Deviations from linear theory for fluctuations below the supercritical primary bifurcation to electroconvection

Dynamics of fluctuations below a stationary bifurcation to electroconvection in the planar nematic liquid crystal N4

We fitted C(k,tau,epsilon) proportional to exp([-sigma(k,epsilon)tau] to time-correlation functions C(k,tau,epsilon) of structure factors S(k,t,epsilon) of shadowgraph images of fluctuations below a supercritical bifurcation at V(0)=V(c) to electroconvection of a planar nematic liquid crystal in the presence of a voltage V=sqrt[2]V(0)cos((2pift) [k=(p,q) is the wave vector and epsilon identical with V(2)(0)/V(2)(c)-1]. There were stationary oblique (normal) rolls at small (large) f. Fits of a modified Swift-Hohenberg form to sigma(k,epsilon) gave f-dependent critical behavior for the minimum decay rates sigma(0)(epsilon) and the correlation lengths xi(p,q)(epsilon).

Onset of patterns in an oscillated granular layer: continuum and molecular dynamics simulations

We study the onset of patterns in vertically oscillated layers of frictionless dissipative particles. Using both numerical solutions of continuum equations to Navier-Stokes order and molecular dynamics (MD) simulations, we find that standing waves form stripe patterns above a critical acceleration of the cell. Changing the frequency of oscillation of the cell changes the wavelength of the resulting pattern; MD and continuum simulations both yield wavelengths in accord with previous experimental results. The value of the critical acceleration for ordered standing waves is approximately 10% higher in molecular dynamics simulations than in the continuum simulations, and the amplitude of the waves differs significantly between the models. The delay in the onset of order in molecular dynamics simulations and the amplitude of noise below this onset are consistent with the presence of fluctuations which are absent in the continuum theory. The strength of the noise obtained by fit to Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in fluid convection experiments, and is comparable to the noise found in experiments with oscillated granular layers and in recent fluid experiments on fluids near the critical point. Good agreement is found between the mean field value of onset from the Swift-Hohenberg fit and the onset in continuum simulations. Patterns are compared in cells oscillated at two different frequencies in MD; the layer with larger wavelength patterns has less noise than the layer with smaller wavelength patterns.

Melting and heating of two-dimensional Coulomb clusters in dusty plasmas

The heating and melting of two-dimensional dust clusters with one additional particle in the lower layer has been investigated experimentally in a gas discharge. The full dynamical properties of the system during the entire phase transition were determined in terms of the spectral power densities of the crystal modes. A two-step melting transition is identified when the gas pressure in the discharge is reduced: first, a sudden increase of the dust temperature takes place due to an instability of the lower-layer particle resulting in a hot crystalline state of the cluster, and second, the actual transition into a fluid state is observed at a decisively lower gas pressure.

Direct transition to electroconvection in a homeotropic nematic liquid crystal

We present an experimental and theoretical investigation of a variant of electroconvection using an unusual nematic liquid crystal in an isotropic configuration (homeotropic alignment). The significance of the system is a direct transition to the convecting state due to the negative conductivity anisotropy and positive dielectric anisotropy. We observe at onset rolls or squares depending on the frequency and amplitude of the applied ac voltage with a strong signature of the zigzag instability. Good agreement with calculations based on the underlying hydrodynamic theory is found. We also construct an extended Swift-Hohenberg model which allows us to capture complex patterns like squares with a quasiperiodic modulation.

Dynamics of laser-induced electroconvection pulses

We first report that, for planar nematic 4-methoxy-benzilidene-4-butylaniline (MBBA), the electroconvection threshold voltage has a nonmonotonic temperature dependence, with a well-defined minimum, and a slope of about -0.12 V/degrees C near room temperature at 70 Hz. Motivated by this observation, we have designed an experiment in which a weak continuous-wave absorbed laser beam with a diameter comparable to the pattern wavelength generates a locally supercritical region, or pulse, in dye-doped MBBA. Working 10-20 % below the laser-free threshold voltage, we observe a steady-state pulse shaped as an ellipse with the semimajor axis oriented parallel to the nematic director, with a typical size of several wavelengths. The pulse is robust, persisting even when spatially extended rolls develop in the surrounding region, and displays rolls that counterpropagate along the director at frequencies of tenths of Hz, with the rolls on the left (right) side of the ellipse moving to the right (left). Systematic measurements of the sample-voltage dependence of the pulse amplitude, spatial extent, and frequency show a saturation or decrease when the control parameter (evaluated at the center of the pulse) approaches approximately 0.3. We propose that the model for these pulses should be based on the theory of control-parameter ramps, supplemented with new terms to account for the advection of heat away from the pulse when the surrounding state becomes linearly unstable. The advection creates a negative feedback between the pulse size and the efficiency of heat transport, which we argue is responsible for the attenuation of the pulse at larger control-parameter values.

