*3.2. Relative Thermodynamic Stability among the Anhydrous Solid Forms*

Form I was known to be the most stable form at r.t., and slurries of mixtures of I + II and I + III resulted in complete transformation into form I. The main question to solve is whether two polymorphs are monotropically (one form is more stable than the other at any temperature) or enantiotropically (a transition temperature exists, below and above which the stability order is reversed) related, and for an enantiotropic system, where transition temperature lies.

Table 1 summarizes the physicochemical data of the three modifications of BL obtained from the thermal analysis experiments. According to the Heat of Fusion Rule of Burger and Ramberger [22], forms I and II are enantiotropically related due to the higher enthalpy of fusion and lower melting point of form I. An estimation of the transition temperature between an enantiotropic pair of polymorphs can be done by using the treatment of Yu [23]. This method uses the temperatures and enthalpies of fusion to calculate the Gibbs free energy difference at the melting temperature of the lower melting form and to extrapolate to other temperatures.

$$T\_{trs} = \frac{\Delta H\_{fus,2} - \Delta H\_{fus,1} + k\Delta H\_{fus,1} \left(T\_{fus,1} - T\_{fus,2}\right)}{\frac{\Delta H\_{fus,2}}{T\_{fus,2}} - \frac{\Delta H\_{fus,1}}{T\_{fus,1}} + k\Delta H\_{fus,1} \ln\left(\frac{T\_{fus,1}}{T\_{fus,2}}\right)}\tag{1}$$

**Table 1.** Physicochemical data of anhydrous forms of BL.


A value of 0.003 was used for the factor *k*, which was empirically determined and allows a good approximation of the heat capacity differences in the majority of cases [23]. Using this equation, we calculated a value of 137 ◦C for the I/II pair.

It was not possible to obtain the value of the enthalpy of fusion of form III as this form transformed while heating in the DSC analysis. Regarding the relative thermodynamic stability between the pair I/III, form I is more stable than III at r.t., being this last one the lower melting form. Therefore, forms I and III must be monotropically related at least from r.t., being form I the most stable one.

As for the pair II/III, without knowing the enthalpy of fusion of form III, it was not possible to use the Heat of Fusion Rule of Burger and Ramberger; therefore, we applied the solvent mediated transformation method [24]. This method is based on the relationship between solubility and stability of crystal forms, i.e., the less stable form will also be the most soluble at given conditions of temperature and pressure. If crystals of both forms are mixed with a saturated solution of the product, the most stable form will grow at the expense of the less stable one. Thus, a mixture of the two modifications was stirred in ethanol at r.t., and after one hour, pure form I was recovered. Hence, a reduction of time was required and a suspension of II and III was stirred for 15 min. Comparing the PXRD between the initial and the final mixture, a decrease in the intensity of X-ray diffraction peaks attributed to form II (compared to form III) was observed, while form I did not appear. This final mixture was suspended again in ethanol for 10 min, and form II decreased again; however, form I began to appear in the mixture, so the transformation of II into III no longer could be studied. We concluded that form III was more stable than form II at r.t., and as III is the lower melting form, an enantiotropy relation could be inferred for the II/III pair. Without having data of the melting enthalpy of form III, it was not possible to calculate the transition temperature between II and III.

Energy diagrams such as those proposed by Burger and Ramberger [22] give considerable insight into polymorphic systems. Based on physicochemical and solvent mediated transformations data, a semi-schematic energy/temperature diagram was constructed in order to display the thermodynamic relationship of the anhydrous polymorphs at different temperatures (Figure 4).

**Figure 4.** Semi-schematic energy/temperature diagram of anhydrous forms of BL.
