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In microrheology, the local rheological properties, such as the viscoelasticity of a complex fluid, are inferred from the free or forced motion of embedded colloidal probe particles. Theoretical machinery developed for forced-probe microrheology of colloidal suspensions focused on either constant-force (CF) or constant-velocity (CV) probes, while in experiments, neither the force nor the kinematics of the probe is fixed. More importantly, the constraint of CF or CV introduces a difficulty in the meaningful quantification of the fluctuations of the probe due to a thermodynamic uncertainty relation. It is known that, for a Brownian particle trapped in a harmonic potential well, the product of the standard deviations of the trap force and the particle position is dkBT in d dimensions, with kBT being the thermal energy. As a result, if the force (position) is not allowed to fluctuate, the position (force) fluctuation becomes infinite. To allow the measurement of fluctuations in theoretical studies, in this work, we consider a microrheology model in which the embedded probe is dragged along by a moving harmonic potential so that both its position and the trap force are allowed to fluctuate. Starting from the full Smoluchowski equation governing the dynamics of N hard active Brownian particles, we derive a pair Smoluchowski equation describing the dynamics of the probe as it interacts with one bath particle by neglecting hydrodynamic interactions among particles in the dilute limit. From this, we determine the mean and the variance (i.e., fluctuation) of the probe position in terms of the pair probability distribution. We then characterize the behavior of the system in the limits of both weak and strong trap. By taking appropriate limits, we show that our generalized model can be reduced to the well-studied CF or CV microrheology models. [ABSTRACT FROM AUTHOR]

A simple model for the rheological behavior of concentrated colloidal dispersions is developed. For a suspension of Brownian hard spheres there are two contributions to the macroscopic stress: a hydrodynamic and a Brownian stress. For small departures from equilibrium, the hydrodynamic contribution is purely dissipative and gives the high-frequency dynamic viscosity. The Brownian contribution has both dissipative and elastic parts and is responsible for the viscoelastic behavior of colloidal dispersions. An evolution equation for the pair-distribution function is developed and from it a simple scaling relation is derived for the viscoelastic response. The Brownian stress is shown to be proportional to the equilibrium radial-distribution function at contact, g(2;[lowercase_phi_synonym]), divided by the short-time self-diffusivity, D0s([lowercase_phi_synonym]), both evaluated at the volume fraction [lowercase_phi_synonym] of interest. This scaling predicts that the Brownian stress diverges at random close packing, [lowercase_phi_synonym]m, with an exponent of -2, that is, η’0 ∼ η(1 - [lowercase_phi_synonym]/[lowercase_phi_synonym]m)-2, where η’0 is the steady shear viscosity of the dispersion and η is the viscosity of the suspending fluid.Both the scaling law and the predicted magnitude are in excellent accord with experiment. For viscoelastic response, the theory predicts that the proper time scale is a2/D0s, where a is the particle radius, and, when appropriately scaled, the form of the viscoelastic response is a universal function for all volume fractions, again in agreement with experiment. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian stress. When the length scale characterizing the interparticle forces is comparable to the particle size, the theory predicts that there is only a quantitative... [ABSTRACT FROM AUTHOR]

The rheology during the start-up and cessation of simple shear flow has been investigated for near hard-sphere colloidal suspensions. Simulations augmented by theoretical analysis are used to determine how the non-Newtonian stress development and relaxation depend on the microstructure. Accelerated Stokesian dynamics (ASD) and Brownian dynamics (BD) simulations are used for 0.05=Pe=500 in concentrated freely flowing suspensions; the Péclet number defining the ratio of shear to thermal motion is Pe = 3πηγa³/kT with η the suspending fluid viscosity, γ the shear rate, and kT the thermal energy. Theoretical predictions based on the Smoluchowski equation for dilute suspensions are made, and these are primarily used for comparison with results from BD simulations in which hydrodynamic interactions are neglected. For suspensions with hydrodynamics, simulations by ASD are used to probe start-up and flow cessation over a large range of Pe; these studies focus on solid volume fraction ϕ = 0:4, with more limited examinations at other ϕ. The use of both BD and ASD simulations allows us to discriminate hydrodynamic interaction effects on the suspension rheology. The Brownian stresses computed by either method exhibit overshoots of their steady state value during the start-up of shear flow. The overshoots occur at strain amplitudes which depend on Pe, and the overshoot is described by a model based on extension of the concept of cage-breaking from glass dynamics. Results from the relaxation of a sheared suspension show that the distortion of the pair distribution function from its equilibrium form has a fast radial relaxation and a slow angular relaxation. The various rheometric functions (relative viscosity; first and second normal stress differences) are found to respond on different timescales, reflecting their different dependences on the flow-induced structure. A re-examination of steady shear flow allows us to find normal stress differences which tend properly toward zero at small Pe, unlike prior work; the discrepancy is found to be due to finite size scaling, as small simulations used in prior work resulted in excessively large normal stress responses at small Pe. [ABSTRACT FROM AUTHOR]

In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics (SEASD), a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions. SEASD is based on the framework of Stokesian Dynamics (SD) with extension to compressible solvents, and uses the Spectral Ewald (SE) method [Lindbo and Tornberg (2010) [29] ] for the wave-space mobility computation. To meet the performance requirement of dynamic simulations, we use Graphic Processing Units (GPU) to evaluate the suspension mobility, and achieve an order of magnitude speedup compared to a CPU implementation. For further speedup, we develop a novel far-field block-diagonal preconditioner to reduce the far-field evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio and Brady (2003) [39] . We extensively discuss implementation and parameter selection strategies in SEASD, and demonstrate the spectral accuracy in the mobility evaluation and the overall O ( N log ⁡ N ) computation scaling. We present three computational examples to further validate SEASD and SEASD-nf in monodisperse and bidisperse suspensions: the short-time transport properties, the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear Brownian rheology. Our validation results show that the agreement between SEASD and SEASD-nf is satisfactory over a wide range of parameters, and also provide significant insight into the dynamics of polydisperse colloidal suspensions. [ABSTRACT FROM AUTHOR]

