Long-time Diffusion Coefficient Research Articles

Brownian dynamics of confined rigid bodies.

We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.

Effective temperatures and the breakdown of the Stokes-Einstein relation for particle suspensions.

The short- and long-time breakdown of the classical Stokes-Einstein relation for colloidal suspensions at arbitrary volume fractions is explained here by examining the role that confinement and attractive interactions play in the intra- and inter-cage dynamics executed by the colloidal particles. We show that the measured short-time diffusion coefficient is larger than the one predicted by the classical Stokes-Einstein relation due to a non-equilibrated energy transfer between kinetic and configuration degrees of freedom. This transfer can be incorporated in an effective kinetic temperature that is higher than the temperature of the heat bath. We propose a Generalized Stokes-Einstein relation (GSER) in which the effective temperature replaces the temperature of the heat bath. This relation then allows to obtain the diffusion coefficient once the viscosity and the effective temperature are known. On the other hand, the temporary cluster formation induced by confinement and attractive interactions of hydrodynamic nature makes the long-time diffusion coefficient to be smaller than the corresponding one obtained from the classical Stokes-Einstein relation. Then, the use of the GSER allows to obtain an effective temperature that is smaller than the temperature of the heat bath. Additionally, we provide a simple expression based on a differential effective medium theory that allows to calculate the diffusion coefficient at short and long times. Comparison of our results with experiments and simulations for suspensions of hard and porous spheres shows an excellent agreement in all cases.

Nonmonotonic diffusion of particles among larger attractive crowding spheres.

We study the diffusive motion of particles among fixed spherical crowders. The diffusers interact with the crowders through a combination of a hard-core repulsion and a short-range attraction. The long-time effective diffusion coefficient of the diffusers is found to depend nonmonotonically on the strength of their attraction to the crowders. That is, for a given concentration of crowders, a weak attraction to the crowders enhances diffusion. We show that this counterintuitive fact can be understood in terms of the mesoscopic excess chemical potential landscape experienced by the diffuser. The roughness of this excess chemical potential landscape quantitatively captures the nonmonotonic dependence of the diffusion rate on the strength of crowder-diffuser attraction; thus, it is a purely static predictor of dynamic behavior. The mesoscopic view given here provides a unified explanation for enhanced diffusion effects that have been found in various systems of technological and biological interest.

Swimming path statistics of an active Brownian particle with time-dependent self-propulsion

Typically, in the description of active Brownian particles, a constant effective propulsion force is assumed, which is then subjected to fluctuations in orientation and translation, leading to a persistent random walk with an enlarged long-time diffusion coefficient. Here, we generalize previous results for the swimming path statistics to a time-dependent, and thus in many situations more realistic, propulsion which is a prescribed input. We analytically calculate both the noise-free and the noise-averaged trajectories for time-periodic propulsion under the action of an additional torque. In the deterministic case, such an oscillatory microswimmer moves on closed paths that can be much more complicated than the commonly observed straight lines and circles. When exposed to random fluctuations, the mean trajectories turn out to be self-similar curves which bear the characteristics of their noise-free counterparts. Furthermore, we consider a propulsion force which scales in time t as ∝tα (with α = 0,1,2, …) and analyze the resulting superdiffusive behavior. Our predictions are verifiable for diffusiophoretic artificial microswimmers with prescribed propulsion protocols.

Brownian particles on rough substrates: Relation between intermediate subdiffusion and asymptotic long-time diffusion

Brownian particles in random potentials show an extended regime of subdiffusive dynamics at intermediate times. The asymptotic diffusive behavior is often established at very long times and thus cannot be accessed in experiments or simulations. For the case of one-dimensional random potentials with Gaussian distributed energies, we present a detailed analysis of experimental and simulation data. It is shown that the asymptotic long-time diffusion coefficient can be related to the behavior at intermediate times, namely, the minimum of the exponent that characterizes subdiffusion and hence corresponds to the maximum degree of subdiffusion. As a consequence, investigating only the dynamics at intermediate times is sufficient to predict the order of magnitude of the long-time diffusion coefficient and the time scale at which the crossover from subdiffusion to diffusion occurs, i.e., when the long-time diffusive regime and hence thermal equilibrium is established. Inversely, theoretical predictions derived for the asymptotic long-time behavior can be used to quantitatively characterize the intermediate behavior, which hardly has been studied so far.

