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.