S. M. Tschopp1 and J. M. Brader11Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland(Dated: March 26, 2024)The superadiabatic dynamical density functional theory (superadiabatic-DDFT) is a promising new method for the study of colloidal systems out-of-equilibrium. Within this approach the viscous forces arising from interparticle interactions are accounted for in a natural way by treating explicitly the dynamics of the two-body correlations. For bulk systems subject to spatially homogeneous shear we use the superadiabatic-DDFT framework to calculate the steady-state pair distribution function and the corresponding viscosity for low values of the shear-rate. We then consider a variant of the central approximation underlying this superadiabatic theory and obtain an inhomogeneous generalization of a rheological bulk theory due to Russel and Gast. This paper thus establishes for the first time a connection between DDFT approaches, formulated to treat inhomogeneous systems, and existing work addressing nonequilibrium microstructure and rheology in bulk colloidal suspensions.
Note that the radial function f (r) is obtained from numerical integration of equation (37) and that r = (cid:112)x2 + y2. The predictions of the Russel-Gast-type closure are generally very similar to those obtained from the superadiabaticDDFT approach of the previous section, although deviations become apparent on closer inspection.
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The results for gsup at the lowest bulk density considered, b = 0.1, are very close to the known low-density limit, for which f (r) = 1/2r2, with accumulation and depletion at contact within the compressional and extensional quadrants, respectively. As the bulk density is increased we observe the emergence of packing oscillations. We recall that superadiatic-DDFT predicted that the radial function f appearing in equation (23) has a contact value which remains independent of the bulk density. The analogous quantity within the present approximation is the product f (r) geq(r) appearing on the right hand-side of equation (42). We find that the contact value of this product does exhibit a nontrivial dependence on b and, as we will see, this causes the low-shear viscosity generated by equation (40) to deviate from that predicted by superadiabatic-DDFT.
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In Fig. 5 we show the pair distribution function on both the extensional axis, along which the particles get pulled apart by the shear, and on the compressional axis, along which the particles get pressed together. On the 7 (42) FIG. 5. Pair distribution function. We show the full nonequilibrium pair distribution function of hard-disks along the extensional axis, y = x, and the compressional axis, y = x. Results are given from both the superadiabatic-DDFT and the Russel-Gast-type approximations, for b = 0.7. The additional black curve indicates the equilibrium radial distribution function for comparison. The dimensionless shear-rate is chosen to be d2/2D0 = 1.
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The reduction in amplitude of the peaks is slightly more pronounced within the RusselGast-type approximation. In both cases the radial position of the second and higher-order peaks shifts to larger values of r, consistent with the fact that particles are being pulled away from each other by the shear flow. In contrast, on the compressional axis the height of all maxima increase and the radial locations of the second and higher-order peaks are shifted to smaller values of r, since the particles are being pushed closer together by the shear flow. The two theories again make similar predictions, although the Russel-Gast-type approximation yields a slightly larger increase in peak height than the superadiabatic-DDFT. The general similarity of the predictions from the two approximation schemes for the nonequilibrium microstructure is reassuring,
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