Time Dependence Of Mean Square Displacement Research Articles

An exactly solvable model for non-Fickian transport in dynamically heterogeneous media

Abstract Diffusion, observed in various condensed phases, finds its theoretical background in Einstein’s theory of Brownian motion, characterized by the linear time-dependence of mean square displacement (MSD) denoting Fickian behavior and the Gaussian distribution of particle displacement. Nevertheless, diverse systems exhibit either non-linear, non-Fickian time-dependence of the MSD or non-Gaussian displacement distribution. Montroll and Weiss’s continuous-time random walk (CTRW) model and the stochastic diffusivity (SD) model have provided insights into anomalous diffusion phenomena and Fickian-yet-non-Gaussian transport in dynamically heterogeneous environments, respectively. Building upon these approaches, Song et al developed a generalized transport equation with an environment-dependent diffusion kernel, providing a quantitative explanation for non-Fickian MSD and non-Gaussian displacement distribution. Based on the generalized transport equation, this study introduces an exactly solvable model for a non-Gaussian displacement distribution, accommodating arbitrary time profiles in its MSD, including both Fickian and non-Fickian behaviors. Our findings confirm the model’s capability in describing such transport processes. Furthermore, the proposed model unifies the CTRW model under fast environmental fluctuations and the SD model under Fickian time dependencies, making it suitable for understanding tracer particle motion within explicit solvent or complex media.

Position-Dependent Segmental Relaxation in Bottlebrush Polymers.

Segmental dynamics of specifically labeled poly(propylene oxide), PPO, based bottlebrush polymers, PNB-g-PPO, were studied using quasi-elastic neutron scattering. The focus was set to different parts of the side chains to investigate the hypothetical gradual relaxation behavior within the side chains of a bottlebrush polymer. Different sections of the side chains were highlighted for QENS via sequential polymerization of protonated and deuterated monomers to allow the study of the relaxation behavior of the inner and outer parts of the side chain separately. A comparison of these two parts reveals a slowdown due to the grafting process happening across the different regions. This is seen for the segmental relaxation time as well as on the time-dependent mean-square displacement. Additionally, the non-Gaussian parameter, α, shows a decreasing difference from Gaussian behavior with the distance to the backbone. Altogether, this leads to the conclusion that gradual relaxation behavior exists.

Diffusing Wave Microrheology in Polymeric Fluids.

Recently, there has been interest in determining the viscoelastic properties of polymeric liquids and other complex fluids by means of Diffusing Wave Spectroscopy (DWS). In this technique, light-scattering spectroscopy is applied to highly turbid fluids containing optical probe particles. The DWS spectrum is used to infer the time-dependent mean-square displacement and time-dependent diffusion coefficient D of the probes. From D, values for the storage modulus G'(ω) and the loss modulus G''(ω) are obtained. This paper is primarily concerned with the inference of the mean-square displacement from a DWS spectrum. However, in much of the literature, central to the inference that is said to yield D is an invocation g(1)(t)=exp(-2q2X(t)2¯) of the Gaussian Approximation for the field correlation function g(1)(t) of the scattered light in terms of the mean-square displacement X(t)2¯ of a probe particle during time t. Experiment and simulation both show that the Gaussian approximation is invalid for probes in polymeric liquids and other complex fluids. In this paper, we obtain corrections to the Gaussian approximation that will assist in interpreting DWS spectra of probes in polymeric liquids. The corrections reveal that these DWS spectra receive contributions from higher moments X(t)2n¯, n>1, of the probe displacement distribution function.

Configurational temperature in active matter. I. Lines of invariant physics in the phase diagram of the Ornstein-Uhlenbeck model.

This paper shows that the configurational temperature of liquid-state theory, T_{conf}, defines an energy scale, which can be used for adjusting model parameters of active Ornstein-Uhlenbeck particle (AOUP) models in order to achieve approximately invariant structure and dynamics upon a density change. The required parameter changes are calculated from the variation of a single configuration's T_{conf} for a uniform scaling of all particle coordinates. The resulting equationsare justified theoretically for models involving a potential-energy function with hidden scale invariance. The validity of the procedure is illustrated by computer simulations of the Kob-Andersen binary Lennard-Jones AOUP model, showing the existence of lines of approximate invariance of the reduced-unit radial distribution function and time-dependent mean-square displacement.

