4.4.3. Diffusion Coefficient

As mentioned above, the independent *S* values analysis was not entirely consistent with the crystal morphology results. Therefore, another factor, the diffusion capacity of solvent molecules, was introduced to look into the solvent diffusion effects on crystal growth. Based on the well known Einstein relationship [46], the diffusion coefficient (*D*) of the solvent molecules was defined as the derivative of the mean square displacement (MSD) with respect to time [47]:

$$D = \frac{1}{6} \lim\_{t \to \infty} \frac{d}{dt} \sum\_{i=1}^{N} \langle \left| r\_i(t) - r\_i(0) \right|^2 \rangle,\tag{8}$$

where *N* stands for the number of solvent molecules and *r*i(*t*) represents the position of the molecule *i* at time *t*.

Stronger interactions existed between the solvents and crystal faces with larger *D* values due to an increasing number of solvent molecules diffusing to the interface [21]. As listed in Table 5, the *D* values were listed in the following sequence: (1 0 −1) > (0 0 2) > (1 0 1) > (1 1 −1) > (0 1 1) > (1 1 0) in isopropanol, (1 0 −1) > (0 0 2) > (1 1 −1) > (0 1 1) > (1 1 0) > (1 0 1) in methyl acetate, and (1 0 −1) > (1 0 1) > (0 0 2) > (1 1 −1) > (1 1 0) > (0 1 1) in ethyl acetate. Compared with those of isopropanol, the *D* values of methyl acetate and ethyl acetate were relatively large, indicating stronger interactions with the crystal surfaces which were consistent with the values of *E*int in the corresponding solvent systems. Obviously, the diffusion coefficient on the (1 0 −1) face was the largest one among the six crystal faces for all the three kinds of solvent molecules. This indicated that more solvent molecules gathered on the (1 0 −1) face and strong interactions existed between the solvent molecules and the crystal face although the adsorption areas provided by this face were small with the minimum S value. Thus, for the factors affecting the growth of the (1 0 −1) face, the solvent–crystal interface interaction was more dominant than the surface roughness. As a result, the (1 0 −1) face manifested a relatively slow growth of the crystal face and the largest area in real morphology. Apart from the large *D* value on (0 0 2) faces, the *D* values on crystal faces were in reasonable agreement with the experimental crystal area results, as solute molecules were more likely to adsorb on the faces with smaller *D* values, on which solvent molecules take less growth active sites [48]. For example, the (1 1 0) and (0 1 1) faces disappeared in ethyl acetate with smaller *D* values (2.11 <sup>×</sup> 10−<sup>9</sup> m2 s−<sup>1</sup> and 1.94 <sup>×</sup> <sup>10</sup>−<sup>9</sup> <sup>m</sup><sup>2</sup> <sup>s</sup><sup>−</sup>1, respectively), while the (1 1 0) and (0 1 1) face with small *<sup>D</sup>* values (0.67 <sup>×</sup> <sup>10</sup>−<sup>9</sup> <sup>m</sup><sup>2</sup> <sup>s</sup>−<sup>1</sup> and 0.69 <sup>×</sup> <sup>10</sup>−<sup>9</sup> m2 <sup>s</sup><sup>−</sup>1, respectively) did not exist in the final morphology in isopropanol. Despite the high *D* values on (0 0 2) faces, it was easier for solute molecules to adsorb on the crystal surface than it was for the solvent molecules to form a new layer of catechol crystal because their |*E'*att| values were larger than the corresponding |*E*s| values.


**Table 5.** The diffusion coefficient (D)<sup>1</sup> of the solvent molecules on the different catechol crystal faces.

<sup>1</sup> All *D* values are in 10−<sup>9</sup> m2 s<sup>−</sup>1.
