Scalar Order Parameter Research Articles

Magnetoviscosity and orientational order parameters of dilute ferrofluids

The linear and nonlinear rheological behavior of dilute ferrofluids is determined from an underlying kinetic model and the dependence of the viscosity coefficient on the scalar orientational order parameters is obtained. In case of uniaxial symmetry, the antisymmetric contribution to the hydrodynamic stress tensor is of the same form as in the classical Ericksen–Leslie theory of uniaxial nematic liquid crystals and the linear magnetoviscosity is found to coincide with earlier results obtained by the so-called effective field method. While the assumption of uniaxial symmetry is fulfilled exactly in the limit of strong vorticity and weak magnetic field, the exact result for the linear magnetoviscosity shows corrections due to contributions from biaxial symmetry. Measures for the deviations from uniaxial symmetry are introduced and the generalization of the stress tensor in case of biaxial symmetry is obtained. The investigations are accompanied by numerical simulation of the kinetic equation and reveal that the assumption of uniaxial symmetry seems to be a good approximation for most values of the magnetic field and vorticity.

On the classification of equilibrium superfluid states with scalar and tensor order parameters

A classification of the equilibrium states of superfluid liquids with scalar and tensor order parameters, based on the concept of quasi-averages, is presented. The condition of unbroken symmetry is generalized to inhomogeneous equilibrium states. The admissible conditions of spatial symmetry are found in terms of integrals of the motion. A connection between these symmetry conditions and the helicoidal structure of the spin and spatial anisotropy vectors is established. Under certain restrictions it is shown that the equilibrium structure of the order parameter can be represented as a product of a homogeneous part of the order parameter and a part that depends on the spatial coordinates.

Supervised learning with restricted training sets: a generating functional analysis

We study the dynamics of supervised on-line learning of realizable tasks in feed-forward neural networks. We focus on the regime where the number of examples used for training is proportional to the number of input channels N. Using generating function techniques from spin glass theory, we are able to average over the composition of the training set and transform the problem for N to infinity to an effective single pattern system, described completely by the student autocovariance, the student-teacher overlap and the student response function, with exact closed equations. Our method applies to arbitrary learning rules, i.e. not necessarily of a gradient-descent type. The resulting exact macroscopic dynamical equations can be integrated without finite-size effects up to any degree of accuracy, but their main value is in providing an exact and simple starting point for analytical approximation schemes. Finally, we show how, in the region of absent anomalous response and using the hypothesis that (as in detailed balance systems) the short-time part of the various operators can be transformed away, one can describe the stationary state of the network succesfully by a set of coupled equations involving only four scalar order parameters.

Fluctuation-dominated phase ordering driven by stochastically evolving surfaces: depth models and sliding particles.

We study an unconventional phase ordering phenomenon in coarse-grained depth models of the hill-valley profile of fluctuating surfaces with zero overall tilt, and for hard-core particles sliding on such surfaces under gravity. We find that several such systems approach an ordered state with large scale fluctuations which make them qualitatively different from conventional phase ordered states. We consider surfaces in the Edwards-Wilkinson (EW), Kardar-Parisi-Zhang (KPZ) and noisy surface-diffusion (NSD) universality classes. For EW and KPZ surfaces, coarse-grained depth models of the surface profile exhibit coarsening to an ordered steady state in which the order parameter has a broad distribution even in the thermodynamic limit, the distribution of particle cluster sizes decays as a power-law (with an exponent straight theta), and the scaled two-point spatial correlation function has a cusp (with an exponent alpha=1/2) at small values of the argument. The latter feature indicates a deviation from the Porod law which holds customarily, in coarsening with scalar order parameters. We present several numerical and exact analytical results for the coarsening process and the steady state. For linear surface models with a dynamical exponent z, we show that alpha=(z-1)/2 for z<3 and alpha=1 for z>3, and there are logarithmic corrections for z=3, implying alpha=1/2 for the EW surface and 1 for the NSD surface. Within the independent interval approximation we show that alpha+straight theta=2. We also study the dynamics of hard-core particles sliding locally downward on these fluctuating one-dimensional surfaces, and find that the surface fluctuations lead to large-scale clustering of the particles. We find a surface-fluctuation driven coarsening of initially randomly arranged particles; the coarsening length scale grows as approximately t(1/z). The scaled density-density correlation function of the sliding particles shows a cusp with exponents alpha approximately 0.5 and 0.25 for the EW and KPZ surfaces. The particles on the NSD surface show conventional coarsening (Porod) behavior with alpha approximately 1.

