DNA computing to solve NP-complete search problems.

https://science.sciencemag.org/content/sci/278/5337/446.full.pdf

DNA Solution of the Maximal Clique Problem

Many interesting and important materials such as polymers, gels, and biomaterials are viscoelastic; when responding to an external stress, they both store and dissipate energy. This behavior is quantified by the complex shear modulus, Gv, which provides insight into the material’s microscopic dynamics. Typically, Gv is measured by applying oscillatory strain to a sample and measuring the resulting stress. Recently a new method, called microrheology, has been developed which determines Gv from the thermal motion of microscopic tracer particles embedded in the material [1,2]. Microrheology offers significant potential advantages: it provides a local probe of Gv in miniscule sample volumes and can do so at very high frequencies. While microrheology provides an accurate measure of Gv for simple systems, its validity in common complex systems is far from certain. If the tracers locally modify the structure of the medium, or sample only pores in an inhomogeneous matrix, then bulk rheological properties will not be determined. Such subtle effects currently limit many interesting applications of microrheology. In this Letter, we introduce a new formalism, which we term “two-point microrheology,” based on measuring the cross-correlated thermal motion of pairs of tracer particles to determine Gv. This new technique overcomes the limitations of single-particle microrheology. It does not depend on the size or shape of the tracer particle; moreover it is independent of the coupling between the tracer and the medium. We demonstrate the validity of this approach with measurements on a highly inhomogeneous material, a solution of the polysaccharide guar. Two-point microrheology correctly reproduces results obtained with a mechanical rheometer, whereas single-particle microrheology gives erroneous results. We also compare ordinary and two-point microrheology of F-actin [2–4], a semiflexible biopolymer constituent of the cytoskeleton. Different results are obtained with the two techniques, suggesting that earlier interpretations of F-actin microrheology should be reexamined.

/pdf/two-point-microrheology-of-inhomogeneous-soft-materials-5f5ib5yttc.pdf

Two-point microrheology of inhomogeneous soft materials.

We present an optical realization of a thermal ratchet. Directed motion of Brownian particles in water is induced by modulating in time a spatially periodic but asymmetric optical potential. The net drift shows a maximum as a function of the modulation period. The experimental results agree with a simple theoretical model based on diffusion.

Optical Thermal Ratchet

We develop a multiple particle tracking technique for making precise, localized measurements of the mechanical microenvironments of inhomogeneous materials. Using video microscopy, we simultaneously measure the Brownian dynamics of roughly one hundred fluorescent tracer particles embedded in a complex medium and interpret their motions in terms of local viscoelastic response. To help overcome the inherent statistical limitations due to the finite imaging volume and limited imaging times, we develop statistical techniques and analyze the distribution of particle displacements in order to make meaningful comparisons of individual particles and thus characterize the diversity and properties of the microenvironments. The ability to perform many local measurements simultaneously allows more precise measurements even in systems that evolve in time. We show several examples of inhomogeneous materials to demonstrate the flexibility of the technique and learn new details of the mechanics of the microenvironments that small particles explore. This technique extends other microrheological methods to allow simultaneous measurements of large numbers of probe particles, enabling heterogeneous samples to be studied more effectively.

/pdf/investigating-the-microenvironments-of-inhomogeneous-soft-16hprxv4yn.pdf

Peter D. Kaplan

Papers

Monodisperse Double Emulsions Generated from a Microcapillary Device

DNA Solution of the Maximal Clique Problem

Two-point microrheology of inhomogeneous soft materials.

Optical Thermal Ratchet

Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking