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Shear advection of a fluctuation with initial wavevector in the -direction, , and advected wavevector at later time . While is the wavelength in the -direction at , at later time , the corresponding wavelength in the (negative) -direction obeys . At all times, is perpendicular to the planes of constant fluctuation amplitude. Note that the magnitude increases with time. Brownian motion, neglected in this sketch, would smear out the fluctuation.
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Steady-state incoherent intermediate scattering functions as functions of accumulated strain for various shear rates ; the data were obtained by Besseling et al. (2007) in a colloidal hard-sphere dispersion at packing fraction (at ) using confocal microscopy; the wavevector points in the vorticity direction and has (at the peak of ). The effective Péclet numbers were estimated with the short-time self-diffusion coefficient at this concentration by van Megen et al. (1998). ISHSM calculations with separation parameter at (PY- peaking at ), and for strain parameter , are compared to the data for the values labeled. The yielding master function at lies among the experimental data curves which span , but discussion of the apparent systematic trend of the experimental data would require ISHSM to better approximate the shape of the final relaxation process.
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Steady-state incoherent intermediate scattering functions measured in the vorticity direction as functions of accumulated strain for various shear rates ; data from molecular dynamics simulations of a supercooled binary Lennard-Jones mixture below the glass transition at (Varnik, 2006). These collapse onto a yield scaling function at long times. The wavevector is (at the peak of ). The quiescent curve, shifted to agree with the one at the highest , shows aging dynamics at longer times outside the plotted window. The apparent yielding master function from simulation is compared to the ones calculated in ISHSM for glassy states at or close to the transition (separation parameters as labeled) and at nearby wavevectors (as labeled). ISHSM curves were chosen to match the plateau value , while strain parameters at (solid line) and at (dashed line) were used.
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(Isotropic) distortion of the structure factor under shear within ISHSM approximation in glass states. The limiting values for are shown at the transition (, black line), and at (red/gray line) corresponding to the confocal data in Fig. 2 at ; for the latter , the increase in with shear rate is shown (blue dot-dashed line); strain parameter was chosen. The upper thin dashed lines indicate the equilibrium structure factors from PY approximation for these densities (with corresponding color code). The inset shows the complete distorted structure factors at (red solid line for ; blue dashed-dotted line at ) compared to the equilibrium ones (black dashed line); spline interpolation through the data on the grid of the main figure was used. At the difference for cannot be resolved.
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Integrands for the parameter in stability equation (84); the one labeled results for the memory function in Eq. (81), and the one labeled results for ISHSM approximation (C1), setting to recover in both cases. For comparison the integrand leading to the exponent parameter is included as dotted line.
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A microscopic approach is presented for calculating general properties of interacting Brownian particles under steady shearing. We start from exact expressions for shear-dependent steady-state averages, such as correlation and structure functions, in the form of generalized Green–Kubo relations. To these we apply approximations inspired by the mode coupling theory (MCT) for the quiescent system, accessing steady-state properties by integration through the transient dynamics after startup of steady shear. Exact equations of motion, with memory effects, for the required transient density correlation functions are derived next; these can also be approximated within an MCT-like approach. This results in closed equations for the nonequilibrium stationary state of sheared dense colloidal dispersions, with the equilibrium structure factor of the unsheared system as the only input. In three dimensions, these equations currently require further approximation prior to numerical solution. However, some universal aspects can be analyzed exactly, including the discontinuous onset of a yield stress at the ideal glass transition predicted by MCT. Using these methods we additionally discuss the distorted microstructure of a sheared hard-sphere colloid near the glass transition, and consider how this relates to the shear stress. Time-dependent fluctuations around the stationary state are then approximated and compared to data from experiment and simulation; the correlators for yielding glassy states obey a “time-shear-superposition” principle. The work presented here fully develops an approach first outlined previously [Fuchs and Cates, Phys. Rev. Lett.89, 248304 (2002)], while incorporating a significant technical change from that work in the choice of mode coupling approximation used, whose advantages are discussed.
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