5680 Mergell and Everaers Macromolecules,Vol.34.No.16.2001 22 the gene satz for nd-linked ne and net 0 0204 0.6 0.8 1. of constrai allel -Eins ment into Flor nd single chain Rou the radically different.we can th hink of two res k with ident a phent the con 严款 der the contributions from the A and B confinements.While eqs 34 3 th :0a owever contr m01 than to the a sh n us ed in ref 14. T am orks s ss-linkin To our kn Antant point.which holds for all three models. Shy that del ry out caleu lations with strengths functions of the t is not a priori the er whet he the ben e tub IIL Tube Models cular coy He h a syste s der efrom more fundame info tu the network the and by levant ca results ere the strain diffe was ad the ions ea 31 of n nodel A and To model to cross-linking.they use 135 and model B to be the ap priate choic ial ement nder traube and Rubinstein and Panyukov usly. bot nt simul G=+ V>i (48) he a hat in order to preserv differen where in contrast to ref 44 the various e no lor free parar et rs but dep 9 of the ugh e P22 on sing on Ir mo Hketopoimtoutaposibiesourdebroisioeoietd tailed dthe contributions from the A and B confinements. While eqs 34 and 38 are reproduced in the limits γBp ) 0 and γAp ) 0, respectively, eq 47 reflects the fact that a mode can never contribute more than kBT to the shear modulus. Thus, for γBp ) 1 (respectively γAp ) 1) the pth mode contributes this maximum amount indepen￾dent of the value of γAp (respectively γBp). An important point, which holds for all three models, is that it is not possible to estimate the confinement contribution to the shear modulus from the knowledge of the absolute strength lA, lB of the confining potentials alone. Required is rather the knowledge of the relative strengths γAp,γBp which in turn are functions of the network connectivity. G. Discussion. It is not a priori clear, whether entanglement effects are more appropriately described by model A or model B. While model A has the benefit of simplicity, Ronca and Allegra proposed model B,16 because it leads (on length scales beyond the tube diameter) to the conservation of intermolecular contacts under strain. Similar conclusions were drawn by Hei￾nrich and Straube25 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 em￾pirical approach seems to be the safest option. Fortu￾nately, the evidence provided by experiments36 and by simulations14 points into the same direction. Since details of the interpretation of the relevant experiments are still controversial (see section III.D.3), we concentrate on simulation results where the strain dependence of approximate eigenmodes of the phantom model was measured directly.14 Figure 1 shows a comparison of data obtained for defect-free model poly￾mer networks to the predictions eq 31 of model A and eq 36 of model B. The result is unanimous. We therefore believe eq 35 and model B to be the appropriate choice for modeling confinement due to entanglements. The shear modulus of an entangled network should thus be given by44 where in contrast to ref 44 the various γp are no longer free parameters but depend through eq 22 on a single parameter: the strength l of the confining potential, which is assumed to be homogeneous for all monomers. The difficulty of this formal solution of the generalized constrained fluctuation model for polymer networks is hidden in the use of the generalized Rouse modes of the phantom model, which are difficult to obtain for realistic connectivities.46,47 A useful ansatz for end-linked net￾works is a separation into independent Flory-Einstein respectively Rouse modes for the cross-links and net￾work strands.14,44 In fact, the simulation results pre￾sented in Figure 1 are based on such a decomposition. For randomly cross-linked networks with a typically exponential strand length polydispersity, the separation into Flory-Einstein and single-chain Rouse modes ceases to be useful. In this case, we can think of two radically different strategies. • To keep the network connectivity in the analysis. For example, there is no principle reason why the methods presented by Sommer et al.47 and Everaers14 could not be combined, to investigate the strain depen￾dence of constrained generalized Rouse modes in com￾puter simulations. Note, however, that this completely destroys the self-averaging properties of the approxima￾tion used in ref 14. Analytic progress in the evaluation of, for example, eq 38 for the entanglement contribution to the shear modulus requires information on the statistical properties of the eigenvalue spectra of net￾works generated by random cross-linking. To our knowl￾edge, the only available results were obtained numeri￾cally by Shy and Eichinger.48 Note that model C is irrelevant, if one is able to carry out calculations with the proper network eigenmodes. • To average out the connectivity effects in tube models for polymer networks.15 In the second part of the paper, we will consider linear chains under the influence of two types of confinement: network con￾nectivity and entanglements. III. Tube Models In SANS experiments of dense polymer melts, it is possible to measure single chain properties by deuter￾ating part of the polymers.49 If such a system is first cross-linked into a network and subsequently subjected to a macroscopic strain, one can obtain information on the microscopic deformations of labeled random paths through the network.49 To interpret the results, they need to be compared to the predictions of theories of rubber elasticity. Unfortunately, for randomly cross￾linked networks it is quite difficult to calculate the relevant structure factors even in the simplest cas￾es.12,50,51 Because the cross-link positions on different precursor chains should be uncorrelated, Warner and Edwards15 had the idea to consider a tube model, where the cross-linking effect is “smeared out” along the chain. To model confinement due to cross-linking, they used (in our notation) model A, since this ansatz reproduces the essential properties of phantom models (affine deformation of equilibrium positions and deformation independence of fluctuations). In contrast, Heinrich and Straube25 and Rubinstein and Panyukov43 treated con￾finement due to entanglements using model B. Obvi￾ously, both effects are present simultaneously in poly￾mer networks. In the following, we will develop the idea that in order to preserve the qualitatively different deformation dependence of the two types of confinement, they should be treated in a “double tube” model based on our model C. Before entering into a detailed discussion, we would like to point out a possible source of confusion related Figure 1. Excitation of constrained modes parallel and perpendicular to the elongation at λ ) 1.5 as a function of the mode degree of confinement 0 e γ e 1. The dashed (dotted) lines show the predictions eq 31 of model A and eq 36 of model B respectively for generalized Rouse modes of a phantom network with identical connectivity. The symbols represent the result of computer simulations of defect-free model polymer networks.14 The investigated modes are single-chain Rouse modes for network strands of length N ≈ 1.25Ne. G ) Gph + kBT V ∑ p γp 2 (48) 5680 Mergell and Everaers Macromolecules, Vol. 34, No. 16, 2001