Active microrheology of hydrodynamically interacting colloids: Normal stresses and entropic energy density

A single Brownian “probe” particle is driven by an external force through a colloidal suspension and its motion studied to elucidate the relative impacts of external, Brownian, and interparticle forces on the suspension stress. As the probe moves through the suspension, distortions to and relaxation of the particle arrangement give rise to nonequilibrium stress. The shape of the distorted microstructure is set by the strength of the external force, F₀, relative to the entropic restoring force, kT/a_th, of the suspension, and by the balance of microscopic forces between the constituent particles. The former is given by the Péclet number, $Pe≡F0/(2kT/ath)$⁠, where kT is the thermal energy and a_th is the thermodynamic size of the particles. The latter comprise external, Brownian, and interparticle forces, and the sensitivity of each to flow strength Pe is set by the dimensionless repulsion range, $κ≡(ath−a)/a$⁠, where a is the hydrodynamic size of the particles. The total stress comprises hydrodynamic and entropic contributions which manifest as Brownian, interparticle, and external force-induced stress. To analyze the influence of these forces on structure and suspension stress as they evolve with flow strength, we formulate and solve a Smoluchowski equation analytically in the dual limits of weak and strong external force and hydrodynamic interactions, and numerically for arbitrary values of Pe and κ. Nonequilibrium statistical mechanics are then utilized to compute elements of the stress tensor. Owing to the axisymmetric geometry of the microstructure about the line of the external force, only the diagonal elements are nonzero. When hydrodynamic interactions are negligibly weak, only the hard-sphere interparticle force matters regardless of the flow strength, and the results of Zia and Brady [J. Rheol. 56(5), 1175–1208 (2012)] are recovered whereby normal stresses scale as Pe² and Pe in the limits of weak and strong forcing, respectively. That is, entropic forces dominate suspension stress regardless of the value of Pe when hydrodynamic interactions are weak. As the repulsion range κ shrinks, hydrodynamic interactions begin to play a role: When forcing is weak, Brownian disturbance flows provide the dominant contribution to suspension stress, but as Pe increases, the external force-induced stress takes over to dominate the total stress. Interestingly, the total suspension stress decreases as the strength of hydrodynamic interactions increases, regardless of the value of Pe. That is, hydrodynamic interactions suppress suspension stress. Owing to the dependence of hydrodynamic interactions on particle configuration, this stress suppression varies with flow strength: At low Pe, the stress scales as Pe² and the suppression is quantitative, whereas at high Pe, the stress scales as Pe^δ, where 1 ≥ δ ≥ 0.799 for hydrodynamic interactions spanning from weak to strong. We identify the origin of such suppression via an analysis of pair trajectories: While entropic forces—interparticle repulsion and Brownian motion—destroy reversible trajectories, hydrodynamic interactions suppress structural asymmetry and this underlies the suppression of the nonequilibrium stress. We relate the stress to the energy density: Hydrodynamic interactions shield particles from direct collisions and promote fore-aft and structural symmetry, resulting in reduced entropic energy storage.