Random Crosslinking

We will consider here cases in which linear chain molecules are randomly crosslinked. Therefore, we will deal with the crosslinked state with maximum probability given the various chain lengths and number of crosslinks. 

To clarify further, a crosslinked polymer is schematically shown in Fig. I. In crosslinked polymers, the primary polymer is the polymer in which all crosslinks are attached. Therefore, in the case of random crosslinking, the pol)mier chains used as raw material are the primary polymers. The primary polymer is coimected with another primary polymer via a crosslink. At this point, two crosslink points are formed with one crosslink because the crosslink point is defined as a unit where three chains merge. The crosslink density (p) is defined as the number of crosslink points divided by the number of chemical repeat units. Therefore, the crosslink density of the crosslinked polymer in Fig. 1 is p = 4/12 = 0.333. Also, the crosslink density of the primary polymer A is p = 1/4 = 0.25. The definition adopted here was proposed by Flory [8] and is the one most widely used. However, it is important to understand this definition because there are occasions where the crosslink density is defined by the number of crosslinks (crosslink points/2) as in the case of crosslinking by radiation. The probability of each chemical repeat unit being the crosslink point in the random crosslinking system is constant and is p in Flory s definition of crosslink density. Thus far the 

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The most investigated examples are to be formd in the precipitation of polyelectrolytes by metal ions. Here, networks are formed by the random crosslinking of linear polymer chains, and the theory requires some modification. The condition for the formation of an infinite network is that, on average, there must be more than two crosslinks per chain. Thus, the greater the length of a polymer chain the fewer crossUnks in the system as a whole are required. 

The theory of gelation (Flory, 1953,1974) has been summarized in Section 2.2.3. This theory regards gelation as the consequence of the random crosslinking of linear polymer chains to form an infinite three-dimensional network. The phenomenon is, of course, well illustrated by examples drawn from the gelation of polycarboxylic acids by metal ions. 

The SANS experiments of Clough et al. (21) on radiation crosslinked polystyrene are presented in Figure 9, and appear to fit the phantom network model well. However, these networks were prepared by random crosslinking, and the calculations given are for end-linked networks, which are not truly applicable. 

Randomly - Crosslinked PDMS. The polydimethylsiloxane (PDMS) used to make random networks was obtained from General Electric. Membrane osmometry showed to be 430,000 g/g-mole. The polymer was mixed with various amounts of a free-radical crosslinking agent, dicumylperoxide (Di-Cup R, Hercules Chemical Co.). Samples were then pressed into sheets and crosslinking was effected by heating for 2 h at 150°C in a heated press. Mc values were calculated using equation 2, and are included in Table I. 

Randomly - Crosslinked PB and PI. Polybutadiene (Diene 35 NFA, Firestone Tire and Rubber Co.) and cis-polyisoprene (Natsyn 2200, Goodyear Tire and Rubber Co.) were crosslinked with dicumyl-peroxide, as for PDMS. Mc values were also calculated by means of equation 2. They are given for PI in Table I and are listed for PB in reference 2. 

Inhomogeneities in a real network may occur either because of a continuous distribution of molecular weight between crosslinks or due to the regions of different average molecular weights (as may be the case in randomly crosslinked networks). 

Two types of networks were prepared (i) randomly crosslinked polybutadiene, and (ii) model urethane networks, (a) polybutadiene based, and (b) poly(e-caprolactone) based. The randomly crosslinked networks were prepared from polybutadiene (Duragen 1203 obtained from General Tire and Rubber Co.) crosslinked with di-cumyl peroxide. Specifications of the as obtained polybutadiene are given in Table I. Polybutadiene was purified by dissolving in benzene and precipitating in methanol. Precipitated polybutadiene was redissolved in benzene. Seven different weights of dicumyl... 

Fig. 16. Molar mass dependencies of the intrinsic viscosity [rf] for the same samples as shown in Fig. 15 (end-linked PS-stars [94] and randomly crosslinked polyesters [92,93,95]...

Random crosslinking reactions lead quite efficiently to network formation, requiring that the individual chains have relatively open configurations which... 

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

To derive the condition for the gel point in random crosslinking of existing chains of arbitrary molecular weight distribution, let us consider a primary chain selected at random having P monomer units. If one unit of the selected chain happens to be crosslinked, the probability that this unit is a part of the selected chain is equal to the weight fraction wP of the P-mer. The expected number of additional crosslinked units in the P-mer is then q(P — 1). The mean expected number of additional crosslinked units in a chain is... 

Random crosslinking is equivalent to random polycondensation of RAt with jR A2 monomers, if the distribution of primary chains is the so-called "most probable (55). 

If the sol fraction is not negligible, the fraction of crosslinked units in the gel is higher than in the whole system. For random crosslinking of primary chains with an arbitrary molecular weight distribution, the crosslinking density in the gel qb is related to the overall crosslinking density q by (55) ... 

