We discuss issues concerning the reconstruction of attractors from experimental chaotic time-series data using Takens’s method of delays [in Proceedings of the Warwick Symposium, 1981, edited by D. A. Rand and L. S. Young (Springer, New York, 1981)]. The focus of this paper is on the selection of appropriate lag-time and embedding-dimension values with an emphasis on the relationship between those parameters and data-measurement considerations. We are particularly interested in the effect that low-pass filtering has on the appearance and measured properties of reconstructed attractors. Empirical results are presented using data measured from a laboratory fluidized bed and from data generated by integrating the Lorenz [J. Atmospheric Sci. 20, 130 (1963)] and Franceschini [Physica 6D, 285 (1983)] models of chaotic dynamic systems.

Small-amplitude Boussinesq convection in a plane layer with rigid, conducting boundaries rotating uniformly about a vertical axis is studied. A horizontally unbounded layer is modeled by periodic boundary conditions and the effect of mean-flow suppression by distant sidewalls is considered. An exact linear stability calculation partitions parameter space into regions of stationary and oscillatory convective onset. In the stationary regime, the critical Taylor number and critical angle for the onset of the Küppers-Lortz instability are determined as a function of the Prandtl number σ. Of the two competing two-dimensional patterns in the oscillatory regime, traveling waves are the preferred platform for 0.442<σ<0.677. For σ<0.442 standing waves are preferred at onset for small rotation rates while traveling waves are preferred for larger rotation rates.

Micellar phases of the sodium dodecyl (lauryl) sulfate (SLS)–water–decanol system have been studied by x-ray scattering in the isotropic (I) phase, with emphasis on the I→hexagonal (${\mathit{H}}_{\mathrm{\alpha }}$) and I→nematic-cylindrical (${\mathit{N}}_{\mathit{c}}$) lyotropic liquid-crystal phase transitions. Analysis of the scattering curves is made through modeling of the product P(q)S(q), where P(q) is the micellar form factor and S(q) is the intermicellar interference function, calculated from screened Coulombic repulsion in a mean spherical approximation. Results show that micelles grow more by decanol addition near the I→${\mathit{N}}_{\mathit{c}}$ transition (anisometry ν≃3) than by increased amphiphile concentration in the binary system near the I→${\mathit{H}}_{\mathrm{\alpha }}$ phase transition (ν≃2.4). These results compare well with recent theories for isotropic–liquid-crystal phase transitions.

A discrete model is presented to describe the dynamics of nematic liquid crystals for the case where topological defects dominate the spatial pattern. A numerical study is given of the annihilation kinetics of the defects in the two-dimensional nematic system with ${\mathit{P}}^2$ symmetry. The structure factor is found to obey a scaling law S(k,t)=〈k${\mathrm{〉}}_{\mathit{t}}^2$g(k/〈k${\mathrm{〉}}_{\mathit{t}}$) where the first moment 〈k${\mathrm{〉}}_{\mathit{t}}$ varies as 〈k${\mathrm{〉}}_{\mathit{t}}$∼${\mathit{t}}^{\mathrm{-}0.42}$. The asymptotic power-law tail g(x)∼${\mathit{x}}^{\mathrm{-}4.5}$ is found.

The effect of combined intrinsic size and charge polydispersity and hydrodynamic interaction on the short-time dynamics in unimodal and bimodal suspensions of charged macroparticles is investigated. The first cumulant of the dynamic-scattering function is calculated based on a bimodal Schulz distribution for the particle sizes and charges. The influence of the solvent-mediated hydrodynamic interaction between the macroparticles is approximately described by a far-field expansion in terms of reciprocal distance ${\mathit{r}}^{\mathrm{-}1}$, including terms up to order ${\mathit{r}}^{\mathrm{-}11}$. The partial static-structure functions, which are required as an input, are obtained using the multicomponent-hypernetted-chain approximation. Our results are compared with data from dynamic-light-scattering experiments on binary mixtures of polystyrene spheres and on unimodal suspensions of charged silica particles. It is shown that both the intrinsic polydispersity and the hydrodynamic interaction significantly affect the measured first cumulant.

