For the nucleation induction period with PTFE membrane, under high
supersaturation,
\(\ln t_{\text{ind}}=\ 0.0933\left(\text{lns}\right)^{-2}+\ 3.2500\),\(R\ =\ 0.9859\) (27)
Under low supersaturation,
\(\ln t_{\text{ind}}=\ 0.0188\left(\text{lns}\right)^{-2}+\ 4.3727\),\(R\ =\ 0.9856\) (28)
According to the classical nucleation
theory49, the
intersection of these two lines should be the demarcation point of
homogeneous nucleation and heterogeneous nucleation. The interfacial
tension of thiourea at the corresponding temperature can be obtained
from the heterogeneous nucleation slope and homogeneous nucleation
slope. Thus, the solid-liquid interface tension of thiourea in the
solution with PTFE membrane involved can be obtained, \(\gamma=3.391\)mJ/m2.
Similarly, for the nucleation induction period with PES membrane, under
high supersaturation:
\(\ln t_{\text{ind}}=\ 0.0464\left(\text{lns}\right)^{-2}+3.7544\),\(R=0.9688\) (29)
Under low supersaturation:
\(\ln t_{\text{ind}}=\ 0.0077\left(\text{lns}\right)^{-2}+\ 4.5629\),\(R=0.9647\) (30)
The solid-liquid interface tension of thiourea in the solution with PTFE
membrane can be obtained,\(\ \gamma=2.221\) mJ/m2.
Surface tension required for heterogeneous nucleation on the PTFE hollow
fiber membrane is 3.391 mJ/m2, while PES hollow fiber
membrane is 2.221 mJ/m2. This is also consistent with
the contact angles analyzed results in Figure 7, PES membrane possesses
a higher affinity to the crystallization solution. Therefore, according
to Eq. (2) to (6), heterogeneous nucleation barrier of PES hollow fiber
membranes will be lower than that of PTFE membrane, which means PES MACC
will possess higher nucleation rate B p under the
same supersaturation degree. The pre-exponential factor can be obtained
from the Eq. (10). With the different contact angles and surface
tension, the ratio of the heterogeneous nucleation rate between two
kinds of membranesB p-PES/B p-PTFE is around
3.60.
While, the number of nuclei can be calculated from the total mass of the
crystal product, the mass distribution of crystals. And with the known
duration using membrane module to generate nuclei, we can evaluate the
nucleation rate, semiquantitatively. The experimental data showed that
the nucleation rate of the PES membrane was 14.3
#/mm3·s, the one of the PTFE membrane was 4.0
#/mm3·s, and the ratio is around 3.57, which was
accompanied with the former theoretical research results of nucleation
kinetics.