A hard-sphere fluid is in sedimentation equilibrium

PHYSICS · Multi-part · 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

A hard-sphere fluid is in sedimentation equilibrium, with gravitational length

\ell_g = k_BT/\Delta m\,g

. Write its osmotic pressure as

\Pi v_p/k_BT = \phi Z(\phi)

, with

\phi = nv_p

. ((a)) Starting from hydrostatic balance,

\mathrm{d}\Pi/\mathrm{d}z = -n\,\Delta m\,g

, show that height and volume fraction are related by

\frac{\mathrm{d}z}{\ell_g} = -\frac{1}{\phi}\,\frac{\mathrm{d}(\phi Z)}{\mathrm{d}\phi}\,\mathrm{d}\phi,

and check that

Z = 1

gives back the exponential profile. ((b)) For the Carnahan-Starling fluid, find the height, in units of

\ell_g

, over which

\phi

falls from

0.45

0.05

. (You may confirm by differentiating that

\ln\phi + (3-\phi)/(1-\phi)^3

is an antiderivative of the integrand you need.) ((c)) Compare with an ideal dispersion falling over the same range of

\phi

, explain in one sentence why the dense layer is thicker, and give the layer's thickness in micrometres for

\ell_g = 0.50\,\mu\mathrm{m}