We study the motion of a colloidal particle as it is driven by an oscillating external force of arbitrary amplitude and frequency through a colloidal dispersion. Large amplitude oscillatory flows (LAOFs) are examined predominantly from a phenomenological perspective in which experimental measurements inform constitutive models. Here, we investigate a LAOF from a microstructural perspective by connecting motion of the probe particle to the material response while making no assumptions a priori about how stress relaxes in the material. The suspension exerts nonconservative, hydrodynamic forces on the probe, while distortions in the particle configuration exert conservative forces: Brownian and interparticle forces, for example. The relative importance of each of these contributions to particle motion evolves with the degree of displacement from equilibrium. When the force on the probe is weak, the linear microviscoelasticity of the suspension is probed [see, e.g., Khair and Brady, J. Rheol. 49, 1449–1481 (2005)]. When oscillation rate is slow, the steady microrheology is probed [see, e.g., Squires and Brady, Phys. Fluids 17, 073101 (2005); Khair and Brady, J. Fluid Mech. 557, 73–117 (2006)]. This article develops a micromechanical model that recovers these limiting cases and then uses the same model to reveal the microrheology of colloidal dispersions deformed by a probe driven with arbitrary force amplitude and frequency. A chief result of this work is the discovery of a regime in which the resistance to motion of the probe particle is on average weaker than the resistance the probe experiences when deformed by high frequency oscillation. This hypoviscous effect arises when the reciprocating motion of the probe particle opens a channel free of other particles which is thus less resistive to probe motion. This effect is most apparent under the conditions of strong forces, rapid oscillation, and large extent of deformation.
The authors thank Professors Eric Furst, Norman, Wagner and Itai Cohen for helpful discussions about LAOS. Support from NASA is gratefully acknowledged by JWS (Grant Nos. NAG3-2832 and NNX07AD02G).
II. MODEL COLLOIDAL DISPERSION
III. A MICROSTRUCTURAL VIEW OF LARGE AMPLITUDE OSCILLATION
A. The pair Smoluchowski equation
B. Coupling of structural modes
IV. NONLINEAR MICROVISCOELASTICITY
A. A review of active microrheology
B. Probe velocity in response to fixed amplitude, oscillatory external force
C. Solution methodologies
1. Legendre polynomial expansion
2. Finite-difference solution
A. Asymptotic limits
1. Weak forcing (), linear response
2. Rapid oscillation (), linear response
3. Slow oscillation (), steady response
B. Arbitrary force amplitude and oscillation rate
1. Interpretation of a Lissajous-Bowditch curve (L-B): Microrheology
2. A Pipkin diagram for microrheology
C. Large-amplitude oscillations (): The hypoviscous regime
D. Interpretation of large amplitude oscillatory deformation experiments
E. A brief note on scaling results for higher concentrations
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