2.1 Nucleation induction period with membrane interfacial involved.
The nucleation induction period \(t_{\text{ind}}\) is defined as the time interval from the solution reaches the supersaturated state to the solid particles appear and is detected. \(t_{\text{ind}}\) is commonly introduced and measured as a critical crystallization parameter to evaluate the crystallization operation, which can be affected by factors such as supersaturation, stirring strength and external interfacial, physical field, etc.13,36,37The presence of external crystal seeds usually shorten the induction period.
It is generally believed that the induction period is inversely proportional to the primary nucleation rate \(B_{p}\) of the crystal, namely38,39,
\(t_{\text{ind}}\propto{B_{p}}^{-1}\) (1)
The initial nucleation rate equation expressed by the Arrhenius reaction rate can be written as:
\(B_{p}=Aexp[-\frac{16\pi\gamma^{3}{V_{m}}^{2}}{3\kappa^{3}T^{3}(lns)^{2}}]\)(2)
Where \(A\) is an exponential factor; \(V_{m}\) is a molar volume;\(\kappa\) is Boltzmann constant, and γ is the surface tension of the crystallization solution. Thus, the induction period\(t_{\text{ind}}\) can be expressed as,
\(\frac{16\pi\gamma^{3}{V_{m}}^{2}+K}{3\kappa^{3}T^{3}(lns)^{2}}\) (3)
where \(K\) is a system dependent constant.
Therefore, under constant temperature conditions, \(\ln t_{\text{ind}}\)and \((lns)^{-2}\) follows the linearly relationship, and the slope of the line is defined as \(\alpha\),
\(\alpha=\frac{16\pi\gamma^{3}{V_{m}}^{2}}{3\kappa^{3}T^{3}}\) (4)
The surface tension of heterogeneous nucleation is directly impacted by the hydrophilic characteristics between the membrane surface and the crystallization system. As the case of complete non-affinity between the crystalline solid and the foreign solid surface (corresponding to the complete non-wetting in liquid-solid systems), the contact angle\(\theta\) = 180°. The overall free energy of nucleation is the same as that required for homogeneous nucleation. For the case of partial affinity (the partial wetting of a solid with a liquid), 0<\(\theta\)<180°, which indicates that nucleation is easier to achieve because the overall excess free energy required is less than that for homogeneous one. For the case of complete affinity (complete wetting), \(\theta\) =0, and the free energy of nucleation is equal to zero. Therefore, the solid-liquid surface tension γ in the presence of the introduced membrane interface can be obtained from the above formula.
\(\gamma=\left[f\frac{3\alpha\kappa^{3}T^{3}}{16\pi{V_{m}}^{2}}\right]^{\frac{1}{3}}\)(5)
where f is coefficient related to the contact angle, and expressed as followed,
\(f=\frac{\left(2+cos\theta\right)\left(1-\text{cosθ}\right)^{2}}{4}\)(6)
In principle, the induction period will decrease with the declining surface tension. Thus, solid-liquid surface tension γ is an important thermodynamic parameter for the nucleation and growth process of crystal, especially for the novel crystallization process involved heterogenous interface (MACC, e.g.), which will be discussed in the later section.