We consider a lattice-gas model of particles with internal orientational degrees of freedom. In addition to antiferromagnetic nearest-neighbor (NN) and next-nearest-neighbor (NNN) positional interactions we also consider NN and NNN interactions arising from the internal state of the particles. The system then shows positional and orientational ordering modes with associated phase transitions at $T_p$ and $T_o$ temperatures at which long-range positional and orientational ordering are, respectively, lost. We use mean-field techniques to obtain a general approach to the study of these systems. By considering particular forms of the orientational interaction function we study coupling effects between both phase transitions arising from the interplay between orientational and positional degrees of freedom. In mean-field approximation coupling effects appear only for the phase transition taking place at lower temperatures. The strength of the coupling depends on the value of the long-range order parameter that remains finite at that temperature.

For repetitive experiments performed on a deterministic system with initial states restricted to a certain region in phase space, the relative frequency of an event has a definite value insensitive to the preparation of the experiments only if the initial states leading to that event are distributed uniformly in the prescribed region. Mechanical models of coin tossing and roulette spinning and equal a priori probability hypothesis in statistical mechanics are considered in the light of this principle. Probabilities that have arisen from uniform distributions of initial states do not necessarily submit to Kolmogorov’s axioms of probability. In the finite-dimensional case, a uniform distribution in phase space either in the coarse-grained sense or in the limit sense can be formulated in a unified way.

A theory is presented for the transient process of decay from the state of marginal stability occurring in a subcritical pitchfork bifurcation. We introduce an approximation for the stochastic paths that allows the calculation of a generating function for the statistics of the escape time. The time-dependent moments and transient fluctuations are calculated from the distribution of passage times by a scaling transformation that is associated with initial non-Gaussian statistics. Our results are in agreement with numerical simulations reported also here.

A class of colored (non-δ-correlated) random matrices composed of two independent systems of random number sequences, each of which is allocated to diagonal and off-diagonal parts of the matrices, is studied as workable models for the analysis of level statistics of sufficiently large degree-of-freedom Hamiltonian systems. In band random-matrix models, the rapid saturation toward the Wigner-type distribution as a function of the bandwidth is observed in spacing distribution patterns, suggesting that colored random matrices are capable of reproducing the non-δ-correlated random property found in level-spacing characteristics of real Hamiltonian systems. Detailed analysis reveals that at least two system parameters characterizing the colored random matrices are necessary for a workable basis of the model. The effect of the variance around the mean value of random number sequences on the spacing distribution is investigated in the small-bandwidth region, and a new type of anomalous distribution with non-Wigner-like distribution patterns is found in the small-variance limit.

We analyze the light-scattering spectrum of a suspension in a viscoelastic fluid under density and velocity gradients. When a density gradient is present, the dynamic structure factor exhibits universality in the sense that its expression depends only on the reduced frequency and the reduced density gradient. For a velocity gradient, however, the universality breaks down. In this last case we have found a transition point from one to three characteristic frequencies in the spectrum, which is governed by the value of the external gradient. The presence of the viscoelastic time scales introduces a shift in the ‘‘critical’’ point.

We study the stability of spatially coherent, time-periodic states in noisy, classical, discrete-time, many-body systems with short-range interactions. Generic stability of periodic k cycles with k>2 can be achieved only by rules carefully constructed to exploit lattice anisotropy and so suppress droplet growth. For ordinary rules which do not utilize spatial anisotropy in this way, periodic k cycles with periods k>2 are metastable rather than stable under generic conditions, losing spatial coherence through nucleation and growth of droplets. The unusual dynamical properties of the periodic states stabilized by anisotropy are described.

