7 results
Impact of polydispersity and confinement on diffusion in hydrodynamically interacting colloidal suspensions
- Emma Gonzalez, Christian Aponte-Rivera, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 925 / 25 October 2021
- Published online by Cambridge University Press:
- 01 September 2021, A35
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We present a computational study of the equilibrium dynamics of a polydisperse hard-sphere colloidal dispersion confined in a spherical cavity. We account for many-body hydrodynamic and lubrication interactions between particles and with the confining cavity utilizing our confined Stokesian dynamics model, expanded here for size polydispersity. We find that, even though the tendency of polydispersity to homogenize structure in a suspension is still present in confinement, strong correlations induced by the cavity resist homogenization. Although seemingly opposite, these two effects have a common driver, which is to maximize configurational entropy of particles in the cavity interior. These structural effects couple with the hydrodynamics to change the particle dynamics: polydispersity weakens lubrication effects near the cavity wall, allowing small (large) particles to diffuse faster (slower) than in a monodisperse suspension. As a small (large) particle gets farther from the wall, polydispersity weakens many-body hydrodynamic couplings, driving diffusivity up (down). While the local cage dynamics dominates short-time self-diffusion, long-time dynamics is also affected. In the concentrated regime, polydispersity and confinement combine to induce radial de-mixing into size-segregated populations. The cavity becomes the most influential ‘nearest neighbour’, setting the length scale of and dynamics within these radial domains. This intermediate length-scale caging makes the angular dynamics insensitive to polydispersity but leads to radial long-time mean-square displacement that changes qualitatively with volume composition. These results hold promise for explaining colloidal-scale physics implicated in the functioning of biological cells, and the engineering of non-living confined colloids where size de-mixing could be useful in the design of encapsulated micro-reactors and therapeutic vesicles.
Faxén formulas for particles of arbitrary shape and material composition
- Benjamin E. Dolata, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 910 / 10 March 2021
- Published online by Cambridge University Press:
- 11 January 2021, A22
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We prove a duality between the functional forms of the Faxén formulas associated with a particle of a given shape and material composition and the corresponding singularity solutions for the velocity disturbances induced by that particle, and extend it to the case of systems with coupled transport processes, enabling the solution of a large family of problems via Faxén methods. Prior approaches to constructing proofs of duality of Faxén formulas and Stokes-flow singularities relied on knowledge of all boundary conditions on all particle surfaces, viz. the Lorentz reciprocal theorem approach. We recognized that, in order to bypass the complexity of boundary conditions one can instead invoke energy methods that give reciprocity between operators rather than between specific stress and velocity fields. We derive reciprocal relations between operators, from which we demonstrate that the Faxén/singularity duality is a consequence of a generalized reciprocal relation between conjugate thermodynamic variables. We use our reciprocal relations to derive expressions for the hydrodynamic force on a sphere of arbitrary composition, the hydrodynamic stresslet exerted on a deformable droplet in an arbitrary velocity field, the phoretic force exerted on a rigid particle in the thin double-layer limit in response to arbitrary externally imposed field and the total stresslet on a charged particle in an arbitrary velocity field, i.e. an electroviscous Faxén law.
Heterogeneous dispersions as microcontinuum fluids
- Benjamin E. Dolata, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 888 / 10 April 2020
- Published online by Cambridge University Press:
- 11 February 2020, A28
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We present a non-local ‘microcontinuum’ constitutive equation describing particle suspensions undergoing heterogeneous flows in the low Stokes and particle Reynolds number regime. Most prior models rely on mesoscale averages of structural dynamics that smear out non-local effects or otherwise place strong restrictions on flow type. We relieve this restriction here by ensemble averaging the exact equations that govern the suspension at the microscale, thereby obtaining explicit structure-property connections. The result is a pointwise, heterogeneous ensemble average of the suspension stress valid for suspended particles of arbitrary shape, size, composition and concentration, as well as arbitrary sources of heterogeneity, including non-uniform shear fields, heterogeneous force fields or spatially varying volume fractions. This non-local model accounts for spatial and temporal variations in the flow structure over arbitrary length and time scales. We express the microcontinuum constitutive equation as a superposition of gradient operators that automatically accounts for flow heterogeneity over any length scale larger than the particle size. Batchelor’s result for a homogeneous suspension is recovered in the limit of zero gradients; non-zero gradient terms provide non-local corrections to the average stress and account for statistical heterogeneity in the suspension. We utilize energy methods to compute the influence of these gradient operators on the pointwise-averaged stress tensor, revealing a deep connection to the microhydrodynamics formalism. We apply this general framework to a dilute suspension of spherical particles and show that the resultant constitutive equations are consistent with experimental observations.
Transient nonlinear microrheology in hydrodynamically interacting colloidal dispersions: flow cessation
- Ritesh P. Mohanty, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 884 / 10 February 2020
- Published online by Cambridge University Press:
- 09 December 2019, A14
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Relaxation of a colloidal dispersion following flow shutoff occurs over distinct time scales, providing a useful method for identifying microscopic forces that are otherwise difficult to detect. Active microrheology facilitates such interrogation for small-scale systems such as biological cells, where a Brownian probe is driven by an external force through the dispersion and its motion is tracked to infer flow properties. An interplay between external forces, Brownian motion, hydrodynamic interactions and interparticle forces deforms the microstructure surrounding the probe, which alters probe speed. Application of the Stokes drag law relates mean probe speed to the effective viscosity of the embedding medium. We present a micromechanical theoretical model for this relationship during nonlinear transient relaxation upon removal of the external force. Upon cessation, contributions to viscosity linear in the force vanish instantly, but hydrodynamics influences subsequent relaxation, as well as the pre-cessation structure that sets the initial condition. Stronger pre-cessation flow drives faster relaxation of both structure and rheology. Non-equilibrium entropic contributions persist after shutoff, decaying over time. Hydrodynamic interactions slow this relaxation by hindering relative Brownian and interparticle displacements. The dissipation of entropically stored energy produces a deterministic force that drives deterministic probe motion even after the external force is removed, giving a natural connection to the chemical potential. Using this idea, we show that probe back-travel gives a direct measure of the osmotic pressure and the time scale over which entropically stored energy is dissipated. Modelling a range of repulsions provides a platform for particle formulation tuned to desired transient response.
