Individual Particle Motion in Colloids: Microviscosity, Microdiffusivity, and Normal Stresses

• Main Author: Zia, Roseanna Nellie
• Format: Others
• Published: 2011
• Online Access: https://thesis.library.caltech.edu/6455/8/Zia_Roseanna_Thesis_2011.pdf; Zia, Roseanna Nellie (2011) Individual Particle Motion in Colloids: Microviscosity, Microdiffusivity, and Normal Stresses. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2743-8W26. https://resolver.caltech.edu/CaltechTHESIS:05262011-141745737 <https://resolver.caltech.edu/CaltechTHESIS:05262011-141745737>

<p>Colloidal dispersions play an important role in nearly every aspect of life, from paint to biofuels to nano-therapeutics. In the study of these so-called complex fluids, a connection is sought between macroscopic material properties and the micromechanics of the suspended particles. Such properties include viscosity, diffusivity, and the osmotic pressure, for example. But many such systems are themselves only microns in size overall; recent years have thus seen a dramatic growth in demand for exploring microscale systems at a much smaller length scale than can be probed with conventional macroscopic techniques. Microrheology is one approach to such microscale interrogation, in which a Brownian “probe” particle is driven through a complex fluid, and its motion tracked in order to infer the mechanical properties of the embedding material. With no external forcing the probe and background 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 probe motion, depends on the strength with which the probe is forced, F ext , compared to thermal forces, kT/b, defining a P´eclet number, P e = F ext /(kT /b), where kT is the thermal energy and b the bath-particle size. Both the mean and the fluctuating motion of the probe are of interest. Recent studies showed that the reduction in mean probe speed gives the effective material viscosity. But the velocity of the probe also fluctuates due to collisions with the suspended particles, causing the probe to undergo a random walk process. 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 P´eclet number. At small Pe Brownian motion dominates and the diffusive behavior of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced “micro-diffusivity” that scales as Dmicro ∼ Da P e 2 φb , where viii φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high P´eclet number the fuctuational motion is still diffusive, and the diffusivity becomes primarily force-induced , scaling as (F ext /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-P e limit.</p> <p>Previous work in microrheology defined a scalar viscosity; however, a tensorial expression for the suspension stress in microrheology was still lacking. The notion that diffusive flux is driven by gradients in particle-phase stress leads to the idea that the microdiffusivity can be related directly to the suspension stress. In consequence, the anisotropy of the diffusion tensor may reflect the presence of normal stress differences in non-linear microrheology. While the particle-phase stress tensor can be determined as the second moment of the deformed microstructure, in this study a connection is made between diffusion and stress gradients, and an analytical expression for particle-phase stress as a function of the microdiffusivity and microviscosity is obtained. The two approaches agree, suggesting that normal stresses and normal stress differences can be measured in active microrheological experiments if both the mean and mean-square motion of the probe are monitored. Owing to the axisymmetry of the motion about a spherical probe, the second normal stress difference is zero, while the first normal stress difference is linear in P e for P e ≫ 1 and vanishes as P e 3 for P e ≪ 1. An additional important outcome is that the analytical expression obtained for stress-induced migration can be viewed as a generalized non-equilibrium Stokes-Einstein relation.</p> <p>Studies of steady-state dispersion behavior reveal the hydrodynamic and microstructural mechanisms that underlie non-Newtonian behaviors (e.g. shear-thinning, shear-thickening, and normal stress differences). But an understanding of how the microstructures evolve from the equilibrium state, and how non-equilibrium properties develop in time is much less well understood. Transient suspension behavior in the near-equilibrium, linear response regime has 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 ix in a medium are captured by start-up and cessation of the imposed deformation in the non-linear regime, where the structure is driven far from equilibrium. Here we study the evolution of stress and microstructure in a colloidal dispersion by tracking transient probe motion during start-up and cessation of a strong flow. For large P e, steady state is reached when a boundary layer (in which advection balances diffusion) forms at particle contact on the timescale of the flow, a/U , where a is the probe size and U its speed. On the other hand, relaxation following cessation occurs over several timescales corresponding to distinct physical processes. For very short times, the timescale for relaxation is set by the diffusion over the boundary-layer thickness. Nearly all stress relaxation occurs during this process, owing to the dependence of the bath-particle drag on the contact value of the microstructure. At longer times the collective diffusion of the bath particles acts to close the wake. In this long-time limit as structural isotropy is restored, the majority of the microstructural relaxation occurs with very little change in suspension stress. Theoretical results are presented and compared with Brownian dynamics simulation. Two regimes of probe motion are studied: an externally applied constant force and an imposed constant velocity. The microstructural evolution is qualitatively different for the two regimes, with a longer transient phase and a thinner boundary layer and longer wake at steady state in the latter case. The work is also compared to analogous results for sheared suspensions undergoing start-up and cessation.</p> <p>The study moves next to investigations of dual-probe microrheology. Motivated by the phenomenon of equilibrium depletion interactions, we study the interaction between a pair of probe particles translating with equal velocity through a dispersion with their line of centers transverse to the external forcing. The character of the microstructure surrounding the probes is determined both by the distance R by which the two probes are separated and by the strength of the external forcing, P e = U a/Db , where U is the constant probe velocity and Db the diffusivity of the bath particles. Osmotic pressure gradients develop as the microstructure is deformed, giving rise to an interactive force between the probes. This force is studied for a range of P e and R. For all separations R > 2a, the probes attract when P e is small. As the strength of the forcing increases, a qualitative change in the interactive force occurs: the probes repel each other. The probe separation R at which the x attraction-to-repulsion transition occurs decreases as P e increases, because the entropic depletion attraction becomes weak compared to the force-induced osmotic repulsion. The non-equilibrium interactive force is strictly repulsive for two separated probes.</p> <p>But non-linear microrheology provides far more than a microscale technique for interrogating complex fluids. In 1906, Einstein published the famous thought experiment in which he proposed that if a liquid were indeed composed of atoms, then the motion of a small particle suspended in the fluid would move with the same random trajectories as the solvent atoms. Combining the theories of kinetics, diffusion, and thermodynamics, he showed that the diffusive motion of a small particle is indeed evidence of the existence of the atom. Perrin confirmed the theory with measurement in 1909. This is a profound conclusion, drawn by simply watching a particle move in a liquid. Here, we follow this example and watch a particle move in a complex fluid—but now for a system that is not at equilibrium. In equilibrium systems, the relationship between fluctuation and dissipation is fundamental to our understanding of colloid physics. By studying fluctuations away from equilibrium, we have discovered an analogous non-equilibrium relation between fluctuation and dissipation—and that the balance between the two is stored in the material stress. A final connection can be made between this stress and energy storage.</p>