Translational and rotational dynamics of colloidal particles in suspension: Effect of shear

We report a generalization of a nonequilibrium thermodynamic theory for the mesoscopic dynamics of radially symmetric interacting particles to anisotropic pairwise interactions and attain the one- and two-particle Fokker-Planck kinetics equations at a low-density limit that provides the translational-rotational coupling of their motion due to hydrodynamic interactions, from which we derived the balance equations of linear, angular momentum, and energy dissipation due to particle interactions and energy interchange with heat bath. In this low-density approximation, an already-known virial expression for the long-time translational collective diffusion coefficient of an orientational isotropic suspension in terms of the fluid equilibrium microstructure is recovered. An external shear flow induces, in the diffusive regime, vorticity effects into the rotational diffusion property of the colloidal particles. They manifest in the appearance of the particle's rotational viscosity due to vortex flow. The Smoluchowski equation that governs the dynamical relaxation of colloid microstructure due to particle's Brownian motion under stationary flow is provided.

Particle dynamics in two-dimensional random-energy landscapes: Experiments and simulations

The dynamics of individual colloidal particles in random potential energy landscapes was investigated experimentally and by Monte Carlo simulations. The value of the potential at each point in the two-dimensional energy landscape follows a Gaussian distribution. The width of the distribution, and hence the degree of roughness of the energy landscape, was varied and its effect on the particle dynamics studied. This situation represents an example of Brownian dynamics in the presence of disorder. In the experiments, the energy landscapes were generated optically using a holographic setup with a spatial light modulator, and the particle trajectories were followed by video microscopy. The dynamics is characterized using, e.g., the time-dependent diffusion coefficient, the mean squared displacement, the van Hove function, and the non-Gaussian parameter. In both experiments and simulations the dynamics is initially diffusive, showing an extended subdiffusive regime at intermediate times before diffusive motion is recovered at very long times. The dependence of the long-time diffusion coefficient on the width of the Gaussian distribution agrees with theoretical predictions. Compared to the dynamics in a one-dimensional potential energy landscape, the localization at intermediate times is weaker and the diffusive regime at long times reached earlier, which is due to the possibility to avoid local maxima in two-dimensional energy landscapes.

Colloidal diffusion over a periodic energy landscape

A two-layer colloidal system is developed for the study of colloidal diffusion over a two-dimensional periodic energy landscape. The energy landscape is made from the bottom layer of colloidal spheres forming a honey-comb crystalline pattern above a glass substrate. The corrugated surface of the bottom colloidal crystal provides a gravitational potential field for the diffusing particles in the top layer. The obtained population probability histogram P(x, y) of the diffusing particles is used to fully characterize the energy landscape U(x, y) via the Boltzmann distribution. The dynamical properties of the diffusing particle, such as its escape time tR and diffusion coefficient D are simultaneously measured from the particle's trajectories. The long-time diffusion coefficients D is found to be in good agreement with the theory for all colloidal samples studied. The experiment demonstrates the applications of this newly constructed colloidal energy landscape.

Dynamics of individual colloidal particles in one-dimensional random potentials: a simulation study

Using Monte Carlo simulations, individual Brownian particles have been investigated in a one-dimensional random energy landscape whose energy levels are selected from a Gaussian distribution. The standard deviation of the distribution determines the roughness of the noise-like potential and was varied in the simulations. After initialization, which was done by an instantaneous or infinitely slow (annealed) quench, the particle dynamics were followed. They were characterized by a number of parameters, such as the mean squared displacement, the time dependent diffusion coefficient, the non-Gaussian parameter, and the van Hove function. The dynamics exhibit different regimes: at very short times superdiffusion, followed by normal diffusion, and subsequently an extended period of subdiffusive dynamics due to localization within the minima of the potential, and finally, after a very slow approach towards the long-time limit, again diffusion with a significantly reduced diffusion coefficient. The long-time diffusion coefficient is consistent with theoretical predictions while no predictions exist for the intermediate times. Nevertheless, over the whole time range, the simulation results are in agreement with recent experimental findings on colloidal particles in a random potential created by a holographic optical setup.

