Chain Networks

Data on crack initiation were also presented by Le May and Kelley [105] but not further analysed. The authors were concerned that influences other than the network chain length could complicate or obscure simple relations. They suspected... 

Networks obtained by anionic end-linking processes are not necessarily free of defects 106). There are always some dangling chains — which do not contribute to the elasticity of the network — and the formation of loops and of double connections cannot be excluded either. The probability of occurrence, of such defects decreases as the concentration of the reaction medium increases. Conversely, when the concentration is very high the network may contain entrapped entanglements which act as additional crosslinks. It remains that, upon reaction, the linear precursor chains (which are characterized independently) become elastically effective network chains, even though their number may be slightly lower than expected because of the defects. 

FIGURE 18.2 Tensile strength of styrene-butadiene rubber (SBR) as a function of network chain density. (From Bueche, F. and Dudek, T.J., Rubber Chem. Tech., 36, 1, 1963.)... 

Now, we consider this phenomenon from the viewpoint of the non-Gaussian behavior of the network chain. As is well known, when we assume the idealized molecular network consisting... 

V and n are the number of network chains per unit volume and the number of the segments in a network chain, respectively... 

So far, we have not introduced a specific model of the polymer network chains. This problem can be rigorously solved for cross-linked polymer networks consisting of phantom chains [13], or even in the more general case of filled networks where the chains interact, additionally, with spherical hard filler particles [15]. 

The 6-function makes sure that if two segments and 2 meet on the huge network chain they can form a permanent constraint R( i) = R( 2)- Hence, this process will produce a network junction of functionality/n = 4, usually realized as sulfur bridges in technical elastomers like, for example, tire treads. 

In POLYM the output data of KINREL are used with compositional information to calculate the number and mass average molecular masses (Rn and Rm, respectively) and number and end-group average functionalities (fp and fg> respectively) in the pre-gel region in all stages. In addition, the network characteristics such as sol fraction, mj, and the number of elastically active network chains per monomer (5), Ng, are calculated in the post-gel regime of stage 3. 

When the butyl rubber was compounded with up to 30 percent of polyisobutylene, which, lacking the unsaturated isoprene units, did not enter into the cross-linking reaction, the tensile strengths were, of course, considerably reduced. They were found nevertheless to be accurately represented by the same equation, (53), provided merely that Sa is taken as the fraction of the composite specimen consisting of network chains subject to orientation. Thus, in this case... 

The structure of a perfect network may be defined by two variables, the cycle rank and the average junction functionality (f>. Cycle rank is defined as the number of chains that must be cut to reduce the network to a tree. The three other parameters used often in defining a network are (i) the number of network chains (chains between junctions) v, (ii) the number of junctions p, and (iii) the molecular weight Mc of chains between two junctions. They may be obtained from and using the relations... 

In a typical elastomer, the number of skeletal bonds in a network chain range from about 100 to 700 [25]. Networks with chains shorter than 100 bonds have... 

The molecular theories of networks to be presented in the following paragraphs are based on the Gaussian picture of the individual network chains. With reference to the form of the distribution function, these theories are referred to as "Gaussian theories". 

The instantaneous vector r joining two junctions at the extremities of a network chain may be written as the sum of a time-averaged mean f and the instantaneous fluctuation Ar from this mean, that is,... 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

Figure 2 Schematic drawing of a slip link, with its possible motions along the network chains specified by the distances a, and its locking into position as a cross-link.

Experimental determinations of the contributions above those predicted by the reference phantom network model have been controversial. Experiments of Rennar and Oppermann [45] on end-linked PDMS networks, indicate that contributions from trapped entanglements are significant for low degrees of endlinking but are not important when the network chains are shorter. Experimental results of Erman et al. [46] on randomly cross-linked poly(ethyl acrylate)... 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

Here, x is the number of repeat units in one network chain, /-q the number of solvent molecules, n2 the total number of network chains in the system, i the number of ionic groups on the chains, v the number of chains, and v20 the volume fraction of chains during the formation of the network. 

When the network chains contain ionic groups, there will be additional forces that affect their swelling properties. Translational entropy of counterions, Coulomb interactions, and ion pair multiplets are forces that lead to interesting phenomena in ion-containing gels. These phenomena were studied in detail by Khokhlov and collaborators [74-77]. The free energy of the networks used by this group is... 

With regard to elastomers of controlled structure, those having unusual distributions of network chain lengths have been of particular interest [88,89]. The most novel elastomer of this type consists of a binary combination of unusually short network chains (molecular weights of a few hundred) and the much longer chains typically associated with elastomeric behavior (molecular weights of ten or twenty thousand). Such a network is sketched in Figure 6. 

Figure 6 A network having a bimodal distribution of network chain lengths. The short chains are arbitrarily shown by heavier lines than the long chains, and the dots represent the crosslinks, typically resulting from the end linking of functionally terminated chains. 

It should be pointed out that there are three requirements for obtaining these improvements. The first is that the ratio Ms/Ml of molecular weights of the short (Ms) and long chains (Ml) be very small (i.e., that their molecular weights be very different). The second is that the short chains be as short as possible for example, a network having network chain molecular weights of 200 and 20,000 g/mol... 

An example of a relevant optical property is the birefringence of a deformed polymer network [246]. This strain-induced birefringence can be used to characterize segmental orientation, both Gaussian and non-Gaussian elasticity, and to obtain new insights into the network chain orientation (see Chapter 8) necessary for strain-induced crystallization [4,16,85,247,248]. 

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