Conditions for the Formation of Infinite Networks

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. 

To examine the significance of this approximation further, it should be noted that a highly branched condensation polymer molecule, such as the one shown in Fig. 61, retains many unreacted functional groups which offer a number of opportunities for reaction between pairs on the same molecule. That intramolecular reaction between them proceeds to an appreciable degree in competition with intermolecular condensa- 

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 

First of all it is necessary to determine the branching coefficient a, w hich is defined as the probability that a given functional group of a branch unit leads via a chain of bifunctional units to another branch unit. In a polymer of the type shown in Fig. 61, a is the probability that an A group selected at random from one of the trifunctional units is connected to a chain the far end of which connects to another trifunctional unit. As will be shown later, both the location of the gel point and the course of the subsequent conversion of sol to gel are directly related to a. 

The probability, oj, that the chain ends in a branch unit regardless of the number,, of pairs of bifunctional units is given by the sum of such expressions having = 0, 1, 2, etc., respectively. That is 

---

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 critical value of a at which the formation of infinite networks becomes possible can be deduced as follows. If the branching unit is ttifunctional, 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 a < Vi, there is less than an even chance that each chain will lead to a branch unit, and thus, to two more chains. Under these circumstances the network cannot possibly continue indefinitely. When a > Vi, a growing chain has better than an even chance to reproduce two new chains. Two such chains will on the average reproduce 4a new chains and so on n chains can be expected to lead to 2na new chains, which is greater than n, when a > V2. Unlimited structures, or what we have called infinite networks, are then possible. Hence a = VSt presents the critical condition for incipient formation od infinite networks in a trifunctional branched system . 

Extrapolation of pj. g to the limit of zero pre-gel intramolecular reaction for given reaction systems shows that post-gel intramolecular reaction always results in network defects, with significant increases in Mg above Mg. Such post-gel intramolecular reaction is characterised as pg g. The variation of pg g with intramolecular-reaction parameters shows that even in the limit of infinite molar mass, i.e. no spatial correlation between reacting groups, inelastic loops will be formed. The formation may be considered as a law-of-mass-action effect, essentially the random reaction of functional groups. Intramolecular reaction under such conditions (p2 ) must be post-gel and may be treated using classical polymerisation theory. 

The branching coefficient as defined by Flory is a = the probability that a certain branched unit will be joined to a second branched unit rather than to a terminal group. For example, for a trifunctional monomer if a = 1/2 the molecule is a continuous chain equivalent in theory to a gel. In this case a = 1/2 is the critical condition defining the start of the formation of an infinite tridimensional network. 

We discuss a solution of molecules ( monomers ) with functionality f > 3 (in general) from each molecule may emanate zero to f bonds to neighboring molecules and thus this molecule may participate in the formation of a large cluster which is called a macromolecule. Two monomers in the same cluster or macromolecule are thus connected directly or indirectly (through other monomers in the same cluster) by such bonds whereas two monomers in two different macromolecules are not connected by such bonds. We denote the number of monomers in one macromolecule by s and then call this macromolecule also an s-cluster an isolated monomer without bonds to its neighbors is thus designated as an 1-cluster with s = 1. (For simplicity, we also call s the mass of the macromolecule, i.e. we set the molecular weight of the monomers equal to unity in the theoretical discussions.) Under certain conditions, an infinite cluster can be formed, i.e. a network which extends from one end of the sample to the other. 

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. 

---