Long-time Diffusion Coefficient Research Articles

Tangentially active polymers in cylindrical channels

We present an analytical and computational study characterizing the structural and dynamical properties of an active filament confined in cylindrical channels. We first outline the effects of the interplay between confinement and polar self-propulsion on the conformation of the chains. We observe that the scaling of the polymer size in the channel, quantified by the end-to-end distance, shows different anomalous behaviours under different confinement and activity conditions. In particular, we report scaling exponents that are markedly different from their passive counterparts. Interestingly, we show that the universal relation, describing the ratio between the end-to-end distance of passive polymer chains in cylindrical channels and in bulk is broken by activity. Finally, we show that the long-time diffusion coefficient under confinement can be rationalised by an analytical model, that takes into account the presence of the channel and the elongated nature of the polymer.

Bridging Macroscopic Diffusion and Microscopic Cavity Escape of Brownian and Active Particles in Irregular Porous Networks.

While irregular and geometrically complex pore networks are ubiquitous in nature and industrial processes, there is no universal model describing nanoparticle transport in these environments. 3D super-resolution nanoparticle tracking was employed to study the motion of passive (Brownian) and active (self-propelled) species within complex networks, and universally identified a mechanism involving successive cavity exploration and escape. In all cases, the long-time ensemble-averaged diffusion coefficient was proportional to a quantity involving the characteristic length scale and time scale associated with microscopic cavity exploration and escape (D ∼ r2/ttrap), where the proportionality coefficient reflected the apparent porous network connectivity. For passive nanoparticles, this coefficient was always lower than expected theoretically for a random walk, indicating reduced network accessibility. In contrast, the coefficient for active nanomotors, in the same pore spaces, aligned with the theoretical value, suggesting that active particles navigate "intelligently" in porous environments, consistent with kinetic Monte Carlo simulations in networks with variable pore sizes. These findings elucidate a model of successive cavity exploration and escape for nanoparticle transport in porous networks, where pore accessibility is a function of motive force, providing insights relevant to applications in filtration, controlled release, and beyond.

Using the Haken-Strobl-Reineker Model to Determine the Temperature Dependence of the Diffusion Coefficient.

Stochastic quantum Liouville equations (SQLE) are widely used to model energy and charge dynamics in molecular systems. The Haken-Strobl-Reineker (HSR) SQLE is a particular paradigm in which the dynamical noise that destroys quantum coherences arises from a white noise (i.e., constant-frequency) spectrum. A system subject to the HSR SQLE thus evolves to its "high-temperature" limit, whereby all the eigenstates are equally populated. This result would seem to imply that the predictions of the HSR model, e.g., the temperature dependence of the diffusion coefficient, have no validity for temperatures lower than the particle bandwidth. The purpose of this paper is to show that this assumption is incorrect for translationally invariant systems. In particular, provided that the diffusion coefficient is determined via the mean-squared-displacement, considerations about detailed-balance are irrelevant. Consequently, the high-temperature prediction for the temperature dependence of the diffusion coefficient may be extrapolated to lower temperatures, provided that the bath remains classical. Thus, for diagonal dynamical disorder the long-time diffusion coefficient, D∞(T) = c1/T, while for both diagonal and off-diagonal disorder, D∞(T) = c1/T + c2T, where c2 ≪ c1. An appendix discusses an alternative interpretation from the HSR model of the "quantum to classical" dynamics transition, whereby the dynamics is described as stochastically punctuated coherent motion.

Universal scaling of the diffusivity of dendrimers in a semidilute solution of linear polymers.

The static and dynamic properties of dendrimers in semidilute solutions of linear chains of comparable size are investigated using Brownian dynamics simulations. The radius of gyration and diffusivity of a wide variety of low generation dendrimers and linear chains in solution follow universal scaling laws independent of their topology. Analysis of the shape functions and internal density of dendrimers shows that they are more spherical than linear chains and have a dense core. At intermediate times, dendrimers become subdiffusive, with an exponent higher than that previously reported for nanoparticles in semidilute polymer solutions. The long-time diffusivity of dendrimers does not follow theoretical predictions for nanoparticles. We propose a new scaling law for the long-time diffusion coefficients of dendrimers which accounts for the fact that, unlike nanoparticles, dendrimers shrink with an increase in background solution concentration. Analysis of the properties of a special case of a higher functionality dendrimer shows a transition from polymer-like to nanoparticle-like behaviour.

