Macromolecules.Vol.34,No.16.200 Tube Models for Rubber-Elastic Systems 5685 IV.Conclusion gene ide show dwards. ns of Eichir Mark. is the ob ked networks. he of the .10 wand 1020 licitly co ider the eous presenc e of tw 达 del (dotted i tion of the amle we belleve to have made some podo f str For e mple.it remains to be shown how the whoegean the ibly important fewauldp one to ob d an exc of th aveneglectedauctliationstnths he lozenge ne anisotrop aractera motio ithere nc only the central part of the more elaboperpe est ap to clarify this point. observed by Straube et al.32 In particular, we agree with Read and McLeish34 that the interpretation of Straube et al.31,32,36 is based on an ad hoc approximation in the calculation of structure functions from model B. In principle, their alternative idea, to investigated the influence of dangling ends on the structure-function within models A and B,34 can be easily extended to model C. Judging from the small differences between the models (Figures 7 and 8) and the results in ref 34, this would probably allow one to obtain an excellent fit of the data and to correctly account for the deformation dependence of the tube.36 However, since the lozenge patterns were also observed in triblock systems where only the central part of the chains was labeled,33 dangling ends seem to be too simple an explanation. At present it is therefore unclear, if the lozenge patterns are a generic effect or if they are due to other artifacts such as chain scission.36,37 Simulations14,28-30 might help to clarify this point. IV. Conclusion In this paper, we have presented theoretical consid￾erations related to the entanglement problem in rub￾ber-elastic polymer networks. More specifically, we have dealt with constrained fluctuation models in general and tube models in particular. The basic idea goes back to Edwards,21 who argued that on a mean￾field level different parts of the network behave, as if they were embedded in a deformation-dependent elastic matrix which exerts restoring forces toward some rest positions. In the first part of our paper, we were able to show that the generalized Rouse modes of the corre￾sponding phantom network without entanglement re￾main eigenmodes in the presence of the elastic matrix. In fact, the derivation of eq 14, which is the Hamiltonian of the exactly solvable constrained mode model (CMM),44 provides a direct link between two diverging develop￾ments in the theory of polymer networks: the ideas of Edwards, Flory, and others on the suppression of fluctuations due to entanglements and the consider￾ations of Eichinger,11 Graessley,45 Mark,46 and others on the dynamics of (micro) phantom networks. An almost trivial conclusion from our theory is the observa￾tion, that it is not possible to estimate the entanglement effects from the knowledge of the absolute strength of the confining potentials alone. Required is rather the knowledge of the relative strength which in turn is a function of the network connectivity in eq 38. Unfortunately, it is difficult to exploit our formally exact solution of the constrained fluctuation model for arbitrary connectivity, since it requires the eigenvalue spectrum of the Kirchhoff matrix for randomly cross￾linked networks. In the second part of the paper we have therefore reexamined the idea of Heinrich and Straube25 to introduce entanglement effects into the Warner￾Edwards model15 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 Panyukov43 we have argued that in contrast to confine￾ment due to cross-linking, confinement due to entangle￾ments 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. The effects are shown to be nonadditive. From the solution of the generalized tube model we have obtained expressions for the mi￾croscopic deformations and macroscopic elastic proper￾ties which can be compared to experiments and simu￾lations. While we believe to have made some progress, we do not claim to have solved the entanglement problem itself. For example, it remains to be shown how the geometrical tube constraint arises as a consequence of the topological constraints on the polymer conforma￾tions. However, even on the level of the tube model, we are guilty of (at least) two possibly important omis￾sions: (i) we have neglected fluctuations in the local strength of the confining potential, and (ii) we have suppressed the anisotropic character of the chain motion parallel and perpendicular to the tube. In the absence of more elaborate theories, computer simulations along the lines of refs 14, 28, 29, 30, and 47 may present the best approach to a quantification of the importance of these effects. Acknowledgment. The authors wish to thank K. Kremer, M. Pu¨ tz, and T. A. Vilgis for helpful discus￾Figure 8. Contour plots of the different structure factors with Rg/dT ) 6: Warner-Edwards model (dashed), Heinrich￾Straube/Rubinstein-Panyukov model (dotted line), “double tube” model with Φ ) 3/4 (solid line). The upper curves correspond to the perpendicular stretching direction. Macromolecules, Vol. 34, No. 16, 2001 Tube Models for Rubber-Elastic Systems 5685