Network infinite

On a molecular scale, the difference between the two classes of materials is rather small. Thermoplastics consist of individual long chain molecules not connected with each other. Addition of a few crosslinks results in an infinite network structure that is the characteristic of thermosets. 

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... 

The presence of polyfunctional units nearly always presents the possibility of forming chemical structures of macroscopic dimensions, to which the term infinite network is appropriately applied (Chap. II). It is at this point that the concept of molecules as the primary chemical entities must be abandoned, for the infinite network reaches the macro-... 

The critical conditions for the formation of infinite networks will be discussed at the outset of the present chapter. Molecular weight distributions for various nonlinear polymers will then be derived. Experimental data bearing on the validity of the theory will be cited also. 

The purpose of the following treatment is to define the conditions under which indefinitely large chemical structures, or infinite networks, will occur. To this end we seek the answer to the question Under what conditions is there a finite probability that an element of the structure selected at random occurs as part of an infinite network In order to simplify the problem, any given molecule such as the one shown in Fig. 61 may be regarded as an assemblage of chains connected together through polyfunctional, or branch, units (trifunctional in... 

The critical value of a at which the formation of an infinite network becomes possible can be deduced as follows If the branching unit is trifunctional, as in Fig. 61, each chain which terminates in a branch unit is succeeded by two more chains. If both of these terminate in branch units, four more chains are reproduced, and so on. If less than an even chance that each chain will lead to a branch unit and thus to two more chains there is a greater than even chance that it will end at an unreacted functional group. Under these circumstances the network cannot possibly continue indefinitely. Eventually termination of chains must outweigh continuation of the network through branching. Consequently, when a < 1/2 all molecular structures must be limited, i.e., finite, in size. 

It is important to note, however, that beyond a = 1/2 by no means all of the material will be combined into infinite molecules. For example, in spite of the favorable probability of branching, a chain selected at random may be terminated at both ends by unreacted functional groups. Or it may possess a branch at only one end, and both of the succeeding two chains may lead to unreacted dead ends. These and other finite species will coexist with infinite networks as long as l/2[Pg.353]

The identification of the gel point with the stage in the polymerization at which infinite networks make their appearance is confirmed by the results cited, and the extension of the assumption of random reaction to polyfunctional systems appears to be warranted. 

Ic. Cross-Linking of Polymer Chains.—Formation of chemical bonds between linear polymer molecules, commonly referred to as cross-linking, also may lead to the formation of infinite networks. Vulcanization of rubber is the most prominent example of a process of this sort. Through the action of sulfur, accelerators, and other ingredients present in the vulcanization recipe, sulfide cross-linkages are created by a mechanism not fully understood (see Chap. XI). Vulcanized rubbers, being typical network structures, are insoluble in all solvents which do not disrupt the chemical structure, and they do not undergo appreciable plastic, or viscous, flow. 

The two conditions stated above do not assure the occurrence of gelation. The final and sufficient condition may be expressed in several ways not unrelated to one another. First, let structural elements be defined in an appropriate manner. These elements may consist of primary molecules or of chains as defined above or they may consist of the structural units themselves. The necessary and sufficient condition for infinite network formation may then be stated as follows The expected number of elements united to a given element selected at random must exceed two. Stated alternatively in a manner which recalls the method used in deriving the critical conditions expressed by Eqs. (7) and (11), the expected number of additional connections for an element known to be joined to a previously established sequence of elements must exceed unity. However the condition is stated, the issue is decided by the frequency of occurrence and functionality of branching units (i.e., units which are joined to more than two other units) in the system, on the one hand, as against terminal chain units (joined to only one unit), on the other. 

The key to the resolution of the apparent contradiction becomes evident upon re-examining the initial derivation which proceeds from Fig. 68. Finite, or bounded, molecular species are implied in the expression for the probability of a specific x-mev configuration thus fx — 2x + l unreacted ends in addition to the one selected at random are prescribed. An infinite network, on the other hand, is terminated only partially by unreacted end groups the walls of the macroscopic container place the ultimate limitation on its extent. Hence the network fraction is implicitly excluded from consideration, with the result that the distribution functions given above are oblivious of it. Failure of to retain the same value throughout the range in a is a... 

The weight fraction of gel calculated for this case from Eq. (45) is plotted against a in Fig. 70. It will be observed that the formation of the infinite network is indicated to commence suddenly at the critical point. This prediction of the theory is abundantly confirmed by the characteristic abruptness with which gel appears in polyfunctional condensations. Also shown in Fig. 70 are the weight fractions Wx of various species as functions of a, calculated according to Eq. (36). 

The molecular distributions for polymers formed by condensations involving polyfunctional units of the type R—A/ resemble those for the branched polymers mentioned above, except for the important modification introduced by the incidence of gelation. The generation of an infinite network commences abruptly at the gel point, and the a-mount of this gel component increases progressively with further condensation. Meanwhile, the larger, more complex, species of the sol are selectively combined with the gel fraction, with the result that the sol fraction decreases in average molecular complexity as well as in amount. It is important to observe that the distinction between soluble finite species on the one hand and infinite network on the other invariably is sharp and by no means arbitrary. 

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. 

Flory (5) states that gelled, insoluble products are formed by the inter-molecular reaction of units which are trifunctional or higher. An infinite network structure is formed however, it is limited only by the volume of the reaction mixture. The monomer size controls the rate of reaction. 

O Saito. Statistical aspects of infinite network formations. Polym Eng Sci 19 234-235, 1979. 

As the number of branching points per chain increases, (when it becomes > 2) an almost infinite network of covalent bonds can be created. Strictly all the system belongs to only one giant macromolecule. The polymer becomes insoluble and infusible it is said to be "crosslinked". 

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