Past exam of the biological physics course of the University of Oxford / 2022 / 5 / e / Solution

The rigid-link freely jointed chain stores extension entirely in orientation. It predicts $⟨z⟩\to L=Nb$ and a divergent force as that limit is approached. Real chemical bonds and bond angles can deform, adding energetic compliance. This permits the measured chain to stretch farther at the same large force and removes the rigid-link vertical asymptote over the experimentally accessible range. Other limitations include segment correlations, solvent effects and eventual failure; a fit alone does not isolate their contributions.

Introduce one axial stretch modulus $S$ with units of force. A useful phenomenological extensible freely jointed chain law adds a linear strain contribution:Equivalently, if each aligned segment is a spring of stiffness $k_s$ in force per length, $S=b{k}_s$ and the additional extension is $Nf/k_s$. At high force, $⟨z⟩\simeq L-N{k}_{B}T/f+Lf/S$; the extra term supplies finite axial compliance. At small force, the compliances add: $d⟨z⟩/df=N{b}^2/(3k_{B}T)+L/S$.

This additive law is a simple requested one-term modification, rather than an exact angular average for harmonic links. Another common approximation is $L\mathcal{L}(fb/(k_{B}T))(1+f/S)$, which couples stretching to orientation. Neither should be extrapolated through bond rupture or claimed to predict a universal finite maximum for an extensible chain. A positive fitted $S$ could improve the high-force extension while leaving the entropic part largely intact.