Constrained fluctuation theories of rubber elasticity: General results and an exactly solvable model

Abstract

Abstract. We present a new model of rubber elasticity where linear forces act to constrain the fluctuations of the eigenmodes of the phantom model. The model allows us to treat the constrained junction and the tube model within the same, transparent formalism, does not require any further approximations, and is particularly suited for the analysis of simulation data for (strained) model polymer networks. As an interesting side result we show that in order for the model to be consistent, the constraints (but not the mean polymer conformations!) have to deform affinely, a severe restriction that might also apply to other models. Complementary, we prove in analogy to the derivation of the virial theorem that introducing constraints into the phantom network Hamiltonian leads to extra terms in addition to the usual Doi-Edwards formulas for the polymer contribution to the stress tensor which vanish only for affinely deforming constraints.

Tube Models for Rubber−Elastic Systems

“…46,47 A useful ansatz for end-linked networks is a separation into independent Flory-Einstein respectively Rouse modes for the crosslinks and network strands. 14,44 In fact, the simulation results presented in Figure 1 are based on such a decomposition. For randomly crosslinked networks with a typically exponential strand length polydispersity, the separation into Flory-Einstein and single-chain Rouse modes ceases to be useful.…”

“…44 In the following we summarize the results and give general expressions for quantities of physical interest such as shear moduli, stress-strain relations, and microscopic deformations.…”

Fluctuating Entanglements in Single-Chain Mean-Field Models

“…Section 3 gives a little history and an explanation of the RA dynamics [51]. We point out that equilibrium models by Heinrich et al [57], Rubinstein and Panyukov [58,59], and Everaers [60] for cross-linked entangled networks have already incorporated the ideas from Ronca and Allegra. The RA dynamics in these network models are based on a decomposition of node fluctuations in the principal strain directions.…”

From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

“…(24). Remarkably, with this procedure the jacobian determinant appearing after the change of variables ν i ,ν α −→ R i , ξ α remains relatively simple.…”

“…The derivation of the path integral solutions of the Fokker-Planck equations becomes however complicated when constraints are added. Various approaches have been proposed in order to cope with constrained stochastic differential equations [8,9,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], but the available path integral formulations for statistical systems out of equilibrium still remain awkward to be used in practical applications.…”