(585b) Microviscosity. Microdiffusivity. Normal Stresses?

In active microrheology, the motion of a probe particle is tracked as it is driven by an external force through a colloidal dispersion.  Most work on microrheology has focused on the average probe motion, with the microviscosity defined theoretically and determined experimentally by application of Stokes' drag law.  Fluctuations in probe motion are also of interest; collisions between probe and bath particles cause velocity fluctuations, scattering the probe from its mean path.  We determined in our recent work that the (long-time) probe scattering is indeed diffusive.  The microdiffusivity, D^micro, is transversely anisotropic, scaling linearly in the volume fraction of bath particles φ (for small φ) for all Peclet numbers, Pe, which gives the strength of probe forcing compared to thermal forces: Pe = F^ext/(kT/b), where kT is the thermal energy and b the bath particle size.  In light of this anisotropy, the notion that self-diffusion is driven by gradients in the particle-phase or osmotic pressure prompts investigation of normal stresses in active microrheology — the anisotropy of the microdiffusivity indicates the presence of normal stress differences.  We take two approaches to determine normal stresses: First, we derive the stresses directly from the deformed microstructure.  Second, a phenomenological approach: via the relationship of the microdiffusivity to particle (osmotic) pressure gradients, &partΣ/&partφ, where Σ is the suspension stress.  Owing to the axisymmetry of the motion about a spherical probe, N₂ = 0, while N₁ is linear in Pe for Pe >> 1 and vanishes when Pe << 1.  The two approaches agree, suggesting that normal stress differences can be measured in active microrheological experiments if both the mean and mean-square motion of the probe are monitored.  While the approach of nonlinear microrheology would now appear to offer a complete rheological tool for interrogating complex fluids at the microscale, the flow produced by single particle forcing is not viscometric and hence cannot formally be defined as rheology.  However, a comparison of results and data obtained via the two techniques -- traditional macrorheology and nonlinear microrheology -- shows strong qualitative, and often quantitative, agreement. Explanation for why this agreement exists, and a basis for how microrheology can be viewed as a rheological tool are put forth.