Compositional effects in the liquid Fe–Ni–C system at high pressure

We performed molecular dynamics simulations based on density functional theory to systematically investigate the Fe–Ni–C system including (1) pure Fe and Ni; (2) binary Fe–Ni, Fe–C, and Ni–C; and (3) ternary Fe–Ni–C liquid compositions at 3000 K and three simulation volumes corresponding to pressure (P) up to 83 GPa. Liquid structural properties, including coordination numbers, are analyzed using partial radial distribution functions. Self-diffusion coefficients are determined based on the atomic trajectories and the asymptotic slope of the time-dependent mean-square displacement. The results indicate that the average interatomic distance between two Fe atoms (rFe–Fe) decreases with P and is sensitive to Ni (XNi) and C (XC) concentration, although the effects are opposite: rFe–Fe decreases with increasing XNi, but increases with increasing XC. Average rFe–C and rNi–C values also decrease with increasing XNi and generally remain constant between the two lowest P points, corresponding to a coordination change of carbon from ~ 6.8 to ~ 8.0, and then decrease with additional P once the coordination change is complete. Carbon clustering occurs in both binary (especially Ni–C) and ternary compositions with short-range rC-C values (~ 1.29 to ~ 1.57 Å), typical for rC-C in diamond and graphite. The self-diffusion results are generally consistent with high-P diffusion data extrapolated from experiments conducted at lower temperature (T). A subset of additional simulations was conducted at 1675 and 2350 K to estimate the effect of T on diffusion, yielding an activation enthalpy of ~ 53 kJ/mol and activation volume of ~ 0.5 cm3/mol.

Simulation of Wetting and Interfacial Behavior of Quaternary Ammonium and Phosphonium Ionic Liquid Nanodroplets Over Face-Centered Cubic Metal Surfaces.

We studied the behavior of quaternary ammonium- and phosphonium-based ([N2225][NTf2] and [P2225][NTf2]) ionic liquid (IL) nanodroplets on copper (Cu) and platinum (Pt) metal surfaces using classical molecular dynamics simulations. Solid-liquid interactions were underpinned by studying the dynamic spreading, contact angle, time-dependent binding energies, mean-square displacements, number and charge density profiles, and orientational distribution of these two IL nanodroplets. In particular, the role and importance of different crystallographic facets of face-centered cubic metal surface [Cu(100), Cu(111), Pt(100), and Pt(111)] were investigated. The results indicate a vast variety of properties that are dictated highly by the structure of different crystallographic facets of the metal substrate. The nature of the facet (e.g., the atomic arrangement symmetry, the extent of closed packed, and the unit cell size) leads to an extended scale spreading of the ILs on the surface with the (111) index but not with the (100) index, while the nature of metals itself plays a role.

Wavepacket dynamics in one-dimensional system with long-range correlated disorder

We numerically investigate dynamical property in the one-dimensional tight-binding model with long-range correlated disorder having power spectrum 1/fα (α: spectrum exponent) generated by Fourier filtering method. For relatively small α<αc(=2) time-dependence of mean square displacement (MSD) of the initially localized wavepacket shows ballistic spread and localizes as time elapses. It is shown that α-dependence of the dynamical localization length determined by the MSD exhibits a simple scaling law in the localization regime for the relatively weak disorder strength W. Furthermore, scaled MSD by the dynamical localization length almost obeys an universal function from the ballistic to the localization regime in the various combinations of the parameters α and W.

Molecular dynamics simulation of carbon dioxide in single-walled carbon nanotubes in the presence of water: structure and diffusion studies

We present a molecular dynamics study on the structure and diffusion of carbon dioxide in single-walled carbon nanotubes (SWCNTs) in the presence of water. We consider (10,10) and (15,15) SWCNTs of nanotube diameter 13.6 and 20.3 Å, respectively, with amounts of pre-adsorbed water equal to 0, 0.025, 0.1, and 0.2 g/cm3. The density of the carbon dioxide in the SWCNTs corresponds to the maximum amount adsorbed at 300 K and 37.6 bar, as previously determined by grand canonical Monte Carlo simulation. We determine the structure of the confined fluids by calculating density distributions in the nanotube axial and radial directions, along with a second-order Legendre polynomial to elucidate their molecular orientations. The diffusion of the confined molecules is then analysed via both the overall time-dependent and space-dependent mean-square displacements in the nanotube axial direction. We find that the systems display distinct axial regions comprised of either pure carbon dioxide or water clusters that have been penetrated by CO2 molecules. The confined molecules form layered structures that strongly depend on the nanotube diameter. However for particular SWCNTs, the shape of the layered structures is not affected by the presence of pre-adsorbed water. The amount of pre-adsorbed water strongly influences the overall diffusion, while the space variation of fluid densities across the nanotubes has a relatively small effect on the space-dependent diffusion.

