Probe response phase space. These are four distinct regimes for colloidal dispersions undergoing large amplitude oscillatory deformation via microrheology: The region of steady linear response for which , the region of linear response for which , the region of steady, nonlinear response for which and the hypoviscous regime where .
The suspension is modeled in the dilute limit by considering the interactions between two particles: A probe particle driven by an external force and a bath particle that is free of external forcing. Both particles have the same hydrodynamic radius, ah , and interact via hard-sphere repulsion when the separation between their centers is 2a.
In a dilute colloidal dispersion, the motion of a probe particle distorts the statistical distribution of the bath particles surrounding it. Here, the probe particle is moving to the right and there is a build-up of bath particles in front (red) and a deficit (dark blue) behind. The force driving the probe through the suspension increases in each image from left to right. The top row shows the structure when hydrodynamic interactions are negligibly weak, as from Squires and Brady (2005) , while the bottom row includes hydrodynamic interactions, as from Khair and Brady (2006) .
Three asymptotic limits of forcing strength (Pe) and oscillation rate (α) with numbers indicating relevant domains: (1) Linear response, ; (2) linear response, and ; (3) steady, nonlinear response, . The three limits are studied in Sec. V A 1–V A 3 , respectively.
Linear susceptibility, χ, relating the averaged probe velocity to the imposed external force, as a function of oscillation frequency, α. Both real ( ) and imaginary ( ) parts are shown, each normalized by the value of at α = 0.
The pseudosteady microvelocity, normalized by the diffusive velocity scale, , and the particle volume fraction, , as a function of the pseudosteady Péclet number . The steady microvelocity normalized on this same scale and plotted as a function of the steady Péclet number, Pe, is indicated by the dashed line [ Khair and Brady (2006) ]. For a given maximum force amplitude, Pe, the pseudosteady microvelocity must recover the same value (i.e., the steady value) for all . This appears to be the case for all the values tested.
Fourier modes of plotted as a function of the Péclet number when α = 0. Each mode is normalized by the value of the first harmonic when Pe = 0. Main plot: Third, fifth, and seventh harmonics. Inset: First harmonic. The first harmonic comprises more than 95% of the total signal.
Examples of Lissajous curves described by the microvelocity. The dotted line reflects how a microstructure acting as a Newtonian fluid would reduce the speed of the probe particle. The microvelocity is linear in and in phase with the oscillatory force. The dashed ellipse reflects how a microstructure acting as a Hookean solid would reduce the speed of the probe particle. The microvelocity is linear in and 90° out of phase with the oscillatory force. The tilted ellipse is representative of a microstructure that acts as a linear viscoelastic fluid, while the nonelliptical curve is indicative of a nonlinear response.
Pipkin diagram: Lissajous curves for a range of forcing strengths (Pe) and oscillation speeds (α). Each Lissajous curve (viscous projection) shows versus and is normalized to fill its sub-box in the Pipkin diagram horizontally and vertically. The color behind each curves corresponds to the scale at the bottom of the figure and represents the slope of the line that connects the points of maximum and minimum force on the curve. We denote this following the notation of Ewoldt et al. (2008) . This viscosity difference is normalized by the solvent viscosity and the particle volume fraction.
Pipkin diagram (hydrodynamic and Brownian contributions shown separately): Lissajous curves from Fig. 9 have been separated here into hydrodynamic and Brownian contributions (black and grey (red online), respectively) to versus and are normalized to fill their respective boxes horizontally and vertically.
The viscous part of the complex viscosity increments and plotted as a function of Pe and α. The increments are identical in the linear-response limit but differ for . Because comprises a sum over all harmonics of (of which the first harmonic is the largest), differences between and arise from the higher (n > 1) response harmonics.
The elastic part of the complex viscosity increment and as a function of Pe and α. The increments are identical in the linear-response limit but differ for . Because comprises a sum over all harmonics of (of which the first harmonic is the largest), differences between and arise from the higher (n > 1) response harmonics. Linear-response behavior is recovered in the limit . In contrast to Fig. 11 , and are plotted against α for various values of Pe.
Lissajous-Bowditch curves for large amplitude oscillatory microrheology under two conditions: (a) Pe = 3, α = 0.3 and (b) Pe = 30, α = 3. A panel of five contour plots of the microstructure surrounding the probe accompanies each figure. Each contour plot gives a snapshot in time of the oscillating structure, indicated by a number in the snapshot which matches the corresponding time in the Lissajous-Bowditch curve. The slope of the dashed line connecting points 1 and 5 indicate the sign of . Videos corresponding to (a) and (b) are provided in the supplementary material.
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