We study the fluctuating motion of a Brownian-sized probe particle as it is dragged by a constant external force through a colloidal dispersion. In this nonlinear-microrheology problem, collisions between the probe and the background bath particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive spread of the probe's trajectory. The influence of the former is determined by the spatial configuration of the bath particles and the force with which the probe perturbs it. With no external forcing the probe and bath particles form an equilibrium microstructure that fluctuates thermally with the solvent. Probe motion through the dispersion distorts the microstructure; the character of this deformation, and hence its influence on the probe's motion, depends on the strength with which the probe is forced, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe = Fext/(kT/b), where kT is the thermal energy and b the bath particle size. It is shown that the long-time mean-square fluctuational motion of the probe is diffusive and the effective diffusivity of the forced probe is determined for the full range of Péclet number. At small Pe Brownian motion dominates and the diffusive behaviour of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced ‘microdiffusivity’ that scales as Dmicro ~ DaPe2φb, where φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high Péclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (Fext/η)φb, where η is the viscosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion longitudinal to the direction of forcing larger in both limits compared to transverse diffusion, but more strongly so in the high-Pe limit. The diffusivity is computed for all Pe for a probe of size a in a bath of colloidal particles, all of size b, for arbitrary size ratio a/b, neglecting hydrodynamic interactions. The results are compared with the force-induced diffusion measured by Brownian dynamics simulation. The theory is also compared to the analogous shear-induced diffusion of macrorheology, as well as to experimental results for macroscopic falling-ball rheometry. The results of this analysis may also be applied to the diffusive motion of self-propelled particles. [ABSTRACT FROM AUTHOR]

We consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particle-tracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes a and b, respectively, and the bath particles are spheres of radii b. The probe’s shape is such that when its major or minor axis is the axis of revolution the excluded-volume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number (Pe), neglecting hydrodynamic interactions. Here, Pe is defined as the non-dimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, a⁁=a/b, thereby delineating the role of the probe’s shape. For a prolate probe, regardless of the value of a⁁, the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small Pe until a second Newtonian plateau is reached as Pe→∞. After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids 17(7), 073101 (2005)] and conventional (macro-)rheological investigations [Bergenholtz et al., “The non-Newtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech. 456, 239–275 (2002)]. For an oblate probe, the microviscosity again transitions between two Newtonian plateaus: for a⁁<3.52 (to two decimal places) the microviscosity at small Pe is greater than at large Pe (again, velocity thinning); however, for a⁁>3.52 the microviscosity at small Pe is less than at large Pe, which suggests it “velocity thickens” as Pe is increased. This anomalous velocity thickening—due entirely to the probe shape—highlights the care needed when designing microrheology experiments with non-spherical probes. [ABSTRACT FROM AUTHOR]

The bulk viscosity of a suspension is defined analogous to that for a pure fluid as the constant of proportionality relating the deviation of the trace of the macroscopic stress from its equilibrium value to the average rate of expansion. In a suspension the equilibrium macroscopic stress is the sum of the thermodynamic pressure of the fluid and the osmotic pressure of the suspended particles. Rigid particles suspended in an expanding fluid cause a disturbance flow that contributes to the total mechanical pressure in the system, thereby changing the effective bulk viscosity. Expressions are derived to compute the effective bulk viscosity for all concentrations and all expansion rates for a system of rigid particles suspended in a uniformly expanding fluid. The expansion flow drives the suspension microstructure out of equilibrium and the thermal motion of the particles tries to restore the equilibrium. The Péclet number, defined as the expansion rate made dimensionless with the Brownian time scale, governs the departure of the microstructure from equilibrium. The contribution to bulk viscosity is determined to second order in volume fraction of particles for all compression rates (all negative Péclet numbers). A “compression thickening” of the suspension is observed at large compression rates. [ABSTRACT FROM AUTHOR]

We study the motion of a Brownian probe particle subjected to a small amplitude oscillatory external force and immersed in a colloidal dispersion, as a theoretical benchmark for particle-tracking microrheology experiments. The application of an external force on the probe drives the microstructure of the dispersion out of equilibrium; opposing this is the Brownian diffusion of the probe and colloidal “bath” particles. The degree to which the microstructure is driven away from equilibrium is given by the nondimensional external force on the probe, or Péclet number, Pe. The nonequilibrium microstructure of the dispersion is calculated for small departures from equilibrium, i.e., to first order in Pe, and to leading order in the bath particle volume fraction, accounting for excluded volume and hydrodynamic interactions between particles. The nonequilibrium microstructure is used to calculate the average velocity of the probe, from which one may infer the complex microviscosity (or modulus) of the dispersion via application of Stokes drag law. The microviscosity is computed over the entire range of oscillation frequencies, thereby determining the linear viscoelastic response of the dispersion. After appropriate scaling, our results are in qualitative agreement with traditional (macro-) rheology studies, suggesting that oscillatory-probe microrheology can be a useful tool to examine the viscoelasticity of hard-sphere colloidal dispersions and perhaps other complex fluids. [ABSTRACT FROM AUTHOR]

A reply to the comments of Cichocki and Felderhof concerning my paper on the rheological behavior of concentrated colloidal dispersions [J. Chem. Phys. 99, 567 (1993)] is given. Differences exist between my scaling predictions as maximum packing is approached for the reduced dynamic viscosity and the dilute analysis of Cichocki and Felderhof. [ABSTRACT FROM AUTHOR]

A foreword to the "Journal of Rheology" is presented.