Calculation of the free energy of polymeric systems in terms of compositional and orientational order parameters

We present a general framework for the calculation of the free energy functional of inhomogeneous polymeric systems in terms of the compositional scalar order parameter φ and the orientational order parameters of second-rank tensor Qij. Our formalism is a straightforward extension of the density functional theory developed by Tang and Freed [J. Chem. Phys. 94, 1572 (1991)] and the contribution of the conformational entropy is given in terms of the gradient expansion. As a demonstration, we give an exact expression for the free energy of Gaussian chains up to second order in the gradients.

Critical phenomena in finite-size binary liquid mixtures with reduced geometry

Correlation characteristics in finite-size binary systems are studied by the Munster method. Using some spatial approximations the first iterations for the direct and the pair correlation functions of scalar order parameter fluctuations near the critical point are found for finite-size systems with geometry of a cylinder. Formulae for the shift of the critical temperature are obtained. The received results are discussed in the terms of scaling theory.

Temperature-induced surface transition in nematic liquid crystals oriented by evaporated SiOx.

Temperature induced surface transitions observed in nematic liquid crystals oriented by obliquely evaporated SiOx are analyzed. It is shown that the experimental data can be interpreted with a theory for the thermal renormalization of the anisotropic part of the surface energy based on the mean field approximation. By assuming that the nematic scalar order parameter is given by the Maier-Saupe theory, the fit is obtained with only one free parameter. The temperature behaviors of the polar and azimuthal angles are in agreement with the proposed model over the whole temperature range.

3D Orientational Order in a Homologous Series of LC Polyesters With Azobenzene Side Groups and Different Lengths of the Alkylene Spacer in the Main Chain

The spatial orientational order of films of a series of LC polymalonates with 4-hexyloxy-4′-nitroazobenzene side groups which differ in the length of the alkylene spacer in the main chain are studied using null ellipsometry and UV/Vis spectroscopy. In the spin-coated films the azobenzene fragments strongly prefer a random in-plane alignment. Uniaxial as well as biaxial order of the azobenzene chromophore were detected in these films irradiating with linearly polarized UV light. Biaxiality is observed in the intermediate stages of irradiation whereas an uniaxial order is realized in the saturated state. Finally, it results in an in-plane structure with its optic axis perpendicular to the electric field vector of the incident light. The components of the tensor of the order parameter were estimated for each homologue as a function of the exposure dose. The scalar order parameters of the photosaturated states decrease monotonously with the increasing length of the alkylene spacer of the LC polymers. Variation of the azobenzene concentration, modification of the liquid crystallinity and peculiarities of the aggregation with the spacer length are considered as the reasons of the effect.

Investigation of Polarization-Induced Orientational Order and Surface Gratings in Liquid Crystalline AZO-Polymers

Polarization-induced orientational order in three different azopolymers is investigated and discussed with respect to the formation of surface relief gratings induced by holographic exposure. For this purpose the birefringence and dichroism induced by different light polarization, namely linear, elliptical or circular, has been studied. To determine the symmetry and the induced amount of orientational order the illuminated region is characterized by the scalar Maier-Saupe order parameter S. Tensor gratings were induced in these materials by holographic exposure with different kinds of polarization patterns and were investigated with polarization and atomic force microscopy. The gratings are characterized by diffraction efficiencies as well as by depth and shape of the surface modulation. Large surface modulations (>2,5μm) and diffraction efficiencies (>90%) for a 5μm thick film were obtained.

The surface action on dimer–monomer liquid crystal system

A microscopical mechanism of the interaction between dimerized nematics of 4, n-alkyloxybenzoic acids and hydrophilic solid surfaces is developed. The liquid crystal molecules: monomers and closed dimers are considered like dipoles and quadrupoles, respectively. It is demonstrated that in the low-temperature and the high-temperature ranges of the nematic phase, the mechanisms of the microscopical interaction are different thus providing different macroscopical effects. The cell thickness and the temperature trends of the scalar order parameter, the effective surface energy and the concomitant anomalies in these trends are discussed on the basis of the proposed microscopical mechanism.