Atomic Structure and Diffusion Properties of Liquid Cu&lt;sub&gt;4&lt;/sub&gt;&lt;sub&gt;5&lt;/sub&gt;Zr&lt;sub&gt;4&lt;/sub&gt;&lt;sub&gt;8&lt;/sub&gt;Al&lt;sub&gt;7&lt;/sub&gt; Metallic Glasses

The microscopic structure and diffusion of liquid Cu45Zr48Al7 alloys were investigated by means of molecular dynamics simulations. We obtain the structure and diffusion properties including partial pair-correlation function, and Voronoi indices and the time-dependent mean-square displacement (MSD). We found that the icosahedra and the distorted icosahedra around Cu, Zr and Al are differentthe and the atom moves quickly Zr, Cu and Al in the order.

Rheology and internal dynamics of colloidal gels from the dilute to the concentrated regime

We explore the relations between dynamical, rheological and structural properties of colloidal gels extending the range previously considered towards higher particle volume fractions . Up to the spectrum of the time-dependent mean-square displacement is shown to be fully described by models accounting for the fractal character of gels. Beyond that, the direct relations between structural and stress-bearing properties used in fractal descriptions do not hold anymore. However, by considering the concepts of passive tracer rheology we show that the macroscopic shear modulus G0 can be derived from the dynamical properties up to even higher volume fractions.

Investigating the micro-rheology of the vitreous humor using an optically trapped local probe

We demonstrate that an optically trapped silica bead can be used as a local probe to measure the micro-rheology of the vitreous humor. The Brownian motion of the bead was observed using a fast camera and the micro-rheology determined by analysis of the time-dependent mean-square displacement of the bead. We observed regions of the vitreous that showed different degrees of viscoelasticity, along with the homogeneous and inhomogeneous nature of different regions. The motivation behind this study is to understand the vitreous structure, in particular changes due to aging, allowing more confident prediction of pharmaceutical drug behavior and delivery within the vitreous humor.

Microrheology with optical tweezers: data analysis

We present a data analysis procedure that provides the solution to a long-standing issue in microrheology studies, i.e. the evaluation of the fluids' linear viscoelastic properties from the analysis of a finite set of experimental data, describing (for instance) the time-dependent mean-square displacement of suspended probe particles experiencing Brownian fluctuations. We report, for the first time in the literature, the linear viscoelastic response of an optically trapped bead suspended in a Newtonian fluid, over the entire range of experimentally accessible frequencies. The general validity of the proposed method makes it transferable to the majority of microrheology and rheology techniques.

Novel algorithm and MATLAB-based program for automated power law analysis of single particle, time-dependent mean-square displacement

