Force–extension relation of cross-linked anisotropic polymer networks

Panayotis Benetatos,Annette Zippelius,Stephan Ulrich

doi:10.1088/1367-2630/14/11/115011

Abstract

Cross-linked polymer networks with orientational order constitute a wide class of soft materials and are relevant to biological systems (e.g.,F-actin bundles). We analytically study the nonlinear force–extension relation of an array of parallel-aligned, strongly stretched semiflexible polymers with random cross-links. In the strong stretching limit, the effect of the cross-links is purely entropic, independent of the bending rigidity of the chains. Cross-links enhance the differential stretching stiffness of the bundle. For hard cross-links, the cross-link contribution to the force–extension relation scales inversely proportional to the force. Its dependence on the cross-link density, close to the gelation transition, is the same as that of the shear modulus. The qualitative behaviour is captured by a toy model of two chains with a single cross-link in the middle.

Highlights

Κ denotes the bending stiffness which is related to the persistence length Lp via Lp = 2κ/((d − 1)kBT ), where d is the dimensionality of the embedding space
The central quantity of interest is the extension of the chain under an applied force f, which in the weakly bending approximation is computed from the thermal fluctuations transverse to the aligning direction: z(L) − z(0) =
Use of the weakly bending approximation requires κ/L T or f L T. This leaves us with one free parameter, x := f L/(κ/L), namely the ratio of work done by the external force to bending energy

Summary

Toy model: two cross-linked chains

Before addressing the full problem of a randomly cross-linked array of aligned chains, we discuss the much simpler case of two strongly stretched chains in two dimensions with one cross-link in the middle, see figure 2. According to the boundary conditions that we use, the eigenfunction representation should be These eigenfunctions diagonalize the Hamiltonian of the weakly bending chain, whereas the cross-link gives rise to a term which is quadratic in the amplitudes but not diagonal. This leaves us with one free parameter, x := f L/(κ/L), namely the ratio of work done by the external force to bending energy This dimensionless quantity can be interpreted as the squared ratio of two lengthscales: the total contour length L to the length √κ/ f over which the boundary conditions penetrate into the bulk (i.e. the size of a link in an effective freely-jointed chain) [19]. We have restored the term due to a finite distance, D, between the chains, which gives rise to a geometric reduction in length due to the cross-link This term is presumably unimportant in a network, where D/L is expected to be small.

Randomly cross-linked network

Conclusions—outlook