Dynamics and delocalization transition for an interface driven by a uniform shear flow

We study the effect of a uniform shear flow on an interface separating the two broken-symmetry ordered phases of a two-dimensional system with nonconserved scalar order parameter. The interface, initially flat and perpendicular to the flow, is distorted by the shear flow. We show that there is a critical shear rate, \gamma_c, proportional to 1/L^2, (where L is the system width perpendicular to the flow) below which the interface can sustain the shear. In this regime the countermotion of the interface under its curvature balances the shear flow, and the stretched interface stabilizes into a time-independent shape whose form we determine analytically. For \gamma > \gamma_c, the interface acquires a non-zero velocity, whose profile is shown to reach a time-independent limit which we determine exactly. The analytical results are checked by numerical integration of the equations of motion.

Scaling of the Euler characteristic, surface area, and curvatures in the phase separating or ordering systems.

We present robust scaling laws for the Euler characteristic and curvatures applicable to any symmetric system undergoing phase separating or ordering kinetics. We apply it to the phase ordering in a system of the nonconserved scalar order parameter and find three scaling regimes. The appearance of the preferred nonzero curvature of an interface separating +/- domains marks the crossover to the late stage regime characterized by the Lifshitz-Cahn-Allen scaling.

The Theory of Quantum Fermi-Liquid Solutions with a Vector Order Parameter

We construct the thermodynamic theory and derive the hydrodynamic equations for a solution of Fermi superfluids with a vector order parameter. The thermodynamic quantities related to spontaneous breaking of phase symmetries and spin rotations are represented in terms of the order-parameter operator. We predict a new entrainment effect related to the mutual influence of magnetic and orbital degrees of freedom. We find the spectra of collective excitations and discuss the distinctions between the states under consideration and those with scalar and vector order parameters.

Thermal renormalization of the anchoring energy of nematic liquid crystals

The temperature dependence of the anchoring energy of a nematic liquid crystal on thermal fluctuations is studied. We consider the weak anchoring case, where the interaction of a nematic molecule on the surface with the substrate is small with respect to the mean field energy due to the other nematic molecules. The analysis is performed by means of a perturbation method in which the expansion parameter is the surface interaction. The presented model is valid for any value of the scalar order parameter. We show that the renormalization of the anchoring coefficients due to thermal fluctuations is proportional to the generalized scalar order parameters. We show also that, at the lowest order in the scalar order parameter, Landau-like theories agree with our mean field approach. An expression for the thermal renormalization of the anchoring coefficients valid in the low temperature region, where the fluctuations are small, is derived. The agreement between our theoretical predictions and the experimental data obtained by other groups is fairly good over a large temperature range.

Ohta-jasnow-kawasaki approximation for nonconserved coarsening under shear

We analytically study coarsening dynamics in a system with nonconserved scalar order parameter, when a uniform time-independent shear flow is present. We use an anisotropic version of the Ohta-Jasnow-Kawasaki approximation to calculate the growth exponents in two and three dimensions: for d=3 the exponents we find are the same as expected on the basis of simple scaling arguments, that is, 3/2 in the flow direction and 1/2 in all the other directions, while for d=2 we find an unusual behavior, in that the domains experience an unlimited narrowing for very large times and a nontrivial dynamical scaling appears. In addition, we consider the case where an oscillatory shear is applied to a two-dimensional system, finding in this case a standard t(1/2) growth, modulated by periodic oscillations. We support our two-dimensional results by means of numerical simulations and we propose to test our predictions by experiments on twisted nematic liquid crystals.

