Polymorphism equilibrium

Phosphorus (like C and S) exists in many allotropic modifications which reflect the variety of ways of achieving catenation. At least five crystalline polymorphs are known and there are also several amorphous or vitreous forms (see Fig. 12.3). All forms, however, melt to give the same liquid which consists of symmetrical P4 tetrahedral molecules, P-P 225 pm. The same molecular form exists in the gas phase (P-P 221pm), but at high temperatures (above 800°C) and low pressures P4 is in equilibrium with the diatomic form P=P (189.5 pm). At atmospheric pressure, dissociation of P4 into 2P2 reaches 50% at 1800°C and dissociation of P2 into 2P reaches 50% at 2800°. 

Present in the next sections are the LDA results for equilibrium structure, pressure-induced transitions and electronic properties of various polymorphs, and the comparative analysis of the results for rutile and anatase that were obtained using LDA and GGA forms of the exchange-correlation potential. 

The fluorite phase is found to be extremely high in energy (it falls outside the energy range of Figure 1). Its equilibrium volume at P=0 would be 27.648 A /mol, and calculated equation of state gives B=287 GPa and B"=4.18. These values make fluorite structure the least compressible of all titanium dioxide polymorphs studied here, but still leaves the observation of a phase with B>500 GPa unexplained. ... 

The theorems of Moutier and Robin apply to evaporation, fusion, polymorphic change, or dissociation of systems in com pletely heterogeneous equilibrium. 

At a certain temperature of transition Tt, the two forms will be in equilibrium, AG will be zero, and m = m2. But at other temperatures the two forms will not be in equilibrium, and if ax > a2 then because AG = Z rin (a2/fli), AG will be negative and polymorph 1 will change spontaneously to polymorph 2 and will therefore be considered the less stable form and vice versa. Studies of the two polymorphic forms of methylprednisolone show that significant differences occur in the apparent solubilities of polymorphic forms and that these may be temperature-related. 

It is well known that many compounds are able to change their physical form whilst suspended in solution. For example, a compound of interest may change from one polymorphic form to another, while different crystalline aggregations of the same compound can have different solubility profiles. Impurities can mask the true solubility, and aggregation in solution can also change the thermodynamic equilibrium. Finally, errors which have been published in the literature data may in fact magnify from publication to publication. 

The isolation of crystalline products having mixed polymorphic compositions (often referred to as concomitant polymorphism) remains a topic of interest, even though the phase rule predicts that a system at equilibrium consisting two components (solvent + solute) and three phases (solution + Form I + Form II) is uni variant. Hence, for crystallizations performed at a fixed pressure (typically atmospheric) the system becomes nonvariant and genuine equilibrium can exist at only one temperature. Therefore, concomitant products must be obtained under nonequilibrium conditions. Flexibility in molecular conformation was attributed to the concomitant polymorphs of a spirobicyclic dione [34] and of 3-acetylcoumarin [35],... 

It is important to ascertain whether the solid phase of the solute changes during equilibration to produce a different polymorph or solvate, by analyzing the solid phase (using either chemical or thermal analysis, or x-ray diffraction). If a solid-solid phase transition occurs during equilibration, the measured equilibrium solubility will be that of the new solid phase of the solute. Methods of circumventing this problem have been proposed and evaluated [26]. 

It should be emphasized that the solubility differences between polymorphs or solvates will be maintained only when a less stable form cannot convert to the most stable form. When such conversion can take place, the equilibrium solubility of all forms will approach a common value, namely that of the most stable form at room temperature. 

Table 7 Equilibrium Solubilities, in mg/ml, of Phenylbutazone Polymorphs at Ambient Temperature in Different Solvent Systems...

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. 

From a plot of the saturation states of the silica polymorphs (Fig. 23.7), the fluid s equilibrium temperature with quartz is about 100 °C. Quartz, however, is commonly supersaturated in geothermal waters below about 150 °C and so can give erroneously high equilibrium temperatures when applied in geothermometry (Fournier, 1977). Chalcedony is in equilibrium with the fluid at about 76 °C, a temperature consistent with that suggested by the aluminosilicate minerals. 

To keep our discussion simple for the moment, we suppress the silica polymorphs tridymite and chalcedony. In the calculation results (Fig. 26.1), the silica concentration gradually decreases from its initial value and, as in the previous calculation, approaches equilibrium with quartz after about half a year. 

Fig. 26.7. Variation in silica concentration (top) and saturation indices (log Q/K) of the silica polymorphs (bottom) over the course of the reaction path shown in Figure 26.6. The dashed lines in the top diagram show Si02(aq) concentrations in equilibrium with quartz, cristobalite, and amorphous silica.

The third law of thermodynamics can be verified experimentally. The stable rhombic low-temperature modification of sulfur transforms to monoclinic sulfur at 368.5 K (p = 1 bar). At that temperature, Ttrs, the two polymorphs are in equilibrium and the standard molar Gibbs energies of the two modifications are equal. We therefore have... 

Pressure-induced amorphization of solids has received considerable attention recently in physical and material sciences, although the first reports of the phenomenon appeared in 1963 in the geophysical literature (actually amorphization on reducing the pressure [18]). During isothermal or near isothermal compression, some solids, instead of undergoing an equilibrium transition to a more stable high-pressure polymorph, become amorphous. This is known as pressure-induced amorphization. In some systems the transition is sharp and mimics a first-order phase transition, and a discontinuous drop in the volume of the substance is observed. Occasionally it is strictly not an amorphous phase that is formed, but rather a highly disordered denser nano-crystalline solid. Here we are concerned with the situation where a true amorphous solid is formed. 

FBRM provides a real time chord length distribution within the crystallizer. The technique is excellent for determining the onset of nucleation and in following general growth rate and rate transitions associated with polymorphic transformation, reaching equilibrium and attrition effects. 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

For the purpose of this case study we will select Isopropyl alcohol as the crystallization solvent and assume that the NRTL-SAC solubility curve for Form A has been confirmed as reasonably accurate in the laboratory. If experimental solubility data is measured in IPA then it can be fitted to a more accurate (but non predictive) thermodynamic model such as NRTL or UNIQUAC at this point, taking care with analysis of the solid phase in equilibrium. As the activity coefficient model only relates to species in the liquid phase we can use the same model with each different set of AHm and Tm data to calculate the solubility of the other polymorphs of Cimetidine, as shown in Figure 21. True polymorphs only differ from each other in the solid phase and are otherwise chemically identical. 

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