Frequency Dependent Friction Coefficient Research Articles

Rate theory of gas-liquid nucleation: Quest for the elusive quantitative accuracy.

The task of a first principles theoretical calculation of the rate of gas-liquid nucleation has remained largely incomplete despite the existence of reliable results from unbiased simulation studies at large supersaturation. Although the classical nucleation theory formulated by Becker-Doring-Zeldovich about a century ago provides an elegant, widely used picture of nucleation in a first-order phase transition, the theory finds difficulties in predicting the rate accurately, especially in the case of gas-to-liquid nucleation. Here, we use a multiple-order parameter description to construct the nucleation free energy surface needed to calculate the nucleation rate. A multidimensional non-Markovian (MDNM) rate theory formulation that generalizes Langer's well-known nucleation theory by using the Grote-Hynes MDNM treatment is used to obtain the rate of barrier crossing. We find good agreement of the theory with the rate obtained by direct unbiased molecular dynamics simulations-the latter is feasible at large supersaturation, S. The theory gives an experimentally strong dependence of the rate of nucleation on supersaturation, S. Interestingly, we find a strong influence of the frequency-dependent friction coefficient at the barrier top. This arises from multiple recrossings of the barrier surface. We find that a Markovian theory, such as Langer's formulation, fails to capture the rate quantitatively. In addition, the multidimensional transition state theory expression performs poorly, revealing the underlying role of the friction coefficient.

The Memory Function and Chaos-Induced Friction in the Chaotic Henon-Heiles System

A non-Markovian linear stochastic equation for the momentum p y (t) is derived for the purpose of clarifying transport processes in the chaotic Henon-Heiles system with the aid of the Mori projection operator formalism. For the time correlation function C y (t) = of the coordinate y(t), this leads to an integrable linear evolution equation. Then, the memory function γ(t) enables us to define a frequency-dependent chaos-induced friction coefficient of the system, γ(iω). We show that this friction coefficient is related to the time correlation function Φ(t) of a nonlinear force f(t), which can be computed numerically. Thus, in the case that the total energy is E = 1/6, it turns out that the structure of the frequency-dependent friction coefficient γ r (ω) consists of three sharp peaks at frequencies ω = 0, 0.859 and 1.891. This leads to a three-term approximation of the memory function, γ(t), with a correlation time τ r ∼ 5T (with T = 2π). It is also shown that the structure of the power spectrum I y (ω) of y(t) consists of four sharp peaks at frequencies ω = 0, 0.500, 0.797 and 1.000. This leads to a four-term approximation of the time correlation function C y (t) with a correlation time τ M ∼ 6T. The frequencies and line widths of the sharp peaks of I y (ω) are given by the friction coefficient γ(iω).

Backtracking of a sphere slowing down in a viscous compressible fluid

It is shown on the basis of the linearized Navier-Stokes equations that a sphere set in motion by a sudden impulse, and slowing down in a viscous compressible fluid, can temporarily reverse its velocity at a late stage of the motion before finally slowing down with a positive velocity. For the effect to occur, the fluid needs to be sufficiently compressible, and the bulk viscosity needs to be sufficiently small relative to the shear viscosity. More generally, the velocity decays nonmonotonically, without necessarily passing through zero. The results are derived from the frequency-dependent friction coefficient, and hold for general boundary condition on the surface of the sphere.

Anomalous diffusion of a particle in an aging medium

We report new results about the anomalous diffusion of a particle in an aging medium. For each given age, the quasi-stationary particle velocity is governed by a generalized Langevin equation with a frequency-dependent friction coefficient proportional to | ω| δ−1 at small frequencies, with 0< δ<2. The aging properties of the medium are encoded in a frequency-dependent effective temperature T eff. ( ω). The latter is modelized by a function proportional to | ω| α at small frequencies, with α<0, thus allowing for the medium to have a density of slow modes proportionally larger than in a thermal bath. Using asymptotic Fourier analysis, we obtain the behaviour at large times of the velocity correlation function and of the mean square displacement. As a result, the anomalous diffusion exponent in the aging medium appears to be linked, not only to δ as it would be the case in a thermal bath, but also to the exponent α characterizing the density of slow modes.

Microscopic description of vibrational energy relaxation in supercritical fluids: On the dominance of binary solute–solvent contributions

The representation of the frequency dependent friction coefficient ia a four-particle two-time correlation function is used to analyze the applicability of collisional and hydrodynamical models of vibrational energy relaxation (VER). The solute–solvent binary dynamics is separated from collective equilibrium correlations by means of Green's functions. The collective contributions manifest themselves mainly ia the solute–solvent radial distribution function (RDF), which reflects peculiarities of the particular solvent thermodynamical (e.g., supercritical) state. The binary dynamics is also closely related to many-body equilibrium correlations, as initial conditions sample microscopic system states in the vicinity of the solute which are the most important for VER. VER rates along a close to critical isotherm are calculated on the basis of the breathing sphere model and the Douglas approximation for force–time correlation functions, while Monte Carlo simulations are used for calculating RDFs. The results are compared with molecular dynamics simulations at low, intermediate and high densities. It is shown that at near-critical conditions as well as far from the critical point the key contribution to VER comes from the short and intermediate time behavior of the force–time correlation function. In configuration space only short range binary solute–solvent correlations are important. Analytical estimations, Monte Carlo and molecular dynamics simulations clearly show that the dynamics of VER can only be understood on the basis of a detailed description of local solute–solvent interactions and correlations.

