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Force-induced diffusion in suspensions of hydrodynamically interacting colloids
- N. J. Hoh, R. N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 795 / 25 May 2016
- Published online by Cambridge University Press:
- 20 April 2016, pp. 739-783
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- Article
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We study the influence of hydrodynamic, thermodynamic and interparticle forces on the diffusive motion of a Brownian probe driven by a constant external force through a dilute colloidal dispersion. The influence of these microscopic forces on equilibrium self-diffusivity (passive microrheology) is well known: all three act to hinder the short- and long-time self-diffusion. Here, via pair-Smoluchowski theory, we explore their influence on self-diffusion in a flowing suspension, where particles and fluid have been set into motion by an externally forced probe (active microrheology), giving rise to non-equilibrium flow-induced diffusion. The probe’s motion entrains background particles as it travels through the bath, deforming the equilibrium suspension microstructure. The shape and extent of microstructural distortion is set by the relative strength of the external force $F^{\mathit{ext}}$ to the entropic restoring force $kT/a_{\mathit{th}}$ of the bath particles, defining a Péclet number $\mathit{Pe}\equiv F^{\mathit{ext}}/(2kT/a_{\mathit{th}})$; and also by the strength of hydrodynamic interactions, set by the range of interparticle repulsion ${\it\kappa}=(a_{\mathit{th}}-a)/a$, where $kT$ is the thermal energy and $a_{\mathit{th}}$ and $a$ are the thermodynamic and hydrodynamic sizes of the particles, respectively. We find that in the presence of flow, the same forces that hinder equilibrium diffusion now enhance it, with diffusive anisotropy set by the range of interparticle repulsion ${\it\kappa}$. A transition from hindered to enhanced diffusion occurs when diffusive and advective forces balance, $\mathit{Pe}\sim 1$, where the exact value is a sensitive function of the strength of hydrodynamics, ${\it\kappa}$. We find that the hindered to enhanced transition straddles two transport regimes: in hindered diffusion, stochastic forces in the presence of other bath particles produce deterministic displacements (Brownian drift) at the expense of a maximal random walk. In enhanced diffusion, driving the probe with a deterministic force through an initially random suspension leads to fluctuations in the duration of probe–bath particle entrainment, giving rise to enhanced, flow-induced diffusion. The force-induced diffusion is anisotropic for all $\mathit{Pe}$, scaling as $D\sim \mathit{Pe}^{2}$ in all directions for weak forcing, regardless of the strength of hydrodynamic interactions. When probe forcing is strong, $D\sim \mathit{Pe}$ in all directions in the absence of hydrodynamic interactions, but the picture changes qualitatively as hydrodynamic interactions grow strong. In this nonlinear regime, microstructural asymmetry weakens while the anisotropy of the force-induced diffusion tensor increases dramatically. This behaviour owes its origins to the approach toward Stokes flow reversibility, where diffusion along the direction of probe force scales advectively while transverse diffusion must vanish.
Hydrodynamic diffusion in active microrheology of non-colloidal suspensions: the role of interparticle forces
- N. J. Hoh, R. N. Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 785 / 25 December 2015
- Published online by Cambridge University Press:
- 16 November 2015, pp. 189-218
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Hydrodynamic diffusion in the absence of Brownian motion is studied via active microrheology in the ‘pure-hydrodynamic’ limit, with a view towards elucidating the transition from colloidal microrheology to the non-colloidal limit, falling-ball rheometry. The phenomenon of non-Brownian force-induced diffusion in falling-ball rheometry is strictly hydrodynamic in nature; in contrast, analogous force-induced diffusion in colloids is deeply connected to the presence of a diffusive boundary layer even when Brownian motion is very weak compared with the external force driving the ‘probe’ particle. To connect these two limits, we derive an expression for the force-induced diffusion in active microrheology of hydrodynamically interacting particles via the Smoluchowski equation, where thermal fluctuations play no role. While it is well known that the microstructure is spherically symmetric about the probe in this limit, fluctuations in the microstructure need not be – and indeed lead to a diffusive spread of the probe trajectory. The force-induced diffusion is anisotropic, with components along and transverse to the line of external force. The latter is identically zero owing to the fore–aft symmetry of pair trajectories in Stokes flow. In a naïve first approach, the vanishing relative hydrodynamic mobility at contact between the probe and an interacting bath particle was assumed to eliminate all physical contribution from interparticle forces, whereby advection alone drove structural evolution in pair density and microstructural fluctuations. With such an approach, longitudinal force-induced diffusion vanishes in the absence of Brownian motion, a result that contradicts well-known experimental measurements of such diffusion in falling-ball rheometry. To resolve this contradiction, the probe–bath-particle interaction at contact was carefully modelled via an excluded annulus. We find that interparticle forces play a crucial role in encounters between particles in the hydrodynamic limit – as they must, to balance the advective flux. Accounting for this force results in a longitudinal force-induced diffusion $D_{\Vert }=1.26aU_{S}{\it\phi}$, where $a$ is the probe size, $U_{S}$ is the Stokes velocity and ${\it\phi}$ is the volume fraction of bath particles, in excellent qualitative and quantitative agreement with experimental measurements in, and theoretical predictions for, macroscopic falling-ball rheometry. This new model thus provides a continuous connection between micro- and macroscale rheology, as well as providing important insight into the role of interparticle forces for diffusion and rheology even in the limit of pure hydrodynamics: interparticle forces give rise to non-Newtonian rheology in strongly forced suspensions. A connection is made between the flow-induced diffusivity and the intrinsic hydrodynamic microviscosity which recovers a precise balance between fluctuation and dissipation in far from equilibrium suspensions; that is, diffusion and drag arise from a common microstructural origin even far from equilibrium.