Recursive Approach for Average Properties

Molecular weight distributions in step-growth polymerization described so far in this chapter are all based on the probability approach of Flory [1,15] and Stockmayer [19]. Starting with the assumptions of equal reactivity of functional groups and no intramolecular reactions, they used combinatorial arguments to derive expressions for the distribution of all species as a function of the reaction extent and then used these distributions to calculate the average properties (Mn, Mw, and PDI). For cases of practical importance these distribution functions become quite complex [19]. 

The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E Y) its expectation (or average value) and E Y A) is conditional expectation, given that the event A has occurred. P A) is the probability that event A occurs. Then the law of total probability for expectation is [31]  

This law is discussed in most introductory books on probability theory. 

The simplest step-growth linear polymer is of the AB type. Typical examples of AB polymerization are the step polymerization of HORCOOH and H2NRCOOH. For example, the polymer from HORCOOH is 

Picking an A group at random, we define the in direction from the chosen A toward the B group of the same mer unit. Out is then the opposite direction from the chosen A toward the remainder of the chain on the A side of the mer. Similar definitions apply to the in and out directions associated with B groups. 

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In addition to being a simpler method for obtaining the average properties such as My, and Mn compared to the Flory and similar approaches [31], the recursive approach also more easily allows an evaluation of the effect of unequal reactivity and unequal structural unit molecular weights on the average properties [27,28,33]. 

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