Noise, coherent fluctuations, and the onset of order in an oscillated granular fluid

We study fluctuations in a vertically oscillated layer of grains below the critical acceleration for the onset of ordered standing waves. As onset is approached, transient disordered waves with a characteristic length scale emerge and increase in power and coherence. The scaling behavior and the shift in the onset of order agrees with the Swift-Hohenberg theory for convection in fluids. However, the noise in the granular system is an order of magnitude larger than the thermal noise in the most sensitive convecting fluid experiments to date; the effect of the granular noise is observable 20% below the onset of order.

Modulated structures in electroconvection in nematic liquid crystals

Motivated by experiments in electroconvection in nematic liquid crystals with homeotropic alignment we study the coupled amplitude equations describing the formation of a stationary roll pattern in the presence of a weakly damped mode that breaks isotropy. The equations can be generalized to describe the planarly aligned case if the orienting effect of the boundaries is small, which can be achieved by a destabilizing magnetic field. The slow mode represents the in-plane director at the center of the cell. The simplest uniform states are normal rolls, which may undergo a pitchfork bifurcation to abnormal rolls with a misaligned in-plane director. We present a new class of defect-free solutions with spatial modulations perpendicular to the rolls. In a parameter range where the zigzag instability is not relevant these solutions are stable attractors, as observed in experiments. We also present two-dimensionally modulated states with and without defects which result from the destabilization of the one-dimensionally modulated structures. Finally, for no (or very small) damping, and away from the rotationally symmetric case, we find static chevrons made up of a periodic arrangement of defect chains (or bands of defects) separating homogeneous regions of oblique rolls with very small amplitude. These states may provide a model for a class of poorly understood stationary structures observed in various highly conducting materials ("prechevrons" or "broad domains").

Electroconvection in nematic liquid crystals with positive dielectric and negative conductivity anisotropy

Electroconvection in an unusual nematic compound with strongly positive dielectric anisotropy and negative anisotropy of the conductivity is investigated. For homeotropic alignment, where one has a direct transition to rolls or squares depending on the frequency of the applied voltage, we present a quantitative theory. From the comparison we infer values for some viscosities, which are rather unusual, but not unreasonable in view of the vicinity of the nematic-smectic transition. For planar alignment, electroconvection sets in above a splay Freedericksz transition with "parallel rolls," which is also captured by the theory.

Boundary effects on the nonequilibrium structure factor of fluids below the Rayleigh-Bénard instability

We consider a horizontal fluid layer between two rigid boundaries, maintained in a stationary thermal nonequilibrium state below the convective Rayleigh-Bénard instability. We derive an explicit expression for the nonequilibrium structure factor in a first-order Galerkin approximation valid for negative and positive Rayleigh numbers R up to the critical Rayleigh number R(c) associated with the appearance of convection. The results obtained for rigid boundaries by the Galerkin-approximation method are compared with exact results previously derived for the case of free boundaries. The nonequilibrium structure factor exhibits a maximum as a function of the wave number q of the fluctuations. This maximum is associated with a crossover from a q(-4) dependence for larger q to a q(2) dependence for small q. This maximum is present at both negative and positive R, becomes pronounced at positive R and diverges as R approaches the critical value R(c).

Temporal and spatial properties of fluctuations below a supercritical primary bifurcation to traveling oblique-roll electroconvection

We present measurements of thermally-induced oblique-roll traveling-wave (TW) fluctuations below the supercritical primary bifurcation to electroconvection (EC) in the nematic liquid crystal 4-ethyl-2-fluoro-4'-[2-(trans-4-pentylcyclohexyl)ethyl]-biphenyl (I52). First we analyze time sequences of one-dimensional shadowgraph images taken parallel to the director to obtain the TW frequency omega and the fluctuation lifetime tau. Within our resolution we find that omega is independent of epsilon [triple bond] V/V(c)-1 (V is the applied voltage amplitude and V(c) its value at the onset of convection). Contrary to linear theory, the relaxation rate 1/tau remains finite at the bifurcation. Next we present the analysis of temporally uncorrelated two-dimensional shadowgraph images of the fluctuations for several values of the electrical conductivity sigma. We fitted an anisotropic two-dimensional Lorentzian function, corresponding to oblique-roll EC, to the time-averaged structure factors S(k) derived from the images. This yielded information about the components of the mean wave vector k(0) and about the correlation length xi as a function of sigma and epsilon. The angle of obliqueness theta of the roll patterns was independent of sigma but decreased anomalously as epsilon approached zero. The modulus k(0) of k(0) depended on sigma. It also showed an anomalous reduction close to onset. The anomalous epsilon dependence of k(0) and theta disagrees with linear theory, which predicts a smooth, essentially linear dependence on epsilon, and presumably is caused by nonlinear interactions between the fluctuations.

Influence of inlet and bulk noise on Rayleigh-Bénard convection with lateral flow

Spatiotemporal properties of convective fluctuations and of their correlations are investigated theoretically in the vicinity of the threshold for onset of convection in the presence of a lateral through-flow using the full linearized equations of fluctuating hydrodynamics. The effect of external forcing by inlet boundary conditions on the downstream evolution of convective fields is separated from the effect of internal bulk thermal forcing with the use of spatial Laplace transformations. They show how the spatial variation of fluctuations and of their correlations are governed by the six spatial characteristic exponents of the field equations.