Analytical solutions for rotational diffusion in the mean field potential: application to the theory of dielectric relaxation in nematic liquid crystals

A theory of dielectric relaxation in nematics is developed for a molecular dipole moment directed at an arbitrary angle to the molecular long axis. Both exact and simple approximate analytical formulae for the longitudinal and transverse components of the complex dielectric permittivity tensor are obtained for the non-inertial rotational Brownian motion of a molecule in the mean field potential of Maier and Saupe. It appears that both longitudinal and transverse relaxation processes are effectively described by two Debye type mechanisms with corresponding relaxation times and dielectric strengths expressed in terms of the order parameter. The generalization of the theory for an arbitrary axially symmetric mean field potential is given.     

Thermally activated escape rate for the Brownian motion of a fixed axis rotator in an asymmetrical double-well potential for all values of the dissipation

The Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov is used to evaluate the relaxation times and the dynamic susceptibility for the rotational Brownian motion of fixed axis rotators in an asymmetric double-well potential. An expression for the escape rate valid for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and crossover regimes is derived. It is shown that this expression provides a good asymptotic estimate of the inverse of the smallest nonvanishing eigenvalue lambda(1) of the underlying Fokker-Planck operator calculated by using the matrix-continued fraction method. For low barriers, where the Mel'nikov and Meshkov approach is not applicable, analytic equations for the correlation time tau( parallel) of the longitudinal dipole correlation function in the VLD and VHD limits are derived and a simple extrapolating equation valid for all values of the damping is proposed.     

On the growth and shape of sodium taurodeoxycholate micellar aggregates: a spin-label and quasielastic light scattering investigation

Electron spin resonance (ESR) and quasielastic laser scattering (QELS) measurements have been carried out on sodium taurodeoxycholate (NaTDC) micellar aqueous solutions. Computer simulation of the ESR line shape has been used to quantitatively analyze the rotational dynamics of the cholestan-spin label (CSL) dissolved by the NaTDC micellar aggregates as a function of temperature and NaCl concentration. The local reorientation of CSL has been accounted for motionally-averaged g- and A-tensors assuming fast oscillation around the spin-probe long molecular axis. The overall Brownian tumbling of CSL-micelle complexes has been modeled by an axially symmetric rotational tensor. Good agreement with experimental spectra is obtained. Best-fit rotational parameters and QELS data suggest that, in the circumstance of large aggregation, NaTDC micelles have cylindrical shape and micellar growth occurs along the cylinder axis.     

Bridge function for the dipolar fluid from simulation

The exact bridge function of the Lennard-Jones dipolar (Stockmayer) fluid is extracted from Monte Carlo simulation data. The projections g(mnl)(r) onto rotational invariants of the non-spherically symmetric pair distribution function g(r, Ω) are accumulated during simulation. Making intensive use of anisotropic integral equation techniques, the molecular Ornstein-Zernike equation is then inverted in order to derive the direct correlation function c(mnl)(r), the cavity function y(mnl)(r), the negative excess potential of mean force lny|(mnl)(r), and the bridge function b(mnl)(r) projections. b(r, Ω) presents strong, non-universal anisotropies at high dipolar coupling. This simulation data analysis may serve as reference and guide for approximated bridge function theories of dipolar fluids and is a valuable step towards the case of more refined, nonlinear water-like geometries.     

Thermally activated escape rate for the Brownian motion of a fixed axis rotator in a double well potential for all values of the dissipation

The extension of the Kramers theory of the escape rate of a Brownian particle from a potential well to the entire range of damping proposed by Mel'nikov and Meshkov [J. Chem, Phys. 85, 1018 (1986)] is applied to the rotational Brownian motion of fixed axis rotators in a double well cosine potential. The procedure yields an expression for the Kramers escape rate valid for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and crossover regimes. This equation provides a good asymptotic estimate of the correlation time tau per pendicular of the longitudinal dipole moment correlation function calculated by solving the underlying Langevin equation using the matrix-continued fraction method. Moreover, for low barriers, where the Mel'nikov and Meshkov approach is not applicable, analytic equations for tau in the VLD and VHD limits are derived and a simple extrapolating equation that is valid for all values of the damping is proposed.     

Solution of the master equation for Wigner's quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)=ax(2)2+bx(4)4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p). The equilibrium position autocorrelation function, dynamic susceptibility, and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded has a quantum correction depending strongly on the barrier height and is compared with that given analytically by the quantum mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution. Moreover, the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.     

