Confocal microscopy of a sedimented dispersion of PMMA spheres (radius

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Interacting systems, real gases and phase equilibria

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

Problem

Confocal microscopy of a sedimented dispersion of PMMA spheres (radius a=1.0 μma = 1.0\,\mu\mathrm{m}, particle volume vp=4.19×10−18 m3v_p = 4.19\times10^{-18}\,\mathrm{m^3}) in a solvent at 298 K298\,\mathrm{K} gives the volume-fraction profile below, measured at five heights 2 μm2\,\mu\mathrm{m} apart, with z=0z=0 at the bottom of the sedimented column. Data: at z=0,2,4,6,8 μmz = 0, 2, 4, 6, 8\,\mu\mathrm{m} the volume fraction is ϕ(z)=0.45,0.390,0.296,0.141,0.011\phi(z) = 0.45, 0.390, 0.296, 0.141, 0.011 respectively. The mass excess of a sphere is Δm=vpΔρ\Delta m = v_p\Delta\rho, with Δρ=200 kg m−3\Delta\rho = 200\,\mathrm{kg\,m^{-3}}. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}} and kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}. ((a)) Convert the profile to a number density n(z)n(z) at each height. ((b)) Taking the top point (z=8 μmz=8\,\mu\mathrm{m}) to be dilute enough that Π(8 μm)=n(8 μm)kBT\Pi(8\,\mu\mathrm{m}) = n(8\,\mu\mathrm{m})k_BT, use hydrostatic balance dΠ/dz=−nΔmg\mathrm{d}\Pi/\mathrm{d}z = -n\Delta m g and the trapezoidal rule (five points, four equal intervals) to find Π\Pi at the bottom of the column. ((c)) Read off the compressibility factor Z(ϕ=0.45)=Π/(nkBT)Z(\phi=0.45) = \Pi/(nk_BT) at the bottom and compare it with the Carnahan-Starling value at the same ϕ\phi.

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