EFFECT OF HYDRODYNAMICS ON COLLOIDAL SUSPENSION RHEOLOGY DURING FLOW STARTUP AND CESSATION

Abstract

We theoretically study the impact of particle roughness, temperature, and hydrodynamic interactions on the development and relaxation of structure and rheology of colloidal dispersions of Brownian spheres subjected to sudden startup and cessation of flow. In particular, we focus on the disparity in timescales over which microscopic forces - hydrodynamic, Brownian and other interparticle forces - act and drive transient rheology, via the theoretical framework of active microrheology. In active microrheology, a probe particle is driven and its motion is tracked in order to infer the physical properties of the surrounding medium. As the probe moves through the suspension, it distorts the particle arrangement (or the microstructure) around it. The microstructure around the probe evolves in time and reaches a steady state at long times. The steady-state structure and its corresponding rheological behaviors are well understood: the relative importance of microscopic forces evolves with the strength of the external flow, which leads to sustained microstructural asymmetry that in turn gives rise to familiar non-Newtonian behaviors like flow-thinning and flow-thickening. However, the transition from equilibrium to steady flow is sometimes marked by overshoots or oscillations in stress or viscosity that suggest a temporally evolving competition between the rate processes that dissipate and store flow energy. Likewise, the evolution of particle microstructure from equilibrium structure to its steady-state asymmetric structure takes place over observable timescales, suggesting a temporally evolving balance between colloidal-scale forces. Similarly, if the external flow is suddenly removed, the particle gradients in the steady-state structure relax over a finite time. In this dissertation, we study the evolution of structure in time, and the corresponding transient rheology during sudden microrheological flow startup and cessation. During flow startup, the motion of the probe drives its surrounding particles out of equilibrium. Brownian motion of the particles gives rise to an entropic restoring force that acts to restore the equilibrium configuration. The ratio of the external force to the restoring entropic force defines a Péclet number, Pe = Fext/(2kT/ath), where Fext is the magnitude of the external force, ath is the thermodynamic radius of the particles, and kT is the thermal energy of the bath. An excluded annulus shell is utilized to model arbitrary repulsion range between particles, or arbitrary strengths of hydrodynamic interactions between particles. The relative strengths of the entropic and hydrodynamic forces are set by the repulsion range, κ = (ath − a)/a (where a is the hydrodynamic radius of the particles), between a pair of particles in a suspension of hydrodynamically interacting repulsive hard spheres. We formulate and solve a transient Smoluchowski equation that governs the spatiotemporal evolution of the particle microstructure around a probe. The governing equations are solved analytically in the asymptotic limit of weak probe forcing, Pe, where microstructural distortions are indistinguishable from equilibrium Brownian fluctuations. The equations are also solved analytically for arbitrary Pe in the dual limits of weak and strong hydrodynamic interactions to examine the early-time structural evolution during flow startup. For arbitrary strengths of Pe and κ, numerical methods were utilized to obtain the transient structural evolution upon flow startup and cessation. The time-dependent structure was then connected to the transient viscosity of the suspension utilizing statistical mechanics. We find that hydrodynamic forces grow slowly in time relative to conservative entropic forces during flow startup: at early times, hydrodynamic and entropic Brownian viscosities scale quadratically and linearly, respectively, and therefore the Brownian viscosity dominates the total viscous drag at early times. Since the Brownian contribution to the viscosity decreases monotonically with increasing flow strength, the total viscosity flow-thins at early times after flow startup, a trend that ultimately reverses at long times to give the expected flow- thickening at steady state. This behavior owes its origin to the dominance of Brownian diffusion in setting short-time particle dynamics, regardless of Pe, which hinders maturation of the boundary layer. Later, when the relatively slowly growing advective relative motion dominates, particles remain coupled together long enough to produce the long-duration interactions associated with flow-thickening behavior. This lengthening of the duration of pair encounters as hydrodynamic coupling grows stronger delays the attainment of steady-state viscosity. An overshoot in the Brownian contribution to the viscosity signals the final formation of the boundary layer in the downstream region of the probe, giving way to "time-thinning", a post-overshoot decay of viscosity to its steady-state value, where advection ultimately outpaces the ability of Brownian motion to viscously dissipate flow energy. This apparent "yield" behavior is simply a transition from more viscous to less viscous flow set by relative relaxation timescales of the structure, revealing that overshoots can be explained at a pair level as a disparity between entropic and advective transport rates. Time-dependent fixed-Pe hydro-thinning and hydro-thickening behaviors are discussed. Upon flow cessation, the non-Newtonian rheology arising directly from hydrodynamic forces dissipates instantaneously, as expected, while the entropic contributions decay over an observable timescale. This disparity in timescales can be utilized in a flow cessation experiment to decouple the individual microscopic contributions to steady-state rheology. We find that while increasing the pre-cessation flow strength enhances both the structural and rheological relaxation rates, hydrodynamic interactions slow down both relaxation rates by hindering relative particle motion. The dissipation of stored entropic energy in the steady-state deformed structure is explored, and we find that the distorted structure gives rise to an entropic force that drives the probe motion even after the external force on the probe is removed. We find that the probe back travels after flow cessation, which is driven by osmotic pressure, can give a direct measure of the timescale over which entropically stored energy is dissipated.