Roseanna Zia

Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions

In active, nonlinear microrheology, a Brownian “probe” particle is driven through a complex fluid and its motion tracked in order to infer the mechanical properties of the embedding material. In the absence of external forcing, the probe and background particles form an equilibrium microstructure that fluctuates thermally. Probe motion through the medium distorts the microstructure; the character of this deformation, and hence its influence on probe motion, depends on the strength with which the probe is forced, F^(ext), compared to thermal forces, kT/b, defining a Peclet number, Pe = F^(ext)/(kT/b), where kT is the thermal energy and b is the characteristic microstructural length scale. Recent studies showed that the mean probe speed can be interpreted as the effective material viscosity, whereas fluctuations in probe velocity give rise to an anisotropic force-induced diffusive spread of its trajectory. The viscosity and diffusivity can thus be obtained by two simple quantities—mean and mean-square displacement of the probe. The notion that diffusive flux is driven by stress gradients leads to the idea that the stress can be related directly to the microdiffusivity, and thus the anisotropy of the diffusion tensor reflects the presence of normal stress differences in nonlinear microrheology. In this study, a connection is made between diffusion and stress gradients, and a relation between the particle-phase stress and the diffusivity and viscosity is derived for a probe particle moving through a colloidal dispersion. This relation is shown to agree with two standard micromechanical definitions of the stress, suggesting that the normal stresses and normal stress differences can be measured in nonlinear microrheological experiments if both the mean and mean-square motion of the probe are monitored. Owing to the axisymmetry of the motion about a spherical probe, the second normal stress difference is zero, while the first normal stress difference is linear in Pe for Pe≫1 and vanishes as Pe^4 for Pe≪1. The expression obtained for stress-induced migration can be viewed as a generalized nonequilibrium Stokes–Einstein relation. A final connection is made between the stress and an “effective temperature” of the medium, prompting the interpretation of the particle stress as the energy density, and the expression for osmotic pressure as a “nonequilibrium equation of state.”

Single-particle motion in colloids: force-induced diffusion

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 probes 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 probes motion, depends on the strength with which the probe is forced, F^(ext), compared to thermal forces, kT/b, defining a Peclet number, Pe = F^(ext)/(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 Peclet 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 D^(micro) ~ D_aPe^2φ_b, where φ_b is the volume fraction of bath particles and D_a is the self-diffusivity of an isolated probe. At the other extreme of high Peclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (F^(ext)/η)φ_b, 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.

A micro-mechanical study of coarsening and rheology of colloidal gels: Cage building, cage hopping, and Smoluchowski's ratchet

We study via theory and dynamical simulation the evolving structure, particle dynamics, and time-dependent rheological properties of an aging colloidal gel, with a focus on the micro-mechanics that drive coarsening and age-related changes in linear-response behavior. When colloids in suspension attract one another, the attractions can lead to phase separation into particle-rich and particle-poor regions separated by a single interface. But this transition is sometimes interrupted before full separation occurs. With certain particle concentrations and interparticle potentials, the attractions that promote phase separation also inhibit it, frustrating the separation and “freezing in” a nonequilibrium particle configuration, resulting in a space-spanning gel. With attractions on the order of a few kT, gelation can produce nonfractal bicontinuous morphologies. In such “reversible” gels, thermal fluctuations are strong enough to rupture bonds and reform new ones, allowing restructuring of the gel over time. Bu...

Stress development, relaxation, and memory in colloidal dispersions: Transient nonlinear microrheology

