Transport properties of cholesteric liquid crystals studied by molecular dynamics simulation

We have studied the transport properties of a cholesteric liquid crystal by molecular dynamics simulation. The molecules consist of six soft ellipsoids of revolution, the axes of which are perpendicular to the line connecting their centres of symmetry. The angle between the symmetry axes of two adjacent ellipsoids is 7.5°, so the molecules are twisted. At high densities they form a cholesteric phase where their twist axes are oriented around the cholesteric axis and the symmetry axes of the ellipsoids are approximately parallel to the local director. We have been particularly interested in thermomechanical coupling or the Lehmann effect, which arises when a temperature gradient parallel to the cholesteric axis induces a torque that rotates the director. The converse is also possible: rotation of the director can drive a heat current. The thermal conductivity, the twist viscosity, the cross-coupling coefficient between the temperature gradient and the torque, and the cross-coupling coefficient between the director angular velocity and the heat current have been calculated by non-equilibrium molecular dynamics simulation methods (NEMD) and by evaluation of the Green-Kubo relations from equilibrium simulations. Two ensembles have been utilized: the ordinary canonical ensemble and another ensemble where the director angular velocity is constrained to be a constant of motion. All the methods give consistent results for the twist viscosity and the thermal conductivity. The NEMD estimates of the cross-coupling coefficients agree within a relative error of 20%. This is consistent with the Onsager reciprocity relations that state that the two cross-coupling coefficients should be equal. The relative error of the Green-Kubo estimates is about 100% even though the order of magnitude is the same as that of the NEMD estimates.     

Linear Response Theory for Thermally Driven Quantum Open Systems

This note is a continuation of our recent paper [V. Jakšić Y. Ogata, and C.-A. Pillet, The Green-Kubo formula and Onsager reciprocity relations in quantum statistical mechanics. Commun. Math. Phys. in press.] where we have proven the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes in thermally driven quantum open systems. In this note we extend the derivation of the Green-Kubo formula to heat and charge fluxes and discuss some other generalizations of the model and results of [V. Jakšić Y. Ogata and C.-A. Pillet, The Green-Kubo formula and Onsager reciprocity relations in quantum statistical mechanics. Commun. Math. Phys. in press.].     

The Green-Kubo Formula and the Onsager Reciprocity Relations in Quantum Statistical Mechanics

We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our derivation is axiomatic and the key assumptions concern ergodic properties of non-equilibrium steady states.     

Reciprocal relations of transport coefficients in simple materials

The cross effects of viscosity and heat conduction in anisotropic simple materials (solids or liquids) are given in the linear regime, using our dissipation function theory introduced recently. Depending on whether the temperature gradient or the heat flux is used in the dissipation function, we show that two different but unambiguous reciprocal relations between the transport coefficients follow. These are compared and contrasted with the confusing predictions from the Onsager theory, and to the results of rational thermodynamics. The uncertain experimental situation in regard to these reciprocal relations is discussed. Experimental tests are strongly urged.     

The Onsager reciprocity relation and generalized efficiency of a thermal Brownian motor

Based on a general model of Brownian motors, the Onsager coefficients and generalized efficiency of a thermal Brownian motor are calculated analytically. It is found that the Onsager reciprocity relation holds and the Onsager coefficients are not affected by the kinetic energy change due to the particle's motion. Only when the heat leak in the system is negligible can the determinant of the Onsager matrix vanish. Moreover, the influence of the main parameters characterizing the model on the generalized efficiency of the Brownian motor is discussed in detail. The characteristic curves of the generalized efficiency varying with these parameters are presented, and the maximum generalized efficiency and the corresponding optimum parameters are determined. The results obtained here are of general significance. They are used to analyze the performance characteristics of the Brownian motors operating in the three interesting cases with zero heat leak, zero average drift velocity or a linear response relation, so that some important conclusions in current references are directly included in some limit cases of the present paper.     

