Active microrheology of driven granular particles

Rheology of granular particles immersed in a molecular gas under uniform shear flow

Non-Newtonian transport properties of a dilute gas of inelastic hard spheres immersed in a molecular gas are determined. We assume that the granular gas is sufficiently rarefied, and hence the state of the molecular gas is not disturbed by the presence of the solid particles. In this situation, one can treat the molecular gas as a bath (or thermostat) of elastic hard spheres at a given temperature. Moreover, in spite of the fact that the number density of grains is quite small, we take into account their inelastic collisions among themselves in its kinetic equation. The system (granular gas plus a bath of elastic hard spheres) is subjected to a simple (or uniform) shear flow. In the low-density regime, the rheological properties of the granular gas are determined by solving the Boltzmann kinetic equation by means of Grad's moment method. These properties turn out to be highly nonlinear functions of the shear rate and the remaining parameters of the system. Our results show that the kinetic granular temperature and the non-Newtonian viscosity present a discontinuous shear thickening effect for sufficiently high values of the mass ratio m/m_{g} (m and m_{g} being the mass of grains and gas particles, respectively). This effect becomes more pronounced as the mass ratio m/m_{g} increases. In particular, in the Brownian limit (m/m_{g}→∞) the expressions of the non-Newtonian transport properties derived here are consistent with those previously obtained by considering a coarse-grained approach where the effect of gas phase on grains is through an effective force. Theoretical results are compared against computer simulations, showing an excellent agreement.

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

Force renormalization for probes immersed in an active bath

Langevin equations or generalized Langevin equations (GLEs) are popular models for describing the motion of a particle in a fluid medium in an effective manner. Here we examine particles immersed in an inherently nonequilibrium fluid, i.e., an active bath, which are subject to an external force. Specifically, we consider two types of forces that are highly relevant for microrheological studies: A harmonic, trapping force and a constant, "drag" force. We study such systems by molecular simulations and use the simulation data to extract an effective GLE description. We find that within this description, in an active bath, the external force in the GLE is not equal to the physical external force, but rather a renormalized external force, which can be significantly smaller. The effect cannot be attributed to the mere temperature renormalization, which is also observed.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equation by means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equation for the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Enskog kinetic theory of binary granular suspensions: Heat flux and stability analysis of the homogeneous steady state

The Enskog kinetic theory of multicomponent granular suspensions employed previously [Gómez González, Khalil, and Garzó, Phys. Rev. E 101, 012904 (2020)2470-004510.1103/PhysRevE.101.012904] is considered further to determine the four transport coefficients associated with the heat flux. These transport coefficients are obtained by solving the Enskog equation by means of the application of the Chapman-Enskog method around the local version of the homogeneous state. Explicit forms of the heat flux transport coefficients are provided in steady-state conditions by considering the so-called second Sonine approximation to the distribution function of each species. Their quantitative variation on the control parameters of the mixture (masses and diameters, coefficients of restitution, concentration, volume fraction, and the background temperature) is demonstrated and the results show that in general the dependence of the heat flux transport coefficients on inelasticity is clearly different from that found in the absence of the gas phase (dry granular mixtures). As an application of the general results, the stability of the homogeneous steady state is analyzed by solving the linearized Navier-Stokes hydrodynamic equations. The linear stability analysis (which holds for wavelengths long compared with the mean free path) shows that the transversal and longitudinal modes are always stable with respect to long-enough wavelength excitations. This conclusion agrees with previous results derived for monocomponent and (dilute) bidisperse granular suspensions but contrasts with the instabilities found in previous works in dry (no gas phase) granular mixtures.

Fluctuation-dissipation relations far from equilibrium: a case study

Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium systems. Their applicability to non-equilibrium systems is, however, debated. Here, we simulate an active microrheology experiment, in which a spherical colloid is pulled with a constant external force through a fluid, creating near-equilibrium and far-from-equilibrium systems. We characterize the structural and dynamical properties of these systems, and reconstruct an effective generalized Langevin equation (GLE) for the colloid dynamics. Specifically, we test the validity of two FDTs: The first FDT relates the non-equilibrium response of a system to equilibrium correlation functions, and the second FDT relates the memory friction kernel in the GLE to the stochastic force. We find that the validity of the first FDT depends strongly on the strength of the external driving: it is fulfilled close to equilibrium and breaks down far from it. In contrast, we observe that the second FDT is always fulfilled. We provide a mathematical argument why this generally holds for memory kernels reconstructed from a deterministic Volterra equation for correlation functions, even for non-stationary non-equilibrium systems. Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an orthogonality constraint on the stochastic force, which is in fact equivalent to the validity of this Volterra equation. Such GLEs automatically satisfy the second FDT and are unique, which is desirable when using GLEs for coarse-grained modeling.

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.

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.