The motion of a single Brownian particle in a complex fluid can reveal material behavior both at and away from equilibrium. In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. Most work in microrheology has focused on steady behavior and established the relationship between the motion of the probe, the microstructure, and the effective microviscosity of the medium. Transient behavior in the near-equilibrium, linear-response regime has also been studied via its connection to low-amplitude oscillatory probe forcing and the complex modulus; at very weak forcing, the microstructural response that drives viscosity is indistinguishable from equilibrium fluctuations. But important information about the basic physical aspects of structural development and relaxation in a medium is captured by startup and cessation of the imposed deformation in the nonlinear regime, where the structure is driven far from equilibrium. Here, we study theoretically and by dynamic simulation the transient behavior of a colloidal dispersion undergoing nonlinear microrheological forcing. The strength with which the probe is forced, Fext, compared to thermal forces, kT/b, governs the dynamics and defines a Peclet number, Pe = F^ext/(kT/b), where kT is the thermal energy and b is the colloidal bath particle size. For large Pe, a boundary layer (in which unsteady advection balances diffusion) forms at particle contact on the time scale of the flow, a/U, where a is the probe size and U its speed, whereas the wake forms over O(Pe) diffusive time steps. Similarly, relaxation following cessation occurs over several time scales corresponding to distinct physical processes. For very short times, the time scale for relaxation is set by a boundary layer of thickness δ ∼ (a+b)/Pe, and so τ ∼ δ^2/D_r, where Dr is the relative diffusivity between the probe of size a and a bath particle. Nearly all stress relaxation occurs during this time. At longer times, the Brownian diffusion of the bath particles acts to close the wake on a time scale set by how long it takes a bath particle to diffuse laterally across it, τ ∼ (a+b)^2/D_r. Although the majority of the microstructural relaxation occurs during this wake-healing process, it does so with little change in the stress. Also during relaxation, the probe travels backward in the suspension; this recovered strain is proportional to the free energy stored in the compressed particle configuration, an indicator that the stress is proportional to the free energy density stored entropically in the microstructure. Theoretical results are compared with Brownian dynamics simulation where it is found that the dilute theory captures the correct behavior even for concentrated suspensions. Two modes of forcing are studied: Constant force and constant velocity. Results are compared to analogous macrorheology results for suspensions undergoing simple shear flow.

Large amplitude oscillatory microrheology

We study the motion of a colloidal particle as it is driven by an oscillating external force of arbitrary amplitude and frequency through a colloidal dispersion. Large amplitude oscillatory flows (LAOFs) are examined predominantly from a phenomenological perspective in which experimental measurements inform constitutive models. Here, we investigate a LAOF from a microstructural perspective by connecting motion of the probe particle to the material response while making no assumptions a priori about how stress relaxes in the material. The suspension exerts nonconservative, hydrodynamic forces on the probe, while distortions in the particle configuration exert conservative forces: Brownian and interparticle forces, for example. The relative importance of each of these contributions to particle motion evolves with the degree of displacement from equilibrium. When the force on the probe is weak, the linear microviscoelasticity of the suspension is probed [see, e.g., Khair and Brady, J. Rheol. 49, 1449–1481 (2005)]. When oscillation rate is slow, the steady microrheology is probed [see, e.g., Squires and Brady, Phys. Fluids 17, 073101 (2005); Khair and Brady, J. Fluid Mech. 557, 73–117 (2006)]. This article develops a micromechanical model that recovers these limiting cases and then uses the same model to reveal the microrheology of colloidal dispersions deformed by a probe driven with arbitrary force amplitude and frequency. A chief result of this work is the discovery of a regime in which the resistance to motion of the probe particle is on average weaker than the resistance the probe experiences when deformed by high frequency oscillation. This hypoviscous effect arises when the reciprocating motion of the probe particle opens a channel free of other particles which is thus less resistive to probe motion. This effect is most apparent under the conditions of strong forces, rapid oscillation, and large extent of deformation.

Active microrheology: Fixed-velocity versus fixed-force

In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. It is conducted in two limiting forms: either the probe is propelled by a fixed force, as with magnetic tweezers, or it is driven at a fixed velocity, as with optical tweezers. Recent work has shown that the mean probe motion can be interpreted as an effective material viscosity, but that this viscosity depends on whether the fixed-force or fixed-velocity mode is employed. We compute the effective viscosity probed by fixed-velocity active microrheology of a dilute colloidal dispersion. A comparison is made between this new result and the effective viscosity probed in the fixed-force mode. In the absence of hydrodynamic interactions, the particle-phase contributions to the effective viscosity for the two modes differ by exactly a factor of two. A simple scaling argument has been previously advanced to rationalize this difference: in...