A model system for the Lorentz gas can be made [Eder, Chen, and Egelstaff, Proc. Phys. Soc. London 89, 833 (1966); McPherson and Egelstaff, Can. J. Phys. 58, 289 (1980)] by mixing small quantities of hydrogen with an argon host. For neutron-scattering experiments the large H-to-Ar cross section ratio (∼200) makes the argon relatively invisible. Dynamic-structure-factor [S(Q,ω) for ${\mathrm{H}}_2$] measurements at room temperature have been made on this system using the IN4 spectrometer at the Institute Laue Langevin, Grenoble, France. Argon densities between 1.9 and 10.5 atoms/${\mathrm{nm}}^3$ were used for 0.4<Q<5 A${̊}^{\mathrm{-}1}$. Additional measurements were made with a He gas host at densities of 4 and 10.5 atoms/${\mathrm{nm}}^3$; helium is relatively invisible also compared to hydrogen. These experiments are described, and some examples of the results are presented to show the qualitative effects observed. The principle observation is a pronounced narrowing of S(Q,ω) as a function of ω as the argon density is increased. This effect is large at low Q and decreases with increasing Q, and also decreases substantially when helium is used in place of argon. In addition, the shape of S(Q,ω) is more complex than can be accommodated within a simple model, but slightly less complicated than a computer simulation so showing the significance of multiple-collision processes.

In the stability analysis of wide-gap axisymmetric Taylor vortex flow, a new instability is found to supersede the Eckhaus instability for sufficiently high Reynolds numbers. Like the Eckhaus instability it is axisymmetric, but it represents a short-wavelength rather than a long-wavelength modulation of the pattern: It tends to eliminate every other vortex pair. In experiments it is therefore expected to change the wave number of the periodic solution more efficiently than the Eckhaus instability. Its origin is traced back to the resonant interaction of modes with wave numbers q and 2q. The essential features of the bifurcation diagrams connected with this mode interaction can be obtained by considering a degenerate codimension-2 bifurcation.

We experimentally study the coarsening of two-dimensional cellular domain patterns in an increasing bias field. As in soap froths and polycrystalline materials, the coarsening is driven by effective cell surface energy. We find that the domain patterns are distinguished by a nonmonotonic change in the total cell surface energy and by topological evolution that proceeds almost exclusively by cell elimination. Aboav’s law describing topological correlations between neighboring cells is found to hold over a wide range of bias and cell density. We have also evaluated the distribution P(n) of n-sided cells over three decades in cell density. The patterns show a form of partial scaling: over a two-decade drop in cell density, the fraction of all cells that are pentagonal bubble traps (fivefold symmetric domain structures containing trapped magnetic bubbles) remains nearly constant at $f_5$≊0.2 while P(n) changes rapidly. The partial scaling may be related to a novel topological structure in which the bubble traps act as stable fivefold vertices.

The role of thin films in different situations of wetting has been investigated by x-ray reflectivity, a technique that allows an independent determination of their thicknesses, densities, and roughnesses, using a nonvolatile siloxane oil spreading on different kinds of surfaces. The spreading of a microscopic droplet involves at least two stages: the development of a ‘‘tongue’’ of molecular thickness controlled by the spreading parameter and the friction on the substrate is followed by a surface diffusion process. The final stages of spreading are shown to be largely dependent on the polymer-substrate interactions. Increasing the chain length of the polymer slows the process down, but does not lead to a qualitatively different behavior. This study has been complemented with capillary rise experiments in order to measure diffusion constants. The results are discussed using a simple model of surface flow.

Using Landau theory, the possibility that hexagonal structures can exist in cholesteric liquid crystals in an applied field is considered. Both positive and negative dielectric anisotropy systems are studied. It is found that two- and three-dimensional hexagonal phases are thermodynamically stable in different regions of the chirality-temperature field phase diagram when the anisotropy is positive, while for negative anisotropy only a three-dimensional phase occurs. This is in agreement with experimental data. In addition, the theoretical results indicate that two different three-dimensional hexagonal phases, having the same space group (P$6_2$22) but different structure factors, can exist when the anisotropy is positive. Ways of verifying this prediction by optical or NMR studies are considered. Finally, the topological aspects of disclinations in the hexagonal phases are considered. Each of the four stable hexagonal structures is found to have two uniaxial positive and one uniaxial negative defect line. The latter is the only topological disclination when the order parameter is taken to be a uniaxial director rather than a biaxial symmetric tensor.

