Mean Square Displacement Of Particle Research Articles

Theory of Brownian motion in a Jeffreys fluid

We have constructed a kinetic theory of Brownian motion in a rheologically complex medium—a Jeffreys fluid that is characterized by a combination of two viscosity mechanisms: ordinary and delayed. This model is shown to be much better suited for the interpretation of experiments on the microrheology of viscoelastic media than the standard Maxwell model. In particular, no oscillations of the mean-square particle displacement arise in a Jeffreys fluid, which is a nonremovable artifact of the theory of Brownian motion in a Maxwell fluid. The developed approach can to be used also consider the diffusion of particles in other complex fluids whose rheology is described by phenomenological schemes.

Efficient on the fly calculation of time correlation functions in computer simulations

Time correlation functions yield profound information about the dynamics of a physical system and hence are frequently calculated in computer simulations. For systems whose dynamics span a wide range of time, currently used methods require significant computer time and memory. In this paper, we discuss the multiple-tau correlator method for the efficient calculation of accurate time correlation functions on the fly during computer simulations. The multiple-tau correlator is efficacious in terms of computational requirements and can be tuned to the desired level of accuracy. Further, we derive estimates for the error arising from the use of the multiple-tau correlator and extend it for use in the calculation of mean-square particle displacements and dynamic structure factors. The method described here, in hardware implementation, is routinely used in light scattering experiments but has not yet found widespread use in computer simulations.

Near-Membrane Dynamics and Capture of TRPM8 Channels within Transient Confinement Domains

BackgroundThe cold and menthol receptor, TRPM8, is a non-selective cation channel expressed in a subset of peripheral neurons that is responsible for neuronal detection of environmental cold stimuli. It was previously shown that members of the transient receptor potential (TRP) family of ion channels are translocated toward the plasma membrane (PM) in response to agonist stimulation. Because the spatial and temporal dynamics of cold receptor cell-surface residence may determine neuronal activity, we hypothesized that the movement of TRPM8 to and from the PM might be a regulated process. Single particle tracking (SPT) is a useful tool for probing the organization and dynamics of protein constituents in the plasma membrane.Methodology/Principal FindingsWe used SPT to study the receptor dynamics and describe membrane/near-membrane behavior of particles containing TRPM8-EGFP in transfected HEK-293T and F-11 cells. Cells were imaged using total internal reflection fluorescence (TIRF) microscopy and the 2D and 3D trajectories of TRPM8 molecules were calculated by analyzing mean-square particle displacement against time. Four characteristic types of motion were observed: stationary mode, simple Brownian diffusion, directed motion, and confined diffusion. In the absence of cold or menthol to activate the channel, most TRPM8 particles move in network covering the PM, periodically lingering for 2–8 s in confined microdomains of about 800 nm radius. Removing cholesterol with methyl-beta-cyclodextrin (MβCD) stabilizes TRPM8 motion in the PM and is correlated with larger TRPM8 current amplitude that results from an increase in the number of available channels without a change in open probability.Conclusions/SignificanceThese results reveal a novel mechanism for regulating TRPM8 channel activity, and suggest that PM dynamics may play an important role in controlling electrical activity in cold-sensitive neurons.

Biased diffusion in a piecewise linear random potential

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.

Interfacial microrheology as a tool to study viscoelastic transitions in nanoconfined soft matter

We present a method to perform in situ microrheological measurements on monolayers of soft materials undergoing viscoelastic transitions under compression. Using the combination of a Langmuir trough mounted on the inverted microscope stage of a laser scanning confocal microscope we track the motion of individual fluorescent quantum dots partly dispersed in monolayers spread at the air-water interface. From the calculated mean square displacement of the probe particles and extending a well established scheme of the generalized Stokes-Einstein relation in bulk to the interface we arrive at the viscoelastic modulus for the respective monolayers as a function of surface density. Measurements on monolayers of glassy as well as nonglassy polymers and a standard fatty acid clearly show sensitivity of our technique to subtle variations, in the viscoelastic properties of the highly confined materials under compression. Evidence for possible spatial variations of such viscoelastic properties at a given surface density for the fatty acid monolayer is also provided.