Dynamics in dense hard-sphere colloidal suspensions

The dynamic behavior of a hard-sphere colloidal suspension was studied by x-ray photon correlation spectroscopy and small-angle x-ray scattering over a wide range of particle volume fractions. The short-time mobility of the particles was found to be smaller than that of free particles even at relatively low concentrations, showing the importance of indirect hydrodynamic interactions. Hydrodynamic functions were derived from the data, and for moderate particle volume fractions (Φ≤ 0.40) there is good agreement with earlier many-body theory calculations by Beenakker and Mazur [Physica A 120, 349 (1984)]. Important discrepancies appear at higher concentrations, above Φ≈ 0.40, where the hydrodynamic effects are overestimated by the Beenakker-Mazur theory, but predicted accurately by an accelerated Stokesian dynamics algorithm developed by Banchio and Brady [J. Chem. Phys. 118, 10323 (2003)]. For the relaxation rates, good agreement was also found between the experimental data and a scaling form predicted by the mode coupling theory. In the high concentration range, with the fluid suspensions approaching the glass transition, the long-time diffusion coefficient was compared with the short-time collective diffusion coefficient to verify a scaling relation previously proposed by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)]. We discuss our results in view of previous experimental attempts to validate this scaling law [L. Lurio et al., Phys. Rev. Lett. 84, 785 (2000)].

Diffusion of surfactant micelles in fluid and crystal phases

We have investigated diffusion of spherical surfactant micelles having a hard-core and weak repulsive potential in fluid, hexagonally close-packed (HCP), and body-centered cubic (BCC) phases by means of neutron spin echo (NSE) and dynamical light scattering (DLS) techniques. The intermediate scattering functions give short-time and long-time diffusion coefficients. In the fluid phase, the first-order virial coefficients of the normalized short-time self- and collective diffusion coefficients agreed with those of the hard spheres. The long-time collective diffusion coefficient exhibited the characteristic slowing down toward the fluid–HCP transition obeying a power law. At the fluid–HCP transition, the short-time diffusion coefficient increased abruptly, indicating increase of the effective free volume available for the micelles. Simultaneously, the long-time diffusion mode disappeared and a very slow mode characteristic of the crystal phase appeared, which might be related to the viscoelastic nature of the micelle crystals.

Dynamics of hard sphere suspensions using dynamic light scattering and X-ray photon correlation spectroscopy: Dynamics and scaling of the intermediate scattering function

Intermediate scattering functions are measured for colloidal hard sphere systems using both dynamic light scattering and x-ray photon correlation spectroscopy. We compare the techniques, and discuss the advantages and disadvantages of each. Both techniques agree in the overlapping range of scattering vectors. We investigate the scaling behavior found by Segré and Pusey [Phys. Rev. Lett. 77, 771 (1996)] but challenged by Lurio et al. [Phys. Rev. Lett. 84, 785 (2000)]. We observe a scaling behavior over several decades in time but not in the long-time regime. Moreover, we do not observe long-time diffusive regimes at scattering vectors away from the peak of the structure factor and so question the existence of long-time diffusion coefficients at these scattering vectors.

Lysozyme Protein Solution with an Intermediate Range Order Structure

The formation of equilibrium clusters has been studied in both a prototypical colloidal system and protein solutions. The appearance of a low-Q correlation peak in small angle scattering patterns of lysozyme solution was attributed to the cluster-cluster correlation. Consequently, the presence of long-lived clusters has been established. By quantitatively analyzing both the SANS (small angle neutron scattering) and NSE (neutron spin echo) data of lysozyme solution using statistical mechanics models, we conclusively show in this paper that the appearance of a low-Q peak is not a signature of the formation of clusters. Rather, it is due to the formation of an intermediate range order structure governed by a short-range attraction and a long-range repulsion. We have further studied dynamic features of a sample with high enough concentration at which clusters are formed in solution. From the estimation of the mean square displacement by using short-time and long-time diffusion coefficient measured by NSE and NMR, we find that these clusters are not permanent but have a finite lifetime longer than the time required to diffuse over a distance of a monomer diameter.

