Network with Fixed Cross-Links

Idealized Network with Fixed Cross-Links 

The classical theory of rubberlike elasticity specifies for this case that the equilibrium shear modulus in infinitesimal deformations is given by 

The Rouse modes of motion of a molecule whose ends are fixed were examined by Mooney, who found that the relaxation spectrum was the same as that corresponding to equations 1 to 4 with v substituted for p/M except for an additional contribution to the modulus with infinite relaxation time and magnitude vkT. Thus, the relaxation and dynamic moduli are given by 

The quantity Pc refers to a strand (Pc = p/vMo, where Mo is the monomer molecular weight). The long-time and low-frequency limiting values of G t) and G,  

Storage and loss shear moduli, normalized by the equilibrium modulus, plotted logarithmically against frequency for the Mooney modification of the Rouse theory. 

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Idealized Network with Fixed Cross-Links... 

B. If fillers act as additional fixed cross-links of the rubber network, why reinforcing effects are not alike to gum network with additional cross-links ... 

In the ladder model treatment of Blizard, an alternative termination of the line of springs (Fig. 10-3) was considered in which each end, rather than being fixed, is attached to three other such lines, each of these to three more, and so on indefinitely, thus reproducing the connectivity of a tetrafunctional network. This.proyision increases the equilibrium compliance by a factor of 2 (corresponding-fo the factor of (/ — 2)//mentioned in Section I above), and it modifies the frequency dependence, which is now expressed by a rather complicated combination of hyperbolic functions. This frequency dependence of J" is also shown in Fig. 10-7 the maximum is slightly broader than for fixed cross-links (i.e., cross-links with affine deformation). 

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

In a polymer network the chains are connected through cross-links that restrict the motion of the chain ends. Consequently a polymer chain belonging to an elastomeric network can be represented as in Figure 3.6, i.e., with one end fixed at the origin O while the other is confined to a small volume dV = dx dy dz. 

Precipitation reactions are similar in principle to agglutination reactions, the difference being that the antigen is a soluble, molecular species rather than a suspended particle such as a bacterium or erythrocyte. At a certain Ab/Ag ratio, the cross-linked polymeric network of antibody and antigen loses its solubility and precipitates. This is called the precipitin reaction.11 Figure 5.6 shows how the quantity of precipitin produced varies with the quantity of antigen added, for a fixed total antibody concentration. 

Descriptions of networks in terms of cross-link index and density provide no information on the internal architecture, that is, the homogeneity, of the network. Depending on how they are produced, most networks are more or less inhomogeneous, that is, the local density has a distribution. With multifunctional polycondensation, the gel point is most often reached at relatively high yields, and the network formed is quite homogeneous. In addition polymerization, the gel point occurs already at relatively low yields. The polymerization continues around the spatially fixed network structured centers, and, so, densely cross-linked centers are produced within a less densely cross-linked matrix. 

Allyl esters, unsaturated polyesters, as well as some of what are known as vinyl or acrylic esters are cured by free radical addition polymerization. In the case of allyl esters, the monomers, themselves, are cross-linked. On the other hand, unsaturated polyesters are copolymerized with monomers such as styrene or methyl methacrylate. Since the unsaturated polyesters have many main-chain double bonds and the structure of a cross-linked network is fixed after quite low conversions, only a few double bonds actually react. These unconverted double bonds can then react later with atmospheric agents, and so produce poor weathering properties of the crosslinked networks. In addition, the polymerization produces many free chain ends that contribute nothing or even disadvantageously to the mechanical properties. The newly developed vinyl or acrylic esters avoid both of these problems in that the monomers capable of cross-linking only have unsaturated double bonds at the molecular ends (see also Section 26.4. S). 

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