Stability hierarchy between Piracetam forms I, II, and III from experimental pressure–temperature diagrams and topological inferences

For Peer Review Stability hierarchy between piracetam forms I, II, and III from experimental pressure-temperature diagrams and topological inferences Journal: Journal of Pharmaceutical Sciences Manuscript ID: Draft Wiley - Manuscript type: Research Article Date Submitted by the Author: n/a Complete List of Authors: Toscani, Siro; Université de Rennes, Département de Chimie Céolin, Rene; Université Paris Descartes, Faculté de Pharmacie Barrio, Maria; Universitat Politècnica de Catalunya, Física i Enginyeria Nuclear Tamarit, Josep-Lluis; Universitat Politècnica de Catalunya, Física i Enginyeria Nuclear Louer, Daniel; Université de Rennes, Département de Chimie Rietveld, Ivo; Université Paris Descartes, Faculté de Pharmacie Keywords: Calorimetry, Crystal polymorphism, X-ray powder diffractometry, Solid state, Phase diagram, Phase transition, Physical characterization, Physical stability, Preformulation, Thermodynamics John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences For Peer Review 1 Stability Hierarchy between Piracetam Forms I, II, and III from Experimental Pressure-Temperature Diagrams and Topological Inferences Siro Toscani 1,2 , René Céolin 1 , Léon Ter Minassian 1,† , Maria Barrio 3 , Nestor Veglio 3,† , Josep-Lluis Tamarit 3 , Daniel Louër 4 , Ivo B. Rietveld 1,* 1 Laboratoire de Chimie Physique et Chimie Minérale, Faculté de Pharmacie, Université Paris Descartes, 4 avenue de l’observatoire, 75006 Paris, France 2 Département de Chimie – UMR 6226, Faculté des Sciences, Université de Rennes 1, Bâtiment 10B, 263, Avenue du Général Leclerc, 35042 Rennes Cedex, France 3 Grup de Caracterització de Materials (GCM), Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Catalunya, Spain 4 Formerly at Laboratoire de Chimie du Solide et Inorganique Moléculaire, CNRS and Université de Rennes I, France † These two authors have passed away, however they have contributed significantly in the early stages of this study * corresponding author Page 1 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 2 Abstract The trimorphism of the active pharmaceutical ingredient piracetam is a famous case of polymorphism that has been frequently revisited by many researchers. The phase relationships between forms I, II, and III were ambiguous because they seemed to depend on the heating rate of the DSC and on the history of the samples or they have not been observed at all (equilibrium II-III). In the present paper, piezo-thermal analysis and high-pressure differential thermal analysis have been used to elucidate the positions of the different solid-solid and solid-liquid equilibria. The phase diagram, involving the three solid phases, the liquid phase and the vapor phase, has been constructed. It has been shown that form III is the high-pressure, low-temperature form and the stable form at room temperature. Form II is stable under intermediary conditions and form I is the low pressure, high temperature form, which possesses a stable melting point. The present paper demonstrates the strength of the topological approach based on the Clapeyron equation and the alternation rule when combined with high-pressure measurements. Keywords: Calorimetry, Crystal polymorphism, X-ray powder diffractometry, Solid state, Phase diagram, Phase transition, Physical characterization, Physical stability, Preformulation, Thermodynamics Page 2 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 3 1 Introduction 1.1 The stability hierarchy of the three solid forms of piracetam The stability hierarchy of polymorphs is often based on transition temperature measurements by DSC and illustrated by schematic diagrams representing the Gibbs energy as a function of temperature. 1,2 However, it may happen that this approach is incomplete and the trimorphism of piracetam is such an example for which the stability hierarchy remains unsolved. It is known for over a decade that forms II and III both transform endothermically into form I on heating; however, the onset temperatures of the two solid-solid transitions at equilibrium are not known, as they appear to depend on the investigator and on the sample used. Kuhnert-Brandstätter et al. report an energy-temperature diagram in which the transition temperature of form II into form I (T II → I ) is about 348 K when measured by thermomicroscopy or about 377 K when measured by DSC. 3 As the transition temperature of form III into form I (T III → I ) is about 388 K when measured by DSC, which is higher than T II → I , the authors conclude that form III is the more stable phase at room temperature. 