Coherent neutron-scattering experiments on liquid cesium and rubidium near the melting point reveal an anomalous dispersion of collective modes for frequencies ω${\mathrm{\tau }}_{\mathit{M}}$≫1 with ${\mathrm{\tau }}_{\mathit{M}}$ the Maxwell relaxation time for shear stress in the liquid. Within the framework of generalized hydrodynamics, these modes are shown to propagate similar to zero sound in a restricted range of wave numbers around Q=0.4 A${\mathrm{̊}}^{\mathrm{-}1}$. Such a behavior is known from glass-forming systems, but here simple liquid metals near the melting point are shown to exhibit such modes too. The feature of the dispersion relation in this Q region yields quantities known from the theory of elasticity of solids such as the shear modulus ${\mathit{G}}_{\mathrm{\infty }}$ and the Poisson ratio ν. Some evidence is found for the existence of small clusters in the liquid on the picosecond time scale.

The ability of a single positronium atom to form a semimacroscopic bubble in helium is a remarkable manifestation of quantum mechanics. Experimental evidence for bubble formation is provided by a dramatic decrease in the decay rate of the triplet state when the ambient conditions favor its formation. The phenomenon is observed near the critical point of helium and is well explained by a mean-field theory in which the positronium atom occupies the ground state of a local potential well induced by its influence on the average local density. Because of the active role played by the positronium atom in producing this localized state, the process is referred to as self-trapping. Similar experiments on other noble gases also show evidence for self-trapping near the liquid-vapor critical point, but the transition from extended to localized behavior is gradual. Mean-field theories, which ignore fluctuations, are not successful at these higher temperatures, suggesting that statistical fluctuations strongly influence the distribution over states of the light atom. This paper presents a theoretical investigation of the localization of positronium in a dense noble gas which employs the path integral to represent the translational degrees of freedom of the light atom. It demonstrates that a theoretical model which properly accounts for fluctuations is able to predict the main features of the experimental measurements.

We use a cellular-automata (CA) method to determine the frequency-dependent permeability k(ω) of a channel and a disordered porous medium. The CA results for the channel flow agree completely with the analytical solution of the problem. For flow in a porous medium, we calculate k(ω) over a wide range of the frequencies ω and for several values of the porosity. Our CA results for disordered porous media support the relation k(ω)/${\mathit{k}}_0$=f(ω/${\mathrm{\omega }}_{\mathit{c}}$), where ${\mathit{k}}_0$ is the static permeability of the medium, ${\mathrm{\omega }}_{\mathit{c}}$ is a characteristic frequency, and f(x) is a universal function, proposed previously, based on experimental data.

An analysis of partial-structure-factor information on molecular liquids is described which uses the spherical harmonic expansion of the site-site structure factors to extract an estimate of the orientational pair-correlation function between molecules. Recently published neutron-diffraction data on the site-site partials in liquid hydrogen iodide at 210 K are used as the input data to this technique. The results, which are presented as a map of the orientational pair-correlation function, g(r,${\mathrm{\omega }}_1$,${\mathrm{\omega }}_2$), show the occurrence of pronounced relative orientational correlations between molecules at this temperature, even though the same orientations are apparently only weakly correlated with the molecular center of mass. The same spherical harmonic expansion procedure can be applied to a number of other molecular liquids where diffraction data are available.

Colloidal silica spheres bearing grafted octadecyl chains dispersed in hexadecane undergo a sol-gel transition with decreasing temperature. The gelation temperature depends on the volume fraction φ and, perhaps, the particle size. For φ>${\mathrm{\phi }}_{gel}$(T), the value at the transition, the elastic modulus varies as (φ-${\mathrm{\phi }}_{gel}$${)}^{\mathit{s}}$ with the prefactor and exponent independent of temperature. This form resembles prediction from static percolation theories, but the exponent s=3.0±0.5 lies significantly below those expected and the transition volume fraction varies with temperature. The relationship of the gelation transition to dynamic percolation and phase transitions predicted by equilibrium statistical mechanics has also been addressed. Matching the calculated structure factor for adhesive spheres with that measured by static light scattering yields the unknown strength of the interparticle attraction as a function of temperature. Though an imperfect fit introduces considerable uncertainty, this empirical relationship distinguishes the gel transition from both the dynamic percolation threshold and the spinodal associated with the fluid-fluid transition for adhesive spheres. Thus we conclude that gelation in this colloidal dispersion corresponds to a metastable state lying between the fluid-solid phase boundary and the spinodal.

