Tetrafunctional network from

Figure 9. SANS measurements of R /Rt° and RL/R ° for stretched radiation cross-linked polystyrene. is determined by measurements in which the neutron is parallel (iso) and perpendicular (aniso) to the stretching direction. Mc is molecular weight between crosslinks. Theoretical curves 2 and 3 are drawn for tetrafunctional networks. Data from Ref. 21.

Figure 8. Part of a tetrafunctional network formed from an RA t and RBi polymerization corresponding to Mc°, the molar mass between junction points of the perfect network (a). Detail of the chain structure defining Mc° for HDl reacting with an OPPE, n is the number-average degree of polymerization of each arm with respect to oxypropylene units, (b). Part of the chain structure defining v, the number of bonds in the chain forming the smallest ring structure (C), for the reaction system in (b) (29). Reproduced, with permission, from Ref. 21. Copyright 1980, Stein-...

It is shown that model, end-linked networks cannot be perfect networks. Simply from the mechanism of formation, post-gel intramolecular reaction must occur and some of this leads to the formation of inelastic loops. Data on the small-strain, shear moduli of trifunctional and tetrafunctional polyurethane networks from polyols of various molar masses, and the extents of reaction at gelation occurring during their formation are considered in more detail than hitherto. The networks, prepared in bulk and at various dilutions in solvent, show extents of reaction at gelation which indicate pre-gel intramolecular reaction and small-strain moduli which are lower than those expected for perfect network structures. From the systematic variations of moduli and gel points with dilution of preparation, it is deduced that the networks follow affine behaviour at small strains and that even in the limit of no pre-gel intramolecular reaction, the occurrence of post-gel intramolecular reaction means that network defects still occur. In addition, from the variation of defects with polyol molar mass it is demonstrated that defects will still persist in the limit of infinite molar mass. In this limit, theoretical arguments are used to define the minimal significant structures which must be considered for the definition of the properties and structures of real networks. 

Fig. 7. Temperature dependence of Ts at a proton frequency of 88 MHz for tetrafunctional network PDMS specimens with the following numbers of dimethylsiloxane units between network points 2 (1), 3 (2), 6 (3), 9 (4), 30 (5) and for the linear PDMS (6) (reprinted from Ref.72))...

Figure 12. Intrinsic atomic stresses <7n, Vjj. <733 as determined from molecular dynamics simulation of tetrafunctional network model in uniaxial volume deformation and for corresponding melt. (After Ref. [19].)...

Similarly, from the relationship between the number of chains and number of cross-links (Problem 3.4) for a tetrafunctional network, we obtain... 

CROSSLINKED NETWORK MODEL CALCULATIONS Our research has shown that tetrathiol crosslinked norbomene resins form a densely crosslinked, three dimensional network. Recently there has been considerable interest in crosslinked networks from a theoretical and practical point of view (29-31). As part of our study we attempted to analyze the polymer network using the Miller-Macosko formalism as applied by Bauer (32). For the purposes cf this analysis we assumed that the curable formulation was an A2 (ene) B4 (thiol) type system. We also assumed, based on HPLC analysis of EBPA DN and acrylate precursor batches that the norbomene resins was a mixture of oligomers consisting of difunctional olefin (85%) and monofimctional olefin (15%). The thiol crosslinker was assumed to be essentially tetrafunctional. Furthermore, we made the not unreasonable assumption that there would be no thiol-thiol or norbornene-norbornene reactions. In one case, a chain extending... 

Stress-strain relations derived from eq. 10 are of the same form as eqs. 7 and 8 derived from eq. 6. They differ only by a numerical factor arising from replacement of v by 5. For a tetrafunctional network this factor is one-half. Hence, the relationship of the retractive force to the extension for simple elongation of a tetrafunctional phantom network is given by eq. 7 or eq. 8 modified by this 16 20... 

F. 15. Vali of themolecalarwei t Mc(f) between cross-links, as rabxdated from the stress-strain isotherms, for some of the (tetrafunctional) networks prepared from PDMS chains with adjacent vinyl groups The results are shown as a function of the fraction F, of chains having these cross-linking sites segregated at the ends of the chains. The three types of networks were prepared by benzoyl peroxide (O), Si[OSi(CH3)jH]4 ( ), and 2,5-di-methyl-2,5-di(t-butylperoxy) hexane (DBPH) (C)... 

