Microscopic Stress Tensor Research Articles

Microscopic description of nematic liquid crystal viscosity.

The problem of nematic viscosity is shown to be successfully solved within the framework of a microscopic description. The Kuzuu-Doi approach to hydrodynamics [N. Kuzuu and M. Doi, J. Phys. Soc. Jpn. 52, 3486 (1983)] for a system consisting of anisotropic molecules is reviewed. It has been shown that the incorrect form of the microscopic stress tensor that is used by Doi [M. Doi, J. Polym. Sci. 19, 229 (1981)] does not influence final expressions for the viscosity coefficients due to the fact that the external magnetic field is taken into account. Molecular expressions for the Leslie viscosity coefficients are calculated by use of the theory of Osipov and Terentjev [Z. Naturforsch. Teil A 44, 785 (1989); Phys. Lett. A 134, 301 (1989)]. A systematic solution of the differential kinetic equation is presented. The nonequilibrium distribution function obtained from the kinetic equation allows us to transform Osipov-like expressions for the viscosity coefficients exactly into those obtained by Kuzuu and Doi. The above-mentioned approaches are proved to be equivalent. A realistic mean potential is applied to the expressions obtained for the Leslie coefficients for the case of 4-methoxybenzylidene-4\ensuremath{'}-n-butylaniline. A comparison of theoretical results and appropriate experimental data is shown to be very good.

Many-body correlations and the cancellation effect in the Green–Kubo time correlation functions for the shear viscosity

Molecular dynamics simulations of the time-correlation functions that appear in the Green–Kubo expression for the shear viscosity are reported for liquid argon over a wide range of densities. The potential part of the microscopic stress tensor is separated into contributions due to the repulsive and attractive branches of the (12/6) Lennard-Jones potential. The separate correlation functions for these parts of the interactions as well as their cross term are further broken down into two-, three- and four-body parts. Each of these shows the same kind of cancellation effect as is found in simulation studies of dipole-induced dipole light scattering. We compare our results to the light scattering findings and find a very consistent picture of the cancellation effect in these time-correlation functions.

On the Microscopic Stress Tensor for Anisotropic Molecules in Homogeneous Systems

Abstract The flowing nematic liquid crystal is considered in the small velocity gradient approximation. The molecular expression for the stress tensor is improved by use of the procedure of Doi. The symmetric and antisymmetric parts of the microscopic stress tensor are shown to differ by the formfactor in comparison to forms which have been used so far. The consequences for the molecular hydrodynamic theories are considered.

Rotational viscosity coefficients of nematic and smecticC liquid crystals: Statistical theory

The rotational diffusion of a rodlike molecule in a nematic and smecticC liquid crystal is considered in the molecular-field approximation. The microscopic friction constant, which determines the molecular rotation drag, possesses an exponential temperature dependence with the activation energy determined by the isotropic part of the intermolecular interaction energy. The rotational viscous coefficients,γ 1 andγ ⊥ are obtained by averaging of the corresponding microscopic stress tensor with the nonequilibrium distribution function. The additional activation energy, proportional to the corresponding order parameter, appears in the expressions for the rotational viscosity coefficients both in nematics andC smectics.

Relativistic Kohn-Sham formalism and the microscopic stress tensor

After a short review of the Kohn-Sham procedure for a system of relativistic charged particles a relativistic extension of the local density approximation is presented. In contrast to former proposals, it satisfies the requirement of Lorentz invariance. Within this relativistic Kohn-Sham theory the microscopic stress tensor is considered and the corresponding complete energy-momentum tensor is constructed. A local expression is found in accordance with the assumptions about the energy density.

The microscopic stress tensor field in particle systems with many-body interactions

It is argued that the up to now only existing expression for the microscopic stress tensor in the presence of many-body interactions, while being formally acceptable, displays some physical shortcomings. These unpleasant features are remedied by explicitly constructing and symmetrizing a new stress tensor field. With the help of this construction, some recent results on the appearance of extremely long-ranged correlations involving the stress tensor field in systems with spontaneously broken symmetries are generalized.

Reply to "Comment on 'Microscopic stress tensors in quantum systems' "

The theory of microscopic stress tensors is discussed in the context of comments by Nielsen and Martin. Microscopic stress tensors are interpreted as physically observable quantities.Received 5 May 1988DOI:https://doi.org/10.1103/PhysRevB.37.10908©1988 American Physical Society

Comment on "Microscopic stress tensors in quantum systems"

We show that the expressions for microscopic stress fields derived previously by the present authors encompass those given in a recent paper by Folland and are correct, in contradiction to claims made by Folland.

Microscopic stress tensors in atoms.

Calculations are presented for microscopic quantum stress tensors for selected closed-shell atomic systems. The stress tensor in spherically symmetric atoms is expressed as a diagonal dyadic in terms of spherical-polar unit vectors. In equilibrium the balancing of momentum and electric flux implies a relationship between the radial and tangential components of the stress tensor. This relationship is used to express the microscopic pressure as a total differential of a pressure virial and to analyze the effects of finite numerical precision and algorithm errors. Effects of non-self-consistency are illustrated by a comparison of local-density-approximation and Hartree-Fock models for the stress tensor.

Microscopic stress tensors in quantum systems.