Communication: translational Brownian motion for particles of arbitrary shape

A single Brownian particle of arbitrary shape is considered. The time-dependent translational mean square displacement W(t) of a reference point at this particle is evaluated from the Smoluchowski equation. It is shown that at times larger than the characteristic time scale of the rotational Brownian relaxation, the slope of W(t) becomes independent of the choice of a reference point. Moreover, it is proved that in the long-time limit, the slope of W(t) is determined uniquely by the trace of the translational-translational mobility matrix μ(tt) evaluated with respect to the hydrodynamic center of mobility. The result is applicable to dynamic light scattering measurements, which indeed are performed in the long-time limit.     

Theoretical and experimental studies of CH3X-Y2 rotational line shapes for atmospheric spectra modelling: application to room-temperature CH3Cl-O2

The case of symmetric tops CH(3)X (X = Br, Cl, F, …) perturbed by non-polar diatoms Y(2) (Y = N(2), O(2), …) is analysed from the viewpoint of theoretical collisional broadening of their rotational lines observed in atmospheric spectra. A semi-classical approach involving an exponential representation of the scattering operator and exact trajectories governed by the isotropic potential is presented. For the first time the active molecule is strictly treated as a symmetric top and the atom-atom interactions are included in the intermolecular potential model. It is shown for the CH(3)Cl-O(2) system that these interactions contribute significantly to the line width for all values of the rotational quantum numbers J and K. Additional testing of modifications required in the semi-classical formalism for a correct application of the cumulant expansion is performed and it is shown that the use of the cumulant average on the rotational states of the perturbing molecule leads to entirely negligible effects for the not very strongly interacting CH(3)Cl-O(2) system. In order to check the theoretical predictions and to extend the scarce experimental data available in the literature to higher values of the rotational quantum numbers, new measurements of room-temperature O(2)-broadened CH(3)Cl rotational lines are carried out by a photomixing continuous-wave terahertz spectrometer. The experimental line widths extracted with a Voigt profile model demonstrate an excellent agreement with theoretical results up to very high J-values (J = 31, 37, 40, 45, 50).     

Mean first passage times for bond formation for a Brownian particle in linear shear flow above a wall

Motivated by cell adhesion in hydrodynamic flow, here the authors study bond formation between a spherical Brownian particle in linear shear flow carrying receptors for ligands covering the boundary wall. They derive the appropriate Langevin equation which includes multiplicative noise due to position-dependent mobility functions resulting from the Stokes equation. They present a numerical scheme which allows to simulate it with high accuracy for all model parameters, including shear rate and three parameters describing receptor geometry (distance, size, and height of the receptor patches). In the case of homogeneous coating, the mean first passage time problem can be solved exactly. In the case of position-resolved receptor-ligand binding, they identify different scaling regimes and discuss their biological relevance.     

Line of charges in electrolyte solution near a half-space II. Electric field of a single charge

The electric potential of a single charge in electrolyte solution near a dielectric or a semiconducting half-space is determined in closed form when the electrostatics is described by the linear Debye-Hückel (D-H) equation. The electric potential strongly depends on the Debye length of the solution, the substrate-to-solution dielectric constant ratio, and the Debye length of the semiconductor. The technique of Hankel transforms is shown to be a useful tool in solving such axially symmetric boundary value problems for the D-H equation.     

Mean first passage time for the contact between the ends of a chain polymer

We investigate the first passage times for the contact between the ends of a Rouse chain, whose initial separation is greater than a predefined contact distance, sigma, and equilibrium-distributed. An approximate analytic expression for the mean first passage time is obtained and compared with the results of previous theories and Brownian dynamics simulations. We find that the results of the present theory are in better agreement with Brownian dynamics simulation results than those of previously reported theories.     

