Solution (source code)

= Solution

Pendant <alcohols> can form \b[reversible <borate> bridges], schematically $\mathrm{P-O-B(OH)-O-P'}$ and $\mathrm{B(O-P)_3}$, where P and P′ belong to different chains. Tetrahedral <borate> species are also possible depending on pH and water content. A bridge must connect different network strands to act as a <cross-link>; an intrachain loop does not contribute equivalently to elasticity.

Between <cross-links>, stretching reduces the number of accessible chain conformations. A simple incompressible <rubber elasticity> model gives the nominal tensile stress
$$
\boxed{\sigma_{\rm nominal}=G(\lambda-\lambda^{-2}),\qquad G\simeq\nu_e k_BT,}
$$
where $\lambda$ is the extension ratio and $\nu_e$ the density of elastically active strands. The initial response is soft; stronger extension gives increasing stress and ultimately finite-chain extensibility. Reversible <borate> exchange permits <stress relaxation>, rate dependence and a loading–unloading <hysteresis> loop, rather than permanent ideal elasticity at arbitrarily long times.

\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2-polymer-responses.png]
{title=Original schematic rubber stress, ester-block cleavage and amphiphilic assembly}