Gaussian behaviour

As discussed in the last 3 years, polysaccharides behave in solution under a worm like chain [26] the local stiffness of the chain is characterized by a persistance length (Ip) the larger Ip is, the larger the chain deviates from the gaussian behaviour in the usual molecular weight range of these natural polymers [27], This makes difficult to use the relations given in litterature for synthetic... 

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. 

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. 

Tables of this sort are valid for Gaussian coils only. In thermodynamically good solvents the Gaussian behaviour of chain molecules is perturbed by what is called the excluded volume effect 30. The P/(0) function depends on the distribution of mass within the particle and this, in turn, is changed if the volume effect is operative.

These two facts motivated a critical check of the validity of Eq. 4.11 in a wide Q-range [9,105,154,155]. For this purpose the information obtainable from fully atomistic MD simulations was essential. The advantage of MD simulations is that, once they are validated by comparison with results on the real system, magnitudes that cannot be accessed by experiments can be calculated, as for example the time dependence of the non-Gaussian parameter. The first system chosen for this goal was the archetypal polymer PL The analysis of the MD simulations results [105] on the self-motion of the main chain hydrogens was performed in a similar way to that followed with experimental data. This led to a confirmation of Eq. 4.11 beyond the uncertainties for Q<1.3 A (see Fig. 4.15). However, clear deviations from the Q-dependence of the Gaussian behaviour... 

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... 

Fig. 5.25 Result of applying shift factors corresponding to an activation energy of 0.43 eV to the relaxation times observed for the collective dynamics (empty symbol) and the self-correlation (full symbol) of PIB 335 K (circle), 365 K (square), and 390 K (triangles) (reference temperature 365 K). The dotted line through the self-correlation data shows the dependence implying Gaussian behaviour (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

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. 

Various Types of Deviations from 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 ... 

Fig. 22 shows the same data of Fig. 21 plotted as force versus deformation ratio. If the data in compression are taken to follow Gaussian behaviour it is seen that a negative strain-dependent correction is needed... 

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... 

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... 

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