Transition from single-file to two-dimensional diffusion of interacting particles in a quasi-one-dimensional channel

Driven transport of soft Brownian particles through pore-like structures: Effective size method

Single-file transport in pore-like structures constitutes an important topic for both theory and experiment. For hardcore interacting particles, a good understanding of the collective dynamics has been achieved recently. Here, we study how softness in the particle interaction affects the emergent transport behavior. To this end, we investigate the driven Brownian motion of particles in a periodic potential. The particles interact via a repulsive softcore potential with a shape corresponding to a smoothed rectangular barrier. This shape allows us to elucidate effects of mutual particle penetration and particle crossing in a controlled manner. We find that even weak deviations from the hardcore case can have a strong impact on the particle current. Despite this fact, knowledge about the transport in a corresponding hardcore system is shown to be useful to describe and interpret our findings for the softcore case. This is achieved by assigning a thermodynamic effective size to the particles based on the equilibrium density functional of hard spheres.

Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion

In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the nonequilibrium stationary state. Here we highlight a new facet of the TUR by showing that it also bounds the timescale on which a finite system can exhibit anomalous kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a single file confined to a ring as well as a dragged Gaussian polymer chain even when detailed balance is satisfied. Conversely, the TUR bounds the onset of superdiffusion in the active comb model. Remarkably, the fluctuations in a comb model evolving from a steady state behave anomalously as soon as detailed balance is broken. Our work establishes a link between stochastic thermodynamics and the field of anomalous dynamics that will fertilize further investigations of thermodynamic consistency of anomalous diffusion models.

The effect of soft repulsive interactions on the diffusion of particles in quasi-one-dimensional channels: A hopping time approach

Fluids confined to quasi-one-dimensional channels exhibit a dynamic crossover from single file diffusion to normal diffusion as the channel becomes wide enough for particles to hop past each other. In the crossover regime, where hopping events are rare, the diffusion coefficient in the long time limit can be related to a hopping time that measures the average time it takes for a particle to escape the local cage formed by its neighbors. In this work, we show that a transition state theory (TST) that calculates the free energy barrier for two particles attempting to pass each other in the small system isobaric ensemble is able to quantitatively predict the hopping time in a system of two-dimensional soft repulsive disks [U(r_ij)=(σ/r_ij)^α] confined to a hard walled channel over a range of channel radii and degrees of particle softness measured in terms of 1/α. The free energy barrier exhibits a maximum at intermediate values of α that moves to smaller values of 1/α (harder particles) as the channel becomes narrower. However, the presence of the maximum is only observed in the hopping times for wide channels because the interaction potential dependence of the kinetic prefactor plays an increasingly important role for narrower channels. We also begin to explore how our TST approach can be used to optimize and control dynamics in confined quasi-one-dimensional fluids.

Anomalously slow transport in single-file diffusion with slow binding kinetics

We computationally study the effects of binding kinetics to the channel wall, leading to transient immobility, on the diffusive transport of particles within narrow channels, that exhibit single-file diffusion (SFD). We find that slow binding kinetics leads to an anomalously slow diffusive transport. Remarkably, the scaled diffusivity D[over ̂] characterizing transport exhibits scaling collapse with respect to the occupation fraction p of sites along the channel. We present a simple "cage-physics" picture that captures the characteristic occupation fraction p_{scale} and the asymptotic 1/p^{2} behavior for p/p_{scale}≳1. We confirm that subdiffusive behavior of tracer particles is controlled by the same D[over ̂] as particle transport.

Quantifying Discretization Errors in Electrophoretically-Guided Micro Additive Manufacturing

This paper presents process models for a new micro additive manufacturing process termed Electrophoretically-guided Micro Additive Manufacturing (EPμAM). In EPμAM, a planar microelectrode array generates the electric potential distributions which cause colloidal particles to agglomerate and deposit in desired regions. The discrete microelectrode array nature and the used pulse width modulation (PWM) technique for microelectrode actuation create unavoidable process errors-space and time discretization errors-that distort particle trajectories. To combat this, we developed finite element method (FEM) models to study trajectory deviations due to these errors. Mean square displacement (MSD) analysis of the computed particle trajectories is used to compare these deviations for several electrode geometries. The two top-performing electrode geometries evaluated by MSD were additionally investigated through separate case studies via geometry variation and MSD recomputation. Furthermore, separate time-discretization error simulations are also studied where electrode actuating waveforms were simulated. The mechanical impulse of the electromechanical force, generated from these waveforms is used as the basis for comparison. The obtained results show a moderate MSDs variability and significant differences in the computed mechanical impulses for the actuating waveforms. The observed limitations of the developed process model and of the error comparison technique are briefly discussed and future steps are recommended.

