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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.244) C Regular solution theory

(p.244) C Regular solution theory

Source:
Polymer Electronics
Publisher:
Oxford University Press

We consider a mixture of two molecules on a cubic lattice. Our goal is to calculate the entropy of mixing of this binary mixture. The approach here is in contrast to that of Section 7.5.1 , where we calculated the entropy per lattice site. Here we shall calculate the entropy of a lattice of n sites explicitly. For a lattice containing n sites, there are n! configurations for n molecules on this lattice. We have two components, with n A and n B molecules of each species, where n = n A + n B, so ø = nA /n and 1 − ø = nB /n. The entropy of mixing is simply the (natural logarithm of the) number of arrangements of the two molecules compared to the number of possible configurations on the lattice. The number of configurations is simply

C ( n A n B ) = n ! n A ! n B ! = n ! ( φn ) ! ( ( 1 φ ) n ) ! ,
(C.1)

and is illustrated for a 12 × 12 lattice in Fig. C.1 . We recall Stirling's approximation

ln n ! n ln n n ,
(C.2)

which is valid for large n, and use it to calculate the entropy of mixing, ∆Sl for the whole lattice,

Δ S 1 = k B ln ( n ! n A ! n B ! ) = k B n ( ln n φ ln( φn )−(1− φ ) ln((1− φ ) n ) ) .
(C.3)

(The reader is reminded that Boltzmann's relation, S = kB ln Ω, links entropy, S, to the number of configurations, Ω.) We can thus simplify eqn (C.3) to obtain

ΔS k B = φ   ln  φ + ( 1 φ ) ln(1− φ ),
(C.4)

where ∆S = ∆Sl/n. By considering the whole lattice we nevertheless recover eqn (7.19) from eqn (C.4) when N = 1. In Section 7.5.1 we explicitly included the entropy of the pure states, whereas in the calculation outlined in this Appendix, the entropy of the pure states is implicitly included as n!.

We cannot easily extend this theory to a polymer in a solvent. Firstly, it is not possible to replace n A by φn/N because C (nA, nB ) requires n A + nB = n, but φn/N + (1 − φ) nn. Secondly and related, a polymer molecule can have different configurations on the lattice, whereas a (p.245)


                     C Regular solution theory

Fig. C.1 There are 144!/(54! × 90!) Ɉ 1.62 × 1040 different combinations of spheres on this lattice. If we consider the lighter‐shaded spheres, then φ = 0.375. The calculated entropy of mixing using eqn (C.3) is within 5% of the actual entropy of mixing. For bigger lattices this (already small) error will diminish.

solvent just takes up one lattice position with only one configuration, and the additional effect of this polymer configurational (conformational) entropy needs to be accounted for. For these reasons, it is necessary to follow the method presented in Section 7.5.1 when polymers are considered.