Phantom theory

Comparison with Statistical Theory, Small-strain Behaviour. Calculations of theoretical moduli, using the phantom theory and the affine theory as limiting cases, were carried out in order to compare the theoretical predictions with values found experimentally (Table IV). 

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. 

Comparison with Statistical Theory at Moderate Strains. So far we have shown, that a transition between the two limiting classical theories, i.e. affine theory and phantom theory, is possible by a suitable choice of the network microstructure. This argument goes beyond the revised theory by Ronca and Allegra and by Flory, which predicts such a transition as a result of increasing strain, thus explaining the experimentally observed strain dependence of the reduced stress. 

The classical affinity model assumes that the doss-links are immobile with respect to the whole network. The fluctuations of the positions of cross-links induced by thermal motion are taken into account in the phantom model proposed by James and Guth. It suggests that the fluctuations of a given CTOss-link proceed independently of the presence of subchains linked to it, and during such fluctuations the subchains can pass freely through each other like phantoms. The classic phantom theory predicts the shear modulus G as ... 

A second theory of phantom pain suggests that second-order neurons in the dorsal horn of the spinal cord become hyperactive. Spontaneous firing of these neurons causes transmission of nerve impulses to the brain and the perception of pain. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

The expressions given in this section, which are explained in more detail in Erman and Mark [34], are general expressions. In the next section, we introduce two network models that have been used in the elementary theories of elasticity to relate the microscopic deformation to the macroscopic deformation the affine and the phantom network models. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

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 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-...

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

The theory (9) predicts that in simple elongation, the ratio fc/f ph decreases with increasing strain and eventually goes to zero (phantom network). Furthermore, at a =1, the theory holds that... 

According to the Flory theory (9), the predicted range on Aj and A2 lies between one and (1 —2/). The upper limit, unity, corresponds to affine behavior and the lower limit occurs In phantom behavior. Therefore both A3 and A2 have predicted asymptotes of one at high functionalities and should be Independent of vs/V. 

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. 

According to the theory (1,2), for phantom imperfect (cf. also (17)) network the front factor A assumes the value... 

Early theories of Guth, Kuhn, Wall and others proceeded on the assumption that the microscopic distribution of end-to-end vectors of the chains should reflect the macroscopic dimensions of the specimen, i.e., that the chain vectors should be affine in the strain. The pivotal theory of James and Guth (1947), put forward subsequently, addressed a network of Gaussian chains free of all interactions with one another, the integrity of the chains which precludes one from the space occupied by another being deliberately left out of account. Hypothetical networks of this kind came to be known later as phantom networks (Flory, 1964,... 

A subsequent theory [6] allowed for movement of the crosslink junctions through rearrangement of the chains and also accounted for the presence of terminal chains in the network structure. Terminal chains are those that are bound at one end by a crosslink but the other end is free. These terminal chains will not contribute to the elastic recovery of the network. This phantom network theory describes the shear modulus as... 

Eichinger,B.E. Elasticity theory. I. Distribution functions for perfect phantom networks. Macromolecules 5,496-505 (1972). 

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