We study dilute solutions subjected to finite-amplitude periodic velocity-gradient fields of characteristics strength Ω, frequency ω, and vanishing time average. On time scales large compared to the flow period, fluctuations in particle position are equilibriumlike but with an effective diffusivity matrix D̃(λ)==E(λ)D, where D is the no-flow diagonal diffusion matrix and D̃ depends nonlinearly on λ==Ω/ω. We prove that detD̃>detD and that trD̃≥trD for all flows of this class; physically, this implies that the rate of density decay and the rate of growth in size of a diffusing cloud of particles is always increased relative to no flow. This enhancement arises from the interaction of the flow with microscopic fluctuations and is an example of Taylor dispersion. When 1/Ω and 1/ω become comparable to the longest internal particle relaxation time the behavior changes since internal modes are excited. As an example, we consider a dilute polymer solution in a simplified ‘‘Rouse dynamics’’ treatment. Relative to the center of gravity the motion of a monomer decomposes into an ‘‘infinite’’ deterministic part arising from the periodic stretching of the chain plus fluctuations whose long-time form is characterized by the same effective diffusivity D̃ as for the center of gravity. The mean-squared size of the chain is increased, relative to equilibrium, by the same factor trE/3 as the diffusivity. Correlation functions amenable to measurement in dynamical scattering experiments are calculated; these provide direct measurement of the nonlinear D̃.

The phase diagram and the structure of smectic-${\mathit{A}}_{\mathit{d}}$ (Sm-${\mathit{A}}_{\mathit{d}}$) and Sm-${\mathit{A}}_2$ phases of binary mixtures of 4-octyloxy-4’-cyanobiphenyl (8OCB) and 4-n-heptyloxyphenyl-4’-cyanobenzoyloxybenzoate (DB7OCN) have been determined by high-resolution x-ray diffraction. This system had previously been reported by Ratna, Shashidhar, and Raja [Phys. Rev. Lett. 55, 1476 (1985)] to exhibit an incommensurate Sm-${\mathit{A}}_{\mathit{i}2}$ phase for concentrations of 8OCB ranging from 25 to 40 mol %. The phase diagram obtained from the present study is in excellent agreement with the mean-field theory of Prost and co-workers [J. Phys. (Paris) 46, 391 (1985)]. The results of the x-ray study establish that the Sm-${\mathit{A}}_{\mathit{i}2}$ phase is, in fact, a coexistence of the Sm-${\mathit{A}}_{\mathit{d}}$ and Sm-${\mathit{A}}_2$ phases at the first-order phase transition between them. Coexistence of the two phases is observed at all concentrations of 8OCB ranging from 35.1 to 0 mol %. At a fixed temperature in the coexistence region, the system evolves towards one phase at a very slow pace. The coexistence region becomes narrower at lower concentrations. Temperature dependence of the smectic wave vectors indicates the presence of a critical point at (physically inaccessible) negative concentrations of 8OCB. The coexistence behavior at the Sm-${\mathit{A}}_{\mathit{d}}$ to Sm-${\mathit{A}}_2$ transition also conforms to the predictions of the theory and resembles the coexistence curve in the proximity of the liquid-gas critical point.

We report detailed observations of cellular growth at the smectic-A–smectic-B interface of the liquid crystal N-[4’-(n-butoxy)benzylidene]-4-(n-octyl) aniline (4O.8) when it grows along a forbidden orientation of the equilibrium shape. Four growth regimes have been observed at increasing velocity, each of them characterized by cells of a given type: symmetric cells with an angular discontinuity at the tip near the onset of instability, then partly faceted asymmetric cells forming propagating domains, followed by ‘‘chaotic’’ cells, and finally dendrites.

Salt-induced aggregation of polystyrene latex confined to two dimensions on an air-water interface has been studied. Fractal dimensions of the aggregates are determined (i) from relation between the radius of gyration and cluster masses and (ii) by the box method. They both increase with aggregation time. While the fractal dimensions are insensitive to electrolyte concentration, the aggregation rate does depend on it. Our results for aggregation kinetics are in good agreement with the dynamic scaling law ${\mathit{n}}_{\mathit{m}}$(t)∼${\mathit{m}}^{\mathrm{-}2}$f(m/${\mathit{t}}^{\mathit{z}}$), for z=1, for dilute latex dispersions [${\mathit{n}}_{\mathit{m}}$(t) is the number of clusters of size m at time t]. A nonlinear behavior in time, however, is observed for more concentrated latex dispersions. We find bell-shaped cluster-size distributions.