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking

Subdiffusion is commonly observed in liquids with high density or in restricted geometries, as the particles are constantly pushed back by their neighbors. Since this “cage effect” emerges from many-body dynamics involving spatiotemporally correlated motions, the slow diffusion should be understood not simply as a one-body problem but as a part of collective dynamics, described in terms of space–time correlations. Such collective dynamics are illustrated here by calculations of the two-particle displacement correlation in a system of repulsive Brownian particles confined in a (quasi-)one-dimensional channel, whose subdiffusive behavior is known as the single-file diffusion (SFD). The analytical calculation is formulated in terms of the Lagrangian correlation of density fluctuations. In addition, numerical solutions to the Langevin equation with large but finite interaction potential are studied to clarify the effect of overtaking. In the limiting case of the ideal SFD without overtaking, correlated motion with a diffusively growing length scale is observed. By allowing the particles to overtake each other, the short-range correlation is destroyed, but the long-range weak correlation remains almost intact. These results describe nested space–time structure of cages, whereby smaller cages are enclosed in larger cages with longer lifetimes.

Effect of Ionic Strength and Surface Charge Density on the Kinetics of Cellulose Nanocrystal Thin Film Swelling

This work explores cellulose nanocrystal (CNC) thin films (<50 nm) and particle-particle interactions by investigating film swelling in aqueous solutions with varying ionic strength (1-100 mM). CNC film hydration was monitored in situ via surface plasmon resonance, and the kinetics of liquid uptake were quantified. The contribution of electrostatic double-layer forces to film swelling was elucidated by using CNCs with different surface charges (anionic sulfate half ester groups, high and low surface charge density, and cationic trimethylammonium groups). Total water uptake in the thin films was found to be independent of ionic strength and surface chemistry, suggesting that in the aggregated state van der Waals forces dominate over double-layer forces to hold the films together. However, the rate of swelling varied significantly. The water uptake followed Fickian behavior, and the measured diffusion constants decreased with the ionic strength gradient between the film and the solution. This work highlights that nanoparticle interactions and dispersion are highly dependent on the state of particle aggregation and that the rate of water uptake in aggregates and thin films can be tailored based on surface chemistry and solution ionic strength.

Diffusion in quasi-one-dimensional channels: A small system n, p, T, transition state theory for hopping times

Particles confined to a single file, in a narrow quasi-one-dimensional channel, exhibit a dynamic crossover from single file diffusion to Fickian diffusion as the channel radius increases and the particles begin to pass each other. The long time diffusion coefficient for a system in the crossover regime can be described in terms of a hopping time, which measures the time it takes for a particle to escape the cage formed by its neighbours. In this paper, we develop a transition state theory approach to the calculation of the hopping time, using the small system isobaric-isothermal ensemble to rigorously account for the volume fluctuations associated with the size of the cage. We also describe a Monte Carlo simulation scheme that can be used to calculate the free energy barrier for particle hopping. The theory and simulation method correctly predict the hopping times for a two-dimensional confined ideal gas system and a system of confined hard discs over a range of channel radii, but the method breaks down for wide channels in the hard discs' case, underestimating the height of the hopping barrier due to the neglect of interactions between the small system and its surroundings.

Single file dynamics in soft materials

The term single file (SF) dynamics refers to the motion of an assembly of particles through a channel with cross-sections comparable to the particles' diameter. Single file diffusion (SFD) is then the diffusion of a tagged particle in a single file, i.e., under the condition that particle passing is not allowed. SFD accounts for a large variety of processes in nature, including diffusion of colloids in synthetic and natural channels, biological motors along molecular chains, electrons in proteins and liquid helium, ions through membranes, just to mention a few examples. Albeit introduced in 1965s, over the last decade the classical notion of SF dynamics has been generalised to account for a more realistic modelling of the particle properties, file geometry, particle-particle and channel-particle interactions, which paves the way to remarkable applications of the SF model, for instance, in the technology of bio-integrated nanodevices. We provide here a comprehensive review of the recent advances in the theory of SF dynamics with the purpose of spurring further experimental work.

A Model for the Transient Subdiffusive Behavior of Particles in Mucus

In this study we have applied a model to explain the reported subdiffusion of particles in mucus, based on the measured mean squared displacements (MSD). The model considers Brownian diffusion of particles in a confined geometry, made from permeable membranes. The applied model predicts a normal diffusive behavior at very short and long time lags, as observed in several experiments. In between these timescales, we find that the "subdiffusive" regime is only a transient effect, MSD∝τα,α<1. The only parameters in the model are the diffusion-coefficients at the limits of very short and long times, and the distance between the permeable membranes L. Our numerical results are in agreement with published experimental data for realistic assumptions of these parameters. Finally, we show that only particles with a diameter less than 40 nm are able to pass through a mucus layer by passive Brownian motion.