A new modified hypernetted-chain integral equation for highly supercooled soft-sphere fluids (MHNCS) is introduced. In the MHNCS approximation, the bridge function is approximated by an appropriate interpolation of the bridge function of the Percus-Yevick hard-sphere model and the leading term of the elementary diagrams, which was first successfully applied to classical one-component plasmas. We have obtained solutions of the MHNCS equation for highly supercooled fluids with sixth- and twelfth-inverse-power potentials. These results are compared with those of molecular-dynamics or Monte Carlo simulations and the solutions of two well-known thermodynamically consistent integral equations, i.e., the Rogers-Young (RY) equation and the reference hypernetted-chain (RHNC) equation. Below the freezing temperature, the solution of the MHNCS equation is found to show a splitting of the second peak of the pair distribution function (PDF) compatible with the computer simulations, while the RY or RHNC equation shows no such a splitting. Using the PDF thus obtained, thermodynamic and structural properties of the highly supercooled soft-sphere fluids are investigated.

A tilt θ of the principal axis of the optical dielectric tensor linear in a transverse electric field was observed for a surface-stabilized nematic liquid crystal. The electroclinic coefficient dθ/dE was observed to increase rapidly on approaching the nematic–smectic-A phase transition from above. This was not expected, and indicates that the incipient translational order is important to the electroclinic effect. The data are consistent with a critical exponent only over a narrow range of temperature, and in this range the apparent exponent is not that expected from theory. In addition, dielectric data are presented for this transition, and birefringence results are discussed. A simple theoretical discussion involving coupled order parameters is presented, and the experimental results are discussed in light of this model.

We study slow, or reaction-limited, colloid aggregation (RLCA) with both static and dynamic light scattering and develop a self-consistent interpretation of the results. Static light scattering is used to determine the fractal dimension of the clusters and the cutoff mass of the power-law cluster-mass distribution. Using this same cutoff cluster mass, we can predict the shape of the temporal autocorrelation function measured by dynamic light scattering. Good agreement with experiments is obtained provided the effects of rotational diffusion are included. In addition, we determine the ratio of the hydrodynamic radius to the radius of gyration of individual RLCA clusters and find β=1.0. A scaling method is used for the q-dependent first cumulants of the temporal autocorrelation functions to obtain a single master curve for data obtained at different times in the aggregation process. The shape of this master curve is very sensitive to several key features of the process of reaction-limited colloid aggregation. It allows us to unambiguously determine the exponent for the power-law cluster-mass distribution, τ=1.5±0.05. Furthermore, we show that the master curves for three completely different colloids, gold, silica, and polystyrene, are indistinguishable. In addition, the fractal dimensions of their RLCA clusters, as measured by static light scattering, are all $d_f$=2.1±0.05, while the aggregation kinetics for each colloid are exponential. This demonstrates that reaction-limited colloid aggregation is universal, independent of the detailed chemical nature of the colloid system.

A three-dimensional lattice model for surfactant mixtures is presented in this paper. We extend the spin-1 or Blume-Emery-Griffiths model to include orientational degrees of freedom for each molecule. The model is then applied to ternary mixtures of surfactant with equal amounts of water and oil, and to binary mixtures of surfactant and water. For each case, phase diagrams are calculated using both mean-field and Bethe approximations. One interesting feature of this model is that it can exhibit long-period lamellar phases even though a simple Hamiltonian, containing only nearest-neighbor interactions, is used. The model also produces a ‘‘disorder line,’’ which is thought to divide the disordered phase of the ternary mixture into a region of ordinary disordered fluid and a region of microemulsion.