Dynamical Conductivity of the Dilute Lorentz Gas with Spherically Symmetric Scatterers

The dynamical conductivity of the Lorentz gas with spherically symmetric potentials is studied to lowest order in the density of scatterers. The frequency-dependent friction coefficient is calculated from the Fourier transform of the force–force time-correlation function determined by the dynamics of a single scattering process. The corresponding dynamical conductivity varies with frequency on the scale of the inverse collision time. As an example, the conductivity is calculated for a scattering potential of the Maxwell type.

Response to ‘‘Comment on ‘Long-time tails in angular momentum correlations’ ’’ [J. Chem. Phys. 104, 7363 (1996)

~Received 29 December 1995; accepted 22 January 1996!@S0021-9606~96!02216-4#In their comment on our paper, Cichocki and Felderhofpoint out that our conclusion about the shape independenceof the long-time tail in the angular momentum correlationfunction must be incorrect because it contradicts a ‘‘rigor-ous’’ result that they have obtained previously. Moreover,they stress that we have incorrectly stated that their theoreti-cal results are limited to the case that the orientation of theparticle is fixed.To start with the second point: It is obvious from Ref. 1,that Cichocki and Felderhof do not consider the case of largeamplitude reorientational motion. In fact, they say as muchin the preamble to the derivation in Ref. 1 where they statethat they consider the case of ‘‘small amplitude motion.’’ Inour paper, we referred to this case as the limit that the ori-entation is ‘‘fixed.’’ Admittedly, it would have been moreaccurate to speak about an orientation that is ‘‘essentiallyfixed,’’ rather than literally fixed. We thought this to be un-necessarily verbose, because truly fixed orientations areunrealistic—in this limit the angular velocity autocorrelationfunction ~AVACF! does not exist.Let us then compare the two cases of real interest,namely the Cichocki–Felderhof limit of small amplitude an-gular motion and the limit of large amplitude reorientationthat we considered in our paper.Cichocki and Felderhof point out that for small ampli-tude motion, the AVACF does depend on the shape of theparticle, and we completely agree. More interestingly, wewere happy to notice that the theoretical predictions ofCichocki and Felderhof for this limit are in excellent agree-ment with the numerical results that for nonspherical objectsunder conditions where the angular displacement is negli-gible. For small amplitude motion, the shape and frequencydependent friction coefficient is proportional to the memoryfunction of the AVACF and gives a shape dependent tail.Thus, so far as the calculation of the friction coefficient isconcerned, the particle is fixed ~‘‘essentially fixed’’ in ourpresent terminology!. The calculation described in Ref. 1neglects the effect of the particle’s orientational motion onthe frequency dependent rotational friction coefficient. Oursimulations show quite convincingly, that for a particle thatis ‘‘essentially fixed,’’ a shape dependent decay of theAVACF is indeed observed. It is reassuring that our model,which uses a relatively crude lattice representation of a non-spherical object, is in essentially quantitative agreement withthe relevant theoretical predictions. This gives us great en-couragement that we can reliably extend our simulations tothe case of concentrated suspensions of non-spherical ob-jects.Next, let us consider the case discussed in Ref. 2, viz.the effect of reorientation on the dynamics of the particle.Reorientation is important because, on a sufficiently longtimescale, i.e., in the truly asymptotic regime, reorientation isalways important. In Ref. 2 we stressed that the theory con-tained in Ref. 1 assumes that reorientation is negligible andis therefore not appropriate ~let alone ‘‘rigorous’’! in thislimit. However, the microscopic theory contained in Ref. 4should apply in the limit of large-amplitude reorientation. Itpredicts that reorientation changes the decay of the AVACFin such a way that it becomes identical to the result for aspherical object with the same moment of inertia. A carefulmicroscopic analysis produces kinetic equations that, in theBrownian limit, reduce to the Navier–Stokes equations ~plusappropriate boundary conditions!. Not all the mode-couplingor kinetic theory treatments, which have been proposed, givethis limit correctly.

