5676 Mergell and Everaers Macromolecules,Vol.34.No.16.2001 theories of rubber elasticity than rheological data. show that On the theoretical side the approach eralization of volved replica methods to describe the loca des the lin p Gra The very eleg nt vorks and he ideas strands to dom-walk服 on the ar regions in space en the CMM an be used rmally so nble ave tho Edwards tube model hile in turn the recalcula the Sngleparenete the strength ent and confir anglemeatcontnbutiom modulus pe hof the tial ns ary affinel uences.we discuss in the s ond p the intro ensemble square roo of the copic stral mite tang inated syst "double constrained-jun ction without of the fluct ations due to cross-links and d due to suchs the ed-chainstrain more or tuations in Networks of is mention rary ectivity Rubinstein and Pa rclar.the auth A.The Model. The Hamilte of the strate f th om r which ar confining potential by a While tube models are ulated and dis o the ween mhe ble sim am strength e an The problem is entl pace fixed volum d tra sparent mann can be analyzed in modes of a linear ain one of the )or th model(CMM). here ment is led by tior nheta int depen upled the prings and the ictuations. Fo o poten This mode m. we note that the p of data where its the linges In the foll wing上 di ions of defec by writing the equations only for on the res fo ane = ries In particular. >X+uKu the the 2 Flory for de ning potential.D ements are given by the no de's functionalit ndent r eters it s part or a =2 ir st to a fou tub ref44 remained fair f nodes and model and the CMM.respectively. network strands have the same length.theories of rubber elasticity than rheological data.38-40 On the theoretical side, the original approach of Warner and Edwards15 used mathematically rather involved replica methods26 to describe the localization of a long polymer chain in space due to cross-linking. The replica method allows for a very elegant, self￾consistent introduction of constraining potentials, which confine individual polymer strands to random-walk like tubular regions in space while ensemble averages over all polymers remain identical to those of unconstrained chains. Later Heinrich and Straube25,32 recalculated these results for a solely entangled system where they argued that there are qualitative differences between confinement due to entanglements and confinement due to cross-linking. In particular, they argued that the strength of the confining potential should vary affinely with the macroscopic strain, resulting in fluctuations perpendicular to the tube axis which vary only like the square root of the macroscopic strain. Replica calculations provide limited insight into physi￾cal mechanism and make approximations which are difficult to control.12 It is therefore interesting to note that Flory was able to solve the, in many respects similar, constrained-junction model17 without using such methods. Recent refinements of the constrained￾junction model such as the constrained-chain model41 and the diffused-constrained model42 have more or less converged to the (Heinrich and Straube) tube model, even though the term is not mentioned explicitly. Another variant of this model was recently solved by Rubinstein and Panyukov.43 In particular, the authors illustrated how nontrivial, subaffine deformations of the polymer strands result from an affinely deforming confining potential. While tube models are usually formulated and dis￾cussed in real space, two other recent papers have pointed independently to considerable simplifications of the calculations in mode space. Read and McLeish35 were able to rederive the Warner-Edwards result in a particularly simple and transparent manner by showing that a harmonic tube potential is diagonal in the Rouse modes of a linear chain. Complementary, one of the present authors introduced a general constrained mode model (CMM),44 where confinement is modeled by deformation dependent linear forces coupled to (ap￾proximate) eigenmodes of the phantom network instead of a tube-like potential in real space. This model can easily be solved exactly and is particularly suited for the analysis of simulation data, where its parameters, the degrees of confinement for all considered modes, are directly measurable. Simulations of defect-free model polymer networks under strain analyzed in the frame￾work of the CMM14 provide evidence that it is indeed possible to predict macroscopic restoring forces and microscopic deformations from constrained fluctuation theories. In particular, the results support the choice of Flory,17 Heinrich and Straube,25 and Rubinstein and Panyukov43 for the deformation dependence of the confining potential. Despite this success, the CMM in its original form suffers from two important deficits: (i) due to the multitude of independent parameters it is completely useless for a comparison to experiment, and (ii) apart from recovering the tube model on a scaling level, ref 44 remained fairly vague on the exact relation between the approximations made by the Edwards tube model and the CMM, respectively. In the present paper, we show that the two models are, in fact, equivalent. The proof, presented in section IIB is a generalization of the result by Read and McLeish to arbitrary connectivity. It provides the link between the considerations of Eichinger,11 Graessley,45 Mark,46 and others on the dynamics of (micro) phantom networks and the ideas of Edwards and Flory on the suppression of fluctuations due to entanglements. As a consequence, the CMM can be used to formally solve the Edwards tube model exactly, while in turn the independent parameters of the CMM are obtained as a function of a single parameter: the strength of the tube potential. Quite interestingly, it turns out that the entanglement contribution to the shear modulus de￾pends on the connectivity of the network. To explore the consequences, we discuss in the second part the intro￾duction of entanglement effects into the Warner￾Edwards model, which represents the network as an ensemble of independent long paths comprising many strands. Besides recovering some results by Rubinstein and Panyukov for entanglement dominated systems, we also calculate the single chain structure factor for this controversial case.32-37 Finally we propose a “double tube” model to describe systems where the confinement of the fluctuations due to cross-links and due to en￾tanglements is of similar importance and where both effects are treated within the same formalism. II. Constrained Fluctuations in Networks of Arbitrary Connectivity A. The Phantom Model. The Hamiltonian of the phantom model4-6 is given by Hph ) k/2∑〈i,j〉 M rij2, where 〈i, j〉 denotes a pair of nodes i, j ∈1, ... , M which are connected by a polymer chain acting as an entropic spring of strength k ) (3kBT)/〈br2〉, and brij(t) ) bri(t) - brj(t) the distance between them. To simplify the notation, we always assume that all elementary springs have the same strength k. The problem is most conveniently studied using periodic boundary conditions, which span the network over a fixed volume10 and define the equilibrium position RBi ) (Xi, Yi, Zi). A conformation of a network of harmonic springs can be analyzed in terms of either the bead positions bri(t) or the deviations bui(t) of the nodes from their equilibrium positions RBi. In this representation, the Hamiltonian separates into two independent contributions from the equilibrium exten￾sions of the springs and the fluctuations. For the following considerations it is useful to write fluctuations as a quadratic form.11 Finally, we note that the problem separates in Cartesian coordinates R ) x, y, z due to the linearity of the springs. In the following we simplify the notation by writing the equations only for one spatial dimension: Here u denotes a M-dimensional vector with (u)i ≡ (bui)x. K is the connectivity or Kirchhoff matrix whose diagonal elements (K)ii ) fik are given by the node’s functionality (e.g., a node which is part of a linear chain is connected to its two neighbors, so that fi ) 2 in contrast to a four￾functional cross-link with fi ) 4). The off-diagonal elements of the Kirchhoff matrix are given by (K)ij ) - k, if nodes i and j are connected and by (K)ij ) 0 otherwise. Furthermore, we have assumed that all network strands have the same length. Hph ) k 2 ∑ 〈i, j〉 XijR 2 + 1 2 ut K u. (1) 5676 Mergell and Everaers Macromolecules, Vol. 34, No. 16, 2001