Delayed yield in colloidal gels: Creep, flow, and re-entrant solid regimes

We investigate the phenomenon of delayed yield in reversible colloidal gels via dynamic simulation, with a view toward revealing the microscopic particle dynamics and structural transformations that underlie the rheological behavior before, during, and after yield. Prior experimental studies reveal a pronounced delay period between application of a fixed shear stress and the onset of liquidlike flow, a so-called “delay time.” Catastrophic network failure—with sudden, cascading rupture of particle clusters or strands—is the primary model proposed for the structural evolution underlying rheological yield. However, no direct observation of such evolution has been made, owing to the difficulty of obtaining detailed microstructural information during the rapid yield event. Here, we utilize dynamic simulation to examine the microstructural mechanics and rheology of delayed yield. A moderately concentrated dispersion of Brownian hard spheres interacts via a short-range attractive potential of O(kT) that leads to arrested phase separation and the formation of a bicontinuous network of reversibly bonded particles. The linear-response rheology and coarsening dynamics of this system were characterized in our recent work. In the present study, a step shear stress is imposed on the gel, and its bulk deformation, as well as detailed positions and dynamics of all particles, are monitored over time. Immediately after the stress is imposed, the gel undergoes solidlike creep regardless of the strength of the applied stress. However, a minimum or “critical stress” is required to initiate yield: When the imposed stress is weak compared to the Brownian stress, the gel continues to undergo slow creeping deformation with no transition to liquidlike flow. Under stronger stress, creep is followed by a sudden increase in the strain rate, signaling yield, which then gives way to liquidlike viscous flow. The duration of the creep regime prior to yield is consistent with the delay time identified in prior experimental studies, decreasing monotonically with increasing applied stress. However, when the deformation rate is interrogated as a function of strain (rather than time), we find that a critical strain emerges: Yield occurs at the same extent of deformation regardless of the magnitude of the applied stress. Surprisingly, the gel network can remain fully connected throughout yield, with as few as 0.1% of particle bonds lost during yield, which relieve local glassy frustration and create localized liquidlike regions that enable yield. Brownian motion plays a central role in this behavior: When thermal motion is “frozen out,” both the delay time and the critical yield stress increase, showing that Brownian motion facilitates yield. Beyond yield, the long-time behavior depends qualitatively on the strength of the applied stress. In particular, at intermediate stresses, a “re-entrant solid” regime emerges, whereupon a flowing gel resolidifies, owing to flow-enhanced structural coarsening. A nonequilibrium phase diagram is presented that categorizes, and can be used to predict, the ultimate gel fate as a function of imposed stress. We make a connection between these behaviors and the process of ongoing phase separation in arrested colloidal gels.

Pair mobility functions for rigid spheres in concentrated colloidal dispersions: Force, torque, translation, and rotation

