Deviations from Gaussian networks

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)). 

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. 

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). 

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... 

Experimental results on reactions forming tri- and tetrafunctional polyurethane and trifunctional polyester networks are discussed with particular consideration of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction is shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. 

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>... 

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 ... 

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 ... 

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... 

In this example, pi and Oi are the parameters for the hidden units. The final output of the network is a simple weighted sum of outputs from each of the hidden units and their weights. Input to the hidden units is the distance between the input x and their respective centers, p.j. If x is closer to pi than p2 then the output from hidden unit 1 is stronger than for hidden unit 2. Figure 4.3 illustrates this point, with a Gaussian basis function, with a=l. The maximum output from the hidden unit occurs when the distance measure is 0. When the input is 3 standard deviations from the mean p, the center of the function, the output of the hidden unit is practically nil. 

For polymer networks, K /t, and M reflects, on the whole, compressibility. Indeed, under deformation of polymer networks, the energy component partially shows itself, some other deviations from the Gaussian network model have also been discussed. These phenomena arc discussed in the special literature (Birshtein and Ptitsyn, 1964 Dusek and Prins, 1969). 

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