Past exam of the soft condensed matter course of the University of Cambridge / 2012 / Paper 1 / 1 / b

For the three-dimensional worm-like chain, define the bending coefficient ${\kappa }_b$ by $F_b=({\kappa }_b/2)\int |{\partial }_s\mathbf{t}{|}^{2}ds$. Its persistence length is $l_p={\kappa }_b/(k_{B}T)$. Holding that coefficient and the polymer contour length $L$ fixed givesThe temperatures must be absolute. The end-to-end distance of a worm-like chain obeysFor a long flexible chain, $L\gg l_p$, the typical size is $R_{\mathrm{rms}}\simeq \sqrt{2L{l}_p}$, so warming contracts the chain, with $R_{\mathrm{rms}}(T_2)/R_{\mathrm{rms}}(T_1)\simeq \sqrt{T_1/T_2}$. The finite-length formula also decreases as $l_p$ decreases, but the rod limit $L\ll l_p$ changes very little. A large segment count alone does not guarantee $L\gg l_p$. If the material's bending coefficient itself depends on temperature, replace the first ratio by $[{\kappa }_b(T_2)/{\kappa }_b(T_1)]T_1/T_2$; the sign then needs that additional material information. This calculation concerns a generic physical filament, with no biomolecular application.