Ideal network

Some compression tests for gel made with extruded fibres have also be carried out in various conditions (Table 2). The Young moduli were in the range of the values obtained for commercial pectins. There was no marked influence of the severity of the treatment. The large difference between the Young moduli and the G value confirmed that these gels are far from ideal networks. 

The stress-strain response of ideal networks under uniaxial compression or extension is characterized as follows ... 

For an ideal network T is in the numerator of the formula for E, so that E is proportional to the absolute temperature. The log E - T curve thus shows a positive slope (not a straight line because of the log-scale but slightly curved upward). In reality this simple picture is often disturbed by deviations from ideal rubber-elastic behaviour. 

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

Froelieh,D., Crawford,D., Rozek,T., Prins,W. Ideal network behavior of amonically prepared polystyrene gels. Macromolecules 5,100-102 (1972). 

In a recent series of papers, Kilian 9,50 52) proposed a new phenomenological approach to rubber elasticity and suggested a molecular network might be considered as a formelastic fluid the conformational abilities of which were adequately characterized by the model of a van der Waals conformational gas with weak interaction. The ideal network is treated as an ideal conformational gas. According to... 

At the very beginning, it seems worthwhile to put forward a definition of an ideal network, so that we can treat any real network by reference to this definition. An ideal network then, is defined to be a collection of Gaussian chains between /-functional junction points (crosslinks) under the condition that all functionalities of the junction points have reacted with the ends of all and different chains. Furthermore, neither the grouping of chain-ends into crosslinks, nor any external effect, such as interaction with a surrounding diluent, should change the Gaussian statistics of the individual chains. 

Even if completely homogeneous and disordered in the relaxed state, a real network differs from the ideal network, defined in Chapter I. Three types of network defects are commonly considered to be present in polymer networks unreacted functionalities, closed loops, and permanent chain entanglements. Within each group there are several possibilities dependent on the arrangement of chains the effect of defects on the elastic properties of the network is thus by no means simple, as has been stressed e.g. by Case (28). Several possible arrangements are shown in Fig. 1, where only nearest neighbour defect structures have been drawn. 

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. 

Summarizing this section, we see that the molecular description of ideal networks, requires at least two, and in the case of swollen networks at least three parameters ... 

To the authors knowledge there are no dry rubberlike networks which obey the stress-strain relation derived in the preceding section (Eq. III-l 1) over an appreciable range of deformation ratios (e.g. from Ax — 1 to 2), whereas this relation should hold for an ideal network at least over such a range before the finite extensibility of the chains impairs the derivation. 

The reason for these failures may have to be sought in deviations from the ideal network structure, as will be discussed in Chapter IV. 

A further test of the validity of ideal network theory can be obtained through studies of the equilibrium swelling of polymer networks (Eq. 11-22). The maximum amount of information can be extracted by inducing changes in the equilibrium swelling preferably combined with unidirectional stress-strain data (Eqs. III-21, 26 and 27). 

Summarizing this section, we conclude that, from an experimental point of view, the ideal network theory expressed by Eq. (III-9) seems to be obeyed rather well only in the case of swollen networks. The determination of the elasticity parameters is complicated by the presence of a free enthalpy of dilution term. In spite of this, it is possible to conclude that 1/2 is the most likely value for B in Eq. (III-9). The factor A remains undetermined because experimentally the most one can obtain is A g o 2 8. Concerning this memory parameter, a dependence on the stage of dilution during crosslinking definitely exists, (Eq. Ill-19), but its dependence upon the nature of the diluent used in swelling studies needs further investigation. 

Swollen networks usually show a negligible C2 term. If they do exhibit a C2 term, there is reason to believe that the network structure is inhomogeneous, or even heterogeneous due to microsyneresis (see Chapter II, Sections 3 and 4). These observations suggest that the explanation for C2 might be coupled with a certain structuring in the network beyond that implied by the ideal network picture. 

