Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A1

For a narrowly aligned nematic liquid crystal, the scalar nematic order parameter is $S=⟨(3{\cos }^2\theta -1)/2⟩\simeq 1-\frac{3}{2}⟨{\theta }^2⟩$. Hence the small-angle width of nematic orientation isThis is a root-mean-square angular width about the unoriented nematic director. The single value of $S$ does not determine a complete distribution or its full width at half maximum. For an isotropic two-component Gaussian tilt, each transverse component would have standard deviation ${\theta }_{\mathrm{rms}}/\sqrt{2}$.

For a spherical micelle of sodium dodecyl sulfate with $n$ molecules and a sharp tail-filled core of radius $R$, volume and head-group area give $nv=4\pi R^3/3$ and $n{a}_0=4\pi R^2$. Dividing gives the spherical micelle geometryThere is a consistency limit: $R$ exceeds the supplied maximum tail length $1.67\mathrm{nm}$. Equivalently the amphiphile packing parameter is $v/(a_0{l}_c)=0.368>1/3$, outside the strict spherical-core bound. Thus the requested spherical assumption gives these estimates, but the three molecular dimensions cannot all be exact within that ideal sharp-core geometry. Effective head area, core structure or aggregate shape must adjust; silently replacing $R$ by $l_c$ would no longer satisfy both packing equations.

For the polybutadiene chain in 1,4-dioxane, define $R_x$ from the negative end charge toward the positive one. The electric dipole energy is $-eE{R}_x$, so the force conjugate to end separation is $eE$, even though the two endpoint forces are opposite. Balancing it against the entropic spring force givesNumerically,The zero-field rms of just the $x$ component is $60/\sqrt{3}=34.6\mathrm{nm}$, a different comparison. The mean extension is still much smaller than the contour length $Na=6\mu \mathrm{m}$, consistent with the Gaussian ideal chain force law. This is an end-charged ideal polymer in an electric field; the direct endpoint Coulomb interaction is omitted as stipulated.