A Tunable Magnetic Domain Wall Conduit Regulating Nanoparticle Diffusion

We demonstrate a general and robust method to confine on a plane strongly diffusing nanoparticles in water by using size tunable magnetic channels. These virtual conduits are realized with pairs of movable Bloch walls located within an epitaxially grown ferrite garnet film. We show that once inside the magnetic conduit the particles experience an effective local parabolic potential in the transverse direction, while freely diffusing along the conduit. The stiffness of the magnetic potential is determined as a function of field amplitude that varies the width of the magnetic channel. Precise control of the degree of confinement is demonstrated by tuning the applied field. The magnetic conduit is then used to realize single files of nonpassing particles and to induce periodic condensation of an ensemble of particles into parallel stripes in a completely controllable and reversible manner.

General continuum approach for dissipative systems of repulsive particles

We propose a general coarse-graining method to derive a continuity equation that describes any dissipative system of repulsive particles interacting through short-ranged potentials. In our approach, the effect of particle-particle correlations is incorporated to the overall balance of energy, and a nonlinear diffusion equation is obtained to represent the overdamped dynamics. In particular, when the repulsive interaction potential is a short-ranged power law, our approach reveals a distinctive correspondence between particle-particle energy and the generalized thermostatistics of Tsallis for any nonpositive value of the entropic index q. Our methodology can also be applied to microscopic models of superconducting vortices and complex plasma, where particle-particle correlations are pronounced at low concentrations. The resulting continuum descriptions provide elucidating and useful insights on the microdynamical behavior of these physical systems. The consistency of our approach is demonstrated by comparison with molecular dynamics simulations.

Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional (2D) case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional (1D) case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the 2D case, the linear theory gives description of vortical cooperative motion.

From single-file diffusion to two-dimensional cage diffusion and generalization of the totally asymmetric simple exclusion process to higher dimensions

A two-dimensional constrained diffusion model is presented and characterized by numerical simulations. The model generalizes the one-dimensional single-file diffusion model by considering a cage diffusion constraint induced by neighboring particles, which is a more stringent condition than volume exclusion. Using numerical simulations we characterize the diffusion process and we particularly show that asymmetric transition probabilities lead to the two-dimensional Kardar-Parisi-Zhang universality class. Therefore, this very simple model effectively generalizes the one-dimensional totally asymmetric simple exclusion process to higher dimensions.

Confinement without boundaries: anisotropic diffusion on the surface of a cylinder

Densely packed systems of thermal particles in curved geometries are frequently encountered in biological and microfluidic systems. In 2D systems, at sufficiently high surface coverage, diffusive motion is widely known to be strongly affected by physical confinement, e.g., by the walls. In this work, we explore the effects of confinement by shape, not rigid boundaries, on the diffusion of discs by confining them to the surface of a cylinder. We find that both the magnitude and the directionality of lateral diffusion is strongly influenced by the radius of the cylinder. An anisotropy between diffusion in the longitudinal and circumferential direction of the cylinder develops. We demonstrate that the origin of this effect lies in the fact that screw-like packings of mono- and oligodisperse discs on the surface of a cylinder induce preferential collective motions in the circumferential direction, but also show that even in polydisperse systems lacking such order an intrinsic finite size confinement effect increases diffusivity in the circumferential direction.

Diffusion of interacting particles in discrete geometries: Equilibrium and dynamical properties

We expand on a recent study of a lattice model of interacting particles [Phys. Rev. Lett. 111, 110601 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.110601]. The adsorption isotherm and equilibrium fluctuations in particle number are discussed as a function of the interaction. Their behavior is similar to that of interacting particles in porous materials. Different expressions for the particle jump rates are derived from transition-state theory. Which expression should be used depends on the strength of the interparticle interactions. Analytical expressions for the self- and transport diffusion are derived when correlations, caused by memory effects in the environment, are neglected. The diffusive behavior is studied numerically with kinetic Monte Carlo (kMC) simulations, which reproduces the diffusion including correlations. The effect of correlations is studied by comparing the analytical expressions with the kMC simulations. It is found that the Maxwell-Stefan diffusion can exceed the self-diffusion. To our knowledge, this is the first time this is observed. The diffusive behavior in one-dimensional and higher-dimensional systems is qualitatively the same, with the effect of correlations decreasing for increasing dimension. The length dependence of both the self- and transport diffusion is studied for one-dimensional systems. For long lengths the self-diffusion shows a 1/L dependence. Finally, we discuss when agreement with experiments and simulations can be expected. The assumption that particles in different cavities do not interact is expected to hold quantitatively at low and medium particle concentrations if the particles are not strongly interacting.

