Enthalpy difference between polymorphs

The energetics of anatase, brookite, and rutile were measured by high temperature oxide melt calorimetry, first by Navrotsky and Kleppa (1967) and later by Mitsuhashi and Kleppa (1979). The results, as well as those of other studies, scatter significantly (see Table 4). Although some other reasons for these discrepancies were proposed by Mitsuhashi and Kleppa (1979), it now seems more likely that much of the difference could result from different particle size and/or water retained in the samples under calorimetric conditions. What is clear is that the enthalpy differences between polymorphs are small enough for complex crossovers in free energy among the three phases to be likely. Also there is no doubt that rutile is the stable polymorph for macrocrystalline materials. 

Enthalpy differences between polymorphic crystal forms of the same substance [18,19] can be directly measured by calorimetry or by the difference in enthalpies of dissolution. A critical survey of these experiments shows that these enthalpy differences are very small, of the order of 0-10 kj mol-1, as expected since they must be a small fraction of the enthalpy of melting. In many cases the measured value is undistinguishable from experimental noise. These enthalpy differences can be estimated by just taking the differences in lattice energy between pairs of crystal phases whose complete structure is known, but the intrinsic uncertainty of such a calculation is also of the same order of magnitude of the property it tries to simulate. 

If the metastable polymorph has contact with a solvent then it reverts to the stable polymorph with the passage of time, but it may be indefinitely stable at room temperature under dry conditions. The enthalpy differences between polymorphs of organic crystals are generally only a few kJ/mol. Polymorphs of pure materials are generally close-packed i.e. without appreciable cavities in their structures. The density differences between polymorphs are usually about 1 % or less. 

In some instances, distinct polymorphic forms can be isolated that do not interconvert when suspended in a solvent system, but that also do not exhibit differences in intrinsic dissolution rates. One such example is enalapril maleate, which exists in two bioequivalent polymorphic forms of equal dissolution rate [139], and therefore of equal free energy. When solution calorimetry was used to study the system, it was found that the enthalpy difference between the two forms was very small. The difference in heats of solution of the two polymorphic forms obtained in methanol was found to be 0.51 kcal/mol, while the analogous difference obtained in acetone was 0.69 kcal/mol. These results obtained in two different solvent systems are probably equal to within experimental error. It may be concluded that the small difference in lattice enthalpies (AH) between the two forms is compensated by an almost equal and opposite small difference in the entropy term (-T AS), so that the difference in free energy (AG) is not sufficient to lead to observable differences in either dissolution rate or equilibrium solubility. The bioequivalence of the two polymorphs of enalapril maleate is therefore easily explained thermodynamically. 

A number of theoretical approaches can account for the fact that an enthalpy of formation of such a binary oxide or a ternary oxide is large and negative. The stability of a ternary oxide relative to the binary constituent oxides is, however, often small, as demonstrated in Table 7.1 using Mg2SiC>4 as an example [1], The enthalpy differences between the three different polymorphs of Mg2Si04 - olivine, /3-phase and spinel - are less than 2% of the enthalpy of formation of the polymorphs. These enthalpy differences are comparable in magnitude to the enthalpy... 

Fig. 5.11 DSC trace (at 20 °C min ) of polymorph YN of 5-XII (see caption of Fig. 5.9 for definition of terms) showing an exothermic conversion (noted as C on the 5x expanded trace) and subsequent melting (noted as R, trace amount) and Y. The area under C gives an estimate of the enthalpy difference between YN and Y. (From Yu et al. 2000, with permission.) The authors attribute the differences between these traces and that in Fig. 5.10 to the difference in heating rate.

On the basis of solubilities in benzene, Horn and Honigman (1974) estimated the relative thermodynamic stability of the five most commonly recognized polymorphs Sisa = y<8enthalpy difference between the two commercially most significant a and forms as 10.75 kJmoP which is consistent with the results of the solvent-mediated a ft... 

The enthalpy difference between two polymorphic crystal structures a and f) is ... 

If a substance undergoes a transformation from one physical stale to another, such as a polymorphic transition, the fusion or sublimation of a solid, or the vaporization of a liquid, the heat adsorbed hy the substance during the transformation is defined as the latent heat of transformation (transition, fusion, sublimation or vaporization). It is equal in the enthalpy change of the process, which is the difference between the enthalpy of the substance in the two states at (he temperature of the transformation. For the purpose of thcrmochemical calculations, i( is usually reported as a molar quantity with die units of calories (or kilocalories) per mule (or gram formula weight). The symbol L or AH. with a subscript i.f (or in), s. and n is commonly used and the value is usually given at the equilibrium temperature of the transformation under atmospheric pressure, or at 25 C. 

Behme and Brooke (1991) have derived an equation for estimating the solubility ratio of two polymorphs at a given temperature using DSC data on the enthalpy of fiMty) nd the melting temperature of each form. The free energy of the transition orthe natural logarithm of the solubility ratio multiplied byRT (Equation 19.6) was estimated by the difference between the solubility estimates for each form calculated by Equation 19.10 ... 

According to Eq. (16), the difference between the differential heats of solution of two polymorphs is a measure of the heat of transition AH between the two forms. Because enthalpy is a state function (Hess s law), this difference must necessarily be independent of the solvent system used. However, conducting calorimetric measurements of the heats of solution of the polymorphs in more than one solvent provides an empirical verification of the assumptions made. For instance, AH values of two losartan polymorphs were found to be 1.72 kcal/mol in water and 1.76 kcal/mol in dimethylformamide [53]. In a similar study with moricizine hydrochloride polymorphs, AH values of 1.0 kcal/mol and 0.9 kcal/mol were obtained from their dissolution in water and dimethylformamide, respectively [54]. These two systems, which show good agreement, can be contrasted with that of enalapril maleate, where was determined to be 0.51 kcal/mol in methanol and 0.69 kcal/mol in acetone [55]. Disagreements of this order (about 30%) suggest that some process, in addition to dissolution, is taking place in one or both solvents. 

The energy differences calculated with the DFT-D method and QM/atom-atom hybrid approach with polarization are of a reasonable magnitude. However, the a form is known to undergo an exothermic solid-state transition to the /0 polymorph at 150 °C, with a transition enthalpy between 1.9 and 2.9kJmol . The T = 0K order of stability provided by aU of the computational methods is opposite to that seen experimentally at the transition temperature. A reversal in the sign of the enthalpy difference can only... 

If it is further assumed that the heat capacity differences between solid and liquid are zero in the temperature range of interest, which is often a fair approximation. Equation 5.5 follows, where AHmeiu (= Huq-Hi) is the melting enthalpy of polymorph I ... 

A number of other methods are available to elucidate the thermodynamic relationship between polymorphs. Solution calorimetry, for example, can be used to determine the enthalpy difference of two forms (H gijj(A ) — H5gijj(A"), since the enthalpy difference is equal to the difference of the heats of dissolution. 

Whereas the enthalpy of formation of A12Si(>5 from the elements is large and negative, the enthalpy of formation from the binary oxides is much less so. Af ox m is furthermore comparable to the enthalpy of transition between the different polymorphs, as shown for A SiOs in Table 1.5 [3], The enthalpy of fusion is also of similar magnitude. 

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