On the diffusivity of cosmic ray transport

A common assumption in particle transport theory is the assumption of diffusive propagation, corresponding to a linear increase of the mean square displacement of particle orbits with time. In recent years, however, several authors have discovered nondiffusive transport in observations and in test particle simulations. Some analytical theories also predict subdiffusion or superdiffusion of charged particles. In this article, a recently developed test particle code is employed to explore diffusive and nondiffusive transport regimes. For slab, two‐component, and isotropic turbulence the simulations are performed for different particle energies. It is demonstrated that, in most cases, cross‐field transport is indeed subdiffusive. The level of subdiffusivity is sensitively controlled by the turbulence geometry.

Simulation of Lagrangian dispersion using a Lagrangian stochastic model and DNS in a turbulent channel flow

The mean square displacements of fluid particles in a turbulent channel flow at Re τ = 100 are investigated using a modified Langevin equation, and are compared with DNS results. Both the Lagrangian integral time scales directly obtained from DNS and the predicted values using an empirical relation between the Eulerian and the Lagrangian integral time scales are used in the modified Langevin equation to test the effects of integral time scales on the dispersion of particles. The results show that the slight variation of the Lagrangian integral time scale has little influence on the dispersion. The agreement between results of the model equation and those of DNS is satisfactory except the streamwise dispersion for intermediate times (20 < t * < 300), where the results of the model equation are slightly overestimated compared to those of DNS. The cause of such discrepancy is discussed.

Time-reversible molecular dynamics algorithms with bond constraints

Time-reversible molecular dynamics algorithms with bond constraints are derived. The algorithms are stable with and without a thermostat and in double precision as well as in single-precision arithmetic. Time reversibility is achieved by applying a central-difference expression for the velocities in the expression for Gauss' principle of least constraint. The imposed time symmetry results in a quadratic expression for the Lagrange multiplier. For a system of complex molecules with connected constraints the corresponding set of coupled quadratic equations is easily solved by a consecutive iteration scheme. The algorithms were tested on two models. One is a dumbbell model of Toluene, the other system consists of molecules with four connected constraints forming a triangle and a branch point of constraints. The equilibrium particle distributions and the mean-square particle displacements for the dumbbell model were compared to the corresponding functions obtained by GROMACS. The agreement is perfect within statistical error.

A Novel Approach to High Accuracy of Video-Based Microrheology

Video-based particle tracking monitors the microscopic movement of labeled biomolecules and fluorescent probes within a complex cellular environment. Information gained from this technique enables us to extract the dynamic behavior of biomolecules and the local mechanical properties inside the cell from a tracked particle's mean-square displacement (MSD). However, MSD measurements are highly susceptible to static error introduced by noise in the image acquisition process that leads to an incorrect positioning of the particle. Static error can mask the subtle effects from the local microenvironment on the MSD and potentially generate misleading conclusions about the biophysical properties of cells. An approach that greatly increases the accuracy of MSD measurements is presented herein by combining experimental data with Monte Carlo simulations to eliminate the inherent static error. This practical method of static error correction greatly advances particle-tracking techniques.

Particle tracking microrheology of gel-forming amyloid fibril networks

Microrheology is a technique that is increasingly used to investigate the local viscoelastic properties of complex fluids non-invasively, by tracking the motion of micron-sized probe spheres. In this work, passive Particle Tracking Microrheology (PTM) is used to study network formation in the milk protein beta-lactoglobulin at 80 degrees C and pH 2. In these conditions the protein aggregates to form thread-like structures known as amyloid fibrils, which can further aggregate into elastic networks. Using PTM, gels were observed to form at significantly lower concentrations than determined by bulk rheometry, where the oscillatory shear forces may disrupt either fibril or network formation. During incubation, the Mean Square Displacement (MSD) of the probe particles exhibited time-cure superposition, allowing the critical relaxation exponent to be calculated as approximately 0.63, consistent with other biopolymer gels. Combined with the gel-like appearance of the complex modulus at long incubation times, this confirms that a true gel is forming, with physical or chemical crosslinks forming between the fibrils, refining the conclusions of other workers in the field.

Viscoelasticity in Homogeneous Protein Solutions

We probe the transport properties in protein solutions stable with respect to any, solid or liquid, phase separation as a step in the understanding of transport in the cytosol of live cells. We determine the mean-squared displacement of probe particles in the time range 10;{-3}-10 s in solutions of a model protein. The tested solutions exhibit significant elasticity at high frequencies, while at low frequencies, they are purely viscous. We attribute this viscoelasticity to a dense network of weakly-bound chains of protein molecules with characteristic lifetime of 10-100 ms. The found intrinsic viscoelasticity of protein solutions should be considered in biochemical kinetics models.