3 They also mention that “the relative high stability of mod. II can be explained by the small difference of energetic data between mod. II and mod. III.“ Picciochi et al. concluded that “the results of solution and DSC experiments indicate that the stability hierarchy of forms I, II, and III at 298.15 K … is III > II > I“. 4 Maher et al. reported a ∆G - T diagram exhibiting endothermic transitions II → I and III → I below 363 K and below 368 K respectively, thus leading to the conclusion that form III is the more stable form below 368 K. 5 Céolin et al. reported that the endothermic transitions III → I and II → I occur at 392 K and 399 K respectively, thus inferring that form II is the more stable phase below 392 K. 6 Although a tendency can be observed that form III is the more stable phase at room temperature, the large variation of solid-solid transition temperatures in the different papers warrants an alternative approach to determine the stability hierarchy of the polymorphs without ambiguity. In this paper, it will be demonstrated how the stability hierarchy can be inferred from experimental pressure-temperature diagrams. Page 3 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 4 1.2 Literature data 1.2.1 Crystallographic data Three solid forms of piracetam (C 6 H 10 N 2 O 2 , M = 142.16 g mol -1 ) have been structurally characterized under ambient conditions (Table 1 and Table S1 in the Supporting Information), and two more forms as well as two hydrates have been obtained under pressure: form IV (monoclinic, P2 1 /c), 7 form V (triclinic, P-1), 8 a monohydrate, 7 and a dihydrate 8 . From the data in Table 1, it can be inferred that form III is the denser form, while form I is the less dense one. Table 1. Specific volumes (v) from the literature of the three solid forms of piracetam obtained under ambient conditions Form System, space group v /cm 3 g -1 T /K Ref. I Monoclinic P2 1 /n 0.766 1 0.7518 R.T. 150 9 7 II Triclinic P-1 0.7383 0.7094 R.T. 100 10 11 III Monoclinic P2 1 /n 0.7216 0.7321 0.7245 0.7251 0.7058 R.T. R.T. R.T. 290 100 12 10 13 14 11 1.2.2 Calorimetric data The thermal behavior of piracetam polymorphs has been described by several authors (see Table 2). From the data in Table 2, it can be inferred that ∆ II→I h is consistently smaller than ∆ III→I h by 3 to 6 J g -1 . The melting point of form II, T II→L , was reported to be 415.5 K 3 or 415 K 6 , but this information should be considered cautiously because it is based on the optical observation of a change in morphology 3 or the presence of a very small endothermic peak in a few DSC curves 6 . Page 4 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 5 Table 2. Calorimetric data for piracetam forms I, II, and III from the literature T II-I /K ∆ II→I h /J g -1 T III-I /K ∆ III→I h /J g -1 T I-liq /K ∆ I→liq h /J g -1 T III-liq /K ∆ III→liq h /J g -1 Ref. 382 382 23.71 23.92 393 392 26.45 26.73 - 423 - 179.38 - - - - 15 5 348 or 377 21 390 26.7 426 180.7 413 206 3 399 24 392 28 426 180 412 210 6 382.7 22.72 394.4 26.24 425.9 176.6 or 182.9 4 2 Material and Methods 2.1 Samples Piracetam forms II and III of medicinal grade were obtained from Isochem, France. They were used as such after verification by differential scanning calorimetry and highresolution X-ray powder diffraction. Form I, which is not persistent at RT, was obtained from either of these polymorphs by heating above the transition temperature and quenching. 6 2.2 Differential scanning calorimetry Three DSC analyzers were used to determine the temperatures and heats of transition: a TA Instruments (USA) DSC10 Differential Scanning Calorimeter, a Q100 thermal analyzer from TA-Instruments (USA), and a Mettler-Toledo (Switzerland) 822e thermal analyzer. Indium (T fus = 429.75 K, ∆ fus H = 3.267 kJ·mol -1 ) and zinc (T fus = 692.68 K, Δ fus H = 7.320 kJ·mol -1 ) were used as standards for temperature and enthalpy calibration. Specimens were weighed with microbalances sensitive to 0.01 mg and sealed in aluminum pans. 2.3 High resolution X-ray powder diffraction Two series of non-ambient X-ray powder diffraction data were obtained for the thermal expansion study. A Bragg-Brentano optics was used with a Bruker D5005 powder diffractometer, equipped with an Anton Paar high-temperature oven camera HTK1200. Cu Kα 1,2 X-rays were obtained with a diffracted-beam graphite monochromator. Powders were pressed on a Ni grid sample-holder in order to reduce transparency effects. Samples of forms II Page 5 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 6 and III were heated above 400 K and upon cooling form I was observed. It must be noted that volume changes during phase transformations may affect the quality of the sample surface (the volumes per formula units are 4.6% and 3.8% greater for form I than for forms III and II, respectively). The powder data were collected at selected temperatures under air. Previous diffraction studies showed a short life time of samples of form I obtained from samples of forms II and III quenched at room temperature. 