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Polymer Electronics$

Mark Geoghegan and Georges Hadziioannou

Print publication date: 2013

Print ISBN-13: 9780199533824

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199533824.001.0001

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

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

Source:
Polymer Electronics
Publisher:
Oxford University Press

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 ),

M ¯ n = M w,m X n = M w,m 1 p ,
(B.1)

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

N x = n 0 ( 1 p ) 2 p x 1 ,
(B.2)

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

P ( x ) = ( 1 p ) p x 1 .
(B.3)

Here, px −1 is the probability of finding x connected monomers (i.e. x1 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 x1 bonds.

The mass fraction is given by

w x = xN x n 0 = x ( 1 p ) 2 p x 1 ,
(B.4)

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

M ¯ w = M w,m ( 1 p ) 2 x = 1 x 2 p x 1 ,
(B.5)

since Mx = xM wm. We can simplify the sum with the known mathematical relation

x = 1 x 2 p x 1 = 1 + p ( 1 p ) 3 ,
(B.6)

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

A more detailed derivation of this proof is given in several books for example, the text by Young and Lovell (1991). This book presents a generalized introduction to polymers at a level that all readers of this book should find comfortable.