Self-diffusion in two-dimensional hard ellipsoid suspensions

We studied the self-diffusion of colloidal ellipsoids in a monolayer near a flat wall by video microscopy. The image processing algorithm can track the positions and orientations of ellipsoids with subpixel resolution. The translational and rotational diffusions were measured in both the laboratory frame and the body frame along the long and short axes. The long-time and short-time diffusion coefficients of translational and rotational motions were measured as functions of the particle concentration. We observed the nondiffusive crossover region in the intermediate time regime due to the caging of neighboring particles. Both the beginning and the ending times of the intermediate regime exhibit power-law dependence on concentration. The long-time and short-time diffusion anisotropies change nonmonotonically with concentration and reach minima in the semidilute regime because the motions along long axes are caged at lower concentrations than the motions along short axes. The time derivatives of mean-square displacements change linearly with the inverse of time in the intermediate time regimes at various particle densities. This indicates that their relaxation functions decay as 1/t which provides new challenges in theory. The effects of coupling between rotational and translational Brownian motions were demonstrated and the two time scales corresponding to anisotropic particle shape and anisotropic neighboring environment were measured.

Microrheology and Characteristic Lengths in Wormlike Micelles made of a Zwitterionic Surfactant and SDS in Brine

We study the Brownian motion of probe particles embedded in a wormlike micellar fluid made of a zwitterionic surfactant N-tetradecyl-N,N-dimethyl-3-ammonio-1-propanesulfonate (TDPS), sodium dodecyl sulfate (SDS), and salty water to get structural and dynamical information of the micellar network. The motion of the probe particles was tracked with diffusing wave spectroscopy, and the mean square displacement as a function of time for the particles was obtained. This allowed us to obtain the long-time diffusion coefficient for microspheres moving in the micellar network and the cage size where each particle is harmonically bound at short times in that network. The bulk mechanical susceptibility of the fluid determines the response of the probe particles excited by the thermal stochastic forces. As a consequence, the mean square displacement curves allowed us to calculate the elastic (storage) and the viscous (loss) moduli as a function of the frequency. From these curves, spanning a wide frequency range, we estimated the characteristic lengths as the mesh size, the entanglement length, the persistence length, and the contour length for micellar solutions of different zwitterionic surfactant concentration, surfactant ratio ([SDS]/[TDPS]), salt concentration, and temperature. Mesh size, entanglement length, and persistence length are almost insensitive to the change of these variables. In contrast, the contour length changes in an important way. The contour length becomes shorter as the temperature increases, and it presents a peak at a surfactant ratio of ∼0.50-0.55. When salt is added to the solution, the contour length presents a peak at a salt concentration of ∼0.225 M, and in some solutions, this length can reach values of ∼12 μm. Scission energies help us to understand why the contour length first increases and then decreases when salt is added.

Macromolecular dynamics in red blood cells investigated using neutron spectroscopy

We present neutron scattering measurements on the dynamics of haemoglobin (Hb) in human red blood cells (RBCs) in vivo. Global and internal Hb dynamics were measured in the ps to ns time and Å length scales using quasi-elastic neutron backscattering spectroscopy. We observed the cross over from global Hb short-time to long-time self-diffusion. Both short- and long-time diffusion coefficients agree quantitatively with predicted values from the hydrodynamic theory of non-charged hard-sphere suspensions when a bound water fraction of around 0.23 gram H(2)O per gram Hb is taken into account. The higher amount of water in the cells facilitates internal protein fluctuations in the ps time scale when compared with fully hydrated Hb powder. Slower internal dynamics of Hb in RBCs in the ns time range were found to be rather similar to results obtained with fully hydrated protein powders, solutions and Escherichia coli cells.