Molecular motion of a nickel-bis(dithiolato) complex in solution

The molecular motion of the planar bis(maleonitriledithiolato)nickel anion, Ni(mnt)(2)(-), has been studied as a function of temperature using electron spin resonance (ESR) in several polar solvents; they are ethyl alcohol, eugenol, dimethyl phthalate, tri-n-butyl phosphate, tris(2-ethyl-hexyl)phosphate, diglyme, and a dimethylformamide-chloroform mixed solvent. Calculated spectra in agreement with the experimental X-band spectra are obtained using axially symmetric reorientation when the long in-plane axis is the unique (parallel) axis of the rotational diffusion tensor with D parallel/D perpendicular = 3.0-4.0; D parallel and D perpendicular are the diffusion constants for reorientation about the parallel and perpendicular axes, respectively. The reorientational model required for the simulations is either in or close to the Brownian rotational diffusion limit. In the slow motional (low temperature) region, the spectra can be simulated using the glassy g values. As the temperature increases, however, agreement is obtained only if the intermediate g factor, g(y), for the non-axially symmetric Zeeman interaction increases while g(x), g(z), and the motional model remain unchanged; this scheme and others for which gx and g(z) are possibly temperature-dependent are discussed. The values of D perpendicular from the simulations are in general agreement with those from earlier analyses of the width of the central spectral feature. The simulations and width analyses indicate (as do electrochemical, conductivity, and vapor-phase osmometry data) that the paramagnetic species reorienting in solution has a shape similar to that of the Ni(mnt)(2)(-) ion.     

Statistics of Polymer Solutions from Scaling Concepts for Geometry and Relativistic Self-Diffusion in Liquids

Basic results on conformational statistics of polymer solutions are derived from recent scaling concepts for geometry and a relativistic picture for Brownian self-diffusion in liquid media. Any chain conformation is interpreted as a geometrical state affected by its end-to-end dimension, which here denotes the mean deviation between geodesic paths diffusing in the relativistic liquid phase. Statistical polymer length distributions solve an ondulatory equation in non-Euclidean manifolds for coil extension and shape. When length scale is vanishing, the size scaling is found again in terms of parallelism angle rotations. The characteristic chain ratio identifies instead an average metric coefficient, originating topologically from rotational degrees of freedom internal to single molecules.     

Iterative Solution Method for the Linearized Poisson-Boltzmann Equation: Indirect Boundary Integral Equation Approach

An iterative solution scheme is proposed for solving the electrical double-layer interactions governed by the linearized Poisson-Boltzmann equation. The method is based on the indirect integral equation formulation with the double-layer potential kernel of the linearized Poisson-Boltzmann equation. In contrast to the conventional direct integral equation approach that yields Fredholm integral equations of the first kind, the indirect integral equation approach yields well-posed Fredholm integral equations of the second kind. The eigenvalue analysis reveals that the spectral radius of the double-layer integral operator is always less than one. Thus, iterative solution schemes can be successfully implemented for solving the electrical double-layer interactions for very large and complex systems. The utility of the iterative indirect method is demonstrated for several examples which include spherical and spheroidal particles. Copyright 2001 Academic Press.     

Insights into the Kramers’ flux‐over‐population rate for chemical reactions in liquid phases through the matrix transport equation

Kramers’ equation models a chemical reaction as a Brownian particle diffusing over a potential barrier under the influence of medium viscosity. In the case of high viscosity, the equation reduces to a simpler Smoluchowski equation. In this report, we have contrived an equivalent matrix‐transport equation that relates the ordered pair (activity, flux) of the output (activated complex) to that of the input (reactant). With an initial condition of the Dirac delta type placed at the location of the reactant, and a reflecting boundary condition set on the reactant state, and an absorbing boundary condition on the activated complex state, we are able to prove the equality relation between the mean first passage time, , for the diffusion and the inverse of the rate constant, k^?1, for the reaction counterpart. We have also derived , where λ_i is the ith eigenvalue of the Smoluchowski differential operator stipulated with the above‐mentioned boundary conditions. We have also deduced that, in the long time limit, the number of particles remaining inside the diffusion domain decays exponentially with a relaxation time just the same as the concentration of the reactant does for a first‐order reaction system.     

Effect of correlated noises on directed motion

The Langevin equation for a Brownian particle, in contact with a heat bath which offers state dependent friction, is considered to study the directed motion in presence of two external correlated noises. The effects of correlation on transport of the Brownian particle in a symmetric periodic potential is studied and it has been found that the steady state current increases with increase in the degree of correlation. This property suggests that by controlling the degree of correlation one can enhance the current in a properly designed experiment.     

Chemical reaction dynamics within anisotropic solvents in time-dependent fields

The dynamics of low-dimensional Brownian particles coupled to time-dependent driven anisotropic heavy particles (mesogens) in a uniform bath (solvent) have been described through the use of a variant of the stochastic Langevin equation. The rotational motion of the mesogens is assumed to follow the motion of an external driving field in the linear response limit. Reaction dynamics have also been probed using a two-state model for the Brownian particles. Analytical expressions for diffusion and reaction rates have been developed and are found to be in good agreement with numerical calculations. When the external field driving the mesogens is held at constant rotational frequency, the model for reaction dynamics predicts that the applied field frequency can be used to control the product composition.