Nonlinear active micro-rheology in a glass-forming soft-sphere mixture

Eyring equation and fluctuation-dissipation far away from equilibrium

Understanding and managing the influence that either external forces or non-equilibrated environments may have on chemical processes is essential for the current and future development of theoretical chemistry. One of the central questions to solve is how to generalize the transition state theory in order to make it applicable in far from equilibrium situations. In this sense, here we propose a way to generalize Eyring's equation based on the definition of an effective thermal energy (temperature) emerging from the coupling of both fast and slow dynamic variables analyzed within the generalized Langevin dynamics scheme. This coupling makes the energy distribution of the fast degrees of freedom not equilibrate because they have been enslaved to the dynamics of the corresponding slow degrees. However, the introduction of the effective thermal energy enables us to restore an effective adiabatic separation of timescales leading to a renormalization of the generalized fluctuation-dissipation theorem. Hence, this procedure opens the possibility to deal with systems far away from equilibrium. A significant consequence of our results is that Eyring's equation is generalized to treat systems under the influence of strong external forces.

Active microrheology in two-dimensional magnetic networks

We study active microrheology in two-dimensional (2D) magnetic networks. To this end, we use Langevin dynamics computer simulations where single non-magnetic or magnetic tracer particles are pulled through the network structures via a constant force f. Structural changes in the network around the pulled tracer particle are characterized in terms of pair correlation functions. These functions indicate that the non-magnetic tracer particles tend to strongly affect the network structure leading to the formation of channels at sufficiently high forces, while the magnetic tracer particles modify the network structure only slightly. At zero pulling force, f = 0, both non-magnetic and magnetic tracer particles are localized, i.e. they do not show diffusive behavior in the long-time limit. Nevertheless, the friction coefficient, as obtained from the steady-state velocity of the tracer particles, seems to indicate a linear-response regime at small values of f. Beyond the latter linear response regime, the diffusion dynamics of the tracer particles are anisotropic with superdiffusive behavior in force direction. This transport anomaly is investigated via van Hove correlation functions and residence time distributions.

Observation of Strongly Heterogeneous Dynamics at the Depinning Transition in a Colloidal Glass

We study experimentally the origin of heterogeneous dynamics in strongly driven glass-forming systems. Thereto, we apply a well-defined force with a laser line trap on individual colloidal polystyrene probe particles seeded in an emulsion glass composed of droplets of the same size. Fluid and glass states can be probed. We monitor the trajectories of the probe and measure displacements and their distributions. Our experiments reveal intermittent dynamics around a depinning transition at a threshold force. For smaller forces, linear response connects mean displacement, and quiescent mean squared displacement. Mode coupling theory calculations rationalize the observations.

Tracer diffusion in crowded narrow channels

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Time-dependent perpendicular fluctuations in the driven lattice Lorentz gas

We present results for the fluctuations of the displacement of a tracer particle on a planar lattice pulled by a step force in the presence of impenetrable, immobile obstacles. The fluctuations perpendicular to the applied force are evaluated exactly in first order of the obstacle density for arbitrarily strong pulling and all times. The complex time-dependent behavior is analyzed in terms of the diffusion coefficient, local exponent, and the non-Skellam parameter, which quantifies deviations from the dynamics on the lattice in the absence of obstacles. The non-Skellam parameter along the force is analyzed in terms of an asymptotic model and reveals a power-law growth for intermediate times.

Nonequilibrium depletion interactions in active microrheology

Entropic depletion forces arise between mesoscopic bodies that are immersed in a suspension of macromolecules, such as colloid-polymer mixtures. Here we consider the case of a driven colloidal probe in the presence of another, passive colloidal particle, both solvated in an ideal bath of small spherical particles. We calculate the nonequilibrium forces mediated by the depletants on the two colloidal particles within a dynamical superposition approximation (DSA) scheme. In order to assess the quality of this approximation, and to obtain the colloidal microstructure around the driven probe, we corroborate our theoretical results with Brownian dynamics simulations.

Cooperative behavior of biased probes in crowded interacting systems

We study, via extensive numerical simulations, dynamics of a crowded mixture of mutually interacting (with a short-range repulsive potential) colloidal particles immersed in a suspending solvent, acting as a heat bath. The mixture consists of a majority component - neutrally buoyant colloids subject to internal stimuli only, and a minority component - biased probes (BPs) also subject to a constant force. In such a system each of the BPs alters the distribution of the colloidal particles in its vicinity, driving their spatial distribution out of equilibrium. This induces effective long-range interactions and multi-tag correlations between the BPs, mediated by an out-of-equilibrium majority component, and prompts the BPs to move collectively assembling in clusters. We analyse the size-distribution of the self-assembling clusters in the steady-state, their specific force-velocity relations and also properties of the effective interactions emerging between the BPs.

Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas

We consider a tracer particle on a lattice in the presence of immobile obstacles. Starting from equilibrium, a force pulling on the particle is switched on, driving the system to a new stationary state. We solve for the complete transient dynamics of the fluctuations of the tracer position along the direction of the force. The analytic result, exact in first order of the obstacle density and for arbitrarily strong driving, is compared to stochastic simulations. Upon strong driving, the fluctuations grow superdiffusively for intermediate times; however, they always become diffusive in the stationary state. The diffusion constant is nonanalytic for small driving and is enhanced by orders of magnitude by increasing the force.

Origin of the effective mobility in non-linear active micro-rheology

The distinction between the damping coefficient and the effective non-linear mobility of driven particles in active micro-rheology of supercooled liquids is explained in terms of individual and collective dynamics. The effective mobility arises as a collective effect which gives insight into the energy landscape of the system. On the other hand, the damping coefficient is a constant that modulates the effect of external forces over the thermal energy which particles have at their disposition to perform Brownian motion. For long times, these thermal fluctuations become characterized in terms of an effective temperature that is a consequence of the dynamic coupling between kinetic and configurational degrees of freedom induced by the presence of the strong external force. The interplay between collective mobility and effective temperature allows to formulate a generalized Stokes-Einstein relation that may be used to determine the collective diffusion coefficient. The explicit relations we deduce reproduce simulation data remarkably well.

Active microrheology in a colloidal glass

We study the dynamics of a probe particle driven by a constant force through a colloidal glass of hard spheres. This nonequilibrium and anisotropic problem is investigated using a new implementation of the mode-coupling approximation with multiple relaxation channels and Langevin dynamics simulations. A force threshold is found, below which the probe remains localized, while above it the probe acquires a finite velocity. We focus on the localized regime, comparing theory and simulations concerning the dynamics in the length scale of the cage and the properties of the transition to the delocalized regime, such as the critical power-law decay of the probe correlation function. Probe van Hove functions predicted by the theory show exponential tails reminiscent of an intermittent dynamics of the probe. This scenario is microscopically supported by simulations.

Nonlinear response and emerging nonequilibrium microstructures for biased diffusion in confined crowded environments

We study analytically the dynamics and the microstructural changes of a host medium caused by a driven tracer particle moving in a confined, quiescent molecular crowding environment. Imitating typical settings of active microrheology experiments, we consider here a minimal model comprising a geometrically confined lattice system (a two-dimensional striplike or a three-dimensional capillary-like system) populated by two types of hard-core particles with stochastic dynamics (a tracer particle driven by a constant external force and bath particles moving completely at random). Resorting to a decoupling scheme, which permits us to go beyond the linear-response approximation (Stokes regime) for arbitrary densities of the lattice gas particles, we determine the force-velocity relation for the tracer particle and the stationary density profiles of the host medium particles around it. These results are validated a posteriori by extensive numerical simulations for a wide range of parameters. Our theoretical analysis reveals two striking features: (a) We show that, under certain conditions, the terminal velocity of the driven tracer particle is a nonmonotonic function of the force, so in some parameter range the differential mobility becomes negative, and (b) the biased particle drives the whole system into a nonequilibrium steady state with a stationary particle density profile past the tracer, which decays exponentially, in sharp contrast with the behavior observed for unbounded lattices, where an algebraic decay is known to take place.

Understanding the nonlinear dynamics of driven particles in supercooled liquids in terms of an effective temperature

In active microrheology, the mechanical properties of a material are tested by adding probe particles which are pulled by an external force. In case of supercooled liquids, strong forcing leads to a thinning of the host material which becomes more pronounced as the system approaches the glass transition. In this work, we provide a quantitative theoretical description of this thinning behavior based on the properties of the Potential Energy Landscape (PEL) of a model glass-former. A key role plays the trap-like nature of the PEL. We find that the mechanical properties in the strongly driven system behave the same as in a quiescent system at an enhanced temperature, giving rise to a well-characterized effective temperature. Furthermore, this effective temperature turns out to be independent of the chosen observable and individually shows up in the thermodynamic and dynamic properties of the system. Based on this underlying theoretical understanding, we can estimate its dependence on temperature and force by the PEL-properties of the quiescent system. We furthermore critically discuss the relevance of effective temperatures obtained by scaling relations for the description of out-of-equilibrium situations.

