An equilibrium hard-sphere colloidal crystal is compressed osmotically to a volume…

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

An equilibrium hard-sphere colloidal crystal is compressed osmotically to a volume fraction ϕ=0.60\phi = 0.60, above the melting point ϕm=0.545\phi_m=0.545 but below random close packing. The free-volume cell model gives the crystal's compressibility factor as Zcr=11−(ϕ/ϕcp)1/3,Z_{\rm cr}=\frac{1}{1-(\phi/\phi_{\rm cp})^{1/3}}, with ϕcp=0.7405\phi_{\rm cp}=0.7405 the fcc close-packing fraction. ((a)) Estimate ZcrZ_{\rm cr} at ϕ=0.60\phi=0.60. ((b)) For silica spheres of radius a=0.75 μma=0.75\,\mu\mathrm{m} (particle volume vp=43πa3v_p=\tfrac43\pi a^3) at 298 K298\,\mathrm{K}, estimate the osmotic pressure Π=ϕZcrkBT/vp\Pi=\phi Z_{\rm cr}k_BT/v_p needed to hold the crystal at this volume fraction.

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