Past exam of the soft condensed matter course of the University of Cambridge / 2013 / Paper 1 / 1

TODO: user-deferred biological application. The organism-specific comparison and its three requested rescalings are intentionally left unsolved.

The tension is supplied by Stokes drag on the free bead. With bead radius $b=0.5\mu \mathrm{m}$, the stipulated isolated-sphere approximation givesThe trap also balances drag on the trapped bead, but that bead's drag is not an additional force transmitted through the chain. The polymer contour length is $L=Na=6\mu \mathrm{m}$. Use a room-temperature estimate $T\simeq 300\mathrm{K}$, consistent with the water-viscosity approximation imported from part (a); the expressions below retain the dependence on $T$.

For a theta solvent, Gaussian chain elasticity gives $F_{\mathrm{el}}=3k_{B}T{R}^2/(2N{a}^2)$ and henceThis is the mean extension along the force, rather than the unforced end-to-end distance of a polymer chain $a\sqrt{N}=60\mathrm{nm}$. At $300\mathrm{K}$, $fa/(k_{B}T)=0.410$; the finite-link freely jointed chain result $R=L[\coth (fa/k_{B}T)-k_{B}T/(fa)]$ is $0.810\mu \mathrm{m}$, close to the Gaussian estimate.

The good-solvent force-extension scaling changes the elasticity. In the tensile blob regime, a blob has size ${\xi }_t=k_{B}T/f$, contains $g\sim ({\xi }_t/a{)}^{5/3}$ segments, and the aligned string of $N/g$ blobs has lengthOnly the scaling and order of magnitude of the good-solvent value are determined: solvent-dependent swelling amplitudes are not supplied. The nonlinear force law is the essential correction to extrapolating a weak-force coil spring.^[ref]

To expose the unrealistic extrapolation, the unforced good-solvent size estimate is $R_0\sim a{N}^{3/5}=151\mathrm{nm}$. Treating this coil as a linear spring with spring constant $3k_{B}T/R_0^2$ predicts $R\simeq 5.17\mu \mathrm{m}$ at $300\mathrm{K}$, although its required condition $f{R}_0\ll k_{B}T$ fails by a factor of about $103$. Dropping the order-one factor three, as in the rough estimate $k\sim k_{B}T/R_0^2$, even gives $R\sim 15.5\mu \mathrm{m}>L$. An extension beyond the polymer contour length is an artefact of extending the coil's linear response outside its range. The tensile blob estimate avoids that violation but here ${\xi }_t\simeq 1.46\mathrm{nm}$ and $g\sim 4.4$, so it is only marginally within a many-segment-blob regime. finite extensibility of a polymer chain, chain drag and flow disturbance by both large beads further limit quantitative realism. The specified neglect of chain drag is a model assumption, not evidence that it is physically negligible.

For a symmetric Flory-Huggins polymer blend, the mixing free energy per lattice site isThe critical conditions $f^{\prime \prime }=0$ and $d^{3}f/d{\varphi }^3=0$ give ${\varphi }_c=1/2$ and ${\chi }_c=2/N=0.002$. ThereforeSince $d\chi /dT<0$, heating improves miscibility: this is an upper critical solution temperature in the stipulated model. The very high value is an extrapolation of the supplied interaction-parameter fit, not evidence that an intact experimental blend can be brought reversibly to this temperature; chemical stability and the fitted temperature range must be checked before interpreting it as an accessible transition.

At $293.15\mathrm{K}$, $\chi =0.132421$ and $N\chi \simeq 132$, well into strong segregation. For the polymer-blend interface width, state the convention explicitly. In the equal-segment, long-chain square-gradient approximation,Its first integral gives ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$, soThe inverse-central-slope width is consequentlyA convention calling the hyperbolic-tangent scale itself the width instead quotes $0.752\mathrm{nm}$; these describe the same profile. The predicted interface spans only a few statistical segments, so the continuum approximation for gradients is quantitatively marginal. Intrinsic composition width and experimentally broadened width also differ because roughness and finite spatial resolution can broaden a measured profile.