Fig. 29. Schematic representation of a part of a structured network and its theoretical treatment (a) ideal network structure without bundles (b) network with a simple"two chain bundle" (c) theoretical treatment by Volkenstein et al. (174). The separated "bundle illustrates the intactness of the original network chain statistics (d) theoretical treatment by Blokland (14). Each chain has a number of obstructed steps. No relation between the obstructed parts of different chains...

Such networks have been widely used to establish whether the theories of rubber elasticity and of equilibrium swelling are valid. But these theories are based on a number of hypotheses which are obviously far from being fulfilled by the above networks. The so-called ideal networks should obey the following requirements ... 

From Eq. (11), and taking into account the expression v = (M F02)-1, valid for ideal networks, it can be seen that the analytical relationship between the equilibrium swelling degree Q and the molecular weight M of the elastic chains is rather complicated. Several additional points have to be considered ... 

Flory (1941a,b, 1953) and Stockmayer (1943, 1944) laid out the basic relations for establishing the evolution of structure with conversion in nonlinear polymerizations. Their analysis is based on the following assumptions defining an ideal network ... 

It can be stated that networks based on a simple formulation (one monomer reacting with a comonomer), obtained from the step-polymerization process will exhibit a homogeneous structure. This is the case for epoxy-amine networks (the most studied) and polyurethane networks that have been used very often as ideal networks for structure-property correlations. 

The structure of precursors, the number of functional groups per precursor molecule, and the reaction path leading to the final network all play important roles in the final structure of the polymer network. Some thermosets can be considered homogeneous ideal networks relative to a reference state. It is usually the case when networks are prepared by step copolymerization of two monomers (epoxy-diamine or triol-diisocyanate reactions) at the stoichiometric ratio and at full conversion. 

Homogeneous ideal networks, also called closed networks, result from a single-step polymerization mechanism of a stoichiometric mixture of monomers, reacted to full conversion. Many amine-crosslinked epoxies of Tg < 200°C and polyurethanes obtained using a single isocyanate monomer and a single polyol belong to this family. 

With regards to networks other than epoxies, one can encounter problems essentially linked to the nature of crosslinks, e.g., in the following materials which are not necessarily ideal networks but for which the hypothesis of... 

There are several ways to modify the crosslink density of ideal networks. The first one is the use of monomers with the same structure but with different molar masses. Many workers have reported on epoxy networks... 

In the case of unsaturated polyesters, nondegraded samples made from a prepolymer of molar mass M and a styrene mass fraction s have a chain-ends concentration b = [2(1 — s)/M]p, where p is the density. If ve is the actual concentration of elastically active network chains, an ideal network would be obtained by welding each chain end to another one, leading to... 

An analysis of network formation shows that terminal bifunctionality must be very close to the ideal of two functional groups per molecule to produce a high quality vulcanizate. The quality of a vulcanizate can be related to the required number average molecular weight, MR, of a polymer in an equivalent random network (6). MR values can be determined for an idealized network formed by coupling bi- or monofunctional prepolymers—i.e., no nonfunctional molecules—using the expression ... 

The low shear rheology measurements also show a rapid increase in the viscoelastic properties (modulus and zero shear viscosity) with increase of bentonite concentration above the gel point (> 30 g dm bentonite). Several models have been proposed to account for the elastic properties of concentrated dispersions, of which that originally proposed by van den Tempel (25) and later developed by Papenhuizen (26) seems to be the most appropriate for the present system. According to this model, if the interaction energy minimum between adjacent particles is sufficiently negative, a three-dimensional network structure may ensue, giving an elastic component. Various models can be used to represent the three dimensional structure, the simplest of which would be either an ideal network where all particles are... 

In one respect the defects of an amorphous solid are easier to deal with than those of a crystal. Any small deviation in the local structure of the defect in a crystal results in an identifiably different state, resulting in many possible defect structures. More than 50 point defects are known in crystalline silicon and there is probably an even larger diversity of extended defects. In the amorphous material, small differences in local structure which fall within the disorder of the ideal network cannot be resolved meaningfully. Thus one expects fewer separate classes of defects, but with their energy levels broadened out by the disorder, as illustrated schematically in Fig. 4.1. 

---