Polyelectrolytes: Theory, Properties and Applications

Structural Properties and Phase Behavior ... 5

for analyzing the structural properties and phase behavior of partially charged
DCPs in the disordered state. The resolution of Eq. 1 requires the knowledge
of the bare structure matrix S 0 (q) and the interaction matrices. The excluded
volume parameters vij can be expressed in terms of s = 1   the solvent

volume fraction ( being the total polymer volume fraction) and as, bs, ab
the A/solvent, B/solvent and A/B Flory–Huggins interaction parameters,
respectively as

aa 0 as

s

1

vv 2



  



; bb 0 bs

s

1

2 vv



  



;

ab ba 0 as bs ab

s

1

v v v



       



(2)

(v 0 = l^3 is a unit reference volume and l the monomer length, v 0 is
sometimes ignored to ease the notation). In general, the ionization matrix f is
diagonal with elements fa and fb representing the fractional charges of A and B
monomers, respectively. Since we are concerned with charged/neutral DCPs,
we let fa = f, fb = 0 and, assuming DebyeHückel model of electrolyte

solutions gives the simple expression for (^)   22 B
4 πl
q
q


where lB is the
Bjerrum length [26, 27], lB = e^2 / 0 kBT, kB is the Boltzmann constant, T the
absolute temperature,  0 the dielectric constant of vacuum, e the electron

charge (at 20°C lB is near 7Ǻ),is the inverse screening length (Ǻ–^1 ).

Inverting Eq. 1 and noting that S(q) and S 0 (q) are diagonal matrices, give the
partial structure factors as

 





 



 

  



 

0 0 0

aa a bb bb b aa^2 ab ab ab

0 0 0

  ;     ;    ;

  

S q S q S q

DS q v DS q v q f DS q v

S q S q S q

(3)

with

 

 

 

 

 

 

 

0 0 0 2

0 0 02 a b 2 ab

0 a b ab bb aa ab

0 0 0

( ) ;

    

             

    

S q S q S q

S q S S S D v v q f v

S q S q S q

. (4)