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.