Diffusion of a flexible polymer chain in a disordered medium

A theory is presented of the long wavelength dynamics of a flexible polymer chain without excluded volume interactions in a medium composed of randomly distributed static scatterers. The model is applicable to the motion of a polymer chain in a random porous solid. The theory is based on an approximate resummation of an expansion of the frequency-dependent friction coefficient in powers of scatterer density. If the chain is short or the scatterer density is low, the diffusion constant is inversely proportional to the chain length (molecular weight), in accordance with the Rouse model. Above a critical value of the chain length or scatterer concentration, the chain undergoes hindered diffusion with a diffusion constant that is smaller than the Rouse value. If the scatterer size is much smaller than the polymer radius of gyration, this diffusion constant has an inverse quadratic dependence on chain length, as predicted by the reptation model. If the scatterer size is large compared to the radius of gyration, the diffusion constant depends on the inverse cube of the chain length. A similar crossover is predicted for the time dependence of the mean squared displacement of a long chain. At short times, the chain undergoes Rouse diffusion, and at long time it exhibits hindered diffusion. An intermediate time regime of anomalous diffusion is predicted.

Dielectric friction on a moving ion

We consider the continuum model for dielectric friction on an ion moving in a fluid. The ion may have non-zero polarizability and the frequency dependent dielectric constant of the fluid may involve more than one relaxation time. The frequency dependent friction coefficient is calculated to second order in the charge of the ion. We show that the Hubbard-Onsager theory of ionic mobility needs to be corrected.

Memory effects in the energy dissipation for slow collective nuclear motion

The energy dissipation in slow collective nuclear motion is considered as a combined effect of the diabatic production of particle-hole excitations and the subsequent equilibration by two-body collisions. Memory effects due to the long mean free path of the nucleons are treated analytically for an interacting Fermi gas within moving walls leading to a friction kernel (frequency-dependent friction coefficient) in the classical equation of collective motion.

A molecular theory of Stokes–Einstein behavior. I. Translational Brownian motion

By using the Mori generalized Langevin equation an expression for the velocity autocorrelation function of a Brownian particle is obtained. By an appropriate choice of the potential of interaction between the particle and the fluid molecules the frequency dependent friction coefficient reduces to the value found when the slip boundary condition is applied in a hydrodynamic calculation; a different choice gives almost exactly the stick value. The reasons why the previous ’’mode-coupling’’ calculations give an incorrect form for the friction coefficient are discussed.

A molecular theory of Stokes–Einstein behavior. II. Rotational diffusion of a Brownian sphere

The analysis of the translational diffusion coefficient of a Brownian particle, given in the preceding paper, is extended to treat the rotational motion. Agreement with the hydrodynamic value for the frequency dependent friction coefficient is found.

Brownian motion in a fluid close to its' critical point

Abstract The frequency dependent friction coefficient of a spherical Brownian particle of radius a in a compressible fluid close to its critical point is evaluated taking into account the large correlation length ζ of the density fluctuations in the fluid. The expression found depends for finite frequencies on the ratio a /Gx. It is furthermore shown that a negative contribution to the amplitude of the usual t −3/2 long time tail of the velocity autocorrelation function of the particle appears very close to the critical point when ζ ⪢ a .

Frequency-dependent friction in oscillatory laminar pipe flow

Based on the Iberall model of fluid transients, a frequency-dependent friction coefficient is derived for laminar pipe flow of compressible fluids. The analysis, employing Laplace transformation, is of particular utility in the description of fluid motions which are naturally periodic and, hence, where the dependency of skin friction on frequency of excitation can be accounted for by expressing the coefficient in terms of the dominant frequency of the system. The analytical result is applied to the classical example of oscillations of an incompressible fluid in an open-ended U-tube. In distinct contrast to predictions based on the very common, quasi-steady engineering approximation of skin friction effects, the theory is found to be in excellent agreement with actual observations of successive peaks of the column motion over the whole range of frequencies investigated.

Hydrodynamic Theory of the Velocity Correlation Function

AbstractThe velocity correlation function of an atom in a simple liquid is calculated using a frequency dependent version of the Stokes‐Einstein formula. Stokes' law for the frictional force on a moving sphere is generalized to arbitrary frequency, compressibility, and viscoelasticity, with arbitrary slip of the fluid on the surface of the sphere. This frequency dependent friction coefficient is then used in a generalized Stokes‐Einstein formula, and the velocity correlation function is found by Fourier inversion. By using physically reasonable values for viscoelastic parameters, good agreement is obtained with the velocity correlation function determined by Rahman using computer experiments.

Hydrodynamic Theory of the Velocity Correlation Function

The velocity correlation function of an atom in a simple liquid is calculated using a frequency-dependent version of the Stokes-Einstein formula. Stokes's law for the frictional force on a moving sphere is generalized to arbitrary frequency, compressibility, and visco-elasticity, with arbitrary slip of the fluid on the surface of the sphere. This frequency-dependent friction coefficient is then used in a generalized Stokes-Einstein formula, and the velocity correlation function is found by Fourier inversion. By using physically reasonable values for viscoelastic parameters, good agreement is obtained with the velocity correlation function determined by Rahman using computer experiments.