The formulation of detailed models for the dynamics of condensed soft matter including colloidal suspensions and other complex fluids requires accurate description of the physical forces between microstructural constituents. In dilute suspensions, pair-level interactions are sufficient to capture hydrodynamic, interparticle, and thermodynamic forces. In dense suspensions, many-body interactions must be considered. Prior analytical approaches to capturing such interactions such as mean-field approaches replace detailed interactions with averaged approximations. However, long-range coupling and effects of concentration on local structure, which may play an important role in, e.g., phase transitions, are smeared out in such approaches. An alternative to such approximations is the detailed modeling of hydrodynamic interactions utilizing precise couplings between moments of the hydrodynamic traction on a suspended particle and the motion of that or other suspended particles. For two isolated spheres, a set of these functions was calculated by Jeffrey and Onishi [J. Fluid Mech. 139, 261-290 (1984)] and Jeffrey [J. Phys. Fluids 4, 16-29 (1992)]. Along with pioneering work by Batchelor, these are the touchstone for low-Reynolds-number hydrodynamic interactions and have been applied directly in the solution of many important problems related to the dynamics of dilute colloidal dispersions [G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 375-400 (1972) and G. K. Batchelor, J. Fluid Mech. 74, 1-29 (1976)]. Toward extension of these functions to concentrated systems, here we present a new stochastic sampling technique to rapidly calculate an analogous set of mobility functions describing the hydrodynamic interactions between two hard spheres immersed in a suspension of arbitrary concentration, utilizing accelerated Stokesian dynamics simulations. These mobility functions provide precise, radially dependent couplings of hydrodynamic force and torque to particle translation and rotation, for arbitrary colloid volume fraction ϕ. The pair mobilities (describing entrainment of one particle by the disturbance flow created by another) decay slowly with separation distance: as 1/r, for volume fractions 0.05 ≤ ϕ ≤ 0.5. For the relative mobility, we find an initially rapid growth as a pair separates, followed by a slow, 1/r growth. Up to ϕ ≤ 0.4, the relative mobility does not reached the far-field value even beyond separations of many particle sizes. In the case of ϕ = 0.5, the far-field asymptote is reached but only at a separation of eight radii and after a slow 1/r growth. At these higher concentrations, the coefficients also reveal liquid-like structural effects on pair mobility at close separations. These results confirm that long-range many-body hydrodynamic interactions are an essential part of the dynamics of concentrated systems and that care must be taken when applying renormalization schemes.

Far-from-equilibrium sheared colloidal liquids: disentangling relaxation, advection, and shear-induced diffusion.

Using high-speed confocal microscopy, we measure the particle positions in a colloidal suspension under large-amplitude oscillatory shear. Using the particle positions, we quantify the in situ anisotropy of the pair-correlation function, a measure of the Brownian stress. From these data we find two distinct types of responses as the system crosses over from equilibrium to far-from-equilibrium states. The first is a nonlinear amplitude saturation that arises from shear-induced advection, while the second is a linear frequency saturation due to competition between suspension relaxation and shear rate. In spite of their different underlying mechanisms, we show that all the data can be scaled onto a master curve that spans the equilibrium and far-from-equilibrium regimes, linking small-amplitude oscillatory to continuous shear. This observation illustrates a colloidal analog of the Cox-Merz rule and its microscopic underpinning. Brownian dynamics simulations show that interparticle interactions are sufficient for generating both experimentally observed saturations.

Pair mobility functions for rigid spheres in concentrated colloidal dispersions: Stresslet and straining motion couplings

Accurate modeling of particle interactions arising from hydrodynamic, entropic, and other microscopic forces is essential to understanding and predicting particle motion and suspension behavior in complex and biological fluids. The long-range nature of hydrodynamic interactions can be particularly challenging to capture. In dilute dispersions, pair-level interactions are sufficient and can be modeled in detail by analytical relations derived by Jeffrey and Onishi [J. Fluid Mech. 139, 261-290 (1984)] and Jeffrey [Phys. Fluids A 4, 16-29 (1992)]. In more concentrated dispersions, analytical modeling of many-body hydrodynamic interactions quickly becomes intractable, leading to the development of simplified models. These include mean-field approaches that smear out particle-scale structure and essentially assume that long-range hydrodynamic interactions are screened by crowding, as particle mobility decays at high concentrations. Toward the development of an accurate and simplified model for the hydrodynamic interactions in concentrated suspensions, we recently computed a set of effective pair of hydrodynamic functions coupling particle motion to a hydrodynamic force and torque at volume fractions up to 50% utilizing accelerated Stokesian dynamics and a fast stochastic sampling technique [Zia et al., J. Chem. Phys. 143, 224901 (2015)]. We showed that the hydrodynamic mobility in suspensions of colloidal spheres is not screened, and the power law decay of the hydrodynamic functions persists at all concentrations studied. In the present work, we extend these mobility functions to include the couplings of particle motion and straining flow to the hydrodynamic stresslet. The couplings computed in these two articles constitute a set of orthogonal coupling functions that can be utilized to compute equilibrium properties in suspensions at arbitrary concentration and are readily applied to solve many-body hydrodynamic interactions analytically.