Deviations from Gaussian

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). 

The observed deviations from Gaussian stress-strain behaviour in compression were in the same sense as those predicted by the Mooney-Rivlin equation, with modulus increasing as deformation ratio(A) decreases. The Mooney-Rivlin equation is usually applied to tensile data but can also be applied compression data(33). According to the Mooney-Rivlin equation... 

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 deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

The stronger deviations from Gaussian behaviour for PVE, a polymer with large side groups (see Table 1.1), could be caused by the different mobility of the hydrogens linked to the main chain carbons and those in the side groups. This hypothesis could be confirmed by the MD simulations. Distinguishing the... 

Kittel and Abrahams 12S) have predicted an approximately Lorentzian magnetic resonance line shape for a system of spins which are randomly distributed over a small fraction of a large number of possible sites. This effect has been observed in electron spin resonance (124)- Kittel and Abrahams estimate that appreciable deviations from Gaussian shape will occur when the fraction of sites occupied, f, is less than 0.1, in the case of spins of / = H iu a simple cubic lattice with the magnetic field directed... 

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)). 

Various Types of Deviations from Gaussian Behaviour... 

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. 

Another reason for deviations from Gaussian behaviour, even at large N, lies in the finite extensibility of polymer chains. To account for this, one utilizes the complete expression for the partition function qt of a chain of N chain elements, rather than just the Gaussian approximation to it. A very clear exposition of the statistical mechanics of a chain with length rt under tension, can be found in Hill s book (85) yielding... 

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... 

In physics, we are often interested in some quantities that deviate from Gaussian (normal) distribution, since the deviation is exceptional. Indeed, in physics, search for power-law distributions or log-normal distributions has been... 

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