 Solution

Pendant alcohols can form reversible borate bridges, schematically $\mathrm{P}-\mathrm{O}-\mathrm{B}(\mathrm{OH})-\mathrm{O}-{\mathrm{P}}^{\prime }$ and $\mathrm{B}(\mathrm{O}-\mathrm{P}{)}_3$, 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 stresswhere $\lambda$ 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.