In Table 2 are listed some values of n/f calculated from Eq. (B-19) for tetrafunctional networks and by means of the following Rg(M ) dependence obtained by SANS experiments on linear PS molecules ... 

Figure 8. SANS measurements of R /Fg° and R /Rg° versus A of tetrafunctional polydimethylsiloxane networks. Mw and Mn are weight and number average molecular weights before crosslinking, cp is the polymer fraction in solution before cross-linking. Data from Ref. 20.

In order to enable these fluctuations to occur, the network chains are assumed to be "phantom" in nature i.e. their material properties are dismissed and they act only to exert forces on the junctions to which they are attached. With common networks having tetrafunctional junctions, the results of the two approaches differ by a factor of two. Identical results are only obtained from both theories, when the functionality is infinite. From a practical viewpoint, however, a value of about 20 for f can already be equated to infinity because crosslink densities can hardly be obtained with an accuracy better than 10%. 

Thermodynamic Analysis. As reported previously, the storage modulus G of PDMS networks with tetrafunctional crosslinks is independent of frequency between 10 3 and 1 Hz (21). This behaviour which is entirely different from that of vulcanized natural rubber or synthetic polyisoprene networks, was attributed to the lack of entanglements, both trapped and untrapped, in these PDMS networks. Figure 4 shows that G of a network with comb-like crosslinks is also frequency independent within an error of 0.5%. For comparison, two curves for PDMS having tetrafunctional crosslinks are also shown. The flat curves imply that slower relaxations are highly unlikely. Hence a thermodynamic analysis of the G data below 1 Hz can be made as they equal equilibrium moduli. 

Figure 3. Threshold tear energy T . Key O, A, , PDMS networks , A. PB networks , PI networks versus molecular weight Mc between cross-links calculated from Ct. O, , , random cross-linking A, A. trifunctional end-linking , tetrafunctional end-linking.

The deviations from Gaussian stress-strain behaviour for the tetrafunctional polyurethane networks of Figure 9 are qualitatively similar to these found for the trifunctional polyester networks (Z5), and the error bars on the data points for systems 4 and 5 in Figure 9 indicate the resulting uncertainties in Mc/Mc. It is clear that such uncetainties do not mask the increases in Mc/Mc with amount of pre-gel intramolecular reaction. 

Figure 14.6 Destruction of elastically active network chains resulting from a chain scission in the case of tetrafunctional (a) and trifunctional nodes (b). 

Figure 15.3 The decay of the so-called quadrupolar order as a function of the time interval t2 (quadrupolar order is created by a pulse at time tx, then relaxes during a time interval t2 before being observed via a third pulse). Data from three differently crosslinked PDMS networks are presented (M = 9700 g.mol"1 o, tetraallyloxyethylene-crosslinked (tetrafunctional) + crosslinked with bis(allyloxy)-3-[bis(allyloxymethyl]-2, 2-propylene oxide (hexafunctional) A tetravinylcyclosiloxane-crosslinked]. The longitudinal relaxation is plotted for comparison. Experimentally, the quadrupolar relaxation time T1Q equals about 1.33 T1Z...

An elastically active network chain is active in the equilibrium elastic response of the network to deformation. From the topological point of view, an EANC is a chain between two active branch points. An active branch point is a imit from which at least three paths issue to infinity. In the case under consideration, only some of the chemically tetrafunctional diamine units can become active branch points. If the polyepoxide were more than bifunctional, it would also contribute to the number of EANC s. In analogy with Eq. (14), the pgf for the numbo- of bonds with infinite continuation issuing from a diamine unit T,(z) is given by... 

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,... 

Experimental results on reactions forming tri- and tetrafunctional polyurethane and trifunctional polyester networks are discussed with particular consideration of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction is shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. 

For a network generated from stoichiometric quantities of a difunctional polymer and a tetrafunctional cross-linker where cross-linking occurs only at the chain ends the network chain average molecular weight should approximate the prepolymer 1. ... 

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