Soft Matter at the Thermal Scale, Ch Spinodal decomposition and coarsening
Problem
A dilute aqueous suspension of silica spheres of radius a=50nm is destabilised by a non-adsorbing polymer (a short-range depletion attraction, as in Chapter 2's colloid-polymer mixtures). The quench is shallow, so dense-phase droplets first appear one at a time by nucleation over a barrier (Chapter 11), and only afterwards ripen. Take water's viscosity as η=1.0×10−3Pas, the temperature as 298K, the droplet volume as that of a sphere, the volume fraction of single silica particles in the continuous phase far from any droplet as ϕ∞=2×10−3, and the dense-droplet/continuous-phase interfacial tension as γ=0.1μNm−1, an ultralow value typical of a colloidal interface near its critical point.
((a)) Using the Stokes-Einstein relation for the diffusing silica monomers (radius a) that are transported between droplets, calculate the diffusion coefficient D.
((b)) The Lifshitz-Slyozov law gives R3−R03=ωt with ω=98Dϕ∞vmγ/kBT, where vm=34πa3 is the volume of one diffusing unit. Find the capillary length ℓc=2γvm/kBT and check that ℓc≪R0, as the law requires. Taking R0=200nm as the radius of the smallest surviving droplets once ripening sets in, estimate the time for the mean droplet radius to reach 1μm.
((c)) The dense-phase droplets occupy only about 1% of the sample volume and never touch each other. State, with a reason drawn from the viscous-flow argument for connected patterns, whether this system coarsens by the diffusive route above or by the hydrodynamic route, and say what feature of the morphology (not the viscosity) decides the answer.
((d)) Suppose instead the same depletion attraction were made strong enough, and the quench deep enough, to enter the metastable gas-liquid spinodal directly (bypassing nucleation). Explain in one or two sentences, using the idea of the dense phase meeting a glass line, why the pattern could stop coarsening altogether and leave a gel rather than following the growth law of part (b) indefinitely.