Membrane and Desalination Technologies - TCE Moodle Website

Preparation of Polymeric Membranes 67
The phase equilibrium (the binodal curve) can be obtained when the objective function F is
in minimum. The system within the binodal curve is unstable. It will decompose into two
equilibrium phases, which are related by tie lines in the phase diagram.
For the swelling curve BG, since there is no polymer dissolved in the mixture of solvent
and nonsolvent, the objective function F1 is defined as:
F1 ¼ X2
i¼1
Dm ðswellingÞ
i
RT
Dm
!
ðmixture=solvent nonsolventÞ 2
i
RT
: ð15Þ
The unstable liquid–liquid two-phase region is divided into two regions by the spinodal
curve. The metastable region is formed between the spinodal and binodal curves. In this
metastable region, small perturbations will decay, and the phase decomposition can only be
caused by a large perturbation. Within the spinodal curve, any small perturbation will induce
the phase separation of the system. The spinodal curve in a ternary system is defined as (51):
G22G33 ¼ G 2 23 ; ð16Þ
G22, G33 and G23 are the derivatives of DGm with respect to fi, fj
Gij ¼ @DGm=ðVRTÞ
( )
v1: ð17Þ
@fi@fj Here, V is the total volume. The spinodal curve for a ternary system with constant
interaction parameters can be expressed as follows:
1
f 1
þ v1
v2
1
f 2
2g12
1
f 2
þ v1
v3
1
f 3
LINEARIZED CLOUD POINT CURVE CORRELATION
2g13 ¼ 1
’ 1
g12 g13 þ v1
The phase behavior of a ternary polymer solution can be determined by cloud point
measurements and the compositions of the cloud points refer to the position of the binodal.
The correlation of the linearized cloud point curve was obtained for polymer solutions in the
mixture of one solvent and one nonsolvent (52–54). With this correlation, the information
about the thermodynamics of ternary systems can be obtained obviously, which is helpful for
preparing membrane forming systems. The linearized cloud point curve correlation describes
a linear relationship between the nonsolvent concentration relative to the polymer concentration
and the solvent concentration relative to the polymer concentration at the condition of
phase separation, which is shown as (54):
v2
g23
2
:
ð18Þ
ln f1 ¼ b ln
f3 f2 þ a; ð19Þ
f3