Reaction kinetics and the most probable distribution

One of the most important properties in condensation polymers is the polymer s molecular weight. Consider a linear step-growth polymerization of AB-type monomers. For illustrative purpose, let A refer to a hydroxyl group and B to a carboxyhc add group, so that a polyester is formed. The polymerization reaction is  

As the reaction proceeds to a fractional conversion of p for either functional group A or B, the reaction mixture contains polymer molecules with a variety of chain lengths. Thus, the modeling objective here is to quantify the development of polymer chain length distribution (CLD) at different end-group conversions. 

In step-growth polymerization, it is common to use the symbol p (rather than x) to refer to conversion of the limiting functional group, because the conversion is equal to the probability that a particular functional group (e.g., an —OH group) in the original reaction 

Consumption of exactly n monomers to form this polymer chain requires n — 1 independent chain-linking reactions, each with probability p, and also that one hydroxyl group (i.e., the one at the end of the resulting molecule) has not been consumed, which has probability 1 — p. The distribution in Equation 7.4 is called the most probable distribution (MPD) or Schulz-Flory distribution. This same distribution also arises in some chain-growth polymerization systems (e.g.. Equation 2.84 in Chapter 2 is another form of the MPD). 

The probability P(n) corresponds to the mole fraction of n-mer, x(n). At 90% conversion (p = 0.9), the mole fraction of monomer (n = 1) in the reaction mixture is 0.09, the mole fraction of dimer (n = 2) is 0.081, and the mole fraction of trimer (n = 3) is 0.0729, with successively larger chains having smaller and smaller mole fractions. As conversion increases (p 1), the mole fractions of small molecules decrease, and the mole fractions of larger molecules increase, but the species with the largest concentration is always the monomer, so that the number CLD (Equation 7.4) is a decreasing function of chain length. 

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