Enhanced diffusion of tracer particles in nonreciprocal mixtures.

We study the diffusivity of a tagged particle in a binary mixture of Brownian particles with nonreciprocal interactions. Numerical simulations reveal that, for a broad class of interaction potentials, nonreciprocity can significantly increase the long-time diffusion coefficient of tracer particles and that this diffusion enhancement is associated with a breakdown of the Einstein relation. These observations are quantified and confirmed via two different and complementary analytical approaches: (i) a linearized stochastic density field theory, which is particularly accurate in the limit of soft interactions, and (ii) a reduced two-body description, which is exact at leading order in the density of particles. The latter reveals that diffusion enhancement can be attributed to the formation of transiently propelled dimers of particles, whose cohesion and speed are controlled by the nonreciprocal interactions.

Active and passive microrheology with large tracers in hard colloids.

The dynamics of a tracer particle in a bath of quasi-hard colloidal spheres is studied by Langevin dynamics simulations and mode coupling theory (MCT); the tracer radius is varied from equal to up to seven times larger than the bath particles radius. In the simulations, two cases are considered: freely diffusing tracer (passive microrheology) and tracer pulled with a constant force (active microrheology). Both cases are connected by linear response theory for all tracer sizes. It links both the stationary and transient regimes of the pulled tracer (for low forces) with the equilibrium correlation functions; the velocity of the pulled tracer and its displacement are obtained from the velocity auto-correlation function and the mean squared displacement, respectively. The MCT calculations give insight into the physical mechanisms: At short times, the tracer rattles in its cage of neighbours, with the frequency increasing linearly with the tracer radius asymptotically. The long-time tracer diffusion coefficient from passive microrheology, which agrees with the inverse friction coefficient from the active case, arises from the transport of transverse momentum around the tracer. It can be described with the Brinkman equation for the transverse flow field obtained in extension of MCT, but cannot be recovered from the MCT kernel coupling to densities only. The dynamics of the bath particles is also studied; for the unforced tracer the dynamics is unaffected. When the tracer is pulled, the velocity field in the bath follows the prediction of the Brinkman model, but different from the case of a Newtonian fluid.

Interactions enhance dispersion in fluctuating channels via emergent flows

Understanding particle motion in narrow channels can guide progress in numerous applications, from filtration to vascular transport. Thermal or active fluctuations of fluid-filled channel walls can slow down or increase the dispersion of tracer particles via entropic trapping in the wall bulges or hydrodynamic flows induced by wall fluctuations, respectively. Previous studies concentrated primarily on the case of a single Brownian tracer. Here, we address what happens when there is a large ensemble of interacting Brownian tracers – a common situation in applications. Introducing repulsive interactions between tracer particles, while ignoring the presence of a background fluid, leads to an effective flow field. This flow field enhances tracer dispersion, a phenomenon reminiscent of that seen for single tracers in incompressible background fluid. We characterise the dispersion by the long-time diffusion coefficient of tracers numerically and analytically with a mean-field density functional analysis. We find a surprising effect where an increased particle density enhances the diffusion coefficient, challenging the notion that crowding effects tend to reduce diffusion. Here, inter-particle interactions push particles closer to the fluctuating channel walls. Interactions between the fluctuating wall and the now-nearby particles then drive particle mixing. Our mechanism is sufficiently general that we expect it to apply to various systems. In addition, our perturbation theory quantifies dispersion in generic advection–diffusion systems.

Diffusion coefficients of linear trimer particles.

