Friction and diffusion of a Brownian particle in a mesoscopic solvent

The friction and diffusion coefficients of a massive Brownian particle in a mesoscopic solvent are computed from the force and the velocity autocorrelation functions. The mesoscopic solvent is described in terms of free streaming of the solvent molecules, interrupted at discrete time intervals by multiparticle collisions that conserve mass, momentum, and energy. The Brownian particle interacts with the solvent molecules through repulsive Lennard-Jones forces. The decays of the force and velocity autocorrelation functions are analyzed in the microcanonical ensemble as a function of the number N of solvent molecules and Brownian particle mass and diameter. The simulations are carried out for large system sizes and long times to assess the N dependence of the friction coefficient. The decay rates of these correlations are confirmed to vary as N(-1) in accord with earlier predictions. Hydrodynamic effects on the velocity autocorrelation function and diffusion coefficient are studied as a function of Brownian particle mass and diameter.     

Estimating the viscoelastic moduli of complex fluids from observation of Brownian motion of a particle confined to a harmonic trap

A procedure is proposed to estimate the viscoelastic properties of a complex fluid from the behavior of the velocity autocorrelation function of a suspended Brownian particle, trapped in a harmonic potential. The procedure is tested for a model complex fluid with a given frequency-dependent shear viscosity. The analysis shows that the procedure can provide a rather accurate prediction of the viscoelastic properties of the fluid on the basis of experimental data on the velocity autocorrelation function of the trapped Brownian particle in a limited range of time.     

Communication: drift velocity of Brownian particle in a periodically tapered tube induced by a time-periodic force with zero mean: dependence on the force period

We study the drift of a Brownian particle in a periodically tapered tube, induced by a longitudinal time-periodic force of amplitude ∣F∣ that alternates in sign every half-period. The focus is on the velocity dependence on the force period, which is usually considered not tractable analytically. For large ∣F∣ we derive an analytical solution that gives the velocity as a function of the amplitude and the period of the force as well as the geometric parameters of the tube. The solution shows how the velocity decreases from its maximum value to zero as the force period decreases from infinity (adiabatic regime) to zero. Our analytical results are in excellent agreement with those obtained from 3D Brownian dynamics simulations.     

Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion

Brownian motion of a particle situated near a wall bounding the fluid in which it is immersed is affected by the wall. Specifically, it is assumed that an incompressible viscous fluid fills a half-space bounded by a plane wall and that the fluid flow satisfies stick boundary conditions at the wall. The fluctuation-dissipation theorem shows that the velocity autocorrelation function of the Brownian particle can be calculated from the frequency-dependent admittance valid locally. It is shown that the t(-3/2) long-time tail of the velocity relaxation function, valid in bulk fluid, is obliterated and replaced by a t(-5/2) long-time tail of positive amplitude for motions parallel to the wall and by a t(-5/2) long-time tail of negative amplitude for motions perpendicular to the wall. The latter finding is at variance with an earlier calculation by Gotoh and Kaneda.     

Diffusion coefficient of Brownian particle in rough micro-channel

This study examines the feasibility of using of the lattice Boltzmann method to determine how the surface roughness of a quadrate channel affects the diffusion coefficient of Brownian particle(s). The surface was represented by a regular array of spheres. Surface roughness reduced the diffusion coefficient of the Brownian particle(s) because of a change in the velocity autocorrelation function decay and in pressure. Additionally, the neighboring particles increased the diffusion coefficient of Brownian particle.     

LiCl熔盐和玻璃的速度自相关函数和振动频谱

对不同温度下熔融LiCl和LiCl急冷模拟所得分子动力学相轨道的分析表明:虽然模拟系统各个粒子在不同时间的位能、速度和所受力都是不同的,但是,即使在1200K,Li~+离子和Cl~-离子的运动也表现出振荡的特性,温度愈低,振荡特性表现得愈明显。另一方面,模拟也表明落在模拟立方元胞内的粒子只有经过相当长的时间才会迂移到另一个格子。计算启发我们,作为近似似可将LiCl系中离子的运动看作是阻尼振荡和布朗运动的迭加。为了深入研究LiCl系的动力学行为,我们计算了不同温度下LiCl系的速度自相关函数和振动频谱并进行了比较。     

Diffusion of nanoparticles in dense fluids

The diffusion process of a single spherical nanoparticle immersed in a fluid solvent is studied by molecular dynamics simulations. When the nanoparticle mass stays constant, it is shown that, at short times, the decay of the nanoparticle velocity autocorrelation function is strongly modified when the particle diameter increases. It is also shown that, at large times, the characteristic algebraic decay induced by the hydrodynamic correlations between the solvated particle and the solvent presents a scaling behavior depending on the particle diameter.     

