Branched units

The probability that a chain segment is capped at both ends by a branch unit is described by the branching coefficient a. The branching coefficient is central to the discussion of gelation, since whether gelation occurs or not depends on what happens after capping a section of chain with a potential branch point. 

In all calculations the molar masses given in the top of Table I were used. First of all, the effects of variations in the concentration of trifunctional monomers were determined, as exemplified by the nine formulations of Table I and the resulting prepolymer characteristics after full conversion given in Table II. Formulations FIO to F40 result in branched prepolymers, which are cured in the third stage by difunctional monomers. On the other hand, formulations FOO to F04 result in the same linear prepolymer, which is subsequently cured with various mixtures of di- and trifunctional monomers. The number average functionalities of PI (and P2) and of the mixtures of E and F monomers are varied systematically between 2.0 and 2.4. Therefore, the only difference between formulations FjO and FOj is the stage in which the branching units are added. 

The degree of branching can be regarded as the ratio of branched units in the polymer to those in a perfect dendrimer. Thus, the hmiting values are DB=0 for hnear polymers and DB=1 for a perfect dendrimer. Various definitions of DB have been given. If we do not take into account the vinyl group or initiator unit (the core unit ), the DB is defined as... 

From the topology of branched systems with trifunctional branchpoints, for any given molecule the number of branched units is equal to the number of terminal unit minus one. Thus, Eq. 1 can be further simplified to... 

X (number of branched units) (total number of units) - 1... 

Dendrons Containing a Cgg Sphere at Each Branching Unit. 99... 

Fig. 13. Preparation of the branching unit 33. Reagents and conditions (i) f-Butyl 2-hydroxy-acetate, DCC, DMAP, BtOH, CH2CI2,0 °C to room temperature (91%) (ii) C, DBU, I2, toluene, room temperature (30%) (in) DDQ, H2O, CHjClj, room temperature (84%)...

Fig. 19. (a) Divergent and (b) convergent synthesis of dendritic polyrotaxanes with mechanical branching units... 

Fig. 21. Dendritic polyrotaxanes with mechanical branching units containing covalently linked bis-dendrons and a core imit fused to polyether macrocycles...

Alternatively, the structural units of the polymer may be connected together in such a manner as to form nonlinear, or branched, structures of one sort or another. Some, at least, of the structural units must then possess a valency greater than two. A typical nonlinear pol3oner structure in which the branching units represented by Y are trivalent may be indicated as follows ... 

If X is an alkyl group and Y = H, the reference compound XIII is a straight chain hydrocarbon, and the second process above consists in its transformation to a branched unit (XIV) having a single substituent. Hence... 

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... 

Fig. 61). Specifically, a chain is defined as that portion of a molecule between two branch units, or between a branch unit and a terminal unreacted functional group (e.g., OH or COOH). The lengths of the chains will vary, but for the present this variation is unimportant.

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 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. 

If more than one type of branching unit is present, (/—I) must be replaced by the appropriate average, weighted according to the numbers of functional groups attached to the various branched units and the molar amount of each present. The critical condition can be expressed in various ways Eq. (7) is a particularly convenient form for application to condensation polymers. 

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 somewhat analogous complexity distribution for systems containing R—Af branch units is discussed in the Appendixes to this chapter. 

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