Diffusion limited cluster aggregation with irreversible slippery bonds

Irreversible diffusion limited cluster aggregation (DLCA) of hard spheres was simulated using Brownian cluster dynamics. Bound spheres were allowed to move freely within a specified range, but no bond breaking was allowed. The structure and size distribution of the clusters was investigated before gelation. The pair correlation function and the static structure factor of the gels were determined as a function of the volume fraction and time. Slippery bonds led to local densification of the clusters and the gels, with a certain degree of order. At low volume fractions densification of the clusters occurred during their growth, but at higher volume fractions it occurred mainly after gelation. At very low volume fractions, the large-scale structure (fractal dimension), size distribution and growth kinetics of the clusters was found to be close to that known for DLCA with rigid bonds. Restructuring of the gels continued for long times, indicating that aging processes in systems with strong attraction do not necessarily involve bond breaking. The mean-square displacement of particles in the gels was determined. It is shown to be highly heterogeneous and to increase with decreasing volume fraction.

Non-Gaussian statistics and superdiffusion in a driven-dissipative dusty plasma

Particle random motion can exhibit both anomalous diffusion and non-Gaussian statistics in some physical systems. Anomalous diffusion is quantified by a deviation from alpha=1 in a power law for a particle's mean-square displacement, MSD proportional, variant(Deltat)alpha. A deviation from Gaussian statistics for a probability distribution function (PDF) is quantified by fitting to a kappa function or Tsallis distribution, with a fit parameter q. We report an experiment and simulations to test a theory that connects anomalous diffusion and non-Gaussian statistics. In the experiment, a single-layer dusty plasma, which behaved as a two-dimensional (2D) driven-dissipative system, had a non-Gaussian PDF. By adjusting an externally applied laser heating, q was varied over a wide range. A correlation between the deviations from Gaussian statistics and normal diffusion for a 2D liquid was found in the experiment. This correlation indicates a connection between anomalous diffusion and non-Gaussian statistics. However, such a connection is lacking in equilibrium 2D Yukawa liquids, as demonstrated in numerical simulations.

Diffusing wave spectroscopy in Maxwellian fluids

We present a critical assessment of the diffusing wave spectroscopy (DWS) technique for obtaining the characteristic lengths and for measuring the loss and storage moduli of a reasonable well-known wormlike micelle (WM) system. For this purpose, we tracked the Brownian motion of particles using DWS embedded in a Maxwellian fluid constituted by a wormlike micellar solution made of cetyltrimethylammonium bromide (CTAB), sodium salicylate (NaSal), and water. We found that the motion of particles was governed by the viscosity of the solvent at short times and by the stress relaxation mechanisms of the giant micelles at longer times. From the time evolution of the mean square displacement of particles, we could obtain for the WM solution the cage size where each particle is harmonically bound at short times, the long-time diffusion coefficient, and experimental values for the exponent that accounts for the broad spectrum of relaxation times at the plateau onset time found in the (deltar2(t)) vs. time curves. In addition, from the (deltar2(t)) vs. time curves, we obtained G'(omega) and G"(omega) for the WM solutions. All the DWS microreological information allowed us to estimate the characteristic lengths of the WM network. We compare our DWS microrheological results and characteristic lengths with those obtained with mechanical rheometers at different NaSal/CTAB concentration ratios and temperatures.

Microrheology of the Liquid-Solid Transition during Gelation

The viscoelastic properties of physical and chemical polymer gels are characterized through the liquid-solid transition using particle tracking microrheology. Measurements of the probe particle mean-squared displacement are shifted as the extent of gelation increases to generate master curves. From the shift factors, we determine the gel point and critical scaling exponents. Both systems exhibit a critical relaxation exponent n approximately 0.6, where G' approximately G'' approximately omega n for the incipient gel, consistent with the Rouse model of dynamic scaling in the percolation universality class.

Tracer Diffusion in Colloidal Gels

Computer simulations were done of the mean square displacement (MSD) of tracer particles in colloidal gels formed by diffusion or reaction limited aggregation of hard spheres. The diffusion coefficient was found to be determined by the volume fraction accessible to the spherical tracers (phi a) independent of the gel structure or the tracer size. In all cases, critical slowing down was observed at phi a approximately 0.03 and was characterized by the same scaling laws reported earlier for tracer diffusion in a Lorentz gas. Strong heterogeneity of the MSD was observed at small phi a and was related to the size distribution of pores.