In many physical and biophysical studies, single-particle tracking is utilized to reveal interactions, diffusion coefficients, active modes of driving motion, dynamic local structure, micromechanics, and microrheology. The basic analysis applied to those data is to determine the time-dependent mean-square displacement (MSD) of particle trajectories and perform time- and ensemble-averaging of similar motions. The motion of particles typically exhibits time-dependent power-law scaling, and only trajectories with qualitatively and quantitatively comparable MSD should be ensembled. Ensemble averaging trajectories that arise from different mechanisms, e.g., actively driven and diffusive, is incorrect and can result inaccurate correlations between structure, mechanics, and activity. We have developed an algorithm to automatically and accurately determine power-law scaling of experimentally measured single-particle MSD. Trajectories can then categorized and grouped according to user defined cutoffs of time, amplitudes, scaling exponent values, or combinations. Power-law fits are then provided for each trajectory alongside categorized groups of trajectories, histograms of power laws, and the ensemble-averaged MSD of each group. The codes are designed to be easily incorporated into existing user codes. We expect that this algorithm and program will be invaluable to anyone performing single-particle tracking, be it in physical or biophysical systems. Program summaryProgram title: LINSA (acronym: linear segment analysis)Catalogue identifier: AEMD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEMD_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 25 892No. of bytes in distributed program, including test data, etc.: 5 572 780Distribution format: tar.gzProgramming language: MATLAB (MathWorks Inc.) version 7.11 (2010b) or higher, program should also be backwards compatible. Symbolic Math Toolboxes (5.5) is required. The Curve Fitting Toolbox (3.0) is recommended.Computer: Tested on Windows only, yet should work on any computer running MATLAB. In Windows 7, should be used as administrator, if the user is not the administrator the program may not be able to save outputs and temporary outputs to all locations.Operating system: Any supporting MATLAB (MathWorks Inc.) v7.11 / 2010b or higher.Supplementary material: Sample output files (approx. 30 MBytes) are available.Classification: 12External routines: Several MATLAB subfunctions (m-files), freely available on the web, were used as part of and included in, this code: count, NaN suite, parseArgs, roundsd, subaxis, wcov, wmean, and the executable pdfTK.exe.Nature of problem: In many physical and biophysical areas employing single-particle tracking, having the time-dependent power-laws governing the time-averaged meansquare displacement (MSD) of a single particle is crucial. Those power laws determine the mode-of-motion and hint at the underlying mechanisms driving motion. Accurate determination of the power laws that describe each trajectory will allow categorization into groups for further analysis of single trajectories or ensemble analysis, e.g. ensemble and time-averaged MSD.Solution method: The algorithm in the provided program automatically analyzes and fits time-dependent power laws to single particle trajectories, then group particles according to user defined cutoffs. It accepts time-dependent trajectories of several particles, each trajectory is run through the program, its time-averaged MSD is calculated, and power laws are determined in regions where the MSD is linear on a log–log scale. Our algorithm searches for high-curvature points in experimental data, here time-dependent MSD. Those serve as anchor points for determining the ranges of the power-law fits. Power-law scaling is then accurately determined and error estimations of the parameters and quality of fit are provided. After all single trajectory time-averaged MSDs are fit, we obtain cutoffs from the user to categorize and segment the power laws into groups; cutoff are either in exponents of the power laws, time of appearance of the fits, or both together. The trajectories are sorted according to the cutoffs and the time- and ensemble-averaged MSD of each group is provided, with histograms of the distributions of the exponents in each group. The program then allows the user to generate new trajectory files with trajectories segmented according to the determined groups, for any further required analysis.Additional comments: README file giving the names and a brief description of all the files that make-up the package and clear instructions on the installation and execution of the program is included in the distribution package.Running time: On an i5 Windows 7 machine with 4 GB RAM the automated parts of the run (excluding data loading and user input) take less than 45 minutes to analyze and save all stages for an 844 trajectory file, including optional PDF save. Trajectory length did not affect run time (tested up to 3600 frames/trajectory), which was on average 3.2±0.4 seconds per trajectory.

Direct Evidence of a Characteristic Length Scale of a Dynamical Nature in the Boolchand Phase of Glasses

The ac conductivity spectra of xAgI-(1 - x)AgPO(3) fast-ion conducting glasses spanning the flexible, intermediate (isostatically rigid), and stressed rigid phases are analyzed. The rescaled frequency-dependent spectra are mapped into time-dependent mean-square displacements out of which a typical length scale characterizing the spatial extent sqrt[(R(2)(∞))] of nonrandom subdiffusive regions of ionic motions is computed. The latter quantity is studied as a function of AgI compositions, and is found to display a maximum isostatic compositions, providing the first clear evidence of a typical length scale of a dynamical nature when a system becomes isostatically rigid and enters that phase.

Molecular dynamics simulation study of self-diffusion for penetrable-sphere model fluids

Molecular dynamics simulations are carried out to investigate the diffusion behavior of penetrable-sphere model fluids characterized by a finite energy barrier ϵ. The self-diffusion coefficient is evaluated from the time-dependent velocity autocorrelation function and mean-square displacement. Detailed insights into the cluster formation for penetrable spheres are gained from the Enskog factor, the effective particle volume fraction, the mean free path, and the collision frequency for both the soft-type penetrable and the hard-type reflective collisions. The simulation data are compared to theoretical predictions from the Boltzmann kinetic equation and from a simple extension to finite ϵ of the Enskog prediction for impenetrable hard spheres (ϵ→∞). A reasonable agreement between theoretical and simulation results is found in the cases of ϵ∗ ≡ ϵ/k(B)T=0.2, 0.5, and 1.0. However, for dense systems (packing fraction ϕ>0.6) with a highly repulsive energy barrier (ϵ∗ = 3.0), a poorer agreement was observed due to metastable static effects of clustering formation and dynamic effects of correlated collision processes among these cluster-forming particles.