The van Hove distribution function for brownian hard spheres: dynamical test particle theory and computer simulations for bulk dynamics

We describe a test particle approach based on dynamical density functional theory (DDFT) for studying the correlated time evolution of the particles that constitute a fluid. Our theory provides a means of calculating the van Hove distribution function by treating its self and distinct parts as the two components of a binary fluid mixture, with the "self?" component having only one particle, the "distinct" component consisting of all the other particles, and using DDFT to calculate the time evolution of the density profiles for the two components. We apply this approach to a bulk fluid of Brownian hard spheres and compare to results for the van Hove function and the intermediate scattering function from Brownian dynamics computer simulations. We find good agreement at low and intermediate densities using the very simple Ramakrishnan-Yussouff [Phys. Rev. B 19, 2775 (1979)] approximation for the excess free energy functional. Since the DDFT is based on the equilibrium Helmholtz free energy functional, we can probe a free energy landscape that underlies the dynamics. Within the mean-field approximation we find that as the particle density increases, this landscape develops a minimum, while an exact treatment of a model confined situation shows that for an ergodic fluid this landscape should be monotonic. We discuss possible implications for slow, glassy, and arrested dynamics at high densities.     

Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion in a viscous compressible fluid

Brownian motion of a particle situated near a wall bounding the fluid in which it is immersed is affected by the wall. Specifically, it is assumed that a viscous compressible fluid fills a half space bounded by a plane wall, and that the fluid flow satisfies stick boundary conditions at the wall. The fluctuation-dissipation theorem shows that the velocity autocorrelation function of the Brownian particle can be calculated from the frequency-dependent admittance valid locally. The admittance can be found from the linearized Navier-Stokes equations. The t(-3/2) long-time tail of the velocity relaxation function, valid in bulk fluid, is obliterated by the wall and replaced by a t(-5/2) long-time tail of positive amplitude for motions parallel to the wall and by a t(-5/2) long-time tail of negative amplitude for motions perpendicular to the wall. In both cases the amplitude of the t(-5/2) long-time tail turns out to be independent of fluid compressibility and bulk viscosity.     

Note: Aris-Taylor dispersion from single-particle point of view

When a point Brownian particle diffuses in a straight circular tube in the presence of a laminar stationary flow of the liquid, its effective diffusion coefficient along the tube axis increases compared to its value in the absence of flow. The effective diffusion coefficient as a function of the average fluid velocity and the tube radius is given by the Aris-Taylor formula. We give a new derivation of this formula, which is based on consideration of the axial displacement of the particle that moves in the plane normal to the tube axis along a given trajectory. The result is obtained by averaging the displacement and its square over different realizations of the particle trajectory and analyzing the long-time asymptotic behavior of the two moments.     

Spectrum of position fluctuations of a Brownian particle bound in a harmonic trap near a plane wall

The spectrum of position fluctuations of a Brownian particle bound in a harmonic trap near a plane wall is calculated from an approximate result for the Fourier transform of the velocity autocorrelation function. Both a no-slip and a perfect slip boundary condition at the wall are considered. In both cases at low frequency the calculated spectrum differs markedly from recent experimental data. It is suggested that a partial slip boundary condition with a frequency-dependent slip coefficient may explain the experimental results.     

On the short-time limit of ring polymer molecular dynamics

We examine the short-time accuracy of a class of approximate quantum dynamical techniques that includes the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) methods. Both of these methods are based on the path integral molecular dynamics (PIMD) technique for calculating the exact static equilibrium properties of quantum mechanical systems. For Kubo-transformed real-time correlation functions involving operators that are linear functions of positions or momenta, the RPMD and (adiabatic) CMD approximations differ only in the choice of the artificial mass matrix of the system of ring polymer beads that is employed in PIMD. The obvious ansatz for a general method of this type is therefore to regard the elements of the PIMD (or Parrinello-Rahman) mass matrix as an adjustable set of parameters that can be chosen to improve the accuracy of the resulting approximation. We show here that this ansatz leads uniquely to the RPMD approximation when the criterion that is used to select the mass matrix is the short-time accuracy of the Kubo-transformed correlation function. In particular, we show that the leading error in the RPMD position autocorrelation function is O(t(8)) and the error in the velocity autocorrelation function is O(t(6)), for a general anharmonic potential. The corresponding errors in the CMD approximation are O(t(6)) and O(t(4)), respectively.     

