Macromolecules 2001,34.5675-568e 5675 Tube Models for Rubber-Elastic Systems Boris Mergell and Ralf Everaers* Max-Planck-Institut fur Polymerforschung.Pastfach 3148 D-55021 Mainz Germany Received December 29.2000:Revised Manuscript Received April 6.200 d mod which is partially due t n8andpaiatoetoentang properties s well as for the efo ations including structure factors I.Introduction chain structur are the basic structural elemen as long as the system remair Polymer networks vity the ph ton nall nd the mat hich take th f the ne The sit 702'tI magnitude smalle olids.Ma do not enter the Hamil in as nd Joule in the entang ts can be cha racts of r s.Most omit such a deta copic,statis entrop scriptio n favo meric mocma high de are ton-dep osition h may och N AS al t ork i or to path ich the chains ces on their ends springs Th e the ating a p iece of ru as a ands is rticularly the shape ing ical trea alitativ N- anding.a rigor statistical m the comp day.Similar to es.the main ce correctness of the to d In o th its a rand due to the intr rodu dat olution o he edwards tub due to th formatio of nd the are on the inter l tost of the tacts and quenche y contair and defp ensemble averages of static expectation values for the notion and therefore allow for a more detailed test of 0.1021ma002228 Tube Models for Rubber-Elastic Systems Boris Mergell and Ralf Everaers* Max-Planck-Institut fu¨ r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany Received December 29, 2000; Revised Manuscript Received April 6, 2001 ABSTRACT: In the first part of the paper, we show that the constraining potentials introduced to mimic entanglement effects in Edwards’ tube model and Flory’s constrained junction model are diagonal in the generalized Rouse modes of the corresponding phantom network. As a consequence, both models can formally be solved exactly for arbitrary connectivity using the recently introduced constrained mode model. In the second part, we solve a double tube model for the confinement of long paths in polymer networks which is partially due to cross-linking and partially due to entanglements. Our model describes a nontrivial crossover between the Warner-Edwards and the Heinrich-Straube tube models. We present results for the macroscopic elastic properties as well as for the microscopic deformations including structure factors. I. Introduction Polymer networks1 are the basic structural element of systems as different as tire rubber and gels and have a wide range of technical and biological applications. From a macroscopic point of view, rubberlike materials have very distinct visco- and thermoelastic properties.1,2 They reversibly sustain elongations of up to 1000% with small strain elastic moduli which are 4 or 5 orders of magnitude smaller than those for other solids. Maybe even more unusual are the thermoelastic properties discovered by Gough and Joule in the 19th century: when heated, a piece of rubber under a constant load contracts, and conversely, heat is released during stretching. This implies that the stress induced by a deformation is mostly due to a decrease in entropy. The microscopic, statistical mechanical origin of this entropy change remained obscure until the discovery of poly￾meric molecules and their high degree of conformational flexibility in the 1930s. In a melt of identical chains, polymers adopt random coil conformations3 with mean￾square end-to-end distances proportional to their length, 〈br2〉 ∼ N. A simple statistical mechanical argument, which only takes the connectivity of the chains into account, then suggests that flexible polymers react to forces on their ends as linear, entropic springs. The spring constant, k ) (3kBT)/〈br2〉, is proportional to the temperature. Treating a piece of rubber as a random network of noninteracting entropic springs (the phan￾tom model4-6) qualitatively explains the observed be￾havior, includingsto a first approximationsthe shape of the measured stress-strain curves. Despite more than 60 years of growing qualitative understanding, a rigorous statistical mechanical treat￾ment of polymer networks remains a challenge to the present day. Similar to spin glasses,7 the main difficulty is the presence of quenched disorder over which ther￾modynamic variables need to be averaged. In the case of polymer networks,8-10 the vulcanization process leads to a simultaneous quench of two different kinds of disorder: (i) a random connectivity due to the introduc￾tion of chemical cross-links and (ii) a random topology due to the formation of closed loops and the mutual impenetrability of the polymer backbones. Since for instantaneous cross-linking monomer-monomer con￾tacts and entanglements become quenched with a probability proportional to their occurrence in the melt, ensemble averages of static expectation values for the chain structure etc. are not affected by the vulcanization as long as the system remains in its state of preparation. For a given connectivity the phantom model Hamil￾tonian for noninteracting polymer chains formally takes a simple quadratic form,4-6 so that one can at least formulate theories which take the random connectivity of the networks fully into account.11-13 The situation is less clear for entanglements or topological constraints, since they do not enter the Hamiltonian as such but divide phase space into accessible and inaccessible regions. In simple cases, entanglements can be charac￾terized by topological invariants from mathematical knot theory.8,9 However, attempts to formulate topologi￾cal theories of rubber elasticity (for references see ref 14) encounter serious difficulties. Most theories there￾fore omit such a detailed description in favor of a mean￾field ansatz where the different parts of the network are thought to move in a deformation-dependent elastic matrix which exerts restoring forces toward some rest positions. These restoring forces may be due to chemical cross-links which localize random paths through the network in space15 or to entanglements. The classical theories of rubber elasticity1,16-20 assume that entangle￾ments act only on the cross-links or junction points, while the tube models2,21-26 stress the importance of the topological constraints acting along the contour of strands exceeding a minimum “entanglement length”, Ne. Originally devised for polymer networks, the tube concept is particularly successful in explaining the extremely long relaxation times in non-cross-linked polymer melts as the result of a one-dimensional, curvilinear diffusion called reptation27 of linear chains of length N . Ne within and finally out of their original tubes. Over the past decade, computer simulations14,28-30 and experiments31-33 have finally also collected mount￾ing evidence for the importance and correctness of the tube concept in the description of polymer networks. More than 30 years after its introduction and despite its intuitivity and its success in providing a unified view on entangled polymer networks and melts,2,23-26 there exists to date no complete solution of the Edwards tube model for polymer networks. Some of the open problems are apparent from a recent controversy on the inter￾pretation of SANS data.32-37 Such data constitute an important experimental test of the tube concept, since they contain information on the degree and deformation dependence of the confinement of the microscopic chain motion and therefore allow for a more detailed test of Macromolecules 2001, 34, 5675-5686 5675 10.1021/ma002228c CCC: $20.00 © 2001 American Chemical Society Published on Web 06/28/2001 Downloaded via GUANGDONG UNIV OF TECHNOLOGY on March 14, 2023 at 12:31:24 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles