Friction factors for a lattice of Brownian particles

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Abstract
The resistance to oscillatory motions of arbitrary wavelengths in an infinitely dilute lattice of identical spheres, immersed in a viscous fluid, is calculated from the linearized Navier–Stokes equation to lowest order in fluid inertia and sphere-volume fraction. The application we have in mind is to analyse the hydrodynamic modes in colloidal crystals (a lattice of Brownian particles repelling each other electrically), although other applications are possible. We find that the friction per particle for both compressional and transverse shear modes is close to the Stokes value at short wavelengths, whereas at long wavelengths fluid backflow within the lattice is important and causes the friction to increase for compressional modes. For shear modes, in which backflow is not present, the friction decreases from the Stokes value at short wavelengths to zero at long wavelengths. At sufficiently long wavelengths, when the shear-mode friction becomes small enough, propagating viscoelastic modes are possible in a lattice with elastic forces between spheres. Fluid inertia is most important for long-wavelength transverse motions, since a significant amount of fluid mass gets carried along by each particle. Explicit results for a bcc lattice are presented along with interpolation formulas, and the pertinence of these results to colloidal crystals is discussed. Finally, the effects of constraining walls are explored by considering a one-dimensional lattice near a wall. Backflow imposed by the wall increases the friction factors for the lattice modes, showing that propagating modes are unlikely in colloidal crystals that are confined to a cell thinner than a critical length.

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