It is well known that the bend and twist elastic constants contain a term that diverges at the nematic–smectic-A phase transition according to a simple power law in reduced temperature; i.e., as & where t=(T-$T_{N\mathrm{-}\mathrm{Sm}\mathrm{-}\mathit{A})}$/$T_{N–\mathrm{Sm}\mathrm{-}\mathit{A}}$. We report here a high-resolution light-scattering measurement of the twist constant exponent ${\rho }_2$ (=0.66±0.03) in the four-decade range -6.5t<-2.5. Within experimental uncertainty, this is the correlation length exponent predicted by the well-known three-dimensional XY model, a result originally predicted by de Gennes [Solid State Commun. 10, 753 (1972); Mol. Cryst. Liq. Cryst. 21, 49 (1973)] and later by dislocation-mediated melting models due to Helfrich [J. Phys. (Paris) 39, 1199 (1978)] and Dasgupta and Halperin [Phys. Rev. Lett. 47, 1566 (1981)]. This is the largest value of ${\rho }_2$ reported to date and by comparison with previous results reveals a clear trend of increasing ${\rho }_2$ with increasing nematic range. This trend suggests that smaller values of ${\rho }_2$ observed for previously studied materials, may in fact be effective exponents associated with critical-tricritical crossover behavior but leaves open the question of whether there exists a universal value of ${\rho }_2$.

We propose a model of pattern formation in the growth of snow crystals that takes into account the actual elemental processes relevant to the growth of crystals, i.e., a surface kinetic process for incorporating molecules into a crystal lattice and a diffusion process. This model gives a clear correspondence between the patterns produced and the actual growth conditions such as supersaturation and the diffusion coefficient. Circular patterns due to kinetic roughening, hexagonal patterns, and dendritic patterns are obtained starting from a circular crystal under various growth conditions. We analyze these patterns and discuss the mechanisms of appearance of round patterns, the development of hexagonal patterns, and the formation of dendritic patterns of snow crystals. Finally, it is shown that the dimensionless crystal size with reference to the mean free path of a water molecule plays an important role in the pattern formation of growing snow crystals.

A numerical solution of the continuum equations for nonaligning nematic liquid crystals in cylindrical Couette flow is obtained. The initial orientation of the director is radial and therefore contained in the plane of the shear. The stability of the solution is analyzed for a series of sets of material constants in order to investigate the dependence of the stability threshold with temperature. The results are compared with experiments.

The probability of finding a nearest neighbor at some given distance from a reference point in a many-body system of interacting particles is of importance in a host of problems in the physical as well as biological sciences. We develop a formalism to obtain two different types of nearest-neighbor probability density functions (void and particle probability densities) and closely related quantities, such as their associated cumulative distributions and conditional pair distributions, for many-body systems of D-dimensional spheres. For the special case of impenetrable (hard) spheres, we compute low-density expansions of each of these quantities and obtain analytical expressions for them that are accurate for a wide range of sphere concentrations. Using these results, we are able to calculate the mean nearest-neighbor distance for distributions of D-dimensional impenetrable spheres. Our theoretical results are found to be in excellent agreement with computer-simulation data.

Recent careful studies showed that the phase separation of a system of hard disks with strongly positive nonadditivity predicted by Tenne and Bergmann [J. Chem. Phys. 70, 1952 (1979)] using scaled particle theory is dubious. As exact computer calculation results are still lacking for these systems, we demonstrate here the phase separation by molecular-dynamics calculations with hard soft-sphere mixture (HSS) potentials (the softness of the potential is so small that it can be regarded as a hard potential). The HSS potential represents the hard-sphere (HS) potential with regard to computation of fluid properties to a very good approximation. HSS systems with extremely positive and negative nonadditivity are considered. While the former separate already at a density of only 3.5% of the closed-packed value, the latter remain homogeneous up to very high densities, as expected. Apparently the numerical method used by Tenne and Bergmann fails to account correctly for the phase separation in strongly (positive) nonadditive HS systems, while it gives good results for the Widom-Rowlinson model recently investigated by computer calculations.