Networks with Gaussian Behaviour

In this Chapter we will consider the elasticity and swelling of networks which deviate from our definition of ideal networks only because of a certain number of network defects (see Chapter II, Section 2). We will designate by v the number of elastically effective network chains, which in an ideal network equals the number of chains because of the absence of defects. 

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. 

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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. 

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). 

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 ... 

With the advancement of a curing reaction, the of the resin will increase, but the goal is to quantitatively predict the of a resin as a function of cure conversion. Several models have been proposed to correlate the with the conversion or extent of curing (a). With the increase in conversion, the concentration of reactive functionalities decreases, and crosslinks or junction points are formed, leading to the departure from Gaussian behaviour. Steric hindrance affects chain conformation at high crosslink densities. The models are based on the statistical description of network formation and calculation of the concentration of jrmction points of different functionalities as a function of conversion. However, one issue that complicates the calculation and which is not fully resolved is whether to consider all the junction points or only those which are elastically effective. 

The behaviour of the specimen as a whole follows from an understanding of the behaviour of a representative chain. Imagine it detached from the network with one end placed at the origin of a coordinate system (see Fig. 3.2). It will be assumed that its randomness is exactly reproduced by the Gaussian theory ( 2.9). That is to say, with the passage of time the molecule changes shape in a truly random manner. It follows from eqn (2.N.7.1) that... 

Elastomers cured with two crosslinking systems such as sulphur and the polymerisation products of p-benzoquinone are shown to have much improved overall mechanical properties. Non-Gaussian behaviour of quinone polymer crosslinked elastomers viewed as bimodal networks was studied. The study focused on the effect of ageing time on the reduced stress values of the networks in relation to the elongation of the samples. The study is also extended to cover the possible effect of these bound antioxidants on the onset of the vulcanisation process and the hardness values of the elastomeric networks. 15 refs. 

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). 

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. 

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)... 

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. 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

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 statistical theory is remarkable in that it enables the macroscopic deformation behaviour of an elastomer to be predicted from considerations of how the molecular structure responds to an applied strain. However, it is important to realize that it is only an approximation to the actual behaviour and has significant limitations. Perhaps the most obvious problem is with the assumption that end-to-end distances of the chains can be described by the Gaussian distribution. This problem has been highlighted earlier in connection with solution properties (Section 3.3) where it was shown that the distribution cannot be applied when the chains become extended. It can be overcome to a certain extent with the use of more sophisticated distribution functions, but the use of such functions is beyond the scope of this present discussion. Another problem concerns the value of N. This will be governed by the number of junction points in the polymer network which can be either chemical (crosslinks) or physical (entanglements) in nature. The structure of the chain network in an elastomer has been discussed earlier (Section 4.5). There will be chain ends and loops which do not contribute to the strength of the network, but if their presence is ignored it follows that if all network chains are anchored at two crosslinks then the density, p, of the polymer can be expressed as... 

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