Mean Square Displacement Of Particle Research Articles

In Russian

A method is proposed of determining the subdiffuson equation parameters. The method is based on a single particle tracking experiment data analysis. The combination of random barriers model and random traps model is considered as a model of subdiffusion. Subdiffusion equation related to this model contains two memory functions as a parameters. The first function characterizes the distribution of the heights of barriers, the other one characterizes the distribution of the depths of traps. Within the framework of the model under consideration, expressions are derived for the mean square displacement of particles averaged over time and over the ensemble. Expressions are derived for random walks in unlimited space and in a restricted domain of arbitrary shape with the reflectiing boundary. It has been shown that the derived expressions allow us to determine the memory functions appearing in subdiffusion equation. For this purpose, single particle tracking experiment data should be approximated by using these expressions. The method proposed can be used for modeling the diffusion controlled reactions in disordered media.

Diffusion in a Crowded, Rearranging Environment.

It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))(2)⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))(2)⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.

Lower Bound on the Mean Square Displacement of Particles in the Hard Disk Model

The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions $2n \times 2n$ with arbitrary boundary configuration,and we show that the mean square displacement of particles near the center of the box is bounded from below by $c \log n$. The result generalizes to a large class of models with fairly arbitrary interaction.

Nanoparticle diffusion in crowded and confined media.

We identify distinct mechanisms controlling slowing of nanoparticle diffusion through complex media featuring both rigid geometrical confinement and soft mobile crowders. Towards this end, we use confocal microscopy and single particle tracking to probe the diffusion of 400 nm nanoparticles suspended in Newtonian water, in a Newtonian glycerol/water mixture, or in a non-Newtonian polymer solution through a model porous medium, a packed bed of microscale glass beads. The mobility of nanoparticles, as quantified by the long-time diffusion coefficient extracted from the particle mean-squared displacement, slows as the average pore size of the packed bed media decreases for both Newtonian and non-Newtonian solutions. The distribution of particle displacements is non-Gaussian, consistent with the spatial heterogeneity of the geometrical confinement imposed by the packed bed. The slowing of nanoparticle mobility in all solutions follows the predictions of models that describe hydrodynamic interactions with the packed bed. In non-Newtonian solutions, depletion interactions due to the polymers near the glass beads result in temporary adsorption of particles onto the bead surface, as indicated by a stretched-exponential distribution of residence times. Our results therefore suggest that the confined diffusive dynamics of nanoparticles in polymer solutions is controlled by two competing mechanisms: hydrodynamic interactions between particles and spatial obstacles, which dictate the long-time slowing of diffusion, and depletion interactions between particles and confining walls due to the macromolecules, which control transient adsorption and hence alter the statistics of the short-time motion.

In-situ measurement of viscoelastic properties of fresh cement paste by a microrheology analyzer

A microrheology analyzer was adapted to in-situ follow the development of viscoelastic properties of fresh cement pastes (FCPs) for the first time. It enables a non-disturbing measurement on the FCPs through monitoring the mean square displacement of cement particles, which gives an insight into the elastic and viscous properties of materials from a microstructural point of view. Various parameters including elastic index, macroscopic viscosity index, storage modulus, loss modulus and Maxwell parameters were obtained to quantitatively analyze the viscoelastic properties of FCPs. Results indicate that these parameters show a progressive increase with time at first and then stay stable. The incorporation of superplasticizer significantly decreases these parameters and their growth rates. Moreover, superplasticizer could evidently weaken the elastic feature of the FCP due to its effects of improving the dispersion of cement grains and retarding cement hydration. The effects of superplasticizer are more pronounced at lower water to cement ratio.

Two dimensional colloidal crystals formed by particle self-assembly due to hydrodynamic interaction

Aggregation and crystallization of colloidal parti- cles deposited from suspensions on glass surfaces were stud- ied. Trajectories of individual particles are tracked and record- ed. Statistics over large amount of particles were made to measure the mean square displacement (MSD), , as func- tions of time, t. Isolated particles diffuse normally along the solid surface in a two dimensional random manner. A power law of ~t α is obeyed both by short and long time scale MSDs of particles with neighbors. However, the exponent values are quite different. When the particle area fraction f≤ 80%,theparticlesdiffusenormallyattheshorttimescalewith a retarded diffusion coefficient. While at the long time scale, a superdiffusive behavior of the particles is detected due to col- lective motion of particles. When f>80 %, a spatial confine- ment effect shows in addition. The retarded particle dynamics and the collective particle movements both originate from the many-body hydrodynamic interaction enhanced by the quasi- two dimensional geometric conditions due to the existence of thesubstrateandtheneighborparticles.Ifthesubstratesurface condition is favorable and the hydrodynamic interaction dom- inates, the long-range hydrodynamic interaction can lead par- ticles in dense particle aggregations to self-assemble into long particle chains. This chaining behavior finally results in a phase transition taking place gradually over a large range of the particle area fraction.

