Cross-linked pair of polymer chains under strong tension

We study two cross-linked polymer systems in the strong stretching regime. The first consists of two polymers sharing one endpoint, with the other two endpoints coupled by a harmonic potential. Within the weakly bending approximation, we analyze the tensile elastic response for freely jointed or wormlike chains; for the latter, the approximation applies either at large tension or at moderate tension with large persistence length (rodlike limit). We obtain analytic expressions for the force–extension relation and for the longitudinal and transverse mismatch of the cross-linked endpoints. In the thermodynamic limit, the cross-link does not affect the tensile elasticity, but it significantly suppresses transverse fluctuations, effectively forming a loop structure. The second system is a polymer necklace in the thermodynamic limit, composed of two strongly stretched polymers interconnected by a regular sequence of reversible cross-links. Using an analogy with a two-dimensional system of concatenated Gaussian loops (“Gaussian slinky”), we calculate the mean fraction of cross-linked sites as a function of the tensile force and find weak and strong binding regimes connected by a crossover. For shallow binding potential wells (compared with kBT), we employ a continuum description and exploit the mapping between directed polymers and a two-dimensional quantum particle to analytically determine the crossover behavior and the mean transverse separation between the two polymer chains. Overall, these results clarify that terminal cross-linking under strong tension primarily modifies transverse fluctuations while leaving longitudinal elasticity essentially unchanged, leading to an effectively additive stiffening of the composite tensile response. Moreover, the cross-link-induced modification of transverse fluctuations turns out to be independent of the microscopic polymer model (freely jointed versus wormlike).