Phantom network density

The remaining question is, how the deviations from phantom network theory at high branching densities can be explained. 

Figure 7. Ratio of experimentally observed and theoretically calculated modulus, using phantom network theory with f2 and v2, versus branching density z-...

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. 

Since the DGEBA/DDS networks are tetrafunctional and of stoichiometric composition, the theoretical value of z is 2. Furthermore, the crosslink concentration, c, is simply the DDS molecule concentration. Performing the necessary calculations yields the theoretical M, listed in Table 4. Compared to the experimental M, the theoretical values are very consistent. If it is assumed that the DGEBA/DDS networks are not phantom-like (i.e., A = 1), then the ratio of the theoretical and experimental values may serve as an estimate of the dilation factor, These ratios are listed in Table 4, and show that is approximately unity for all the networks. If the experimental M had been calculated using the actual network densities (instead of q = 1 g/cm), the ratios would be even closer to unity, being reduced by approximately 20 percent. 

These predictions need to be modified because real networks have defects. As shown in Fig. 7.7, some of the network strands are only attached to the network at one end. These dangling ends cannot bear stress and hence do not contribute to the modulus. Similarly, other structures in the network (such as dangling loops) are also not elastically effective. The phantom network prediction can be recast in terms of the number density of elastically effective strands v and the number density of elastically effective crosslinks ii. For a perfect network without defects, the phantom network modulus is proportional to the difference of the number densities of network strands v and crosslinks // = since there are fjl network strands per crosslink ... 

Consider a circular cylindrical gel extended along its axis so that the circular symmetry is maintained. Let ho and h be the initial (reference) and deformed length of the cylinder. Then, the principal deformation ratio along the axis z is kz, kx and ky are deformation ratios for the axes perpendicular to the axis z. The strain energy density function for and ideal phantom network, Wgi, can be expressed as follows [3,4,84] ... 

The simplest swelling measurement is the equilibrium determination of V2 ,. Without knowledge of the network structure (i.e. x and Q, it enables one to determine the range where lies (between obtained by making use of Eq. (45) and by Eq. (48)). The calculated is lower than In a phantom network, junction fluctuations decrease the impact of chain entropy changes. It is therefore necessary to have a phantom network with a higher density of crosslinks or a smaller to counteract this effect and to give the same elastic contribution as in an affine network. 

In order to enable these fluctuations to occur, the network chains are assumed to be "phantom" in nature i.e. their material properties are dismissed and they act only to exert forces on the junctions to which they are attached. With common networks having tetrafunctional junctions, the results of the two approaches differ by a factor of two. Identical results are only obtained from both theories, when the functionality is infinite. From a practical viewpoint, however, a value of about 20 for f can already be equated to infinity because crosslink densities can hardly be obtained with an accuracy better than 10%. 

The moduli, measured at crosslinking temperature T, which are given in the last two columns of Table IV, are abou two to three fold greater than those computed from phantom theory. Except for the samples with the lowest branching densities, the observed values agree satisfactorily with those for an affine network. 

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to the... 

A point worth noting here is that several of the molecular models that will be described in the subsequent sections are Neo-Hookean in form. Normally, dry rubbers do not exhibit Neo-Hookean behavior. As for the Mooney-Rivlin form of strain energy density function, rubbers may follow such behavior in extension, yet they do not behave as Mooney-Rivlin materials in compression. In Fig. 29.2, we depict typical experimental data for a polydimethylsiloxane network [39] and compare the response to Mooney-Rivlin and Neo-Hookean behaviors. The horizontal lines represent the affine and the phantom limits (see Network Models in Section 29.2.2). The straight line in the range A <1 shows the fit of the Mooney-Rivlin equation to the experimental data points. 

As shown in more detail elsewhere, the rubbery-state modulus Er showed a trend of increasing with crosslink density, with measured values lying close to or between the predictions of the affine and phantom chain theories of rubber elasticity [63]. However, we also observe an influence from the chemical composition, with the actual values between these two limits reflecting the intrinsic stiffnesses of the three diisocyanates and hence the molecular mobility at the network junctions [63]. 

Taking this value for CmocL in account reduces the difference with the standard model somewhat. The actual value of the modulus then can reasonably well be described by the affine network model for the LJ system. Including Cmod = 1.5 one gets G% 0.027. Using p = 0.37cr (the density of the elastically active part) one gets within the affine model an effective strand length of Neff 13-14, while the cluster analysis yields Ng/f 11. The soft core model is somewhere in between the affine and the phantom model. 

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