We study the diffusive behavior of linear trimer particles via numerical calculations. First, we utilize hydrodynamic bead-shell calculations to compute the microscopic diffusion coefficients for different particle aspect ratios. These values are then used to obtain continuous empirical formulas for said coefficients. As an application example for the empirical formulas, we perform Brownian dynamics simulations of monolayers consisting of a linear trimer surrounded by colloidal spheres. Here, we obtain empirical formulas for the corresponding long-time diffusion coefficients of the trimer. By comparing our data for the microscopic and long-time diffusion coefficients with known results for spherocylinders, we find that the diffusive behavior of both particle geometries is approximately identical. Based on this observation, we introduce simplified equations for the microscopic diffusion coefficients that can be used for arbitrary short rods that are spheres at the minimum aspect ratios. The calculated equations for the diffusion coefficients can be applied to various further numerical and experimental studies utilizing linear trimer particles.

Dynamics of active particles with translational and rotational inertia

Inertial effects affecting both the translational and rotational dynamics are inherent to a broad range of active systems at the macroscopic scale. Thus, there is a pivotal need for proper models in the framework of active matter to correctly reproduce experimental results, hopefully achieving theoretical insights. For this purpose, we propose an inertial version of the active Ornstein–Uhlenbeck particle (AOUP) model accounting for particle mass (translational inertia) as well as its moment of inertia (rotational inertia) and derive the full expression for its steady-state properties. The inertial AOUP dynamics introduced in this paper is designed to capture the basic features of the well-established inertial active Brownian particle model, i.e. the persistence time of the active motion and the long-time diffusion coefficient. For a small or moderate rotational inertia, these two models predict similar dynamics at all timescales and, in general, our inertial AOUP model consistently yields the same trend upon changing the moment of inertia for various dynamical correlation functions.

Persistent motion of a Brownian particle subject to repulsive feedback with time delay.

Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal particle moving in two dimensions under time-delayed, nonlinear feedback control. Specifically, the particle is subject to a force derived from a repulsive Gaussian potential depending on the difference between its instantaneous position, r(t), and its earlier position r(t-τ), where τ is the delay time. Considering first the deterministic case, we provide analytical results for both the case of small displacements and the dynamics at long times. In particular, at appropriate values of the feedback parameters, the particle approaches a steady state with a constant, nonzero velocity whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times, but the (numerically obtained) velocity autocorrelation still reveals some persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a mixed regime at intermediate times with contributions of both ballistic motion and diffusive translational motion, allowing us to extract an estimate for the effective propulsion velocity in presence of noise. We then analyze the data in terms of exact, known results for the MSD of active Brownian particles. The comparison indeed indicates a strong similarity between the dynamics of the particle under repulsive delayed feedback and active motion. This relation carries over to the behavior of the long-time diffusion coefficient D_{eff} which, similarly to active motion, is strongly enhanced compared to the free case. Finally, we show that, for small delays, D_{eff} can be estimated analytically.

Evidence for growing structural correlation length in colloidal supercooled liquids.

Using video microscopy, we measure the long-time diffusion coefficients of colloidal particles at different concentrations. The measured diffusion coefficients start to deviate from theoretical predictions based on random collision models upon entering the supercooled regime. The theoretical diffusion relation is recovered by assigning an effective mass proportional to the size of structurally correlated clusters to the diffusing particles, providing an indirect method to probe the growth of static correlation length scales approaching the glass transition. This method is tested and validated in the crystallization of mono-disperse colloids in quasi-two-dimensional experiments. The correlation length obtained for a binary colloidal liquid increases by a power law toward a critical packing fraction of ∼0.79. The system relaxation time exhibits a power-law dependence on the correlation length in agreement with dynamical facilitation theories.