6 To reduce the time required for data collection, the diffraction patterns were scanned over the angular range 10-35° (2θ), with a step size of 0.025° (2θ) and 3 s step -1 . Due to the low counting statistics, the accuracy of the results is moderate. The peak positions were evaluated with the software WinPLOTR. 16 For each pattern the better defined diffraction lines were selected and twenty peak positions were input in the indexing program DICVOL14. 17 The cell parameters are least-squares refined, as well as the zero-point position of the patterns. A Debye-Scherrer optics was used with an INEL powder diffractometer equipped with a cylindrical position-sensitive detector (CPS120) containing 4096 channels (0.029° 2θ angular step) and monochromatic Cu Kα 1 (λ = 1.54061 Aǒ) radiation. For the measurements as a function of temperature, a liquid nitrogen 700 series Cryostream Cooler from Oxford Cryosystems (UK) was used. Slightly ground specimens were introduced in a Lindemann capillary (0.5 mm diameter) rotating perpendicularly to the X-ray beam during the experiments to improve the average over the crystallite orientations. For the temperature-dependent measurements in the range from 100 K up to the melting point, the sample temperature was equilibrated for about 10 min followed by an acquisition time of ca. 1 hour. The heating rate in between data collection was 1.33 K min -1 . The powder data were analyzed with the software FULLPROF using pattern matching. 18 2.4 Specific volume of molten piracetam 10.1223 g of piracetam (weighed with a microbalance sensitive to 0.1 mg) was introduced in a cylindrical silica tube with a flat bottom, whose mean inner diameter (8.152 mm) was obtained with a DIGI-MET ® 3-point alesometer with a precision of 0.003 mm (see below Table S7, Supporting Information). After being sealed under vacuum, the tube was suspended in a XU 75/300 oven from Climats (France) whose inner temperature, controlled at ±1 K, was allowed to vary slowly between successive measurements in the 390 - 470 K range. The height of the melt was measured through Page 6 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 7 the window of the door of the oven with a cathetometer sensitive to 2 μm. 2.5 Sublimation-condensation experiment A specimen of form III was placed at one end of a cylindrical silica tube (29 cm length), which was sealed under vacuum. It was introduced in a horizontal furnace so that the specimen-containing end was maintained at about 414 K and the other end was outside the furnace at room temperature. Crystals grew by condensation of the vapor phase within 6 days (Figure 1). Figure 1. Sublimation-condensation of piracetam crystals in a silica tube subjected to a temperature gradient for 6 days. The initial sample (form III) was placed in the small container on the left-hand side and maintained at 414 K. 2.6 Piezo-thermal analysis (PTA) An in-house constructed piezo-thermal analyzer, whose operating principle and design have been described elsewhere, 19-22 was used to record the thermal effects associated with phase transitions investigated isothermally as a function of pressure. Pressure was gradually decreased under isothermal conditions at a rate of 20 or 50 MPa h -1 from a maximum pressure of 600 MPa down to atmospheric pressure. The measurement range for the temperature of the equipment is 298 to 490 K. The weight of the specimens (about 15 mg) was determined with a microbalance sensitive to 0.01 mg. It is important to point out that PTA allows precise entropy change measurements as a function of pressure because the fluxmeter of the device records enthalpy changes at constant temperature. 