Crosslinked strand density

The mechanical properties of ionomers are generally superior to those of the homopolymer or copolymer from which the ionomer has been synthesized. This is particularly so when the ion content is near to or above the critical value at which the ionic cluster phase becomes dominant over the multiplet-containing matrix phase. The greater strength and stability of such ionomers is a result of efficient ionic-type crosslinking and an enhanced entanglement strand density. 

The surface tension F of the void ceiling that appears in the capOlarity equation (Eq. (12)) is the key quantity to be specified in understanding the effects of network strand density v on the craze widening stress. For the moment suppose this network is comprised entirely of crosslinked drains. Then to create a surface requires the scission of a certain number of strands per unit area, a geometricaUy necessary strand loss, which is given by (1/2) vd. The energy required to create this surface is then... 

Fig. 11a. A summary of the dominant mode of plastic deformation observed in crosslinked PS films as a function of the strand density v and the temperature at which the deformation was carried out. The open squares, half-filled squares and filled squares represent crazing only, crazing plus shear, and shear only, respectively (From Ref. courtesy of J. Mat. Sci. (Chapman and Hall), b The temperature dependence of the shear yield stress Oy and the crazing stress S (for two values of v)...

A conclusion as to the effect of crosslinking on thermal expansion is not possible. Clearly, the polymers with many crosslinks and with short strands expand less than the uncrosslinked materials when heated. However, this effect cannot exclusively be attributed to the presence of crosslinks. It may just as well originate from the increased density of the crosslinked materials which was shown to be responsible for the increase in the moduli. 

The activation volume of the three polymers turned out to be v 2 nm3, independent of their crosslink density. In the crosslinked polymer A the strands are short and about five of them fit into the activation volume. In contrast, one strand of polymer E requires a volume five times larger than the activation volume ... 

THIS CROSSLINK DENSITY IS CALLED THE WEIGHTED CONCENTRATION OF" EFFECTIVE STRANDS. ... 

The active strand concentration v is obtained from measurements of the initial modulus using Eq.(7.2) with g = 1. Values of v for samples at the same crosslink density but with different primary molecular weights are extrapolated to 1/M=0, giving vc + ve. Values of vc + ve obtained at two or more crosslink densities provide ve, and hence Me, by extrapolation. An advantage of this method is that only relative values of crosslink density must be known absolute values of vc are not required. If absolute values of vc are known, the g factor can be evaluated as well. 

In a crosslinked polymer the densities of network strands, entangled and crosslinked, are additive, i.e.. 

Fig. 9. Plot of the true suain ratio in craze and deformation zones showing the transition from crazing to shear deformation as a function of network strand (entangled + crosslinked) density v. The open squares and open diamonds represent uncrossiinked homopolymers and copolymers, the open triangles and hexagons represent uncrossiinked blends of PS and PPO and the filled triangles and circles represent crosslinked PS (After Ref. courtesy of J. Polym. Sd.-Polym. Phys. (Wiley))...

These predictions need to be modified because real networks have defects. As shown in Fig. 7.7, some of the network strands are only attached to the network at one end. These dangling ends cannot bear stress and hence do not contribute to the modulus. Similarly, other structures in the network (such as dangling loops) are also not elastically effective. The phantom network prediction can be recast in terms of the number density of elastically effective strands v and the number density of elastically effective crosslinks ii. For a perfect network without defects, the phantom network modulus is proportional to the difference of the number densities of network strands v and crosslinks // = since there are fjl network strands per crosslink ... 

Therefore, it is well established that topological entanglements dominate and control the modulus of polymer networks with long network strands. The Edwards tube model explains the non-zero intercept in plots of network modulus against number density of strands (see Figs 7.11 and 7.12). The modulus of networks with very long strands between crosslinks approaches the plateau modulus of the linear polymer melt. The modulus of the entangled polymer network can be approximated as a simple sum. 

The order parameter of nematic networks is obtained as a function of temperature and the crosslink density. For nematic networks crosslinked at the isotropic phase, the nematic to isotropic transition is lower than that of the constituent nematic polymers which are the same length as the strands in the networks, while the transition temperature increases for the... 

The elastic energy contribution t/ei describes the rubber elasticity of the crosslinked polymer chains and is proportional to the cross-Unk density, Co. the number density of elastic strands in the undeformed polymer network. We use the Flory model [29] to specify Ugi ... 

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