Tetrafunctional cross-links networks with

A 2.830 g sample of a poly(methyl methacrylate) (PMMA) network immersed in ethyl acetate (EtAc) reaches swelling equilibrium after 24 h, the final weight being 7.336 g. The cross-linking reaction was carried out with a tetrafunctional cross-linking agent in the undiluted state (v2s = ) Calculate M,.. 

To form a cross-link network with tetrafunctional branch points, it is necessary to have one branch point and one cross-link bond per primary chain. With trifunctional branching, on the other hand, two branch points are required per primary molecule. 

To form a cross-link network with tetrafunctional branch points, it is... 

If the tetrafunctional cross-links to which are connected four clmins leading to the network are suppressed, the network (first-stage) is formed only with the crosslinks having three chains connected to the mesh. Each chain end is connected to this type of cross-link, whose number density is therefore... 

A typical network studied in this regard might have been tetrafunctionally cross-linked in the undiluted state (U2s = 1.00), and exhibit an equilibrium degree of swelhng characterized by V2m = 0.100 in a solvent having a molar volume V = SOcm mor (8.00 x 10" mm mol ) and an interaction parameter with the polymer corresponding to xi = 0-30. Calculate the network-chain density [9]. 

For the nonlinear step growth case above, eiTg, the crosslink density must be related to p. A relevant model, based on calculating the probabilities of finite chains being formed, has been published For the reaction of A -1- 2B2 (e.g., tetra-functional amine -b difunctional epoxy), A4 is considered to be an effective cross-linking site if three or more of its arms lead out to the infinite network. The probability of finding an effective crosslink is related to one minus the probability of a randomly chosen A leading to the start of a finite chain, which in turn is related to the extent of reaction. Application of this procedure to the system of Fig. 15 has been presented in detail The more complicated reaction of a tetrafunctional amine with a trifunctional epoxy was also considered. ... 

As a numerical example, the affine shear modulus of a perfect poly (dimethylsiloxane) tetrafunctional network of density p — 0.97 g/cm and Me = 11,300 g/mol is Gaff = 0.212 x 10 N/m. However, imperfections such as chain ends exist in typical networks, as illustrated in Figure 6. The following correction can be made to account for this circumstance (16). Before the cross-linking reactio it is assumed that Vm chains of length are present in the melt, with... 

Fig. 1. Percent gel and distribution of cross-link density between sol and gel versus the log of the total cross-link density for the cross polymerization of primary chains having a most probable distribution. Two curves representing networks with junction functionalities of 4 (solid) and 24 (dashed) are shown with arrows pointing to the left. In each case M 358,000. The other solid lines represent the gelation behavior of a tetrafunctional system and have arrows pointing to the right.

On the other hand if the network is produced by long chains linked at a T-shaped junction point (as may occur with the polyurethane rubbers for example) the molecular weight per junction point is 1-5 times the molecular weight between cross-links (Fig. 3A. 1(b)). For a tetrafunctional junction point the ratio of the two values will be 2 0/1 0 (Fig. 3A.l(c)). 

Fig. 1.6. Sketches of some simple, perfect networks having (a) tetrafunctional and (b) trifunctional cross-links (both of which are indicated by the dots) [25]. (Reproduced with permission copyright 1982, Rubber Chem. Technol.)...

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). 

Fig. 4. (A) A central tetrafunctional junction surrounded by four other topologically neighboring junctions and a number of spatially neighboring junctions. (B) Schematic drawing of a slip link, with its possible motions along the network chains specified by the distances a, and its locking into position as a cross-link.

In polymer networks, the region of space pervaded by a given chain is shared with many other chains and junctions. A tetrafunctional, statistically cross-linked cis-l,4-polyisoprene network, with a typical average molecular weight between junctions of 5000 g mol" contains of the order of 5 x 10 junctions cm"The characteristic ratio for the a s-l,4-polyisoprene chain is 5.0 and, according to equation (28), the radius of the volume occupied by such a chain will be 56 A. From... 

Most of the pertinent experimental results have been obtained using PDMS model networks and typical results obtained on such tetrafunctional networks in the author s laboratory are shown in Fig. 1. In it, values of A4 are presented as a function of the volume fraction V2c of polymer in the system during the cross-linking procedure. If the reaction were incomplete in the undiluted state, A4 should show a marked dependence on V2C through its effect on the concentrations and mobilities of the reactants. The values of A4 show no trend whatever with decrease in V2c, which argues strongly that the end-... 

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)... 

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