Gaussian network behaviour

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

It must be concluded that chain ordering may be a reality at least in a number of networks and should be taken into account as a possible source of deviations from Gaussian network behaviour (see (IV-3)). 

All dry networks and many swollen networks exhibit deviations from the Gaussian network behaviour discussed in the previous chapter. These deviations may have various causes ... 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

The stress-strain behaviour of models such as that of Figure 11.16 can be explored by solving the associated equations using numerical techniques. In the work of Sweeney et al. on PET fibres [62], a model similar to that of Figure 11.16 but with the Eyring dashpot restrained by a Gaussian network, was solved in this way. The strain at which yield occurs, the general shape of the stress-strain curve, and the stability of the deformation were predicted and found to compare well with experiment. 

The paper first considers the factors affecting intramolecular reaction, the importance of intramolecular reaction in non-linear random polymerisations, and the effects of intramolecular reaction on the gel point. The correlation of gel points through approximate theories of gelation is discussed, and reference is made to the determination of effective functionalities from gel-point data. Results are then presented showing that a close correlation exists between the amount of pre-gel intramolecular reaction that has occurred and the shear modulus of the network formed at complete reaction. Similarly, the Tg of a network is shown to be related to amount of pre-gel intramolecular reaction. In addition, materials formed from bulk reaction systems are compared to illustrate the inherent influences of molar masses, functionalities and chain structures of reactants on network properties. Finally, the non-Gaussian behaviour of networks in compression is discussed. 

The networks studied were prepared from reactions carried out at different initial dilutions. Aliquots of reaction mixtures were transferred to moulds, which were maintained at the reaction temperature under anhydrous conditions, and were allowed to proceed to complete reaction(32). Sol fractions were removed and shear moduli were determined in the dry and equilibrium-swollen states at known temperatures using uniaxial compression or a torsion pendulum at 1Hz. The procedures used have been described in detail elsewhere(26,32). The shear moduli(G) obtained were interpreted according to Gaussian theory(33 34 35) to give values of Mc, the effective molar mass between junction points, consistent with the affine behaviour expected at the small strains used(34,35). 

The deviations from Gaussian stress-strain behaviour introduce uncertainties into the values of Mc/M discussed previously in this paper. However, such uncertainties have been shown to be of secondary importance compared with the ranges of Mc/Mc values found for networks from different reaction systems(25,32). 

Figure 11 shows plots according to equation(lO) of stress-strain data for triol-based polyester networks formed from the same reactants at three initial dilutions (0% solvent(bulk), 30% solvent and 65% solvent). Only the network from the most dilute reactions system has a strictly Gaussian stress-strain plot (C2 = 0), and the deviations from Gaussian behaviour shown by the other networks are not of the Mooney-Rivlin type. As indicated previously, they are more sensibly interpreted in terms of departures of the distribution of end-to-end vectors from Gaussian form. 

The deviations from Gaussian stress-strain behaviour for the tetrafunctional polyurethane networks of Figure 9 are qualitatively similar to these found for the trifunctional polyester networks (Z5), and the error bars on the data points for systems 4 and 5 in Figure 9 indicate the resulting uncertainties in Mc/Mc. It is clear that such uncetainties do not mask the increases in Mc/Mc with amount of pre-gel intramolecular reaction. 

Interesting deviations from Gaussian stress-strain behaviour in compression have been observed which related to the Me of the networks formed, rather than their degrees of swelling during compression measurements. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

According to our definition the end-to-end distances of the network chains have an a priori probability distribution which is Gaussian. The effect of finite extensibility of the chains will be postponed to Chapter IV, because it is a special aspect of non-Gaussian behaviour. 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

The qualitative, overall effect of small N is a stress in the deformed network which is larger than the corresponding Gaussian stress. This effect is, however, very small in normally crosslinked rubbers, whereas these rubbers exhibit large deviations from Gaussian behaviour. The tentative conclusion must therefore be that the deviations from Gaussian behaviour at moderate extensions cannot be caused by short chains only. 

The above analysis was based upon a consideration of deviations from Gaussian behaviour of isolated chains. In reality we are concerned with network chains. This introduces a restriction on the conformational... 

As a consequence of this conclusion, we are immediately faced with the necessity of looking for other explanations of the deviations from Gaussian behaviour than anisotropic excluded volume effects. We, therefore, turn to the suggestion of further structuring in the network made originally by Gee, and worked out subsequently by Volkenstein, Gotlib and Ptitsyn (774), and more recently by Blokland (74). 

In addition to the above experimental point, one can raise a theoretical objection against the way in which Volkenstein et al. introduce the effect which the structure in a network has on its elastic behaviour. In their theory the Gaussian chain statistics are left unchanged in spite of the fact that the chain molecules run through bundles. Such a decoupling of chain statistics and bundles is unwarranted. In Fig. 29 c a schematic representation of the approach of Volkenstein et al. to a structured network is given. Only a two chain network is drawn, although it should, of course, be remembered that in reality a bundle structure will comprise parts of many molecules. 

Although at the moment no adequate theory exists which relates the structure of a network to its elastic behaviour, we may venture to hypothize that the deviation from Gaussian behaviour, as e.g. measured by C2/Cj, is indeed related to the structure. It is of interest to note that... 

It has been pointed out repeatedly that the elastic behaviour of virtually all real networks in the unswollen state deviates appreciably from Gaussian behaviour. Often these deviations depend on the history... 

The theoretical approach for determining the deformation behaviour of a network due to swelling or due to a mechanical force (stress-strain measurements, compression experiment) is based on a hypothetical phantom network. A phantom network is, by definition, a network with the fictitious property that chains and junctions can move freely through one another without destroying the cormectivity of the network. Usually, the network chains behave as Gaussian chains. Within the phantom network model, three network types can be distinguished ... 

---