Silica spheres of radius and density are dispersed

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter

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

Problem

Silica spheres of radius

a = 750\,\mathrm{nm}

and density

\rho_p = 2000\,\mathrm{kg\,m^{-3}}

are dispersed, dilutely and non-interactingly, in water (

\rho_s = 1000\,\mathrm{kg\,m^{-3}}

, viscosity

\eta = 0.89\,\mathrm{mPa\,s}

) at

T = 298\,\mathrm{K}

and allowed to reach sedimentation equilibrium in a closed column. Take

k_BT = 4.11\times10^{-21}\,\mathrm{J}

and

g = 9.8\,\mathrm{m\,s^{-2}}

. Stokes drag on a sphere,

\zeta = 6\pi\eta a

, may be quoted. ((a)) Using the terminal sedimentation velocity set by Stokes drag together with Fick's law for the diffusive counter-flux, derive the equilibrium density profile

n(z)

from the requirement that the net particle flux vanish at every height. Show what relation between

D_0

and

\zeta

this profile must satisfy if it is to agree with the profile obtained instead from constancy of the chemical potential. ((b)) Evaluate the gravitational length

\ell_g

for these spheres. ((c)) Find the ratio

n(500\,\mathrm{nm})/n(0)

between two points

500\,\mathrm{nm}

apart in the column.

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