Randomly Crosslinked Chains

Network structures are still determined by nodes and strands when long chains are crosslinked at random, but the segmental spacing between two consecutive crosslinks, along one chain, is not uniform in these systems which are currently described within the framework of bond percolation, considered within the mean field approximation. The percolation process is supposed to be developed on a Cayley tree [15, 16]. Polymer chains are considered as percolation units that will be linked to one another to form a gel. Chains bear chemical functions that can react with functions located on crosslinkers. The functionality of percolation units is determined by the mean number f of chemical functions per chain and the gelation (percolation) threshold is given by pc = (f-1) 1. The 

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. 

Consequently, they are closely related to one another. The NMR-elasticity relationship is illustrated from the linear dependence of the relaxation rate, %c, on the modulus of elasticity of the gel, E ( 0.5 MPa)  

Similarly, considering the swelling effect of vulcanised poly(butadiene), induced by a good solvent, the relaxation rate has been shown to obey the simple equation  

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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. 

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. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

More definitive evidence of enzymatic attack was obtained with 1 1 copolymers of e-caprolactone and 6-valerolactone crosslinked with varying amounts of a dilactone (98,99). The use of a 1 1 mixture of comonomers suppressed crystallization and, together with the crosslinks, resulted in a low-modulus elastomer. Under in vitro conditions, random hydrolytic chain cleavage, measured by the change in tensile properties, occurred throughout the bulk of the samples at a rate comparable to that experienced by the other polyesters no weight loss was observed. However, when these elastomers were implanted in rabbits, the bulk hydrolytic process was accompanied by very rapid surface erosion. Weight loss was continuous, confined to the... 

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. 

Elastomers exhibit this behavior due to their unique, crosslinked structure (cf. Section 1.3.2.2). It has been found that as the temperatme of an elastomer increases, so does the elastic modulus. The elastic modulus is simply a measme of the resistance to the uncoiling of randomly oriented chains in an elastomer sample under stress. Application of a stress eventually tends to untangle the chains and align them in the direction of the stress, but an increase in temperatme will increase the thermal motion of the chains and make it harder to induce orientation. This leads to a higher elastic modulus. Under a constant force, some chain orientation will take place, but an increase in temperatme will stimulate a reversion to a randomly coiled conformation and the elastomer will contract. 

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 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,... 

Since the crosslinking density a is the probability of any unit to be cross-linked, the number of additional crosslinked units on an, v-mcric chain, crosslinked to the first chain, is expected to be equal to a. multiplied by the number of remaining crosslinkable groups (.v— 1). Thus, the expected number of additional chains crosslinked to the. v-meric chain, which is crosslinked to the first randomly chosen chain, is a(s— 1). Averaging over all chain sizes will give the average expectance of additional chains v. 

An average expectance of additional chains v, which is greater than unity allows for a non-zero probability of the randomly selected crosslinked chain belonging to an infinitely large system [79,83]. This gives the critical value of crosslinking density ac corresponding to the gel point. 

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])...

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...

Examples of vulcanization (a) endlinking (b) random crosslinking. Different lines represent different chains. 

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