Crosslinks elastically effective

This equation applies to networks with defects as well, and hence is more general than Eq. (7.41), but care must be taken to only include elastically effective strands and crosslinks. Elastically effective strands are the ones that deform and store elastic energy upon network deformation. Elastic-ally effective crosslinks are those that connect at least two elastically effective strands. ... 

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. 

It is possible to calculate a number of different kinds of "effective" crosslink densities. Bauer et al have used a quantity they termed the "elastically effective crosslink density " (Cel) correlate cure with solvent resistance and other physical properties of coatings (7-10). The correlation was basically empirical. Formally, the is a calculation of the number of functional groups attached to the infinite network for which there are at least two other paths out to the network on the given polymer or crosslinker. Thus, chains with only one or two paths to the infinite network are excluded. The following expression can be written for... 

PRINT "THE OLD ELASTICALLY EFFECTIVE CROSSLINK DENSITY EXPRESSION OF BAUER" 150 PRINT "AND BUDDE. THE SECOND CALCULATES A CROSSLINK DENSITY WHICH IN" THEORY SHOULD BE PROPORTIONAL TO THE RUBBERY ELASTIC MODULUS."... 

PRINT "THE OLD ELASTICALLY EFFECTIVE CROSSLINK DENSITY - CEL 3330 PRINT... 

In this case, the crosslinking density is the sum of the effective crosslinking density and the secondary cyclization, since both crosslinkages are elastically effective. The calculation conditions are the same as Figure 1 except that the number of secondary cycles formed per effective crosslinkage T) is 20. 

Figure 2. Elastically effective crosslink density versus bake temperature for a 17 minute bake. Low solids 0 experimental values, model values High solids 1 ----- ...

The crosslink density of a polymer network determines the number of elastically effective chains. Some of the chains are tied to a network and... 

Differences in Network Structure. Network formation depends on the kinetics of the various crosslinking reactions and on the number of functional groups on the polymer and crosslinker (32). Polymers and crosslinkers with low functionality are less efficient at building network structure than those with high functionality. Miller and Macosko (32) have derived a network structure theory which has been adapted to calculate "elastically effective" crosslink densities (4-6.8.9). This parameter has been found to correlate well with physical measures of cure < 6.8). There is a range of crosslink densities for which acceptable physical properties are obtained. The range of bake conditions which yield crosslink densities within this range define a cure window (8. 9). 

Swelling data indicate that crosslink density in the continuous phase of the 70 30 and 60 A0 networks is high. Crosslink densities were estimated from the data in Table III by the method of Hill and Kozlowski ( ). Results were for 80 20, "Vg = 10" moles of elastically effective network chains/cm for 70 30, Vg = 2.5 x lO" chains/cm for 60 AO, Vg = A.3 x 10" chains/cm. These estimates suggest that the crosslink densities are within the range reported for conventional, highly crosslinked acrylic HMMM and polyester HMMM enamels (19,20). 

Model networks are tridimensional crosslinked polymers whose elastically effective network chains are of known length and of narrow molecular weight distribution. The techniques used to synthesize such networks are derived from those developed for the synthesis of star shaped macromolecules, whereby the initiator used must be bifunctional instead of monofunctional. ... 

The first step involves preparation of a bifunctional "living" precursor, of known molecular weight and low polydispersity. Crosslinking can be achieved either by the addition of stoichiometric amounts of a multifunctional electrophilic deactivator, or by adding small a small amount of a bifunctional monomer (such as DVB or ethylene glycol dimethacrylate), the polymerization of which will be initiated by the carbanionic sites of the precursor. In either case, the precursor chains become the elastically effective chains of the networks. The experimental conditions... 

Scanlan has suggested another criterion (282). An effective network junction point is a crosslink in which at least three of the four strands radiating from it lead independently to the network. A crosslink with only two strands anchored to the network simply continues an active strand a crosslink with only one anchored strand is part of a dangling end and can make no elastic contribution at equilibrium. An elastically effective strand is therefore one which joins two effective network junction points. Accordingly, the total number of active strands is simply one half the number of gel-anchored strands radiating from effective junction points ... 

Table 7.1. Concentrations of elastically effective strands according to the Flory and Scanlan criteria for random crosslinking of monodisperse primary chains...

Concentration of elastically effective strands in crosslinked network (Part 7). 

Free chain ends (unreacted functionalities) reduce the number of active network chains in a network compared with the same network without free ends. Disregarding possible presence of loops and entanglements, C — 1 C crosslinks are necessary according to Flory (55) to connect C chains into one giant macromolecule. Additional crosslinks will be elastically effective. Their number is given by... 

Fig. 13. The effect of a crosslink at P on the mechanical behaviour of a two chain network between fixed points is mathematically equivalent to a fixed point on one of the chains, thus leading to three rather than four elastically effective chains [Duiser and Staverman, Chompff and Duiser 43, 32, 33)]...

To predict macrosyneresis in the case of dissolved polymer chains being crosslinked it is necessary to know the change in the number of elastically effective network chains as a result of changing the number of chemical crosslinks and the effect of v on (r2),. and possibly on y. Moreover, if the molecular weight of primary chains is finite and the sol fraction is important, a complete description of the phase separation requires treating the system as a multicomponent one, containing a network phase and a multicomponent diluent with branched polymer species. If additional crosslinking is carried on in an extracted network the latter complication can be eliminated. 

They should consist of elastically effective chains only. An elastically effective chain should connect two different crosslinks, and two such crosslinks should be tied by only one elastic chain. This means that the gel should contain no defects such as pendant chains (one end of which only is connected with a crosslink), loops (chains linked at both ends to the same crosslink), or double connections. Physical crosslinks (permanent entanglements) should be prohibited, too. 

The functionality of the crosslinks should be known, and constant throughout the geL The functionality is the number of elastically effective network chains which are tied to one given crosslink. 

It thus seems that there is no direct link between volumetric and elastic properties in the glassy state and that the anomalous density variations cannot be attributed to a crosslink density effect, either direct (on molecular packing) or indirect (through internal antiplasticization as discussed below). It seems reasonable to correlate this behavior with the presence of unreacted epoxides. The density would be (in the systems under consideration) a continuously increasing function of the amine/epoxide ratio, owing to the... 

Three-component silicone formulations (crosslinkable silicone + crosslinking agent + catalyst) are used to obtain pronounced elastic effects, but single-component products have also been introduced in the meanwhile for various areas of application. 

In most real systems, energy and entropy changes can occur. The elasticity of an ideal network is entropy controlled. In this picture stresses are caused by the chain orientation. From the theory of rubberlike elasticity it can be shown that the shear modulus of an ideal network depends on the number of elastically effective cahins between the crosslinks (19) Gq = v k-T where v means the number of elastically effective chains in unit volume. 

Dependence of the Shear Modulus on the Concentration. The experimental results of Figure 3a show that the plateau values increase with the detergent concentration. Unfortunately, we were not able to reach the rubber plateau for all concentrations for lack of the frequency range. From the theory of networks it is possible to calculate the number of elastically effective chains between the crosslinks from the shear modulus Gq of the rubber plateau (12). If the network... 

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 ... 

An interesting sub-dass of lonomers are telechelic lonomers, of which there are several noteworthy examples. The term "telechelic" indicates that the ions are attached exclusively at the chain termini and that every chain end contains an ionic moiety. Such placement provides a network free of dangling chain ends, and minimizes melt viscosity, since the telechelic polymer molecular weight is of the same order as the elastically effective molecular wei t between crosslinks. This is in direct contrast to "random"... 

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