Fluidization of a vertically vibrated two-dimensional hard sphere packing: A granular meltdown

We report measurements of the fluidization process in vertically vibrated two-dimensional granular packings. An initially close packed granular bed is exposed to sinusoidal container oscillations with gradually increasing amplitude. At first the particles close to the free surface become mobile. When a critical value of the forcing strength is reached the remaining crystal suddenly breaks up and the bed fluidizes completely. This transition leads to discontinuous changes in the density distribution and in the root mean square displacement of the individual particles. Likewise the vertical center of mass coordinate increases by leaps and bounds at the transition. It turns out that the maximum container velocity v0 is the crucial driving parameter determining the state of a fully fluidized system. For particles of various sizes the transition to full fluidization occurs at the same value of v 2 0/gd, where d is the particle diameter and g is the gravitational acceleration. A discontinuous fluidization transition is only observed when the particles are highly elastic.

Stochastic simulation of Lagrangian trajectories in near-wall turbulence

A dissipation-conditioned stochastic model for the simulation of marked fluid-particle trajectories is formulated that reproduces the extreme intermittency of Lagrangian accelerations and Lagrangian velocity increments observed in recent direct numerical simulations (DNS) of near-wall turbulence [Lee, Yeo, and Choi, Phys. Rev. Lett. 92, 144502 (2004); Choi, Yeo, and Lee, Phys. Fluids 16, 779 (2004)]. Model agreement with data from the DNS for the mean-square displacements of marked fluid particles is good and significantly better than that which can be obtained with stochastic models formulated in terms of the mean rather than the fluctuating rate of dissipation of turbulent kinetic energy. The model accounts, in part, for the observed extreme intermittency of rotations of the Lagrangian velocity vector.

Einstein, Perrin, and the reality of atoms: 1905 revisited

We have repeated Perrin’s 1908 experiment for the determination of Avogadro’s number by determining the mean square displacement of small particles undergoing Brownian motion. Our apparatus differs from Perrin’s by the use of a CCD camera and is much less tedious to perform. We review Einstein’s 1905 analysis of Brownian motion and Langevin’s alternative derivation of the Einstein equation for the mean square displacement. We also show how Einstein’s thinking was a reflection of his belief in the validity of molecular-kinetic theory, a validity not universally recognized 100 years ago.

Interplay of order-disorder phenomena and diffusion in rigid binary alloys in the presence of vacancies: Monte Carlo simulations

Transport phenomena are studied for a binary $(AB)$ alloy on a rigid square lattice with nearest-neighbor attraction between unlike particles, assuming a small concentration ${c}_{v}$ of vacancies $V$ being present, to which $A$ $(B)$ particles can jump with rates ${\ensuremath{\Gamma}}_{A}$ $({\ensuremath{\Gamma}}_{B})$ in the case where the nearest-neighbor attractive energy ${ϵ}_{AB}$ is negligible in comparison with the thermal energy ${k}_{B}T$ in the system. This model exhibits a continuous order-disorder transition for concentrations ${c}_{A},{c}_{B}=1\ensuremath{-}{c}_{A}\ensuremath{-}{c}_{V}$ in the range ${c}_{A,1}^{\mathit{crit}}\ensuremath{\leqslant}{c}_{A}\ensuremath{\leqslant}{c}_{A,2}^{\mathit{crit}}$, with ${c}_{A,1}^{\mathit{crit}}=(1\ensuremath{-}{m}^{*}\ensuremath{-}{c}_{V})∕2$, ${c}_{A,2}^{\mathit{crit}}=(1+{m}^{*}\ensuremath{-}{c}_{V})∕2$, $m\ensuremath{\ast}\ensuremath{\approx}0.25$, the maximum critical temperature occurring for $c\ensuremath{\ast}={c}_{A}={c}_{B}=(1\ensuremath{-}{c}_{V})∕2$---i.e., ${m}^{*}=0$. This phase transition belongs to the $d=2$ Ising universality class, demonstrated by a finite-size scaling analysis. From a study of mean-square displacements of tagged particles, self-diffusion coefficients are deduced, while applying chemical potential gradients allows the estimation of Onsager coefficients. Analyzing finally the decay with time of sinusoidal concentration variations that were prepared as initial condition, also the interdiffusion coefficient is obtained as function of concentration and temperature. As in the random alloy case (i.e., a noninteracting $ABV$ model) no simple relation between self-diffusion and interdiffusion is found. Unlike this model mean-field theory cannot describe interdiffusion, however, even if the necessary Onsager coefficients are estimated via simulation.