# (p.242) B Dispersity in step‐growth polymerization

# (p.242) B Dispersity in step‐growth polymerization

Calculating the dispersity of a polymer mixture is possible to obtain analytically for polymers formed from condensation reactions (or more generally for polymers formed in step‐growth reactions), as is often the case for semiconducting polymers. The dispersity is defined as the ratio of the weight average and number average molecular mass of the polymer. The number average molecular mass is easily defined, because it is essentially the Carothers equation (eqn 6.3 ),

where *M*
_{wm} is the molecular weight of a monomer, and reference to the time dependence has been omitted.

The total number of chains with *x* monomers is given by

where *n*
_{0} = *n*(0). This equation (B.2) is the product of the number of molecules remaining, given by *n*
_{0} (1 − *p*) and the probability *P* (*x*) that a molecule containing *x* monomers exists, given by

Here, *p ^{x}
*

^{−1}is the probability of finding

*x*connected monomers (i.e.

*x*−

*1*bonds) and 1 −

*p*is the probability that the next reactive group remains unreacted. The pre‐factor 1 −

*p*is important because polymers with more than

*x*monomers will also satisfy the requirement for

*x*−

*1*bonds.

The mass fraction is given by

and, using eqn 6.6 , the weight average molecular weight by

since *M _{x}
* =

*xM*

_{wm}. We can simplify the sum with the known mathematical relation

which means that by combining eqns (B.1), (B.5), and (B.6)
we obtain eqn (6.8)
,
*D*
_{M}
= 1 + *p*.