In ideal random crosslinking polymerization or crosslinking of existing chains, the reactivity of a group is not influenced by the state of other groups all free functionalities, whether attached or unattached to the tree, are assumed to be of the same reactivity. For example, the molecular weight distribution in a branched polymer does not depend on the ratio of rate constants for formation and scission of bonds, but only on the extent of reaction. Combinatorial statistics can be applied in this case, but use of the p.g.f. simplifies the mathematics considerably. 

In the following, we will briefly outline the use of the link p.g.f. (l.p.g.f.) for the calculation of the gel point in /-functional polycondensation without and with cyclization including the f.s.s.e. In Chapter II, section 2.2 we will consider an application in connection with the number of elastically active network chains in random polycondensates or in a collection of randomly crosslinked chains. 

In the random crosslinking of existing chains, the primary chains are placed in the root and nodes and each repeat unit can bear part of a crosslink (Fig. 7). In this case ve is the number of elastically active chains per primary chain. 

Fig. 8. Random crosslinking of monodisperse chains and aPoisson distribution of primary chains ve jy 1 Poisson distribution and monodisperse chains, Pn- oo 2 monodisperse chains jP = 10.Pci/P 3 monodisperse chain and Poisson distribution, Pn-> oo 4 Poisson distribution, Pn = 10 5 monodisperse chains P = 10. The dotted lines show the gel point and full conversion for Pn = 10 [Dobson and Gordon (4 )]...

Fig. 9. Random crosslinking of a most probable distribution of primary chains. mlv- 1 Pn - 2 Pn = 10 Pei/Pn 3 P - oo, 4 Pn = 10. The dotted lines show...

Fig. 10. The fraction r0/yei for random crosslinking of high molecular weight primary chain distribution. 1 monodisperse or Poisson, 2 most probable, 3 self-convoluted random [Dobson and Gordon (4/)]...

Deuterium NMR is very sensitive to orientational behavior and order there are a number of papers dealing with constrained polymeric networks. For example, 2H NMR (in both, solid state and solution) is used in the study of the orientational order generated in uniaxially strained rubbers as a function of the crosslink density. Two sets of rubbers (model end-linked silicone rubbers and randomly crosslinked diene networks) were investigated directly (on perdeuterated silicone labelled chains) and indirectly, via C6D6 as an NMR probe for diene rubbers 45). 

Consider the case of polymer chains with an arbitrary distribution of chain lengths and with a number of potentially reactive sites equal to the degree of polymerization (every monomeric unit of the polymer chain constitutes one potentially reactive site). The random crosslinking (vulcanization) of these linear primary chains may be considered a stepwise homopolymerization of Afl species, where the functionality of every species is directly proportional to its molar mass,... 

More precisely, the magnetic relaxation depends on the variable of gelation, i.e., the density of crosslinks, and is closely related to the modulus of elasticity, E, on the one hand and to the swelling ratio, Qm, on the other hand. Long polybutadiene chains are currently randomly crosslinked, using sulfur they can serve to illustrate the NMR approach to the characterisation of vulcanised polymers. It has been shown that the... 

One important feature of randomly crosslinked chains is that they can be highly strained and/or swollen they recover their initial shape when the stress is interrupted and/or after deswelling. Neither the dependence of the modulus of elasticity, E, upon the variable e nor that of the swelling ratio, Qm, have been exactly predicted, until now. Nevertheless, these two physical quantities as well as the rate %c are functions of the variable, e. 

Figure 8.5 Linear dependence of the NMR standard parameter, cc, on the modulus of elasticity E measured on randomly crosslinked polydimethyl siloxane (PDMS) chains. NMR measurements were performed without any sample deformation (redrawn from [19])...

Considering calibrated gels or randomly crosslinked chains, the effect of linkage of monomeric units on NMR is well detected in spite of the purely statistical definition of the segmental spacing between crosslinks in vulcanised chains. 

General Presentation End-linked versus Randomly Crosslinked Networks... 

Deuterated poly( 1,4-butadiene) (PB) networks have also been investigated [18-22], The linear precursor chains Mn from 75000 to 116000 g.mol 1) are homogeneously, selectively deuterated in the 1,4 positions, and randomly crosslinked with different types or amounts of crosslinkers. Star precursor chains deuterated at the center have also been used. 

Figure 15.8 2H NMR spectra obtained in a randomly crosslinked, deuterated PB network. Precursor chain molecular weight 115700 g.mol"1, 1.2% crosslink agent, the average molecular weight between junctions is 27400 g.mol 1 (as determined by swelling experiments) or 11600 g.mol"1 (elastic measurements). The smooth curves are fits with a most probable distribution of chain lengths with number average molecular weight Mc = 11600 and with non-Gaussian chain statistics...

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