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Preparation of Polymeric Membranes 63
reported that the glass transition temperature depression can be described well using the
Kelley–Bueche equation (36). The Kelley–Bueche equation was based on the free volume
concept (36):
Tg ¼ Rf1Tg1 þ f2Tg2 ; ð6Þ
Rf1 þ f2 where the subscripts 1 and 2 denote the diluent and the polymer, respectively. Tg is the glass
transition temperature of diluted polymer. Tgi and fi are the glass transition temperature and
the volume fraction of the component i. The glass transition temperature of a polymer can be
found in literature. The estimate of a solvent can be obtained according to an empirical
equation (37):
g ¼ Tm þ Tb
; ð7Þ
Tg þ Tb
where Tm, Tb, and Tg are the melting temperature, boiling temperature and glass transition
temperature of one solvent, respectively. The coefficient of g in Eq. (7) is almost a constant of
1.15, which changes from 1.09 to 1.30 for some solvents (37). A modified formula is given by
(38),
g ¼ Tm þ Tb
Tg þ Tb
The coefficient of g is 1.36, which deviates from 1.30 to 1.41.
GELATION POINT
þ 0:6 Tb Tm
: ð8Þ
Tb þ Tm
The gelation mechanism for a binary solution of an amorphous polymer was defined by
Berghmans (34). The gelation arises in the system characterized by the intersection of the
binodal and the glass transition boundary in a temperature–concentration phase diagram. The
Berghmans point (B point) is the intersection between the binodal curve and the glass
transition boundary of the system. So the gelation results from the liquid–liquid phase
separation arrested by the vitrification of the polymer-rich phase. By cooling down a
homogeneous solution noted as A in Fig. 2.17, the solution separates into two phases at
point M after it crosses the binodal curve. Upon further cooling, the composition of the
polymer-rich phase finally reaches point B, and the polymer-rich phase is vitrified. Consequently
the structure of the demixed solution is fixed. The time from the original phase
separation (point M) to the final vitrification of the polymer-rich phase (point B) is defined as
the gelation time (tgel). If the cooling speed is not infinitely slow, a porous glass will be
formed. The original porous structure is created by the liquid–liquid demixing and the final
porous structure is related to the vitrification of the polymer solution. The liquid–liquid
demixing will cease as soon as gelation starts.
VITRIFICATION AND COARSENING PHENOMENON
When the phase separation (point M) occurs in the metastable region (II, III) between the
binodal and spinodal curves, the mechanism of the phase separation is commonly referred to