Modifying the neutron diffusion equation using spatial fractional operators and developed diffusion coefficients

Normal diffusion process is quite valid for media without production and absorption. In these media particles move by scattering reaction and mean square displacement of particles is linear function of time (〈r(t)2〉∼t). In the nuclear reactors both production and absorption are present and consequently normal diffusion fails to describe neutrons behavior. In this circumstance sub or super diffusion (also called anomalous diffusion) phenomena are occurred. We have developed a new scheme of diffusion equation which is established on the spatial–fractional derivatives, neutron fractional diffusion equation (NFDE). The NFDE uses an arbitrary order of differential operators and it can model anomalous diffusion in the more accurate manner. The second drawback of diffusion equation is related to high absorption (or production) regions where a steep spatial gradient is existent. Based on the Fick's law, if gradient of flux is very large the neutron current will exceed the scalar flux (|J→|>ϕ). In order to dump this effect, the flux limited diffusion theory is used to calculate diffusion coefficient. The proposed model is validated against experimental measurements of DIMPLE reactor.

Effects of surface topography on the collective diffusion of interacting adsorbed particles

Collective diffusion of particles with repulsive nearest-neighbor interactions on bivariate surfaces is studied through Monte Carlo simulation, in the framework of the Kubo–Green theory. Shallow and deep adsorbing sites form l×l patches distributed at random or in chessboard-like ordered domains on a two-dimensional square lattice. The influence of the energetic correlation and the lateral interactions on the jump and collective diffusion coefficients are analyzed by simulating the coverage fluctuations in the grand canonical ensemble and the mean-square displacements of particles in the canonical ensemble. The combination of topography and lateral coupling is shown to produce interesting effects such as different filling regimes as well as strong effects on the coverage dependence of the transport coefficients.

Particle-resolved dynamics during multilayer growth ofC60

Using large-scale kinetic Monte Carlo (KMC) simulations, we investigate the nonequilibrium surface growth of the fullerene ${\text{C}}_{60}$. Recently we have presented a self-consistent set of energy barriers that describes the nucleation and multilayer growth of ${\text{C}}_{60}$ for different temperatures and adsorption rates in quantitative agreement with experiments [Bommel et al., Nat. Commun. 5, 5388 (2014)]. We found that ${\text{C}}_{60}$ displays lateral diffusion resembling colloidal systems, however it has to overcome an atomlike energetic step-edge barrier for interlayer diffusion. Here we focus on the particle-resolved dynamics, and the interplay between surface morphology and particle dynamics during growth. Comparing ${\text{C}}_{60}$ growth with an atomlike system, we find significant differences in the evolution of the surface morphology, as well as the single-particle dynamics on the growing material landscape. By correlating the mean-squared displacement of particles with their current neighborhood, we can identify the influence of the different time scales that compete during growth and can pinpoint the differences between the two systems.

Mesoscopic model of temporal and spatial heterogeneity in aging colloids

We develop a simple and effective description of the dynamics of dense hard sphere colloids in the aging regime deep in the glassy phase. Our description complements the many efforts to understand the onset of jamming in low density colloids, whose dynamics is still time-homogeneous. Based on a small set of principles, our model provides emergent dynamic heterogeneity, reproduces the known results for dense hard sphere colloids and makes detailed, experimentally-testable predictions for canonical observables in glassy dynamics. In particular, we reproduce the shape of the intermediate scattering function and particle mean-square displacements for jammed colloidal systems, and we predict a growth for the peak of the χ4 mobility correlation function that is logarithmic in waiting-time. At the same time, our model suggests a novel unified description for the irreversible aging dynamics of structural and quenched glasses based on the dynamical properties of growing clusters of highly correlated degrees of freedom.

Mobility of Nanoparticles in Semidilute Polyelectrolyte Solutions

We measure the mobility of nanoparticles at low concentrations in non-Newtonian semidilute aqueous solutions of high-molecular-weight polyelectrolyte polymers. Using optical microscopy and particle tracking algorithms, we image and track hydrophilic polystyrene nanoparticles of diameter 400 nm moving in aqueous solutions of partially hydrolyzed polyacrylamide of molecular weight 8 000 000 Da and concentration of 0.042—4.2 g/L. The effective diffusivity of the nanoparticles in the semidilute polymer solutions, extracted from the long-time limit of the mean-squared displacement using the Stokes–Einstein relation, is greater than that calculated from the zero-shear-rate viscosity measured using bulk rheology. For concentrations c > 0.42 g/L, the mean-square displacements (MSD) of particles measured as a function of lag time revealed that the particle dynamics are subdiffusive at short time scales and are Fickian on long time scales. The time scale for the crossover from subdiffusive to Fickian dynamics incre...