Onsager relations for transport in inhomogeneous media

The nonlinear equations that describe transport in inhomogeneous media cannot be obtained by a straightforward extension of the known phenomenological equations for homogeneous media. One cannot therefore asserta priori that the Onsager reciprocity relations remain valid. Previously the correct equations have been obtained for three special models using kinetic theory. It is here shown that in these models the Onsager relations do indeed hold, provided that they are formulated with care.     

Statistical mechanics of thermal diffusion

Expressions of the flows of atoms A and B of a binary system in a crystal are derived as the response to the imposed gradients of temperature and chemical potentials. The formulation is done using the pair approximation of the Path Probability Method of irreversible statistical mechanics and atomic migration is assumed to be via the vacancy mechanism. The energy carried by photons (and electrons) under the temperature gradient is assumed to be independent of the atomic flux. For the case near equilibrium, linear relations are derived among the atomic fluxes, the energy flux (associated with atomic flux) and the gradients. The Onsager reciprocal relations are proved to hold among the coefficients, including those related to energy flows. The heat of transport (energy carried by a diffusing atom) and the heat conduction due to atomic flux are thus unambigously derived.     

Derivation of the Onsager principle from large deviation theory

The Onsager linear relations between macroscopic flows and thermodynamics forces are derived from the point of view of large deviation theory. For a given set of macroscopic variables, we consider the short-time evolution of near-equilibrium fluctuations, represented as the limit of finite-size conditional expectations. The resulting asymptotic conditional expectation is taken to represent the typical macrostate of the system and is used in place of the usual time-averaged macrostate of traditional approaches. By expanding in the short-time, near-equilibrium limit and equating the large deviation rate function with the thermodynamic entropy, a linear relation is obtained between the time rate of change of the macrostate and the conjugate initial macrostate. A Green–Kubo formula for the Onsager matrix is derived and shown to be positive semi-definite, while the Onsager reciprocity relations readily follow from time reversal invariance. Although the initial tendency of a macroscopic variable is to evolve towards equilibrium, we find that this evolution need not be monotonic. The example of an ideal Knundsen gas is considered as an illustration.     

Equilibrium structures of planar nematic and cholesteric films in electric fields

Liquid crystal layers subjected to an electric or a magnetic field can have several types of instabilities. This paper reviews recent theoretical studies concerning equilibrium structures of planar layers.

Using the Oseen-Frank elasticity theory, the Freedericksz transition and the transition to static periodic domains of planar nematic and cholesteric films are reconsidered. A perturbation treatment of nonlinear torque balance equations for the director is suitable to derive amplitude equations for the film distortions under the action of a field and to predict the topological features of phase diagrams for equilibrium states. The competition between the Freedericksz effect and the formation of periodic distortions is studied varying material and geometrical parameters. Some results are useful to optimize cholesteric mixtures for application in display devices in such a manner, that the occurrence of periodic domains is avoided.     

Coupled normal heat and matter transport in a simple model system.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsager's reciprocity relations.     

Measurement of the continuous Lehmann rotation of cholesteric droplets subjected to a temperature gradient

In 1900, Otto Lehmann observed the continuous rotation of cholesteric drops when subjected to a temperature gradient. This thermomechanical phenomenon was predicted 68 years later by Leslie from symmetry arguments but was never reobserved to our knowledge. In this Letter, we present an experiment allowing quantitative analysis of the Lehmann effect at the cholesteric-isotropic transition temperature. More precisely, we measure the angular velocity of cholesteric drops as a function of their size and the temperature gradient and we show that applying an electric field can stop the drop rotation. From these observations and a theoretical model we estimate the Lehmann coefficient nu.     

The principle of non-mechanical transport of a liquid crystal in thin capillaries

The non-mechanical principle of transport of a liquid crystal (LC) encapsulated in a narrow cavity between two coaxially arranged cylinders is introduced based on the interaction of the temperature and director field gradients. The temperature gradient is created due to a heat flow from the inner cylinder surface, whereas the temperature on the outer cylinder surface is maintained constant. The director field gradient is caused by the deformation of the planar-oriented LC cavity upon exposure to a double electrostatic layer, which naturally appears at the LC phase-solid interface. The size of the gap between the bounding surfaces, cylinder curvatures, and thermal conditions are determined, which allow initiation of the LC phase flow in the horizontal direction.     

