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 nonlinearmicrorheology 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, F ext , compared to thermal forces, kT/b, defining a Péclet number,, 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 D micro ∼ D a P e 2 φ 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 Péclet 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.
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.
Performance of mesoscale modeling methods for predicting rheological properties of charged polystyrene/water suspensions J. Rheol. 56, 353 (2012) Three-dimensional flow of colloidal glasses J. Rheol. 56, 259 (2012) Emergence of turbid region in startup flow of CTAB/NaSal aqueous solutions between parallel plates J. Rheol. 56, 245 (2012) Nonlinear rheology and yielding in dense suspensions of hard anisotropic colloids J. Rheol. 55, 1069Rheol. 55, (2011 Additional information on J. Rheol. SynopsisThe 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, F ext , compared to thermal forces, kT/b, governs the dynamics and defines a P eclet number, Pe ¼ F ext =ðkT=bÞ, 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 d $ ða þ bÞ=Pe, and so s $ d 2 =D r , where D r 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, s $ ða þ bÞ 2 =D r . Although the majority of the microstructural relaxation occurs during this wakehealing 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 a) Author to who...
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