Functionality branching unit

Note A branch unit from which /linear chains emanate may be termed an /-functional branch unit, e.g., five-functional branch unit. Alternatively, the terms trifunctional, tetrafunctional, pentafunctional, etc. may be used, e.g., pentafimctional branch unit. 

Linear polymers are the most commonly found, and consist of chains of D units endblocked by a variety of functionalized M units. Branched-chain silicones consist mainly of D units, with a D unit being replaced by a T or a Q unit at each point of branching. Cyclic PDMS oligomers are also common and can play a role in adhesion. They are usually found as mixtures of structures going from three siloxy units, to four, five, and higher siloxy units. A whole range of analytical techniques can determine the detailed molecular structures of these materials [20,21],... 

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. 

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 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. 

From here on the deduction of the probability Fn that the arbitrarily selected unreacted A group is part of a molecule containing n branch units proceeds identically with the deduction of Fx for the simple /-functional distribution. Hence... 

The total number of chains, being by definition equal to half the sum of the unreacted functional groups and the functional groups attached to branch units, is... 

Divergent or convergent synthesis of dendrimers which already carry the desired functionality in the branching unit, i.e within the dendrimer skeleton.11 10 11 19 21 221... 

The indices 3 or 4 indicate the functionality of the monomeric units, and n denotes the number of branching units in one branched molecule. This number of branching units is related to the degree of polymerization x by the relationship... 

The value of p 7000 makes the term (1 -a)gP a and thus gives practically the result of Kurata s estimation [ 129]. This observation leads to the conclusion that it is not the total number of branching units in the branched cluster that defines the type of the g dependence on g but very likely it is the functionality of the repeating unit. However, further experiments with/=4 have to be made before a well established statement can be made. 

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