Theory of phantom networks

Remark As previously stated, the above theory is the simplest one. There are other physical approaches of entropic elasticity, especially the theory of phantom networks (Queslel and Mark, 1989), in which the crosslinks freely fluctuate around their mean position. The above relationship then becomes ... 

The theory of phantom network was formulated by James and Guth [24] in the forties. They assumed that chains are Gaussian with the distribution P(r) of the end-to-end vector... 

The theory of phantom networks by James in 1947 and elaborated by Duiser and Staverman in 1965 and by Flory in 1976 is shown to lead generally to the frontfactor equation for the elastic free energy. 

Comparing these assumptions with the theory of phantom networks of Sects. 2 and 3, we see 3 elements of this theory missing ... 

To networks of class 2 as defined by Eq. (40b) neither the theory of phantom networks with fixed junctions of James nor the theory of localised phantom networks as presented in Sects. 2 and 3 is applicable since no junctions are fixed and the network is not in equilibrium withjtself. There is no direct and simple relation between the Kirchhoff matrix and th i -configuration. The i reai-values have been displaced from the corresponding R ph-values by liquid or entanglement forces. The exact nature of these forces in not known. 

A real network can be treated theoretically in two ways. One can start from the theory of phantom networks and introduce three perturbations 1° the constraints imposed on the fluctuations, 2° the distortion of the / p, -configuration or the S pi,-configuration leading to a different i rearConfiguration, 3 the different behaviour of jiuictions with different functionalities. 

Both limits correspond to the classical approaches of phantom network theory. 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

In the secmid part various theories of polymer networks are presented. The affine network model, phantom network, and theories of real networks are discussed. Scattering from polymer chains is also briefly presented. 

The modern theory of real networks now permits a more accurate determination of network structures through use of equations (84),(87), and (95) (187-204). Stress-strain measurements can he analyzed as shown in Figure 17. The phantom modulus thus determined leads to v and Me through equations (75) and (76) (189). Swelling equilibrium data are similarly analyzed through equation (94), with the parameter k given hy equation (87) (189). 

Birefringence of Phantom Networks. This theory is the basis for all theories that deal with birefringence of elastomeric polymer networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network to consist of phantom (ie, non-interacting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network the birefringence induced... 

As discussed briefly in the introduction the elastic and relaxational properties of polymer networks are also expected to be influenced significantly by the presence of entanglements. The classical theories, the phantom network modeP and the affine deformation model, describe the two extreme points of view. In the first, at least in its original form, the network strands and the crosslinks are not subject to any constraint besides connectivity and functionality. The other extreme considers the crosslinks to be fixed in space and deform affinely under deformation. A number of modifications of these theories have been proposed in which the junction fluctuations are partially suppressed. All of these models however consider the network strands as entropic springs. The entropic force, as... 

The number of elastic degrees of freedom is three times the cycle rank. The elasdc behaviour of real networks in particular in the unswollen suite, deviates from that of phantom networks. A recent theory to explain this deviation is based upon the assumption of a difference between pofyfimctiorud and bijwictiorml junctions which is alien to the concept of phantom networks and is pkysicedly not plausible. An alternative theory is presented based upon die concept of constrained elastic degrees of freedom instead of constrained junctions. 

In 1976, Flory has presented an elegant and concise version of the network theory of James of 1974, thereby terminating the controversy between this theory and Flory s earlier one. However, according to Flory the theory of James applies to phantom networks only. This creates the new problem of the relation between the properties of phantom networks and real networks. This problem is the main subject of this paper. 

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. 

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. 

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) ... 

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. 

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... 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

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