Thermodynamics in Polymorphism Research

1. Thermodynamics in Polymorphism Research  Why consider thermodynamic relationships in polymorphism research?  Why and how to draw an Energy-Temperature Diagram?  Examples

2. Many analytical tools are used in Polymorphism Research:  Thermomicroscopy  Differential Scanning Calorimetry  Thermogravimetry  Microcalorimetry / Solution Calorimetry  (N)IR and Raman Spectroscopy  X-Ray diffraction methods  Solid-state NMR spectroscopy  Pycnometry  ..........

3. ...and more or less all the results derived by the methods are used to answer the following questions (I):  How many polymorphs have been crystallized from a given substance?  Which crystal form is thermodynamically stable at „ambient conditions“? And how can it be obtained?

4. ...and or more less all the results derived by the methods are used to answer the following questions (II):  If a substance is polymorphic, are the modifications enantiotropically or monotropically related? And in the case of enantiotropism: Where is the thermodynamic transition point?  Which crystal form(s) is (are) thermodynamically metastable, but durable for a significant amount of time to be considered for a (pharmaceutical) product due to special properties?

5. To avoid erroneous interpretation, it is useful to treat the data as part of a closed system. Is a tool available, that gives the possibility to take the results from different methods and put it in a closed system? Yes, it is the semiquantitative graphical solution of the Gibbs-Helmholtz-Equation: DH = DG -TDS H = H(T) E cp G0 = H0 TS -S G = G(T) T

6. Energy/temperature diagram Fundamental tool for the solution of complex polymorphic systems. Graphical semiquantitative solution of the Gibbs-Helmholtz Equation for polymorphic systems.

7. References  Buerger, M.J. Crystallographic aspects of phase transformations. In Smoluchowski, R., Mayer, J.E. and Weyl, W.A. (eds.), Phase transformation in Solids, John Wiley and Sons, New York, 1951, pp. 183-211.  Burger, A. and Ramberger, R. On the polymorphism of pharmaceuticals and other molecular crystals. I: Theory of Thermodynamic Rules. Mikrochim. Acta II (1979) 259-271.  Burger, A. and Ramberger, R. On the polymorhism of pharmaceuticals and other molecular crystals. II: Applicability of Rhermodynamic Rules. Mikrochim. Acta II (1979) 273-316.  Grunenberg, A., Henck, J.-O. and Siesler, H.W. Theroretical and practical application of energy/temperature diagrams as an instrument in preformulation studies of polymorphic drug substances. Int.J.Pharm. 129(1996)147-158.

8. Essential thermodynamic background (Approximations)Heat Capacity and Enthalpy  Each crystal form has its own heat capacity, which is a function of the enthalpy H and the temperature T. Cp = (¶H / ¶T)p Solids show low compressibility. The heat capacity of solids at constant volume and constant pressure are about the same. The heat capacity increases with increasing temperature since T and H are always positiv. The H isobars of two modifications are parallel. They do not intersect. Their distance (DtrH, transition enthalpy) is directly measurable (DSC). There are no lattice vibrations of ideal crystals at absolute zero. The heat capacity at 0 K is zero.

9. mHI D mHII D trHII-I D tr, II®I Enthalpy isobarsTwo modifications and the melt Enthalpy(H) liq Energy HII HI cp Temperature [K] 0 mp I mp II

10. Essential thermodynamic background (Approximations)Gibbs Free Energy and Entropy  At 0 Kelvin G = H.  The Entropy is the partial derivative of the Free Enthalpy and the Temperature. (¶G / ¶T)p = -S.  Since S is always positive, G decreases with increasing temperature.  The G Isobars of two crystal forms converge and never intersect twice.  The relationship between the Enthalpy H and the Free Enthalpy G is defined by the Gibbs-Helmholtz Equation.

11. Free Enthalpy isobarsTwo polymorphs and the melt Energy Free Energy (G) -S DG G II G I Temperature in K 0 K

12. Free Enthalpy isobars The relative position of the G-Isobars of different modifications can be determined by solubility experiments. Essential thermodynamic background mII: saturation solubility of mod.II in a given solvent mI: saturation solubility of mod.I in the same (as for mod.II) given solvent

13. Energy Free Energy (G) -S DG G II G I Temperature in K 0 K Free Enthalpy isobarsTwo polymorphs and the melt Zoom In

14. DGT2 DGT3 Free Enthalpy isobarsTwo polymorphs and the melt Energy Free Energy (G) DGT1 G II G I Temperature in K 0 K

15. Enantiotropism vs Monotropism Phase transitions of solids can be thermodynamically reversible or irreversible. Modifications, which transform reversibly without passing the liquid or gaseous state are calld enantiotropic polymorphs. If the modifications are not interconvertable under these conditions, the system is monotropic.

