An ink is made of pigment spheres in water at ( )

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

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

Problem

An ink is made of pigment spheres in water at 298 K298\,\mathrm{K} (kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}), with density mismatch Δρ=1500 kg m−3\Delta\rho = 1500\,\mathrm{kg\,m^{-3}} and g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. A formulator wants each pigment particle, as it settles freely, to diffuse its own radius much faster than it sinks that far, and takes this free-particle criterion to be Peg=a/ℓg≤0.1\mathrm{Pe}_g = a/\ell_g \le 0.1. Using Peg=4π3Δρ g a4kBT,\mathrm{Pe}_g = \frac{4\pi}{3}\frac{\Delta\rho\,g\,a^4}{k_BT}, predict the largest pigment radius that meets this requirement, and say whether a pigment of twice this radius would still meet the requirement.

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