Past exam of the soft condensed matter course of the University of Cambridge / 2016 / Paper 1 / 1

Use the amphiphile packing parameter $P=v/(a_0{\ell }_c)$, where $v$ is tail volume, $a_0$ the effective head-group area and ${\ell }_c$ the maximum tail extension. The conventional cylindrical-micelle window is $1/3\lesssim P\lesssim 1/2$. Its upper limit follows directly from cylindrical micelle geometry: a long core of radius $R$ has volume-to-area ratio $R/2$, so $v/a_0=R/2\le {\ell }_c/2$. The spherical packing limit is $P=1/3$; above it a filled spherical core cannot accommodate the assumed head-group area and chain length.

Inverting both positive inequalities givesThus the useful estimate is a head-group area of approximately $0.25$–$0.37\mathrm{n}{\mathrm{m}}^2$. If phase-boundary configurations are excluded, use strict inequalities. These endpoints are geometric morphology crossovers, rather than exact thermodynamic phase boundaries: chain conformations and head-group interactions also affect which micelle minimizes the free energy. The supplied molecular volume is interpreted as the tail-core volume required by this packing model.^[ref]

For an equilibrium contact angle measured through the liquid, Young's equation balances the tangential interfacial forces:The pure-ethanol measurement therefore fixes the solid–vapour surface energy asThe inferred surface energy is $40.7\mathrm{mJ}{\mathrm{m}}^{-2}$, equivalently $40.7\mathrm{mN}{\mathrm{m}}^{-1}$. The equality of these units follows because creating interfacial area costs energy.

For the equal-fraction mixture, the stated linear relations give ${\gamma }_{SL}=20.5\mathrm{mN}{\mathrm{m}}^{-1}$ and ${\gamma }_{LV}=28.5\mathrm{mN}{\mathrm{m}}^{-1}$. Holding the solid–vapour surface energy fixed,The small decrease in ${\gamma }_{SL}$ increases the numerator of Young's equation, but the increase in ${\gamma }_{LV}$ has the larger relative effect and reduces $\cos \theta$. Consequently the contact angle increases and the mixture wets the surface less well. This inference uses equilibrium wetting on an unchanged smooth solid; a changing surface composition or contact-angle hysteresis would require additional measurements.

The ordinary physics is Stokes settling velocity. For a generic isolated rigid sphere of radius $a$, particle mass density ${\rho }_p$ and surrounding-fluid mass density ${\rho }_f$, balance the buoyancy-corrected body force against Stokes law:Here $\eta$ is the dynamic viscosity. If the effective acceleration is approximately constant over the displacement and the terminal regime has been reached, $s=Ut$. In an actual centrifuge $g_{\mathrm{eff}}={\Omega }^{2}r$, so a sufficiently large radial change requires integrating $dr/dt\propto r$ instead of assuming constant $U$. This generic relation does not resolve the deferred biological application.