Network chains number density

Table 1. Reduced segment density and the ratio (molecular mass of a network chain)/(number of statistical segments) for some common rubbers...

In the postgel regime where hydrated gel networks exist, one needs consider the number N i) of polymer chains of type I involved in the network. Their number density is given by (i) = (i) / S2, and their volume fraction by (i) = n (i) (i). 

In a randomly cross-linked network, the number density of chain ends is related to the number-average molecular weight A of the N primary linear chains and to the polymer density p by equation (49). The vertices in the network consist of the junctions and the 2N chain ends. This case is described by equation (50), and combination of equations (43), (46), (49) and (50) yields equation (51). The number of junctons p in the spanning tree is obtained by setting equal to zero in equation (51), resulting in equation (52). The number of junctions which are elastically effective is the difference between p and p and is related to by equation (53). 

To illustrate how the effect of the adsorption on the modulus of the filled gel may be modelled we consider the interaction of the same HEUR polymer as described above but in this case filled with poly(ethylmetha-crylate) latex particles. In this case the particle surface is not so hydrophobic but adsorption of the poly (ethylene oxide) backbone is possible. Note that if a terminal hydrophobe of a chain is detached from a micellar cluster and is adsorbed onto the surface, there is no net change in the number of network links and hence the only change in modulus would be due to the volume fraction of the filler. It is only if the backbone is adsorbed that an increase in the number density of network links is produced. As the particles are relatively large compared to the chain dimensions, each adsorption site leads to one additional link. The situation is shown schematically in Figure 2.13. If the number density of additional network links is JVL, we may now write the relative modulus Gr — G/Gf as... 

The degree of cross-linking can be expressed in terms of cross-links per gram or per unit volume. If C is the moles of cross-links per unit volume, n the number of network chains per unit volume, d the density of cross-linked polymer, and Me the number-average molecular weight of the polymer segments between cross-links, then... 

Expression (1.44) is useful to estimate the number density of the active chains of the network, due to the measured temperature T and shear modulus G. 

Epoxy networks may be expected to differ from typical elastomer networks as a consequence of their much higher crosslink density. However, the same microstructural features which influence the properties of elastomers also exist in epoxy networks. These include the number average molecular weight and distribution of network chains, the extent of chain branching, the concentration of trapped entanglements, and the soluble fraction (i.e., molecular species not attached to the network). These parameters are typically difficult to isolate and control in epoxy systems. Recently, however, the development of accurate network formation theories, and the use of unique systems, have resulted in the synthesis of epoxies with specifically controlled microstructures Structure-property studies on these materials are just starting to provide meaningful quantitative information, and some of these will be discussed in this chapter. 

The mean average molecular mass of the network chains is determined for the elastomer matrix outside the adsorption layer. Contributions to the network structure fi om different types of junctions (chemical junctions, adsorption junctions, and topological hindrances due to confining of chains in the restricted geometry (entropy constraints or elastomer-filler entanglements) are estimated. The major contributions to the total network density are provided by the topological hindrances near the filler surface and by the adsorption junctions. The apparent number of the elementary chain units between the topological hindrances is estimated to be approximately 40-80 elementary chain units. 

There are two parameters used as a measure of cross-Unk density the number of network chains, v, usually expressed as v/ V, where V is the volume of the unstrained network and the number of cross-links (p) per unit volume, p/F. The relationship between p and v is established by knowing the number of chains starting from a particular cross-linking point, (functionality). The two most important types of network are the tetrafunctional (c ) =z 4) and the trifunctional ( = 3). Another characteristic parameter of a network is the number-average molecular weight between cross-links,... 

The number density of network strands determines and the plateau modulus caused by inter-chain entanglements is Cg. This plateau modulus is understood on a molecular level by imagining surrounding chains confining each network strand to an effective tube. 

Vapp is essentially the network chain density in the rubber phase, the number of filler-contributed linkages being much smaller than the number of cross-links, v,. 

If the crosslinks are four-functional, each connects four network chains. Figure 2.9 (point c) shows that when a crosslink is introduced between two network chains, the result is four network chains, i.e. each new crosslink increases the number of network chains by two. Consequently, the crosslink density is N/2. Hence, by Eq. (2.9), is inversely proportional to the crosslink density. 

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