Macroscopic spin and charge transport theory

According to the general principle of non-equilibrium thermodynamics, we propose a set of macroscopic transport equations for the spin transport and the charge transport. In particular, the spin torque is introduced as a generalized `current density' to describe the phenomena associated with the spin non-conservation in a unified framework. The Einstein relations and the Onsager relations between different transport phenomena are established. Specifically, the spin transport properties of the isotropic non-magnetic and the isotropic magnetic two-dimensional electron gases are fully described by using this theory, in which only the macroscopic-spin-related transport phenomena allowed by the symmetry of the system are taken into account.     

Temperature variations in the director orientation and anchoring energy at the surface of cholesteric layers

Slight changes with temperature in the director orientation at the surface of a perfect planar cholesteric liquid crystal (CLC) layer, which are precursors of a jump in the pitch of the cholesteric helix, are observed by measuring the temperature dependence of the optical transmission spectra of the layer. The observed changes in the director orientation are described in the framework of the continuum theory of CLCs, supplemented with allowance for the surface anchoring forces. In particular, the angle of deviation of the director at the surface from the alignment direction at the exact temperature of the jump in pitch is expressed in terms of the anchoring potential. The relation obtained is use to find the anchoring potential in the samples. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 1, 37–42 (10 January 1996)     

Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I

The Boltzmann equations for a formal bimolecular chemical reaction in homogeneous gas phase are transformed into an infinite system of quadratic differential equations, by expanding the distribution functions of the molecules into the Grad polynomials. The properties of these expanded Boltzmann equations reflect the macroscopic laws. In particular they enable the Onsager reciprocity relations to be derived from time-reversal invariance.     

Enskog’s kinetic theory of dense gases for chemically reacting binary mixtures, II: Light scattering and sound propagation

Enskog’s kinetic theory for a symmetric moderately dense reaction A+A?B+B is used to determine Fick’s and Fourier’s law. The transport coefficients of diffusion, thermal-diffusion rate and thermal conductivity are represented graphically for endothermic and exothermic reactions and are analyzed as a function of the activation energy and of the density of the mixture. The Onsager reciprocity relations are numerically investigated and verified. The problems related to sound propagation and light scattering are investigated for such a mixture and it is shown that the influence of chemical reactions on phase velocity, attenuation coefficient and light scattering spectra is more pronounced for rarefied gases although there is a considerable change in these quantities as the mixture becomes denser.     

Transport Properties of a Modified Lorentz Gas

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsager's reciprocity relations to within numerical accuracy as well as the Green–Kubo relations. We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.     

Cholesteric–nematic transitions induced by a shear flow and a magnetic field

The untwisting of the helical structure of a cholesteric liquid crystal under the action of a magnetic field and a shear flow has been studied theoretically. Both factors can induce the cholesteric–nematic transition independently; however, the difference in the orienting actions of the magnetic field and the shear flow leads to competition between magnetic and hydrodynamic mechanisms of influence on the cholesteric liquid crystal. We have analyzed different orientations of the magnetic field relative to the direction of the flow in the shear plane. In a number of limiting cases, the analytic dependences are obtained for the pitch of the cholesteric helix deformed by the shear flow. The phase diagrams of the cholesteric–nematic transitions and the pitch of the cholesteric helix are calculated for different values of the magnetic field strength and the angle of orientation, the flow velocity gradient, and the reactive parameter. It is shown that the magnetic field stabilizes the orientation of the director in the shear flow and expands the boundaries of orientability of cholesterics. It has been established that the shear flow shifts the critical magnetic field strength of the transition. It is shown that a sequence of reentrant orientational cholesteric–nematic–cholesteric transitions can be induced by rotating the magnetic field in certain intervals of its strength and shear flow velocity gradients.