Non-equilibrium pair interactions in colloidal dispersions
- Benjamin E. Dolata, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 836 / 10 February 2018
- Published online by Cambridge University Press:
- 12 December 2017, pp. 694-739
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We study non-equilibrium pair interactions between microscopic particles moving through a model shear-thinning fluid. Prior efforts to model pair interactions in non-Newtonian fluids have largely focused on constitutive models derived from polymer-chain kinetic theories focusing on conformational degrees of freedom, but neglecting the details of microstructural evolution beyond a single polymer length scale. To elucidate the role of strong structural distortion in mediating pair interactions in Brownian suspensions, we formulate and solve a Smoluchowski equation describing the detailed evolution of the particle configuration between and around a pair of microscopic probes driven at fixed velocity by an external force through a colloidal dispersion. To facilitate analysis, we choose a model system of Brownian hard spheres that do not interact hydrodynamically; while simple, this ‘freely draining’ model permits insight into connections between microstructure and rheology. The flow induces a non-equilibrium particle density gradient that gives rise to both viscous drag and an interactive force between the probes. The drag force acts to slow the centre-of-mass velocity of the pair, while the interactive force arising from osmotic pressure gradients can lead to attraction or repulsion, as well as deterministic reorientation of the probes relative to the external force. The degree to which the microstructure is distorted, and the shape of that distortion, depend on the arrangement of the probes relative to one another and their orientation to the driving force. It also depends on the magnitude of probe velocity relative to the Brownian velocity of the suspension. When only thermal fluctuations set probe velocity, the equilibrium depletion attraction is recovered. For weak forcing, long-ranged interactions mediated via the bath-particle flux give rise to entropic forces on the probes. The linear response is a viscous drag that slows forward motion; only the weakly nonlinear response can produce relative motion–attraction, repulsion or reorientation of the probes. We derive entropic coupling tensors, similar in ethos to pair hydrodynamic tensors, to describe this behaviour. The structural symmetry that permits this analogy is lost when forcing becomes strong, revealing instabilities in system behaviour. Far from equilibrium, the interactive force depends explicitly on the initial probe separation, orientation and strength of forcing; widely spaced probes interact through the distorted microstructure, whereas the behaviour of closely spaced probes is largely set by excluded-volume effects. In this regime, a pair of closely spaced probes sedimenting side-by-side tend to attract and reorient to permit alignment of their line-of-centres with the flow, while widely spaced probes fall without reorienting. Our results show qualitative agreement with experimental observations and provide a potential connection to the observed column instability in shear-thinning fluids.
Equilibrium structure and diffusion in concentrated hydrodynamically interacting suspensions confined by a spherical cavity
- Christian Aponte-Rivera, Yu Su, Roseanna N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 836 / 10 February 2018
- Published online by Cambridge University Press:
- 11 December 2017, pp. 413-450
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The short- and long-time equilibrium transport properties of a hydrodynamically interacting suspension confined by a spherical cavity are studied via Stokesian dynamics simulations for a wide range of particle-to-cavity size ratios and particle concentrations. Many-body hydrodynamic and lubrication interactions between particles and with the cavity are accounted for utilizing recently developed mobility and resistance tensors for spherically confined suspensions (Aponte-Rivera & Zia, Phys. Rev. Fluids, vol. 1(2), 2016, 023301). Study of particle volume fractions in the range $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.40$ reveals that confinement exerts a qualitative influence on particle diffusion. First, the mean-square displacement over all time scales depends on the position in the cavity. Additionally, at short times, the diffusivity is anisotropic, with diffusion along the cavity radius slower than diffusion tangential to the cavity wall, due to the anisotropy of hydrodynamic coupling and to confinement-induced spatial heterogeneity in particle concentration. The mean-square displacement is anisotropic at intermediate times as well and, surprisingly, exhibits superdiffusive and subdiffusive behaviours for motion along and perpendicular to the cavity radius respectively, depending on the suspension volume fraction and the particle-to-cavity size ratio. No long-time self-diffusive regime exists; instead, the mean-square displacement reaches a long-time plateau, a result of entropic restriction to a finite volume. In this long-time limit, the higher the volume fraction is, the longer the particles take to reach the long-time plateau, as cooperative rearrangements are required as the cavity becomes crowded. The ordered dynamical heterogeneity seen here promotes self-organization of particles based on their size and self-mobility, which may be of particular relevance in biophysical systems.
Single-particle motion in colloids: force-induced diffusion
- ROSEANNA N. ZIA, JOHN F. BRADY
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- Journal:
- Journal of Fluid Mechanics / Volume 658 / 10 September 2010
- Published online by Cambridge University Press:
- 09 June 2010, pp. 188-210
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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.