Entanglement contribution

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

Modulus data on crosslinked systems would seem to offer the most direct method for studying entanglement effects. Certainly, from the standpoint of molecular modeling, the advantages of equilibrium properties are clear. However, the structural characterization of networks has proven to be very difficult, and without such characterization it is almost impossible to separate entanglement contributions from those of the chemical crosslinks alone. Recent work suggests, however, that these problems are not insurmountable, and some quantitative results are beginning to appear. 

It appears that one can develop a qualitative understanding of the simple flow properties at moderate concentration without going beyond concepts which are already inherent either in the dilute solution theory of polymers or in the properties of particulate suspensions. The dependence of viscosity on c[i ] is believed to reflect a particle-like or equivalent sphere (127) hydrodynamics in solutions of low to moderate concentration. In particular, the experimental facts do not force the consideration of effects which might arise from the permanent connectedness of the polymer backbones. Situations conducive to the entangling of molecules may be present, e.g., overlap of the coils, but either entanglement contributions are small, or else they are overwhelmed by the c[f ] interactions. 

The first two terms on the right of Eq. (7.30) will be recognized as the independent strand contribution to the entropy. The topological or entanglement contribution is then... 

The total pair-wise entanglement contribution is simply the sum of dS for strand pairs with all possible sets of internal coordinates. In a network of N strands there will be N(N — l)/2 such pairs. However, classification is only significant for pairs that are relatively close. Pairs separated by more than a few radii of gyration will belong to unentangled class exclusively. This assures that the total entropy change will be proportional only to N. 

Fig. 13. Dependence of the reduced modulus G, (mol/cm ) and of the trapped entanglement contribution A (mol/cm ) on r for the cured DGEBA-PGE-DDM systems q measurements in the...

The total free energy of deformation per unit volume is the sum of Wi and W2. For large N, the entanglement contribution to network elasticity becomes... 

This result differs somewhat from the expression obtained using the Doi-Edwards model (Eq.40), and it gives a larger departure from neo-Hookean behavior for uniaxial extension (A q>endix II and Fig. 9). In the limit of small deformations the entire contribution to stress comes from the first term in Eq. 62. The entanglement contribution to the infinitesimal shear modulus is predsely the same as the Doi-Edwards expression for the plateau modulus (Eq. 37)... 

Both Eqs (7.64) and (7.65) reduce to Eq. (7.48) in the small deformation limit (A 1). This simple additivity separates the crosslink and entanglement contributions to the stress and hence allows them to be determined... 

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