Micro-heterogeneity of cellulosic fiber biopolymer prepared from corn hulls

Z-trim is a zero calorie cellulosic fiber biopolymer produced from corn hulls. The micro-structural heterogeneities of Z-trim biopolymer were investigated by monitoring the thermally driven displacements of well-dispersed micro-spheres via video fluorescence microscopy named multiple-particle tracking (MPT). By comparing the distribution of the time-dependent mean-square displacement (MSD) of polystyrene micro-spheres embedded in three concentrations of Z-trim, we found that the degree of heterogeneity of the suspensions increased dramatically within a narrow range of concentrations. The ensemble-averaged MSD of 0.5 g/100 g Z-trim biopolymer exhibited a power-law behavior that scaled log-linearly with time lag. This behavior was similar to that of homogeneous aqueous glycerol solution. But the MSD distribution for Z-trim was wider and more asymmetric than for glycerol. Increasing Z-trim concentration caused more deviation of micro-spheres colloidal dispersion from homogeneity as detected by the MPT.

Micro-heterogeneity and micro-rheological properties of high-viscosity oat β-glucan solutions

Soluble fibre β-glucan is one of the key dietary materials in the healthy food products known for reducing serum cholesterol levels. The micro-structural heterogeneity and micro-rheology of high-viscosity oat β-glucan solutions were investigated by monitoring the thermally driven displacements of well-dispersed microspheres via video fluorescence microscopy. By comparing the distribution of the time-dependent mean-square displacement (MSD) and ensemble-averaged MSD of polystyrene microspheres imbedded in four concentrations of β-glucan solutions, we found that the solutions exhibited perfectly homogeneous behaviour at ⩽1%, but showed a certain degree of heterogeneity at 2%. Micro-rheology investigation revealed that β-glucan solutions displayed nearly perfect viscous behaviour at ⩽1%, but the property changed into viscoelastic at 2%. Both micro-structural heterogeneity and micro-rheological property shifts occurred over a small concentration range, between 1% and 2%, of β-glucan.

Fractional Diffusion Equation Approach to the Anomalous Diffusion on Fractal Lattices

A generalized fractional diffusion equation (FDE) is presented, which describes the time-evolution of the spatial distribution of a particle performing continuous time random walk (CTRW) on a fractal lattice. For a case corresponding to the CTRW with waiting time distribution that behaves as <TEX>$\psi(t) \sim (t) ^{-(\alpha+1)}$</TEX>, the FDE is solved to give analytic expressions for the Green’s function and the mean squared displacement (MSD). In agreement with the previous work of Blumen et al. [Phys. Rev. Lett. 1984, 53, 1301], the time-dependence of MSD is found to be given as < <TEX>$r^2(t)$</TEX> > ~ <TEX>$t ^{2\alpha/dw}$</TEX>, where <TEX>$d_w$</TEX> is the walk dimension of the given fractal. A Monte-Carlo simulation is also performed to evaluate the range of applicability of the proposed FDE.

Multiple-Particle Tracking Measurements of Heterogeneities in Solutions of Actin Filaments and Actin Bundles

One of the central functions of actin cytoskeleton is to provide the mechanical support required for the establishment and maintenance of cell morphology. The mechanical properties of actin filament assemblies are a consequence of both the available polymer concentration and the actin regulatory proteins that direct the formation of higher order structures. By monitoring the displacement of well-dispersed microspheres via fluorescence microscopy, we probe the degree of spatial heterogeneity of F-actin gels and networks in vitro. We compare the distribution of the time-dependent mean-square displacement (MSD) of polystyrene microspheres imbedded in low- and high-concentration F-actin solutions, in the presence and absence of the F-actin-bundling protein fascin. The MSD distribution of a 2.6- μM F-actin solution is symmetric and its standard deviation is similar to that of a homogeneous solution of glycerol of similar zero-shear viscosity. However, increasing actin concentration renders the MSD distribution wide and asymmetric, an effect enhanced by fascin. Quantitative changes in the shape of the MSD distribution correlate qualitatively with the presence of large heterogeneities in F-actin solutions produced by increased filament concentration and the presence of actin bundles, as detected by confocal microscopy. Multiple-particle tracking offers a new, quantitative method to characterize the organization of biopolymers in solution.

Particle dynamics in the random barrier model: Monte Carlo simulations at low temperatures

The low-temperature particle dynamics in the framework of the random barrier model are studied with the help of a self-developed Monte Carlo algorithm. The time-dependent mean-square displacement of the particles is, thereby, derived without using any physical approximations. Two main results are obtained. (i) In the time regime of anomalous diffusion, the mean-square displacement cannot be scaled with the diffusion coefficient of the particles. (ii) The apparent activation energy for the long-range particle transport increases slightly with temperature. These results are discussed in terms of percolation theory.