Membrane and Desalination Technologies - TCE Moodle Website

66 J. Ren and R. Wang
and no polymer can be founded in the mixture. In region IV, a polymer solution reaches its
glass state or swelling state.
THERMODYNAMIC DESCRIPTION OF THE POLYMER SOLUTION
According to Flory-Huggins theory (50), the Gibbs free energy of mixing (DGm) for a
ternary component system containing a polymer, a solvent and a nonsolvent, can be
expressed with:
with
DGm
RT
¼ X3
i¼1
ni ln fi þ g12ðu2Þn1f 2 þ g13n1f 3 þ g23ðv2Þn2f 3; ð9Þ
u2 ¼ f1 ;
f1 þ f2 v2 ¼
(9a)
f2 ;
f2 þ f3 (9b)
where the subscripts 1, 2, and 3 denote nonsolvent, solvent, and polymer, respectively, ni and
Fi are the number of mole and the volume fraction of component i, gij is binary interaction
parameter between component i and j.
The chemical potential of the individual component i is expressed as:
Dm1 RT ¼ ln f1 sf2 rf3 þð1þ g12f2 þ g13f3Þð1 f1Þ sg23f2f3 g 0 12f2; ð10Þ
sDm2 RT ¼ s ln f2 f1 rf3 þðsþ g12f1 þ sg23f3Þð1 f2Þ g13f1f3 þ g 0 12f1 þ sg 0 23 f 3; ð11Þ
rDm3 RT ¼ r ln f3 f1 sf2 þðr þ g13f1 þ sg23f2Þð1f3Þ g12f1f2 rg 0 23f2 ð12Þ
with g0 ij being as follows:
; (10a)
@u2
g 0 23 ¼ v2ð1 v2Þ @g23
; (11a)
@v2
where Dmi is the chemical potential of component i and s ¼ v1=v2, r ¼ v1=v3. vi is the partial
molar volume of component i.
The requirements for the phase equilibrium (the binodal curve) are:
g 0 12 ¼ u2ð1 u2Þ @g12
Dm i ðpolymer richÞ ¼ Dmi ðpolymer poorÞ ; i ¼ 1; 2; 3 ; ð13Þ
The objective function F is defined as:
F ¼ X3
i¼1
ðpolymer richÞ
Dmi RT
Dm
!
ðpolymer poorÞ 2
i
RT
: ð14Þ