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

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 $L=Na=6\mu \mathrm{m}$. TODO: no numerical temperature is specified for this part; the following temperature-dependent expressions are the results, and $300\mathrm{K}$ is an explicitly chosen room-temperature illustration.

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.