Structure and anisotropic dynamics of stimuli responsive colloidal ellipsoids at the nearest neighbor length scale

Stimuli-responsive self-assembly of (an) isotropic colloids has resulted in a plethora of self-assembled structures with potential applications in fabricating smart materials. A lack of detailed understanding of the interplay between these self-assembled structures and the resulting dynamics has often impeded the exploitation of their full potential. Herein, we have unveiled the relationship between the field-driven self-assembled structures and the corresponding collective dynamics at the nearest neighbor length scale using X-ray photon correlation spectroscopy and magnetic colloidal ellipsoids. We demonstrate that the effective long-time collective diffusion coefficient, Deff(q), scales with the inverse of the scattered intensity for various stimuli-responsive self-assembled phases. At high-volume fraction, the system approaches a kinetically arrested state. The anisotropic slowdown of Deff(q) hints towards the formation of an oriented glass. Our approach opens new avenues for exploring the stimuli-responsive dynamics of strongly interacting colloidal systems with diverse shapes and properties.

Dynamics of active particles with space-dependent swim velocity.

We study the dynamical properties of an active particle subject to a swimming speed explicitly depending on the particle position. The oscillating spatial profile of the swim velocity considered in this paper takes inspiration from experimental studies based on Janus particles whose speed can be modulated by an external source of light. We suggest and apply an appropriate model of an active Ornstein Uhlenbeck particle (AOUP) to the present case. This allows us to predict the stationary properties, by finding the exact solution of the steady-state probability distribution of particle position and velocity. From this, we obtain the spatial density profile and show that its form is consistent with the one found in the framework of other popular models. The reduced velocity distribution highlights the emergence of non-Gaussianity in our generalized AOUP model which becomes more evident as the spatial dependence of the velocity profile becomes more pronounced. Then, we focus on the time-dependent properties of the system. Velocity autocorrelation functions are studied in the steady-state combining numerical and analytical methods derived under suitable approximations. We observe a non-monotonic decay in the temporal shape of the velocity autocorrelation function which depends on the ratio between the persistence length and the spatial period of the swim velocity. In addition, we numerically and analytically study the mean square displacement and the long-time diffusion coefficient. The ballistic regime, observed in the small-time region, is deeply affected by the properties of the swim velocity landscape which induces also a crossover to a sub-ballistic but superdiffusive regime for intermediate times. Finally, the long-time diffusion coefficient decreases as the amplitude of the swim velocity oscillations increases because the diffusion is mainly determined by those regions where the particles are slow.

A model-system of Fickian yet non-Gaussian diffusion: light patterns in place of complex matter.

Fickian yet non-Gaussian Diffusion (FnGD), widely observed for colloidal particles in a variety of complex and biological fluids, emerges as a most intriguing open issue in Soft Matter. To fully monitor FnGD and advance its understanding, recording many trajectories over a large time range is crucial, which makes experiments challenging. Here we exploit a recently introduced experimental model of finely tunable FnGD: a quasi-2d system of Brownian beads in water moving in a heterogeneous energy landscape generated by a static and spatially random optical force field (speckle pattern). By performing experiments at different optical power, we succeed in monitoring the evolution as well as the precursors of FnGD. Fickian scaling of the mean square displacement is always attained after a subdiffusive regime while the displacement distributions keep on being non-Gaussian, which allows for measuring a characteristic length- and time-scale for the onset of FnGD, ξf and tf. We find that ξf stays constant, whereas tf grows as the inverse of the long-time diffusion coefficient tf ∝ D-1 for increasing the optical power. Deviations from the standard Gaussian shape of the displacement distribution are neatly characterized on a broad range of times, focusing on the excess probability at small displacements and on the decay-length of the distinctive exponential tails. Such deviations are fully built in the subdiffusive regime and, at the FnGD onset, grow with the optical power. As time goes on, the small-displacement probability narrows and the exponential tails progressively break up, with a tendency to recover the Gaussian behaviour. Overall, both subdiffusion and FnGD become more marked and persistent on increasing the optical power, suggesting a strict relation between these two regimes. As clearly demonstrated by our results, the adopted model-system represents a privileged stage for in-depth study of FnGD and opens the way to unveil the nature of this phenomenon through finely tuned and well-controlled experiments.

Data-driven criterion for the solid-liquid transition of two-dimensional self-propelled colloidal particles far from equilibrium.