The surface order parameter and anchoring energy in nematics made by hydrogen bonded molecules

The temperature and the cell thickness trends of the scalar order parameter for the dimerized nematic with short-range smectic order of 4, n-nonyloxybenzoic acid (NOBA) are indicated by FT-FIR spectroscopy. The temperature dependence of this parameter shows an anomaly, in respect to the dependence typical for the classical nematics, at a definite temperature T ∗ in the nematic range. The scalar order parameter increases nonlinearly at cell thickness ( d) decreasing between 20 and 6 μm and goes to saturation when d→∞, while below 6 μm down to 2 μm it decreases tending to a finite small value. It is indicated that the order parameter increasing of 0.035 with the thickness decreasing from 20 to 6 μm is comparable to order increasing induced by a stabilizing electric or magnetic fields; the order parameter decreasing at small cell thickness is an analog of a disordering induced by destabilizing fields. The anomalies in the temperature and cell thickness trends of the scalar order parameter are explained by the dynamics of the hydrogen bonding of the monomers in dimers stimulated by temperature and surface action variations. The relation between the surface order parameter square and the anchoring energy is indicated to be a linear dependence near the nematic–isotopic phase transition temperature. The trend of the anchoring energy on the cell thickness is explained by a modification of the Rapini–Papoular surface energy as an effective one, adding one dielectric part stimulated by phenomena as: existence of a double electric layer, thickness dependence of the surface charge density and surface polarization.

Shear-induced biaxiality in nematic polymers

The role of biaxiality in the dynamics of a nematic liquid crystal under shear flow is systematically examined within a rigid-rod model. The non-linear partial differential equation that governs the evolution of the orientational distribution function is solved numerically by using a Galerkin method on an expansion in spherical harmonics. Examples are given for various types of solutions and different values of the free parameters. The limit of low shear rates is further examined analytically on the basis of a relaxation equation for the alignment tensor of the second rank. Several qualitative features of the different types of solutions are derived. In the important case of in-plane tumbling at low shear rates a remarkable behaviour is found in the numerical calculations: namely, whenever the scalar order parameter reaches its equilibrium value and the in-plane angle is either zero or π/2, the evolution traverses a uniaxial state. An explanation is also given for this fact within our analytical model.

Novel surface state in a class of incommensurate systems

We study the Landau model of the class of incommensurate systems with a scalar order parameter where the modulated phase is driven by a gradient-squared term with negative coefficient. For example, theoretical studies of cholesteric liquid crystals in a field (electric or magnetic) suggest that such an modulated phase should exist at high chirality. The bulk phase diagram in the presence of a bulk external field which couples linearly to the order parameter exhibits a modulated phase inside a loop in the temperature-field plane, and a homogeneous phase outside. On analyzing the same model for a semi-infinite system, we find a surprising result; the system exhibits surface states in a region where the bulk phase is homogeneous (but close to the modulated region). These states are very different from the well-known surface states induced either by a surface field or by enhanced interactions at the surface, for they exist and are energetically favored even when the sole effect of the surface is to terminate the bulk, as expressed by free boundary conditions taken at the surface. Near the surface, the surface-state order parameter is very different from the bulk value (in fact, it has the opposite sign). When the temperature or the bulk field are varied to move away from the modulated state, we find a surface phase transition at which the surface states become energetically unfavorable, though they continue to exist as metastable states. We then study how a surface field changes the surface phase diagram.

Concentration dependence of the scalar order parameter in liquid-crystalline systems with variable molecular shape

A theoretical model taking into account the photoisomerization influence on the nematic scalar order parameter is presented. An extension of the Maier-Saupe model is used to investigate the concentration dependence of the scalar order parameter. The evolution of the concentration of cis-trans isomers and of the scalar order parameter with the time of exposure to the illumination is investigated. Furthermore, it is shown that for sufficiently long exposition of the mesophase to the illumination the nematic phase can disappear. This phase transition takes place at a critical concentration of cis constituents.

Correlation properties, critical parameters and critical light scattering in finite-size systems

Abstract Correlation characteristics are studied by the Munster method in finite-size systems. The first iterations are found for the direct and pair correlation functions of scalar order parameter fluctuations (e.g., density fluctuations for a single-component liquid) near the critical point for finite-size systems with geometry of a plane-parallel layer and cylinder. Formulae for the correlation length and shifts of critical parameters are obtained. Integratal intensity of electromagnetic waves is studied for critical opalescence for spatially finite systems near the critical point.