Single-file and normal diffusion of magnetic colloids in modulated channels

Diffusive properties of interacting magnetic dipoles confined in a parabolic narrow channel and in the presence of a periodic modulated (corrugated) potential along the unconfined direction are studied using Brownian dynamics simulations. We compare our simulation results with the analytical result for the effective diffusion coefficient of a single particle by Festa and d'Agliano [Physica A 90, 229 (1978)] and show the importance of interparticle interaction on the diffusion process. We present results for the diffusion of magnetic dipoles as a function of linear density, strength of the periodic modulation and commensurability factor.

Diffusion of interacting particles in discrete geometries

We evaluate the self-diffusion and transport diffusion of interacting particles in a discrete geometry consisting of a linear chain of cavities, with interactions within a cavity described by a free-energy function. Exact analytical expressions are obtained in the absence of correlations, showing that the self-diffusion can exceed the transport diffusion if the free-energy function is concave. The effect of correlations is elucidated by comparison with numerical results. Quantitative agreement is obtained with recent experimental data for diffusion in a nanoporous zeolitic imidazolate framework material, ZIF-8.

Everlasting effect of initial conditions on single-file diffusion

We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one-dimensional channel (a single-file model). In particular, we examine the influence of initial conditions on the dynamics of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamics, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e., a closed system that attains equilibrium independent of the initial state).

Noisy zigzag transition, fluctuations, and thermal bifurcation threshold

We study the zigzag transition in a system of particles with screened electrostatic interaction, submitted to a thermal noise. At finite temperature, this configurational phase transition is an example of noisy supercritical pitchfork bifurcation. The measurements of transverse fluctuations allow a complete description of the bifurcation region, which takes place between the deterministic threshold and a thermal threshold beyond which thermal fluctuations do not allow the system to flip between the symmetric zigzag configurations. We show that a divergence of the saturation time for the transverse fluctuations allows a precise and unambiguous definition of this thermal threshold. Its evolution with the temperature is shown to be in good agreement with theoretical predictions from noisy bifurcation theory.

Tunable diffusion of magnetic particles in a quasi-one-dimensional channel

The diffusion of a system of ferromagnetic dipoles confined in a quasi-one-dimensional parabolic trap is studied using Brownian dynamics simulations. We show that the dynamics of the system is tunable by an in-plane external homogeneous magnetic field. For a strong applied magnetic field, we find that the mobility of the system, the exponent of diffusion, and the crossover time among different diffusion regimes can be tuned by the orientation of the magnetic field. For weak magnetic fields, the exponent of diffusion in the subdiffusive regime is independent of the orientation of the external field.

Two-particle distribution and correlation function for a 1D dusty plasma experiment

Experimentally measured velocities are used to obtain the one- and two-particle distribution functions f(1) and f(2) and the two-particle correlation function g(2)≡f(2)-f(1)f(1). The fluctuating velocities of interacting charged microparticles were recorded by tracking their motion while they were immersed in a dusty plasma. The phase space was reduced by having only two particles in a harmonic one dimensional confining potential. In statistical theory, g(2) is usually said to be dominated by the randomness of collisions, but here we find that it is dominated by collective oscillatory modes.

Single-file diffusion in periodic energy landscapes: the role of hydrodynamic interactions

We report on the dynamical properties of interacting colloids confined to one dimension and subjected to external periodic energy landscapes. We particularly focus on the influence of hydrodynamic interactions on the mean-square displacement. Using Brownian dynamics simulations, we study colloidal systems with two types of repulsive interparticle interactions, namely, Yukawa and superparamagnetic potentials. We find that in the homogeneous case, hydrodynamic interactions lead to an enhancement of the particle mobility and the mean-square displacement at long times scales as t(α), with α=1/2+ε and ε being a small correction. This correction, however, becomes much more important in the presence of an external field, which breaks the homogeneity of the particle distribution along the line and, therefore, promotes a richer dynamical scenario due to the hydrodynamical coupling among particles. We provide here the complete dynamical scenario in terms of the external potential parameters: amplitude and commensurability.

Single-file diffusion in a box: effect of the initial configuration

This paper studies the single-file diffusion process on a linear chain of identical pointlike particles with multiple-site occupation confined in a one-dimensional box. The particles are noninteracting, except that double occupancy is forbidden. When particles are confined in a finite box, the final stage is saturation. By means of combinatorial analysis, an exact numerical evaluation of the saturation values for both the mean-square displacement (MSD) of a tracer particle and the center of mass of the system are obtained. Different initial distributions of particles are introduced. The time dependence of the MSD is obtained by means of Monte Carlo simulations. The values of the MSD for the tracer particles as well as the center of mass of the system depend on the size of the particle, the size of the box, and the initial distribution. Moreover, the transient regime depends on the initial distribution. In fact, the crossover from normal to subdiffusive regime is observed for random and alternate initial distributions, while superdiffusive diffusion appears for any stacked initial distributions. In all cases, it is shown that the collisions between particles do not determine the time exponent of the MSD. A simple expression for the transient regime is also obtained for the especial case of random initial distribution.