Nearly hard spheres of radius and density excess over the solvent are dispersed…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter · Viscous flow and dynamical similarity

Soft Matter at the Thermal Scale, Ch Brownian particles, sedimentation and the hard-sphere fluid

Problem

Nearly hard spheres of radius a=0.50 μma = 0.50\,\mu\mathrm{m} and density excess Δρ=100 kg m−3\Delta\rho = 100\,\mathrm{kg\,m^{-3}} over the solvent are dispersed uniformly at volume fraction ϕ0=0.020\phi_0 = 0.020 in a column of height H=2.0 mmH = 2.0\,\mathrm{mm}, then left at 298 K298\,\mathrm{K} (kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}) until the column reaches sedimentation equilibrium, with the top of the column dilute. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. Treat the fluid with the Carnahan-Starling equation of state and take the simulated coexistence pressure Πvp/kBT=6.06\Pi v_p/k_BT = 6.06 from the chapter. ((a)) Using hydrostatic balance and the fact that no particles are lost, show that the osmotic pressure at the bottom is Π(0)=Π(H)+ϕ0 Δρ g H\Pi(0) = \Pi(H) + \phi_0\,\Delta\rho\,g\,H whatever the equation of state. Argue that Π(H)\Pi(H) is negligible here, so that the reduced pressure at the bottom is Π(0)vp/kBT≈ϕ0H/ℓg\Pi(0)v_p/k_BT \approx \phi_0H/\ell_g. ((b)) Evaluate this reduced pressure for the column. Is the bottom of the sediment fluid, or does it contain crystal? If it is fluid, find its volume fraction. ((c)) At the same ϕ0\phi_0, find the smallest column height for which crystal would appear at the bottom in equilibrium.

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