16. mHI D mHII D H I trHII-I H D II tr, II®I Enantiotropism Energy Mod.I  Mod.II G II G I 0 Temperature [K] mp I mp II

17. Mod.II  Mod.I endo heating cooling Mod.II  Mod.I

18. mHII D mHI H D II trHII-I H D I Monotropism Energy Mod.II  Mod.I G II G I 0 Temperature [K] mp I mp II

19. Phase diagram vs Energy/Temperature diagram Henck, J.-O., Kuhnert-Brandstätter, M. J.Pharm.Sci. 88 (1999) 103-108

20. Burger-Ramberger RulesHeat-of-fusion rule • If the higher melting form has the lower heat of fusion the two forms are most probable enantiotropic • otherwise they are monotrotropic Power Compensation DSC

21. Burger-Ramberger RulesHeat-of-transition rule • If an endothermal transition is observed at some temperature it may be assumed that there is a transition point below it, i.e. the two forms are related enantiotropically.

22. Burger-Ramberger RulesHeat-of-transition rule • If an exothermal transition is observed at a given temperature it may be assumed that there is no transition point below it, i.e. the two forms are related monotropically, (or the transition temperature is higher).

23. Burger-Ramberger RulesDensity and Infrared rule • If a modification has a lower density than another one, then it may be assumed that at absolute zero this crystal form is less stable. • If the first absorption band in the infrared spectrum of a hydrogen-bonded molecular crystal is higher for a modification than for the other one, that form may be assumed to have the larger entropy.

24. Enantiotropism vs Monotropism

25. Example: Polymorphism of Nimodipine1 Two modifications, which can be obtained in macroscale at room temperature showing the following data: Mod.I Mod.II Melting point [°C] 124 116 Heat of fusion [kJ mol-1] 39 46 True density [g cm-3] 1.27 1.30 Calculated density Xray [g cm-3] 1.271 1.303 Solubility in water at 25°C [mg/100mL] 0.036 0.018 1Grunenberg, A., Keil, B. and Henck, J.-O. Int. J. Pharmaceutics 118 (1995) 11-21.

26. mHI D mHII D H I trHII-I H D II tr, II®I Nimodipine Energy G II ? G I 0 Temperature [K] mp I mp II

27. Where is the thermodynamic transition point? Using the following equation Tp can be estimated: and k = 0.005 Yu, L., J. Pharm. Sci. 84 (1995) 966-974; Henck, J.-O., Ph.D.Thesis 1996

28. Example: Polymorphism of Nimodipine1 Mod.I Mod.II Melting point [°C] 124 116 Heat of fusion [kJ mol-1] 39 46 Tpcalc. = 82 °C Tpexp. = 88 ± 8 °C (by slurry conversion experiments)

29. mHI D mHII D H I trHII-I H D II tr, II®I Nimodipine Energy G II ~88°C G I 0 Temperature [K] mp I mp II

30. 2 Modifications Mod.I - Mod.II E or M E : Enantiotropism M : Monotropism 3 Modifications Mod.I - Mod.II E E E M M E M M Mod.II - Mod.III E E M M E M E M Mod.I - Mod.III E M M M E E M E G-isobar intersections3 2 1 0 2 2 1 1 ?

31. H H I liq H III tr, II®I Mod.I - Mod.II M Mod.II - Mod.III M Mod.I - Mod.III E Energy X G III G G II I 0 Temperature [K] mp I mp II

32. H H I liq H III tr, II®I Mod.I - Mod.II M Mod.II - Mod.III M Mod.I - Mod.III E Energy X G III G G II I 0 Temperature [K] mp I mp II

33. H H I liq H III tr, II®I Mod.I - Mod.II M Mod.II - Mod.III M Mod.I - Mod.III E Energy X G III G G II I 0 Temperature [K] mp I mp II

34. Example: Tedisamil Dihydrochloride heating II I III  cooling I III heating II I Henck, J.-O., Finner, E. and Burger, A., J. Pharm. Sci.89 (2000) 1151-1159.

35. Mod.I - Mod.II E Mod.I - Mod.III E Mod.II - Mod.III E Henck, J.-O., Finner, E. and Burger, A., J. Pharm. Sci.89 (2000) 1151-1159. Temperature [°C]

36. Example: Sulfamethoxidiazin Burger, A., Ramberger, R. and Schulte, K., Analyse des polymorphen Systems von Sulfamethoxidiazin. Arch. Pharm.313 (1980) 1020-1028.

37. Sulfamethoxidiazin Burger, A., Ramberger, R. and Schulte, K., Analyse des polymorphen Systems von Sulfamethoxidiazin. Arch. Pharm.313 (1980) 1020-1028.

38. Summary (I) What is necessary to draw an E/T diagram?: - use arbitrary units for the E and T axis. - order of relative stability of the polymorphs at 0 Kelvin. - order of relative stability of the polymorphs at higher temperature (melting point).

39. Summary (II) The E/T diagram is a useful tool to interpret data obtained in polymorphism research, because: - it is a graphical solution, that presents the thermodynamic relationships of polymorphs in one picture and helps to avoid erroneous interpretations. - it is a helpful tool to design experiments to obtain a desired polymorph.

40. Acknowledgements Prof. Joel Bernstein Dr. Alfons Grunenberg Dr. Arkady Ellern Prof. Jack Dunitz Prof. Roland Boese Humboldt Foundation, Bonn (Germany)