We establish an explicit data-driven criterion for identifying the solid-liquid transition of two-dimensional self-propelled colloidal particles in the far from equilibrium parameter regime, where the transition points predicted by different conventional empirical criteria for melting and freezing diverge. This is achieved by applying a hybrid machine learning approach that combines unsupervised learning with supervised learning to analyze a huge amount of the system's configurations in the nonequilibrium parameter regime on an equal footing. Furthermore, we establish a generic data-driven evaluation function, according to which the performance of different empirical criteria can be systematically evaluated and improved. In particular, by applying this evaluation function, we identify a new nonequilibrium threshold value for the long-time diffusion coefficient, based on which the predictions of the corresponding empirical criterion are greatly improved in the far from equilibrium parameter regime. These data-driven approaches provide a generic tool for investigating phase transitions in complex systems where conventional empirical ones face difficulties.

Microscopic diffusion coefficients of dumbbell- and spherocylinder-shaped colloids and their application in simulations of crowded monolayers.

We explore the diffusion properties of colloidal particles with dumbbell and spherocylinder shapes using a hydrodynamic bead-shell approach and additional Brownian dynamics (BD) simulations. By applying the bead-shell method, we determine empirical formulas for the microscopic diffusion coefficients. A comparison of these formulas and established experimental and theoretical results shows remarkable agreement. For example, the maximum relative discrepancy found for dumbbells is less than 5%. As an application example of the empirical formulas, we perform two-dimensional (2D) BD simulations based on a single dumbbell or spherocylinder in a suspension of spheres and calculate the resulting effective long-time diffusion coefficients. The performed BD simulations can be compared to quasi-2D systems such as colloids confined at the interface of two fluids. We find that the effective diffusion coefficient of translation mostly depends on the sphere area fraction ϕ, while the effective diffusion coefficient of rotation is influenced by the aspect ratio and ϕ. Furthermore, the effective rotational diffusion constant seems to depend on the particle shape with the corresponding implementation of the interactions. In the resolution limit of our methods, the shape-dependent differences of the microscopic diffusion coefficients and the long-time diffusion constant of translation are negligible in the first approximation. The determined empirical formulas for the microscopic diffusion coefficients add to the knowledge of the diffusion of anisotropic particles, and they can be used in countless future studies.

Dynamics of Flexible Viruses in Polymer Solutions

We examine the coupling of the dynamics of flexible viral nanoparticles to the dynamics of comparably sized polymer chains. Using fluorescence microscopy, we quantify the dynamics of three filamentous viruses, potato virus M (PVM), M13, and pf1, that are suspended in semidilute solutions of partially hydrolyzed polyacrylamide. The dynamics of the viral nanoparticles are approximately diffusive on accessible time and length scales, but the distributions of displacements are non-Gaussian and exhibit increasingly extended tails as the aspect ratio of the viruses or the polymer concentration is increased. The long-time diffusion coefficients do not collapse onto a universal curve based on existing models for rodlike or spherical nanoparticles that are comparably sized to the polymer chains. Instead, the diffusivities appear to collapse as a function of the ratio of the polymer correlation length and a length scale intermediate between the virus radius and length, indicating that the hydrodynamic coupling to the polymer dynamics is affected by the virus anisotropy and flexibility.

On the Time Transition Between Short- and Long-Time Regimes of Colloidal Particles in External Periodic Potentials

