Thermodynamics of polymorphs
The relative thermodynamic stability of two polymorphs is determined by the difference in their Gibbs free energy (ΔG=G2-G1) which also can be represented by the following equation
ΔG = ΔH – TΔS
The enthalpy term, ΔH, corresponds to the lattice energy difference, while the entropy term derives from the difference in lattice vibrations and disorder between the two polymorphs. Entropy of a perfect crystalline solid is zero at the absolute zero temperature (T=0) and the Gibbs free energy is therefore equal to the entropy at 0 Kelvin. The TS term increases more rapidly with increasing temperature than the H term and the Gibbs free energy consequently decreases with increase temperature.
Gibbs free energy as a function temperature varies between polymorphs (and solids in general). The polymorph with lowest Gibbs free is the thermodynamically more stable. If the curves intersect at a certain temperature below the melting points of the polymorphs the system is called enantiotropic. In this case one of the polymorphs is more stable below this transition temperature and the other above this temperature. The phenomenological manifestation of enantiotropy is that there can be a reversible transition from one phase (polymorph) to the other without going through the gas or liquid phase.
A system is called monotropic if one polymorph is thermodynamically mote stable at all temperatures the melting point.
Determination and knowledge of the relative thermodynamic stability of polymorphs, i.e. enantiotropic or monotropic nature of the relationship between polymorphs is very important from a pharmaceutical perspective and if the polymorphs are enantiotropically related the transition temperature should be determined.
A set of relative simple estimates (the name rule is misleading) of the relative stability of polymorphs has been developed by Tammann (1926) and expanded by Burger and Ramberger (1979) and latter by Yu (1995) and Grunenberg et al (1996).
- The heat of transition rule (HTR)
- The heat of fusion rule (HFR)
- Entropy of fusion rule
- Heat capacity rule
- Density rule
- Infrared rule
The heat of transition rule is use full and is according to my experience often a good prediction of the true relationship between polymorphs. HTR, propose that if an endothermic polymorphic conversion is observed at a temperature (T), then a transition temperature (Tt) exists at or below the observed temperature. If an exothermic conversion is observed at T, then the transition temperature does not exist below T (which implies monotropic or enantiotropic relationship with Tt > T). The HFR propose that if the higher melting polymorph has a lower heat of fusion, than the two forms are enatiotropes, and if the higher melting polymorph has the higher heat of fusion, than the two forms are monotopes. The HFR should be used with care as it is often difficult to separate and quantitatively determine the heat of fusion of the individual polymorphs.
The other “rules” can be found in the original papers and in various reviews articles and books. The Density rule and Infrared rule have many exceptions.
The overall conclusion is that these “rules” only should be used as guidance and the thermodynamic relationship between polymorphs must be determined experimentally.