A microrheological study of hydrogel kinetics and micro-heterogeneity.

The real-time dynamic heterogeneity of the gelation process of the amino acid derivative Fmoc-tyrosine (Fmoc-Y) is studied using particle tracking microrheology. To trigger gelation, glucono-δ-lactone (GdL) is added, which gradually lowers the p H over several hours. The onset of self-assembly in the system is signified by a sharp drop in the mean-squared displacement of embedded particles, a phenomenon that is found to correlate with the p H of the system reaching the pK(a) of Fmoc-Y. The gel point is identified and found to be dependent on the GdL concentration. Analysis of embedded probe particle dynamics allows the heterogeneity of the sample to be quantified, using three metrics: the heterogeneity ratio (HR), the non-Gaussian parameter of the van Hove correlation function (N and the bin distribution of the mean-squared displacement (MSD) of single particles (f(z)). Results from the three techniques are found to be approximately comparable, with increases in heterogeneity observed in all samples for incubation times t(w) = 0-3 hours. The final heterogeneity in all samples is found to be remarkably low compared to other systems previously reported in the literature.

Mean-Square Displacement of Particles in Slightly Interconnected Polymer Networks

Structural and viscoelastic properties of slightly interconnected polymer networks immersed in a solvent have been studied in two cases: when the polymer network is building up and when the polymer network is shrinking stepwise in a controlled way. To accomplish this goal, the mean square displacement (MSD) of embedded microspheres in the polymer network was measured as a function of time, with diffusive wave spectroscopy. Particle motion was analyzed in terms of a model, based on a Fokker-Planck type equation, developed for describing particles in Brownian motion within a network that constrain their movement. The model reproduces well the experimental features observed in the MSD vs t curves. The variation of the parameters describing the structure of the network can be understood as the polymerization comes about, and also after the successive volume contractions. In addition, from the MSD curves, the complex shear moduli were obtained in a wide range of frequencies when the network is building up, and at the different shrinking states of the network. Our microrheological results give an insight about the dynamics of embedded particles in slightly interconnected networks, which were also compared with similar results for polymers without interconnections and polymer gels.

Mesoscopic Model of Aging in Colloids

A model of dense hard sphere colloids building on simple notions of particle mobility and spatial coherence is presented and shown to reproduce results of experiments and simulations for key quantities such as the intermediate scattering function, the particle mean-square displacement and the χ 4 mobility correlation function. All results are explained by two emerging and interrelated dynamical properties: i) a rate of intermittent events, quakes , which decreases as the inverse of the system age t , leading to μ q ( t w , t ) α log( t / t w) as the average number of quakes occurring between the ‘waiting time’ t w and the current time t ; ii) a length scale characterizing correlated domains, which increases linearly in log t .

Enhanced diffusion of tracer particles in dilute bacterial suspensions

Swimming bacteria create long-range velocity fields that stir a large volume of fluid and move around passive particles dispersed in the fluid. Recent experiments and simulations have shown that long-time mean-squared displacement of passive particles in a bath of swimming bacteria exhibits diffusive behaviour with an effective diffusion coefficient significantly larger than its thermal counterpart. A comprehensive theoretical prediction of this effective diffusion coefficient and the understanding of the enhancement mechanism remain a challenge. Here, we adapt the kinetic theory by Lin et al., J. Fluid Mech., 2011, 669, 167 developed for 'squirmers' to the bacterial case to quantitatively predict enhanced diffusivity of tracer particles in dilute two- and three-dimensional suspensions of swimming bacteria. We demonstrate that the effective diffusion coefficient is a product of the bacterial number density, their swimming speed, a geometric factor characterising the velocity field created by a single bacterium, and a numerical factor. We show that the numerical factor is, in fact, a rather strong function of the system parameters, most notably the run length of the bacteria, and that these dependencies have to be taken into account to quantitatively predict the enhanced diffusivity. We perform molecular-dynamics-type simulations to confirm the conclusions of the kinetic theory. Our results are in good agreement with the values of enhanced diffusivity measured in recent two- and three-dimensional experiments.

