Macromolecules.Vol.34,No.16.2001 Tube Models for Rubber-Elastic Systems 5683 及=2强切+&2 (77) √42)+g 25 2 o月=2-1gc月+(1-1gc-1(78) 1.5 金 1 4 G=2c2+G0 N/No (79) S.The 4GB2+G ubinst anvukov mod 2G=C'+6cc2+4G (4Cg2+G2 (80 2C2 4GB4 (4C。2+C32 81) 15 hboSewParteanoftheDieiereantTbeModelsh 12 45 6 toto of the average strand length Mbetween cros lnk ponding phantoer -6==-% (82) 女09 42=f2N 07 (83) 60.6 .60.70.80.9 总云J 4 fo oney-Rivn -6-6- (84) Straube/Rubing d42=N (85) an ays sma an that in a highly cross-linked ne typical fluc s a ons = (86) re ponse.Figure show dre gous results for the Mooney in para eters to binstein (87) igure 4 shows the C an C are a function of the entanglement contribution d Again, eq 78 only holds for moderate strains. Shear modulus and the Mooney-Rivlin parameters are given by D. Comparison of the Different Tube Models. In the following, we compare the predictions of the differ￾ent models for the microscopic deformations and the macroscopic elastic properties from two different points of view. 1. As a function of the network connectivity, i.e., the ratio of the average strand length Nc between cross-links to the melt entanglement length Ne. For this purpose, we identify GA with the shear modulus of the corre￾sponding phantom network Gph: where we use f ) 4 for our plots. Similarly, we choose for GB a value of the order of the melt plateau modulus Ge: 2. Assuming that the system is characterized by a certain tube diameter dTC or shear modulus GC, we discuss its response to a deformation as a function of the relative importance 0 e Φ e 1 of the cross-link and the entanglement contribution to the confinement where Φ is of the order (1 + (Ne/Nc)2)-1. 1. Elastic Properties. Figure 2 shows the shear modulus dependence on the ratio of the network strand length Nc to the melt entanglement length Ne. As expected GC crosses over from Gph for short strands to Ge in the limit of infinite strand length. For comparison we have also included the prediction of Rubinstein and Panyukov, Gph + Ge. The shear moduli predicted by our ansatz are always smaller than this sum. In particular, we find G ) Gph for Nc , Ne. The physical reason is that in a highly cross-linked network the typical fluc￾tuations are much smaller than the melt tube diameter. As a consequence, the network does not feel the ad￾ditional confinement and the entanglements do not contribute to the elastic response. Figure 3 shows analogous results for the Mooney-Rivlin parameters C1 and C2 again in comparison to the predictions of Rubinstein and Panyukov. Note that C2 is not predicted to be strand length independent. Figure 4 shows the reduced force in the Mooney￾Rivlin representation for different entanglement con￾tributions Φ to the confinement. For moderate elonga￾tions up to λ ≈ 2 the curves are well represented by the Mooney-Rivlin form. For a given shear modulus, C1 and C2 are a function of the entanglement contribution Φ Figure 2. Langley plot of the shear modulus. The solid line corresponds to the “double tube” model, the dotted line to the Heinrich-Straube/Rubinstein-Panyukov model and the dashed line to the phantom model. Ne represents the entanglement length and Nc the cross-link length. Figure 3. Plot of the parameters 2C1 and 2C2 of the Mooney￾Rivlin equation f(λ-1) ) 2C1 + 2C2λ-1 for the Rubinstein￾Panyukov model (dotted) and the “double tube” model (solid). Figure 4. Mooney-Rivlin representation of the reduced force for different values of Φ (from top to bottom: the Phantom model (dashed line, Φ ) 0), the “double tube” model (solid lines, Φ ) 1/3, 1/2, 3/4) and the Heinrich-Straube/Rubinstein￾Panyukov model (dotted line, Φ ) 1)). gC(λ) ) 2gB 2 (λ) + gA 2 x4gB 2 (λ) + gA 2 (77) σT(λ) ) (λ2 - 1)gC(λ) + (1 - λ-1 )gC(λ-1/2) (78) GC ) 2GB 2 + GA 2 x4GB 2 + GA 2 (79) 2C1 ) GA 4 + 6GB 2 GA 2 + 4GB 4 (4GB 2 + GA 2 ) 3/2 (80) 2C2 ) 4GB 4 (4GB 2 + GA 2 ) 3/2 (81) 1 4 Fb2 kBT dTA 2 ) GA ) Gph ) (1 - 2 f) FkBT Nc (82) dTA 2 ) f - 2 4f b2 Nc (83) 1 8 Fb2 kBT dTB 2 ) GB ) Ge ) 3 4 Fb2 kBT Ne (84) dTB 2 ) 1 6 b2 Ne (85) Φ ) dTC 4 dTB 4 (86) 1 - Φ ) dTC 4 dTA 4 (87) Macromolecules, Vol. 34, No. 16, 2001 Tube Models for Rubber-Elastic Systems 5683