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Fluid Dynamics Seminar

Monday, Feb. 27, 2012, 4:00 PM
Cullimore, Room 611
New Jersey Institute of Technology

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

Roseanna Zia

Department of Mechanical & Aerospace Engineering Princeton University

Abstract

Colloidal dispersions play an important role in nearly every aspect of life—from blood to biofuels to nano-therapeutics. In the study of 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. Complex fluids encompass such diverse systems as biofilms, hydrogels, and the interior of the cell. Many such systems are themselves only microns in size; 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; here, one studies the motion of a single Brownian 'probe' particle as it travels through a complex medium. At equilibrium, the probe undergoes thermal displacements which can be connected to the material properties of the suspending medium by the Stokes-Einstein relation. This 'passive' microrheology reveals near-equilibrium properties, but by its very nature is limited in scope to the linear response of the material. Alternatively, in nonlinear microrheology the probe is actively forced through the medium, and dynamic response behavior studied. Probe motion through the medium 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, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe=Fext/(kT/b), where kT is the thermal energy and b the characteristic microstructural length scale. The mean probe speed can be interpreted as the effective material viscosity, whereas fluctuations in probe velocity give rise to an anisotropic force-induced diffusive spread of its trajectory. The viscosity and diffusivity can thus be obtained by two simple quantities—mean and mean-square displacement of the probe. The notion that diffusive flux is driven by stress gradients leads to the idea that the stress can be related directly to the microdiffusivity, and thus the anisotropy of the diffusion tensor reflects the presence of normal stress differences in nonlinear microrheology. In this study a connection is made between diffusion and stress gradients, and a relation between the particle-phase stress and the diffusivity and viscosity is derived for a probe particle moving through a colloidal dispersion. This relation is shown to agree with two standard micromechanical definitions of the stress, suggesting that normal stresses and normal stress differences can be measured in nonlinear 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 Pe for Pe >> 1 and vanishes as Pe4 for Pe << 1. The expression obtained for stress-induced migration can be viewed as a generalized non-equilibrium Stokes-Einstein relation. A final connection is made between the stress and an "effective temperature" of the medium, prompting the interpretation of the particle stress as the energy density, and the expression for osmotic pressure as a non-equilibrium "equation of state."