We present experimental results of the effective Raman density of states showing that water both in normal and supercooled states has a fracton behavior. We have unambiguously determined a crossover frequency ${\mathrm{\omega }}_{\mathrm{co}}$ between the phonon and fracton regimes. The obtained spectral results and the temperature behavior of ${\mathrm{\omega }}_{\mathrm{co}}$ are consistent with the correlated-site percolation model indicating the existence of four-bonded patches whose spatial extent increases with decreasing temperature. This picture for the water system is confirmed from the spectral results of two different water-alcohol mixtures; the first one with ethanol, which acts as a ‘‘patch-breaking’’ impurity, and the second one with butoxyethanol, which can be considered as a ‘‘patch-increasing’’ impurity.

A free-energy model for mixtures of charged particles interacting through the Yukawa repulsive potential is derived analytically from the Onsager-type exact lower bound for the free energy of the system. It takes the form of a ‘‘nonlinear mixing rule,’’ which relates the configurational free energy of the mixture to that of the one-component Yukawa system. This scaling law holds, under broad assumptions, for the exact Madelung (i.e., asymptotic strong-coupling) energy term for the fluid; within the hypernetted-chain approximation, it is very accurate also for conditions of intermediate and weak coupling. The physics relating this mixing rule with the well-known volume-additivity rule for the Thomas-Fermi mixing of elements is revealed.

We studied the coarsening behavior of two lyotropic liquid-crystal systems by static light scattering. The samples were quenched from the isotropic phase into either the nematic phase or a region of coexistence between nematic and isotropic phases. In the coexistence region, we observed, in both two and three dimensions, Porod power-law tails of the scattering intensity. Such a behavior is described by S(q)∼${\mathit{q}}^{\mathrm{-}(\mathit{d}+1)}$ in the limit of large wave vectors q, where S is the scattering intensity, q is the wave vector, and d is the dimension of the system. In addition, the nematic phases displayed novel power-law scaling behavior at large q, namely, S(q)∼${\mathit{q}}^{\mathrm{-}\mathit{u}}$, where u=4 in two dimensions and u=6 in three dimensions. These results will be compared to recent theoretical predictions.

We present a rheological study of the gel transition in silica gels resulting from the hydrolysis-condensation of a silicon alkoxide. The complex shear modulus is measured in a large frequency range (${10}^{\mathrm{-}2–}$${10}^2$ rad/s). The exponents for the viscosity s=0.7±0.1, for the elasticity t=2.0±0.1, for the dynamic crossover frequency z=2.9±0.2, and for the scaling-frequency power law Δ=0.72±0.03 are in agreement with the scalar percolation model in a restricted region around the gelation time (3%). Beyond, there is a crossover to a vectorial percolation regime characterized by a much higher elasticity exponent t’=3.6±0.1.

We show that the effect of surfactants on Ostwald ripening can be reduced to the problem of Ostwald ripening with a time-dependent surface tenison. Since the latter can be mapped onto a constant-surface-tension case with a nonlinear time transformation, we can systematically study the effect of surfactants without much difficulty. As a result, we find that the scaled distribution function of droplet size remains the same, but the average domain size no longer obeys the power-law growth ${\mathit{t}}^{1/3}$. Furthermore, we show that the average domain size and the total number of droplets saturate at very late times. The former is inversely proportional to the surfactant density at the interfaces, while the latter is proportional to the square (cube) of the surfactant density at the interfaces in two dimensions (three dimensions). We also find that different growths for different surfactant densities at the interface can be written in a crossover scaling form. These results qualitatively confirm the recent results of Laradji et al. [J. Phys. A 44, L629 (1991)].

We present a theoretical analysis of the formation of a dendritic array in a directional-solidification experiment. Our calculation contains three sequential ingredients: acceleration of an initially flat interface and the concomitant buildup of a solutal boundary layer in front of it; onset of a morphological instability, triggered by thermal fluctuations, producing a relatively finely spaced array of dendritic tips; coarsening of this array and final selection of a steady-state primary spacing. For sufficiently large growth speeds, where the resulting dendrites interact with each other weakly, we find–with no adjustable parameters–good agreement with the experiments of Trivedi and Somboonsuk [Acta Metall. 33, 1061 (1985)] and Somboonsuk, Mason, and Trivedi [Metall. Trans A 15A, 967 (1984)].