Transport properties of nitrogen in single walled carbon nanotubes

Transport properties including collective and tracer diffusivities of nitrogen, modeled as a diatomic molecule, in single walled carbon nanotubes have been studied by equilibrium molecular dynamics at different temperatures and as a function of pressure. It is shown that while the asymptotic decay of the translational and rotational velocity autocorrelation function is algebraic, the collective velocity decays exponentially with the relaxation time related to the interfacial friction. The tracer diffusivity in the nanochannel, which is comparable in magnitude with diffusivity in the equilibrium bulk phase, depends only weakly on the conditions at the fluid-solid interface, whereas the collective diffusivity is a strong function of the hydrodynamic boundary conditions and is found to be three orders of magnitude higher than self-diffusivity in carbon nanotubes and for the comparatively rough surface of the rare-gas tube it is one order of magnitude greater. A relationship between the collective diffusivity and the Maxwell coefficient describing wall collisions is obtained. The transport coefficients appear to be insensitive to the long-range details of the potential function.     

Analysis of particle-wall interactions during particle free fall

In this study, the vertical motion of a particle in a quiescent fluid falling toward a horizontal plane wall is analyzed, based on simplified models. Using the distance between the particle and wall as a parameter, the effects of various forces acting on the particle and the particle motion are examined. Without the colloidal and Brownian forces being included, the velocity of small particles is found to be approximately equal to the inverse of the drag force correction function used in this study as the particle approaches the near-wall region. Colloidal force is added to the particle equation of motion as the particle moves a distance comparable to its size. It is found that the particle might become suspended above or deposited onto the wall, depending on the Hamaker constant, the surface potentials of the particle and wall, and the thickness of the electrical double layer (EDL). For strong EDL repulsive force and weaker van der Waals (VDW) attractive force, the particle will become suspended above the wall at a distance at which the particle velocity is zero. This location is referred to as the equilibrium distance. The equilibrium distance is found to increase with increased in EDL thickness when a repulsive force barrier appears in the colloidal force interaction. For the weak EDL repulsive force and strong VDW attractive force case, the particle can become deposited onto the wall without the Brownian motion effect. The Brownian jump length was found to be very small. Many Brownian jumps would be required in a direction toward the wall for a suspended particle to become deposited.     

Statistical mechanical theory for steady state systems. VIII. General theory for a Brownian particle driven by a time- and space-varying force

A Brownian particle subject to a time- and space-varying force is studied with the second entropy theory for nonequilibrium statistical mechanics. A fluctuation expression is obtained for the second entropy of the path, and this is maximized to obtain the most likely path of the particle. Two approaches are used, one based on the velocity correlation function and one based on the position correlation function. The approaches are a perturbation about the free particle result and are exact for weak external forces. They provide a particularly simple way of including memory effects in time-varying driven diffusion. The theories are tested against computer simulation data for a Brownian particle trapped in an oscillating parabolic well. They accurately predict the phase lag and amplitude as a function of drive frequency, and they account quantitatively for the memory effects that are important at high frequencies and that are missing in the simplest Langevin equation.     

Modeling real dynamics in the coarse-grained representation of condensed phase systems

This work presents a systematic multiscale methodology to provide a more faithful representation of real dynamics in coarse-grained molecular simulation models. The theoretical formalism is based on the recently developed multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B. 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] and relies on the generalized Langevin equation approach and its simpler Langevin equation limit. The friction coefficients are determined in multiscale fashion from the underlying all-atom molecular dynamics simulations using force-velocity and velocity-velocity correlation functions for the coarse-grained sites. The diffusion properties in the resulting CG Brownian dynamics simulations are shown to be quite accurate. The time dependence of the velocity autocorrelation function is also well-reproduced relative to the all-atom model if sufficient resolution of the CG sites is implemented.     