Velocity anomaly of a driven tracer in a confined crowded environment

We consider a discrete model in which a tracer performs a random walk biased by an external force, in a dense bath of particles performing symmetric random walks constrained by hard-core interactions. We reveal the emergence of a striking velocity anomaly in confined geometries: in quasi-1D systems such as stripes or capillaries, the velocity of the tracer displays a long-lived plateau before ultimately dropping to a lower value. We develop an analytical solution that quantitatively accounts for this intriguing behavior. Our analysis suggests that such a velocity anomaly could be a generic feature of driven dynamics in quasi-1D crowded systems.

Microrheology of colloidal systems

Microrheology was proposed almost twenty years ago as a technique to obtain rheological properties in soft matter from the microscopic motion of colloidal tracers used as probes, either freely diffusing in the host medium, or subjected to external forces. The former case is known as passive microrheology, and is based on generalizations of the Stokes-Einstein relation between the friction experienced by the probe and the host-fluid viscosity. The latter is termed active microrheology, and extends the measurement of the friction coefficient to the nonlinear-response regime of strongly driven probes. In this review article, we discuss theoretical models available in the literature for both passive and active microrheology, focusing on the case of single-probe motion in model colloidal host media. A brief overview of the theory of passive microrheology is given, starting from the work of Mason and Weitz. Further developments include refined models of the host suspension beyond that of a Newtonian-fluid continuum, and the investigation of probe-size effects. Active microrheology is described starting from microscopic equations of motion for the whole system including both the host-fluid particles and the tracer; the many-body Smoluchowski equation for the case of colloidal suspensions. At low fluid densities, this can be simplified to a two-particle equation that allows the calculation of the friction coefficient with the input of the density distribution around the tracer, as shown by Brady and coworkers. The results need to be upscaled to agree with simulations at moderate density, in both the case of pulling the tracer with a constant force or dragging it at a constant velocity. The full many-particle equation has been tackled by Fuchs and coworkers, using a mode-coupling approximation and the scheme of integration through transients, valid at high densities. A localization transition is predicted for a probe embedded in a glass-forming host suspension. The nonlinear probe-friction coefficient is calculated from the tracer's position correlation function. Computer simulations show qualitative agreement with the theory, but also some unexpected features, such as superdiffusive motion of the probe related to the breaking of nearest-neighbor cages. We conclude with some perspectives and future directions of theoretical models of microrheology.

Active microrheology of driven granular particles

When pulling a particle in a driven granular fluid with constant force Fex, the probe particle approaches a steady-state average velocity v. This velocity and the corresponding friction coefficient of the probe ζ=Fex/v are obtained within a schematic model of mode-coupling theory and compared to results from event-driven simulations. For small and moderate drag forces, the model describes the simulation results successfully for both the linear as well as the nonlinear region: The linear response regime (constant friction) for small drag forces is followed by shear thinning (decreasing friction) for moderate forces. For large forces, the model demonstrates a subsequent increasing friction in qualitative agreement with the data. The square-root increase of the friction with force found in [Fiege et al., Granul. Matter 14, 247 (2012)] is explained by a simple kinetic theory.

Geometry-induced superdiffusion in driven crowded systems

Recent molecular dynamics simulations of glass-forming liquids revealed superdiffusive fluctuations associated with the position of a tracer particle (TP) driven by an external force. Such an anomalous response, whose mechanism remains elusive, has been observed up to now only in systems close to their glass transition, suggesting that this could be one of its hallmarks. Here, we show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. We present and solve analytically a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. For such nonglass-forming systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a universal response of crowded environments to an external force. These findings are confirmed by numerical simulations of systems as varied as lattice gases, dense liquids, and granular fluids.

Colloidal lattice shearing and rupturing with a driven line of particles

We examine the dynamics of two-dimensional colloidal systems using numerical simulations of a system with a drive applied to a thin region in the middle of the sample to produce a local shear. For a monodisperse colloidal assembly, we find a well-defined decoupling transition separating a regime of elastic motion from a plastic phase where the driven particles break away or decouple from the bulk particles and produce a shear band. For a bidisperse assembly, the onset of a bulk disordering transition coincides with the broadening of the shear band. We identify several distinct dynamical regimes that are correlated with features in the velocity-force curves. As a function of bidispersity, the decoupling force shows a nonmonotonic behavior associated with features in the noise fluctuations, power spectra, and bulk velocity profiles. When pinning is added in the bulk, we find that the shear band regions can become more localized, causing a decoupling of the driven particles from the bulk particles. For a system with thermal noise and no pinning, the shear band region becomes more extended and the average velocity of the driven particles drops at the thermal disordering transition of the bulk system.