## 1 Introduction  
Crystallization is one of the oldest chemical operations to produce, purify, or separate solid products. Membrane crystallization refers to a hybrid membrane separation-crystallization process introduced in 2001 [1; 2; 3; 4; 5]. It denotes an advanced process where crystal nucleation and growth are carried out in a well-controlled pathway by using a porous hydrophobic membrane. In particular, a membrane crystallizer is a system where a membrane is in contact on one side with the feed solution (containing a nonvolatile component to be crystallized) and on the other side with a permeate solution (Figure 1). The gradient of vapor pressure between the two subsystems induces the evaporation of the volatile component (i.e., the solvent) from feed solution and its migration through the porous membrane. The continuous removal of solvent increases solute concentration, thus generating supersaturation. Although the membrane does not act as sieving barrier for the selective transport of specific components, a solute-membrane interaction occurs due to the direct contact of the feed solution with the membrane surface. By considering the interactions between solute and solid substrate in terms of contact angle \(\theta\) (which the crystallizing solution forms with the solid substrate), the reduction of \(\Delta G\) due to heterogeneous nucleation is equal to [6; 7; 8]  
\[\frac{\Delta G_{heter}}{\Delta G_{homog}}=\frac{1}{4}(2+\cos\theta)(1-\cos \theta)^{2}\left[1-\varepsilon\frac{(1+\cos\theta)^{2}}{(1-\cos\theta)^{2}} \right]^{3} \tag{1}\]  
where \(\varepsilon\) is the overall surface porosity defined as the ratio between the area of the pores to the total membrane surface area. In the case of a nonporous system (\(\varepsilon=0\)), when the contact angle is equal to \(180^{\circ}\), then \(\Delta G_{heter}~{}=\Delta G_{homog}\), whereas if the contact angle is equal to \(90^{\circ}\), then \(\Delta G_{heter}~{}=\frac{1}{2}\Delta G_{homog}\). Therefore, we can conclude that the presence of a hydrophobic membrane with \(90^{\circ}<\theta<180^{\circ}\) promotes nucleation. The special effect of the heterogeneous contribution allows crystals to nucleate faster and/or by using a lower initial amount of substance with respect to the usual comparable techniques. An example can be found in [9], where the kinetics of lysozyme crystallization  
Figure 1: Schematic representation of a membrane crystallizer plant: (**A**) membrane module; (**B**) pump; (**C**) crystallizer tank; (**D**) permeate tank. In the box at the top right, the scheme of the basic principle of the phenomena occurring inside the membrane module: solvent is removed from the crystallizing solution under a vapor pressure gradient (P1 > P2, where 1 refers to the feed/crystallizing side, 2 refers to permeate side). carried out with microporous hydrophobic membranes were investigated. [PERSON] et al. [9] found considerably shorter induction periods (ranging between 1.2 and 10 h), compared with those measured by using conventional crystallization techniques, and high values of the growth rate constant at low supersaturation ratios. Their findings clearly demonstrate the effect of the membrane surface as a heterogeneous support for crystal nucleation and growth. Because the relation between \(\Delta G_{heter}\) and \(\Delta G_{homog}\) depends on the contact angle value, different polymeric membrane structures exhibit dissimilar interactions with a liquid phase. It is noteworthy that controlling interfacial forces are also contingent with local chemical environments dispersed in the membrane surface [10]. These kinds of interactions can also affect the induction time of nucleation, resulting in a very short time of crystallization. [PERSON] et al. demonstrated that an increase in the cohesive energy between a protein solution and membrane surface was increased by functionalization of the membrane surface.