Microscopic stress tensors are derived for the local-density-approximation and Hartree-Fock models for quantum systems. Dynamically derived stress tensors appear as elements of a continuity equation for the force density in direct correspondence to Newton's laws of classical physics. Equilibrium stress tensors S, derived from the dynamics of the model satisfy a simple, microscopic relationship, divS=0, balancing electric field and momentum-flux contributions at each point in a quantum system. Stress tensors are well-behaved, integrable functions of position which tend to zero as ${r}^{\mathrm{\ensuremath{-}}4}$ at large distances r from finite Coulombic systems. The microscopic stress tensor and pressure are shown to be consistent with macroscopic expressions for these quantities. Comparisons are made of the present work to related work on microscopic stress.

Atomic and molecular representations of molecular hydrodynamic variables

Formulae connecting the atomic and molecular representations of the microscopic momentum density and stress (pressure) tensor are used to relate corresponding equilibrium time correlation functions and non-equilibrium response functions. It is shown that, while the atomic and molecular stresses are essentially equivalent for calculating transport coefficients, and the appropriate correlation functions become similar as t → ∞, a residual difference between these functions, associated with internal molecular motion and reorientation, may persist at intermediate and long times. It is also demonstrated that the low frequency, atomic and molecular stress responses to a homogeneous oscillatory shear are essentially equivalent. The problem of formulating consistent hydrodynamic theories in the atomic and molecular representations is discussed.

Statistical mechanics of inhomogeneous fluids

The nature of the microscopic stress tensor in an inhomogeneous fluid is discussed, with emphasis on the statistical mechanics of drops. Changes in free energy for isothermal deformations of a fluid are expressible as volume integrals of the stress tensor ‘times’ a strain tensor. A particular radial distortion of a drop leads to statistical mechanical expressions for the pressure difference across the surface of the drop. We find that the stress tensor is not uniquely defined by the microscopic laws embodying the conservation of momentum and angular momentum and that the am­biguity remains in the ensemble average, or pressure tensor, in regions of inhomogeneity. This leads to difficulties in defining statistical mechanical expressions for the surface tension of a drop.

Non-equilibrium molecular dynamics simulation and generalized hydrodynamics of transverse modes in molecular fluids

We present a generalized hydrodynamic theory of transverse motion in molecular fluids. The theory considers the coupled motion of the molecular orientation, symmetric and antisymmetric components of the microscopic stress tensor, molecular spin, and transverse momentum density. We have conducted non-equilibrium molecular dynamics simulations, of the Lennard-Jones diatomic fluid, which are designed to test the way in which this theory describes the time dependent responses of orientation, stress and spin to an applied velocity gradient. A primary aim of the work is the evaluation of the dimensionless shear-orientation parameter which appears in this theory and in various theories of depolarized light scattering and flow birefringence. The systems studied correspond to fluorine at 70 K, fluorine at 120 K and carbon dioxide at 273 K, all at approximately atmospheric pressure. We find that may be estimated by measuring the steady state orientation density induced by a uniform applied velocity gradient; we obtain values of =0·46 (F2/70 K), 0·22 (F2/120 K), and 0·20 (CO2/273 K). The generalized hydrodynamic equations are found to give a satisfactory description of the observed time dependent responses. Coupling with the molecular orientation is observed to have a significant effect upon the form of the stress response at long time, and this in turn affects the accuracy with which molecular fluid viscosities may be estimated via the ‘subtraction’ technique of molecular dynamics.

Theory of depolarized light scattering

We have studied the VH depolarized light scattering spectrum of liquid triphenylphosphite over a wide range of temperature. At high temperature and low viscosity the Rytov doublet corresponding to shear-orientational modes is observed, while at low temperature and high viscosity weak side peaks corresponding to viscoelastic shear waves are seen. The fluctuations of the dielectric tensor ? which give rise to depolarized light scattering have been adequately described over the whole temperature range by a ’’three-variable’’ generalized hydrodynamic theory. In contrast with virtually all previous studies of VH spectra we use the independently measured shear viscosity as a known, rather than adjustable, parameter. As a result of our analysis we can express ?(t) as a linear combination of the molecular reorientation density ?(t) and the microscopic stress tensor ?(t), and we find that the intermolecular scattering represented by ?(t) is important at high as well as low temperature. We are also able to evaluate the autocorrelation function of ?(t) in the low-wave number, long-time limit as the sum of two exponential decays with comparable amplitudes. We include a comparative study of other theories.

Low-frequency depolarized light scattering from fluids

A theory of the low-frequency depolarized-light scattering spectra of fluids of nonspherical molecules is obtained. The intensity of the scattered light is proportional to the polarizability-density autocorrelation function. For small frequency and wave vector the behavior of this correlation function can be obtained using the Selwyn-Oppenheim theory. The pertinent variables are the usual hydrodynamic variables, the angular momentum density and the polarizability density. We obtain explicit general expressions for the sharp components of the VH and HH spectra at low frequencies. These expressions include k and temperature-dependent fine structures which contain dynamic correlation functions between the anisotropic polarizability density and the microscopic stress tensor. When terms of zeroth order in the wave vector and second order in the time derivatives of the polarizability and the spin are larger than terms of O( k 2) we derive simpler expressions.

Theory of transport in moderately dense gases

Calculations of the contributions of the intermolecular potential to the microscopic stress tensor, T/sub ij/, in the correlation-function expression for shear viscosity, in transport theory, are made. An expression for the viscosity of a gas of classical hard spheres is derived. (L.B.S.)