Diffusion Spectrum of Polymer Melt Measured by Varying Magnetic Field Gradient Pulse Width in PGSE NMR

The translational motion of polymers is a complex process and has a big impact on polymer structure and chemical reactivity. The process can be described by the segment velocity autocorrelation function or its diffusion spectrum, which exhibit several characteristic features depending on the observational time scale—from the Brownian delta function on a large time scale, to complex details in a very short range. Several stepwise, more-complex models of translational dynamics thus exist—from the Rouse regime over reptation motion to a combination of reptation and tube-Rouse motion. Accordingly, different methods of measurement are applicable, from neutron scattering for very short times to optical methods for very long times. In the intermediate regime, nuclear magnetic resonance (NMR) is applicable—for microseconds, relaxometry, and for milliseconds, diffusometry. We used a variation of the established diffusometric method of pulsed gradient spin-echo NMR to measure the diffusion spectrum of a linear polyethylene melt by varying the gradient pulse width. We were able to determine the characteristic relaxation time of the first mode of the tube-Rouse motion. This result is a deviation from a Rouse model of polymer chain displacement at the crossover from a square-root to linear time dependence, indicating a new long-term diffusion regime in which the dynamics of the tube are also described by the Rouse model.     

Extracting the time scales of conformational dynamics from single-molecule single-photon fluorescence statistics

The quenching rate of a fluorophore attached to a macromolecule can be rather sensitive to its conformational state. The decay of the corresponding fluorescence lifetime autocorrelation function can therefore provide unique information on the time scales of conformational dynamics. The conventional way of measuring the fluorescence lifetime autocorrelation function involves evaluating it from the distribution of delay times between photoexcitation and photon emission. However, the time resolution of this procedure is limited by the time window required for collecting enough photons in order to establish this distribution with sufficient signal-to-noise ratio. Yang and Xie have recently proposed an approach for improving the time resolution, which is based on the argument that the autocorrelation function of the delay time between photoexcitation and photon emission is proportional to the autocorrelation function of the square of the fluorescence lifetime [Yang, H.; Xie, X. S. J. Chem. Phys. 2002, 117, 10965]. In this paper, we show that the delay-time autocorrelation function is equal to the autocorrelation function of the square of the fluorescence lifetime divided by the autocorrelation function of the fluorescence lifetime. We examine the conditions under which the delay-time autocorrelation function is approximately proportional to the autocorrelation function of the square of the fluorescence lifetime. We also investigate the correlation between the decay of the delay-time autocorrelation function and the time scales of conformational dynamics. The results are demonstrated via applications to a two-state model and an off-lattice model of a polypeptide.     

Complex-time velocity autocorrelation functions for Lennard-Jones fluids with quantum pair-product propagators

We use the pair-product approximation to the complex-time quantum mechanical propagator to obtain accurate quantum mechanical results for the symmetrized velocity autocorrelation function of a Lennard-Jones fluid at two points on the thermodynamic phase diagram. A variety of tests are performed to determine the accuracy of the method and understand its breakdown at longer times. We report quantitative results for the initial 0.3 ps of the dynamics, a time at which the correlation function has decayed to approximately one fifth of its initial value.     

The central cell model: a mesoscopic hopping model for the study of the displacement autocorrelation function

On the mesoscale, the molecular motion in a microporous material can be represented as a sequence of hops between different pore locations and from one pore to the other. On the same scale, the memory effects in the motion of a tagged particle are embedded in the displacement autocorrelation function (DACF), the discrete counterpart of the velocity autocorrelation function (VACF). In this paper, a mesoscopic hopping model, based on a lattice-gas automata dynamics, is presented for the coarse-grained modeling of the DACF in a microporous material under conditions of thermodynamic equilibrium. In our model, that we will refer to as central cell model, the motion of one tagged particle is mimicked through probabilistic hops from one location to the other in a small lattice of cells where all the other particles are indistinguishable; the cells closest to the one containing the tagged particle are simulated explicitly in the canonical ensemble, whereas the border cells are treated as mean-field cells in the grand-canonical ensemble. In the present paper, numerical simulation of the central cell model are shown to provide the same results as a traditional lattice-gas simulation. Along with this a mean-field theory of self-diffusion which incorporates time correlations is discussed.