Shear-rate-dependent transport coefficients in granular suspensions

A recent model for monodisperse granular suspensions is used to analyze transport properties in spatially inhomogeneous states close to the simple (or uniform) shear flow. The kinetic equation is based on the inelastic Boltzmann (for low-density gases) with the presence of a viscous drag force that models the influence of the interstitial gas phase on the dynamics of grains. A normal solution is obtained via a Chapman-Enskog-like expansion around a (local) shear flow distribution which retains all the hydrodynamic orders in the shear rate. To first order in the expansion, the transport coefficients characterizing momentum and heat transport around shear flow are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by using a kinetic model of the Boltzmann equation. To simplify the analysis, the steady-state conditions when viscous heating is compensated by the cooling terms arising from viscous friction and collisional dissipation are considered to get the explicit forms of the set of generalized transport coefficients. The shear-rate dependence of some of the transport coefficients of the set is illustrated for several values of the coefficient of restitution.

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.

Unified rheology of vibro-fluidized dry granular media: From slow dense flows to fast gas-like regimes

Granular media take on great importance in industry and geophysics, posing a severe challenge to materials science. Their response properties elude known soft rheological models, even when the yield-stress discontinuity is blurred by vibro-fluidization. Here we propose a broad rheological scenario where average stress sums up a frictional contribution, generalizing conventional μ(I)-rheology, and a kinetic collisional term dominating at fast fluidization. Our conjecture fairly describes a wide series of experiments in a vibrofluidized vane setup, whose phenomenology includes velocity weakening, shear thinning, a discontinuous thinning transition, and gaseous shear thickening. The employed setup gives access to dynamic fluctuations, which exhibit a broad range of timescales. In the slow dense regime the frequency of cage-opening increases with stress and enhances, with respect to μ(I)-rheology, the decrease of viscosity. Diffusivity is exponential in the shear stress in both thinning and thickening regimes, with a huge growth near the transition.

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.

Transport coefficients of solid particles immersed in a viscous gas

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous, more general, treatment of the drag force [Garzó et al., J. Fluid Mech. 712, 129 (2012)]JFLSA70022-112010.1017/jfm.2012.404, here we take into account contributions to the (scaled) transport coefficients η^{*} (shear viscosity), κ^{*} (thermal conductivity), and μ^{*} (Dufour-like coefficient) coming from the temperature dependence of the (dimensionless) friction coefficient γ^{*} characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction, and the friction coefficient. The results indicate that the effect of the gas phase on η^{*} and μ^{*} is non-negligible (especially in the case of relatively dilute systems) while the form of κ^{*} is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.

Thinning and thickening in active microrheology

When pulling a probe particle in a many-particle system at a fixed velocity, the probe's effective friction, defined as the average external force over its velocity, γ(eff):=〈F(ex)〉/u, first stays constant (linear response), then decreases (thinning), and, finally, increases again (thickening). We propose a three-time-scale picture to describe the thinning and thickening behavior. There are three distinct time scales for the bath particles: diffusion, damping, and single probe-bath collision. The dominating time scales are controlled by the pulling velocity and determine the behavior of the probe's friction. We test and confirm this description with a Langevin dynamics simulation. Microscopically, we find that for computing the effective friction, the Maxwellian distribution of bath particles' velocities fails in the regime of high Reynolds and Peclet numbers. This can be understood based on the microscopic mechanism of thickening obtained in the T=0 limit. The dynamic regimes defined by the ratio of different time scales can explain several observations of thinning and thickening in literature.

Non-Newtonian hydrodynamics for a dilute granular suspension under uniform shear flow

We study in this work a steady shearing laminar flow with null heat flux (usually called "uniform shear flow") in a gas-solid suspension at low density. The solid particles are modeled as a gas of smooth hard spheres with inelastic collisions while the influence of the surrounding interstitial fluid on the dynamics of grains is modeled by means of a volume drag force, in the context of a rheological model for suspensions. The model is solved by means of three different but complementary routes, two of them being theoretical (Grad's moment method applied to the corresponding Boltzmann equation and an exact solution of a kinetic model adapted to granular suspensions) and the other being computational (Monte Carlo simulations of the Boltzmann equation). Unlike in previous studies on granular sheared suspensions, the collisional moment associated with the momentum transfer is determined in Grad's solution by including all the quadratic terms in the stress tensor. This theoretical enhancement allows for the detection and evaluation of the normal stress differences in the plane normal to the laminar flow. In addition, the exact solution of the kinetic model gives the explicit form of the velocity moments of the velocity distribution function. Comparison between our theoretical and numerical results shows in general a good agreement for the non-Newtonian rheological properties, the kurtosis (fourth velocity moment of the distribution function), and the velocity distribution of the kinetic model for quite strong inelasticity and not too large values of the (scaled) friction coefficient characterizing the viscous drag force. This shows the accuracy of our analytical results that allows us to describe in detail the flow dynamics of the granular sheared suspension.