Nonaffine Deformation and Elasticity of Polymer Networks

Abstract

We demonstrate that the origin of the nonlinear elasticity of polymer networks rests in their nonaffine deformations. We introduce the affine length R aff, which separates the solid-like elastic deformations on larger scales from liquid-like nonaffine deformations on smaller scales. This affine length grows with elongation λ as R aff ∼ λ3/2 and decreases upon compression as R aff ∼ λ1/2. The behavior of networks on scales up to R aff is that of stretched or compressed individual chains (we call them affine strands). The affine strands are stretched in the elongation direction and confined and folded in the effective tubes in the compression direction. The fluctuations of affine strands determine the diameters of the confining tubes a, which change nonaffinely with the network deformation a ∼ λ1/2. Our model gives a unified picture of deformations of both phantom and entangled networks and leads to a stress−strain relation that is in excellent agreement with experiments.

Tube Models for Rubber−Elastic Systems

“…In the second part of the paper we have therefore reexamined the idea of Heinrich and Straube 25 to introduce entanglement effects into the Warner-Edwards model 15 for linear, random paths through a polymer network, whose localization in space is modeled by a harmonic tube-like potential. In agreement with Heinrich and Straube, 25 and with Rubinstein and Panyukov 43 we have argued that in contrast to confinement due to crosslinking, confinement due to entanglements is deformation dependent. Our treatment of the tube model differs from previous attempts in that we explicitly consider the simultaneous presence of two different confining potentials.…”

“…Similar conclusions were drawn by Heinrich and Straube 25 and Rubinstein and Panyukov. 43 In the end, this problem will have to be resolved by a derivation of the tube model from more fundamental topological considerations. For the time being, an empirical approach seems to be the safest option.…”

“…43 It corresponds to affinely deforming cavities and leads to a more complex behavior including corrections to the classically predicted stress-strain behavior. Using eq 35, the mean excitations of partially frozen modes as well as the thermal fluctuations, become deformation dependent.…”

“…43 In order to account for the network contribution to the shear modulus, these authors add the phantom network results to eqs 69, 70. This leads to the following relations for the Mooney-Rivlin parameters: 43…”

“…Another variant of this model was recently solved by Rubinstein and Panyukov. 43 In particular, the authors illustrated how nontrivial, sub-affine deformations of the polymer strands result from an affinely deforming confining potential.…”