A dilute suspension of polystyrene latex spheres of radius and density is left in a…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Statistical foundations and the equilibrium ensembles

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

Problem

A dilute suspension of polystyrene latex spheres of radius a=300 nma = 300\,\mathrm{nm} and density ρp=1055 kg m−3\rho_p = 1055\,\mathrm{kg\,m^{-3}} is left in a sealed column of water (ρs=1000 kg m−3\rho_s = 1000\,\mathrm{kg\,m^{-3}}) at T=298 KT = 298\,\mathrm{K} until it reaches sedimentation equilibrium. Treat the particles as smooth, non-interacting spheres (the dilute limit) and take kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J} at this temperature and g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. At equilibrium the total chemical potential of a particle, μ(z)=μ∘+kBTln⁡n(z)+Δm gz\mu(z) = \mu^{\circ} + k_BT\ln n(z) + \Delta m\,g z, does not depend on height zz, where n(z)n(z) is the number density and Δm=43πa3(ρp−ρs)\Delta m = \tfrac43\pi a^3(\rho_p-\rho_s) is the buoyant mass of one sphere. ((a)) Starting from the constancy of μ(z)\mu(z), derive the equilibrium profile n(z)n(z) and identify the gravitational length ℓg\ell_g. Evaluate ℓg\ell_g for this suspension. ((b)) Find the height above the bottom of the column at which the number density has fallen to one tenth of its value at the bottom.

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