Effects of boundaries on structure formation in low-dimensional colloid model systems near the liquid-solid-transition in equilibrium and in external fields and under shear

A brief review focusing on low-dimensional colloidal model systems is given describing both simulation studies and complementary experiments, elucidating the interplay between phase behavior, geometric structures, and transport phenomena. These studies address the response of these very soft colloidal systems to perturbations such as uniform or uniaxial compression, laser fields, randomly quenched disorder, and shear deformation caused by moving boundaries. Binary hard-disk mixtures are studied by Monte Carlo simulation, to investigate ordering on surfaces or in monolayers, modeling the effect of a substrate by an external potential. By weak external laser fields the miscibility of the mixture can be controlled, and the underlying mechanism (laser-induced demixing) is clarified. The stability of various space-filling structures is discussed only for the case where no laser fields are present.Hard spheres interacting with repulsive screened Coulomb or dipolar interaction confined in 2D and 3D narrow constrictions are investigated by Brownian Dynamics simulation. With respect to the structural behavior, it is found that layers or planes throughout the microchannel are formed. The arrangement of the particles is disturbed by diffusion, and can also be modified by an external driving force causing a density gradient along the channel. Then the number of layers or planes gets reduced, adjusting to the density gradient, and this self-organized change of order also shows up in the particle velocities. The experimental work that is reviewed here addresses dipolar colloidal particles confined by gravity on a solid substrate on which a set of pinning sites has been randomly distributed. The dynamics of the system is studied by tracking the trajectories of individual particles, and it is found that the mean square displacements of particles that are nearest neighbors of pinned particles are strongly affected by these defects. The influence of the pinning sites on the order and microscopic mechanism of phase transitions in two dimensions is investigated.

Diffusion, Subdiffusion, and Trapping of Active Particles in Heterogeneous Media

We study the transport properties of a system of active particles moving at constant speed in a heterogeneous two-dimensional space. The spatial heterogeneity is modeled by a random distribution of obstacles, which the active particles avoid. Obstacle avoidance is characterized by the particle turning speed γ. We show, through simulations and analytical calculations, that the mean square displacement of particles exhibits two regimes as function of the density of obstacles ρ(o) and γ. We find that at low values of γ, particle motion is diffusive and characterized by a diffusion coefficient that displays a minimum at an intermediate obstacle density ρ(o). We observe that in high obstacle density regions and for large γ values, spontaneous trapping of active particles occurs. We show that such trapping leads to genuine subdiffusive motion of the active particles. We indicate how these findings can be used to fabricate a filter of active particles.

Spatial heterogeneous distribution of SiOx → SiOx±1 reactions in silica liquid

We have numerically studied the diffusion mechanism in silica liquid following an approach where the diffusion rate is evaluated via the SiO(x) → SiO(x±1) reaction rate υ(react) and the mean square displacement of particles d(react) as a reaction happens. Five models at pressure up to 25 GPa and at a temperature of 3000 K have been constructed by molecular dynamic simulation. When applying pressure to the liquid, υ(react) increases monotonously because the Si-O bond becomes weaker with pressure. Meanwhile d(react) attains a maximum near the point of 10 GPa despite the particles move in a significantly smaller volume. Furthermore, the SiO(x) → SiO(x±1) reactions are spatially heterogeneously distributed in the liquid. Upon low pressure, most reactions happen with a small number of Si particles. This reaction localization causes the diffusion anomaly and dynamics heterogeneity in the liquid. With increasing pressure the diffusion mechanism changes from the heterogeneous spatial distribution of reactions to homogeneous one. The simulation also reveals two distinguished regions with quite different coordination environments where the reaction rate significantly differs from each other. These sets of Si particles migrate in space over time and form regions with so-called "fast" and "slow" Si particles. The result obtained here indicates the coexistence of low- and high-density regions, and supports the concept of polymorphism in silica liquid.

Crossover behavior of stock returns and mean square displacements of particles governed by the Langevin equation

It is found that the mean square log-returns calculated from the high-frequency one-day moving average of US and Taiwan stocks with the time internal τ show ballistic behavior with the exponent for small τ and show diffusion-like behavior with the exponent for large τ. Such a crossover behavior can be well described by the mean square displacements of particles governed by the Langevin equation of motion. Thus, θ and D can be considered, respectively, as the temperature-like and diffusivity-like kinetic parameters of the market, and they can be used to characterize the behavior of the market.

Self-diffusion of wet particles in rotating drums

Axial mixing of wet particles in rotating drums was investigated by the discrete element method with the capillary force explicitly considered. Different flow regimes were observed by varying the surface tension of liquid and keeping other conditions unchanged. The analysis of the concentration and mean square displacement of particles indicated that the axial motion of wet particles was a diffusive process characterised by Fick's law. Particle diffusivity decreased with increasing inter-particle cohesion and drum filling level but increased with increasing drum rotation speed. Two competing mechanisms were proposed to explain these effects. A theoretical model based on the relation between local diffusivity and shear rate was developed to predict particle diffusivity as a function of drum operation conditions. It was also observed that despite the high inhomogeneity of particle flow in rotating drums, the mean diffusivity of flow exhibited a strong correlation with granular temperature, defined as the mean square fluctuating velocity of particles.