2.7 High-pressure differential thermal analysis. The transitions in piracetam have been studied with high-pressure differential thermal analysis (HP-DTA). An in-house constructed HP-DTA, similar to the apparatus previously built by Würflinger 23 with temperature and pressure ranges from 203 to 473 K and 0 to 300 MPa, respectively, was used. Unweighed samples (about 10 to 50 mg) were sealed in cylindrical tin pans and to ensure that in-pan volumes were free from Page 7 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 8 residual air, specimens were mixed with an inert perfluorinated liquid (Galden ® from Bioblock Scientifics, Illkirch, France) before sealing. HP-DTA scans were carried out with a heating rate of 2 K min -1 . In addition, DSC experiments at ordinary pressure (i.e., in standard aluminum pans) with mixtures of piracetam and perfluorinated liquid were carried out to verify that the liquid was inert. 3 Results 3.1 Calorimetric data Two series of experiments were carried out to investigate whether the transition temperatures II → I and III → I depend on the history of the samples (see details in the Supporting Information Tables S2 and S3). It can be seen in Figure 2a that the temperature intervals in which the two solid-solid transitions occur fully overlap at the heating rates of 5 and 10 K min -1 . Hence, aliquots of forms II and III were ground to obtain fine powders with about the same grain size (verified by optical microscopy). Each has been heated at various rates to investigate their dependence on the heating rate and to compare the two samples. From the data, shown in Figure 2b (see also the Supporting Information Table S2-S5), the following has been concluded: (1) the melting temperature of form I is virtually independent of the heating rate, (2) transitions III → I and II → I are heating-rate dependent, and (3) it is inconclusive which of the two forms transforms at the lowest temperature into form I. It is not uncommon to observe solid-solid transitions that are heating rate dependent; 24-26 however a complete overlap in temperature range of two distinct transitions is rare. A statistical analysis of the transition temperatures for the two solid-solid transitions obtained with heating rates of 5 and 10 K min -1 (see Supporting Information Tables S2 to S5) is entirely inconclusive as to which of the two forms, II or III, transforms into form I first: T II→I 390.62 K and T III→I 389.04 K (T II→I : 12 values, min 372.35 K, max 402.82 K, σ = 9.4 K and T III→I : 18 values min 363.33 K, max 406.86 K, σ = 12.5 K). Average values of the transition quantities (compiled in the Tables S2 to S5 in the Supporting Information) have been reported in Table 3. Page 8 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 15 Figure 4. Pressure-temperature phase diagram of the phase transitions of piracetam with experimental data points only. Fusion of form I: squares, fusion of form III: diamonds, III→I: plus sign, II→I PTA : circles, and II→I HP-DTA : triangles. The high-pressure results from the two different methods could be smoothly combined into single expressions with the exception of transition II→I whose HP-DTA curve is steeper and clearly separated from the curve obtained by PTA. Nevertheless, it can be observed that both lines converge at the exact same point (point ‘a’ in Figure 4), where the I-III equilibrium line intersects. It is therefore likely that point ‘a’ is the I-II-III triple point. Because piezothermal analysis is an isothermal method, the temperatures at which transitions are observed are not heating-rate dependent, therefore, it could be suggested that the line ‘a-a 1 ’ in Figure 4 coincides more accurately with the I-II equilibrium line. It should also be noted that, although the whole line ‘b-c’ has been identified as the equilibrium I-III through its enthalpy value, once this line intersects the triple point I-II- III (point ‘a’) it must change its stability ranking. If no hydrostatic pressure is applied (i.e. in a standard DSC) transitions III→I and II→I should ideally be observed at 361.7 K (eq. (7c)) and 373.4 K (eq. (8a)), respectively. Since curve ‘a-a 2 ’ in Figure 4 converges with curves ‘a-a 1 ’ and ‘b-c’ at point ‘a’, it may be tentatively inferred that the influence of the heating rate on the temperature of transition II→I decreases with increasing pressure. Page 15 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 16 It is also clear that the I-III-liq triple point (where the equilibria I-III, I-liq, and III-liq intersect, point ‘b’ in Figure 4) will be located at a lower temperature than that of triple point I-II-liq (Not shown in Figure 4) because equilibrium I-III crosses equilibrium I-liq at a lower temperature than equilibrium