The dynamics of colloidal particles at infinite dilution, under the influence of periodic external potentials, is studied hereviaexperiments and numerical simulations for two representative potentials. From the experimental side, we analyzed the motion of a colloidal tracer in a one-dimensional array of fringes produced by the interference of two coherent laser beams, providing in this way an harmonic potential. The numerical analysis has been performedviaBrownian dynamics (BD) simulations. The BD simulations correctly reproduced the experimental position- and time-dependent density of probability of the colloidal tracer in the short-times regime. The long-time diffusion coefficient has been obtained from the corresponding numerical mean square displacement (MSD). Similarly, a simulation of a random walker in a one dimensional array of adjacent cages with a probability of escaping from one cage to the next cage is one of the most simple models of a periodic potential, displaying two diffusive regimes separated by a dynamical caging period. The main result of this study is the observation that, in both potentials, it is seen that the critical timet*, defined as the specific time at which a change of curvature in the MSD is observed, remains approximately constant as a function of the height barrierU0of the harmonic potential or the associated escape probability of the random walker. In order to understand this behavior, histograms of the first passage time of the tracer have been calculated for several height barriersU0or escape probabilities. These histograms display a maximum at the most likely first passage timet′, which is approximately independent of the height barrierU0, or the associated escape probability, and it is located very close to the critical timet*. This behavior suggests that the critical timet*, defining the crossover between short- and long-time regimes, can be identified as the most likely first passage timet′as a first approximation.

Enhanced Diffusive Transport in Fluctuating Porous Media.

Mass transport within porous structures is a ubiquitous process in biological, geological, and technological systems. Despite the importance of these phenomena, there is no comprehensive theory that describes the complex and diverse transport behavior within porous environments. While the porous matrix itself is generally considered a static and passive participant, many porous environments are in fact dynamic, with fluctuating walls, pores that open and close, and dynamically changing cross-links. While diffusion has been measured in fluctuating structures, notably in model biological systems, it is rarely possible to isolate the effect of fluctuations because of the absence of control experiments involving an identical static counterpart, and it is generally impossible to observe the dynamics of the structure. Here we present a direct comparison of the diffusion of nanoparticles of various sizes within a trackable, fluctuating porous matrix and a geometrically equivalent static matrix, in conditions spanning a range of regimes from obstructed to highly confined. The experimental system comprised a close-packed layer of colloidal spheres that were either immobilized to a planar surface or allowed to fluctuate locally, within the space defined by their nearest neighbors. Interestingly, the effective long-time diffusion coefficient was approximately 35-65% greater in the fluctuating porous matrix than in the static one (depending on the size of the nanoparticle probes), regardless of the geometric regime. This was explained by considering the enhancing effects of matrix fluctuations on the short-time diffusion coefficient and cooperative "gate-opening" motions of matrix particles and nanoparticle probes.

Transport coefficients in dense active Brownian particle systems: mode-coupling theory and simulation results

We discuss recent advances in developing a mode-coupling theory of the glass transition (MCT) of two-dimensional systems of active Brownian particles (ABPs). The theory describes the structural relaxation close to the active glass in terms of transient dynamical density correlation functions. We summarize the equations of motion that have been derived for the collective density-fluctuation dynamics and those for the tagged-particle motion. The latter allow to study the dynamics of both passive and active tracers in both passive and active host systems. In the limit of small wave numbers, they give rise to equations of motion describing the mean-squared displacements (MSDs) of these tracers and hence the long-time diffusion coefficients as a transport coefficient quantifying long-range tracer motion. We specifically discuss the case of a single ABP tracer in a glass-forming passive host suspension, a case that has recently been studied in experiments on colloidal Janus particles. We employ event-driven Brownian dynamics (ED-BD) computer simulations to test the ABP-MCT and find good agreement between the two for the MSD, provided that known errors in MCT already for the passive system (i.e., an overestimation of the glassiness of the system) are accounted for by an empirical mapping of packing fractions and host-system self-propulsion forces. The ED-BD simulation results also compare well to experimental data, although a peculiar non-monotonic mapping of self-propulsion velocities is required. The ABP-MCT predicts a specific self-propulsion dependence of the Stokes–Einstein relation between the long-time diffusion coefficient and the host-system viscosity that matches well the results from simulation. An application of ABP-MCT within the integration-through transients framework to calculate the density-renormalized effective swim velocity of the interacting ABP agrees qualitatively with the ED-BD simulation data at densities close to the glass transition and quantitatively for the full density range only after the mapping of packing fractions employed for the passive system.Graphic abstract