Dynamic simulations of colloids by core-modified dissipative particle dynamics

We develop a core-modified dissipative particle dynamics model of colloidal systems which includes an extra term to counteract depletion forces. Results are presented covering the full range of volume fractions. Radial distribution functions for the suspending fluid are shown to change significantly as the volume fraction of colloid increases. Equilibrium results for the long-time diffusion coefficient behave as expected, but the short-time coefficient is anomalous. The form of the equilibrium stress correlation functions is discussed and the derived Green-Kubo viscosities are compared with expected semiempirical forms. For nonequilibrium shear-field simulations we find that the system temperature is not adequately controlled by the dissipative particle dynamics (DPD) thermostat alone. Results using three alternative auxiliary thermostats are compared; a naive choice leading to a string phase at high shear rate. Using a thermostat based on relative particle velocities, the model reproduced the four classical regions of colloid rheology: a first Newtonian plateau, a shear-thinning region, a second Newtonian plateau, and finally a shear-thickening region at high strain rate. The most unexpected result of this exercise is that the core-modified DPD model without auxiliary thermostat almost exactly follows the same curve despite recording a temperature increase of a factor approximately 2.5 over the range.

Hydrodynamic interactions in rod suspensions with orientational ordering

The effect of hydrodynamic interactions on the diffusion of rods in the nematic phase is studied, both experimentally by time-resolved fluorescence video microscopy and theoretically by mesoscale-hydrodynamics simulations. The aspect ratio of the rods and the relative importance of hydrodynamic interactions—compared to direct interactions—are varied independently. This is achieved in experiments by using charged rod-like viruses (fd-virus) with varying ionic strength, both for the wild-type virus and viruses coated with a brush of polymers. In computer simulations, hydrodynamic interactions are incorporated by a particle-based mesoscopic simulation technique. It is found that translational long-time diffusion coefficients for parallel motion along the nematic director, scaled with the diffusion coefficient at infinite dilution, are significantly affected by hydrodynamic interactions, but are insensitive to the aspect ratio. In contrast, the diffusion anisotropy—defined as the ratio of the diffusion coefficients parallel and perpendicular to the nematic director—shows only a weak dependence on hydrodynamic interactions, but strongly varies with the aspect ratio.

Actin polymerization driven mitochondrial transport in mating S. cerevisiae

The dynamic microenvironment of cells depends on macromolecular architecture, equilibrium fluctuations, and nonequilibrium forces generated by cytoskeletal proteins. We studied the influence of these factors on the motions of mitochondria in mating S. cerevisiae using Fourier imaging correlation spectroscopy (FICS). Our measurements provide detailed length-scale dependent information about the dynamic behavior of mitochondria. We investigate the influence of the actin cytoskeleton on mitochondrial motion and make comparisons between conditions in which actin network assembly and disassembly is varied either by using disruptive pharmacological agents or mutations that alter the rates of actin polymerization. Under physiological conditions, nonequilibrium dynamics of the actin cytoskeleton leads to 1.5-fold enhancement of the long-time mitochondrial diffusion coefficient and a transient subdiffusive temporal scaling of the mean-square displacement (MSD proportional, variant tau (alpha), with alpha = 2/3). We find that nonequilibrium forces associated with actin polymerization are a predominant factor in driving mitochondrial transport. Moreover, our results lend support to an existing model in which these forces are directly coupled to mitochondrial membrane surfaces.

Quasi-two-dimensional diffusion of single ellipsoids: Aspect ratio and confinement effects

We report on video-microscopy measurements of the translational and rotational Brownian motions of isolated ellipsoidal particles in quasi-two-dimensional sample cells of increasing thickness. The long-time diffusion coefficients were measured along the long (Da) and short (Db) ellipsoid axes, respectively, and the ratio, Da/Db, was determined as a function of wall confinement and particle aspect ratio. In three dimensions (3D), this ratio (Da/Db) cannot be larger than 2, but in quasi-two dimensions, wall confinement was found to substantially alter diffusion anisotropy and substantially slow particle diffusion along the short axis compared to 3D.