I-II does. 4.2 Triple point I-II-III and equilibrium curve II-III Triple point I-II-III is the intersection of the I-II and I-III equilibrium curves (point ‘a’ in Figure 4). Thus, by setting eqs. (7c) and (8a), or (7c) and (8b) equal, the coordinates of triple point I-II-III are found to be T I-II-III ≅ 414 K and P I-II-III ≅ 95.5 MPa. Equilibrium II-III must pass through triple point I-II-III too, even though it has not been observed experimentally. To determine the value of its dP/dT slope, one can make use of the fact that the equalities Δ III→II s = Δ III→I s - Δ II→I s and Δ III→II v = Δ III→I v - Δ II→I v hold at the triple point I-II-III. Using mean ∆s and ∆v values calculated with the values compiled in Table S9 in the Supporting Information, one finds Δ III→II s = 0.078 – 0.068 = 0.010 J K -1 g -1 , and Δ III→II v = 0.0385 - 0.0282 = 0.0103 cm 3 g -1 , which results in a dP/dT II-III of 0.97 MPa K - 1 through the Clapeyron equation: dP/dT = Δs/Δv = Δh/(TΔv) (9) Using the triple point as a coordinate on the II-III equilibrium line one finds for its P (/PMa) – T (/K) relationship: P II-III = -306 + 0.97 T (10) At P II-III = 0 MPa, eq. 10 becomes 316 K. This is the temperature for triple point II-III-vap, as the vapor pressure of molecular substances around room temperature generally ranges between a few pascal up to a few hundred pascal and P in megapascal is approximately zero. The slope of eq. 10 has been calculated using the entropy obtained by PTA. If instead one uses the mean enthalpy changes obtained in a standard DSC (Table 3), as would be the case for the topological approach, the entropy difference between forms II and III will be 0.0077 J K -1 g -1 at the triple point temperature of 414 K. Using the volume change obtained by X-ray diffraction v II -v III = 0.00907 cm 3 g -1 at 414 K (eqs. 2 and 3), the slope dP/dT II-III calculated with eq. 9 is 0.85 MPa K -1 , i.e. near the value in eq. 10 and clearly Page 16 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 17 the shallowest slope of all equilibrium curves. The use of specific volumes obtained at ordinary pressure is validated by a reported density increase of approximately 5 % between 0 and 0.5 GPa for form II. 8 Triple point I- II-III is found at about 0.1 GPa, thus it can be safely assumed that the decrease in the specific volumes of the three solid phases is in the order of 1%. 4.3 Triple points I-III-liq and II-III-liq The coordinates of triple point I-III-liq in the P,T diagram of piracetam can be most easily found by setting the equations for the equilibria I-III and III-liq (6c and 7c) equal. These two equilibria are experimentally better defined than the I-liq equilibrium curve. The coordinates of triple point I-III-liq are found to be T I-III-liq = 443.5 K and P I-III-liq = 153 MPa. For the II-III-liq triple point, the intersection of curves II-III and III-liq, the P,T coordinates are found by setting equations 10 (II-III) and 7c (III-liq) equal: T II-III-liq = 436 K and P II-III-liq = 117 MPa. This is lower in temperature as well as in pressure than the coordinates of triple point I-III-liq. 4.4 II-liq melting curve and triple point I-II-liq The melting point of form II (or triple point II-liq-vap if the melting takes place under saturated vapor pressure) has not yet been determined convincingly, as a clear melting peak in the DSC has never been observed. An estimate can be obtained by determining the slope of the II-liq equilibrium at the triple point II-III-liq (which has just been obtained) and by extrapolating the obtained expression of the II-liq equilibrium downwards to its saturating vapor pressure. Approximating the equilibrium curve by a straight line, dP/dT can be determined following the same procedure as used above for the II-III equilibrium line. At triple point II-III-liq, equalities Δ II→liq s = Δ III→liq s - Δ III→II s and Δ II→liq v = Δ III→liq v - Δ III→II v hold. Because both enthalpy and volume changes for the melting of form III appeared to depend slightly on the temperature and pressure (see data in Table S8), their values have been calculated with equations S8a and S8b (derived from Table S8). Δ III→II s = 0.01 J K -1 g -1 (section 4.2) and Δ III→liq s = 0.48 J K -1 g -1 (eq. S8a evaluated at triple point II-III-liq) give Δ II→liq s = 0.47 J K -1 g -1 , while Δ III→II v = 0.01030 cm 3 g -1 (section 4.2) and Δ III→liq v = 0.10535 cm 3 g -1 (eq. S8b evaluated at triple point II-III-liq) result in Δ II→liq v = 0.09505 cm 3 g -1 . Thus, dP/dT II→liq = 0.47/0.09505 = 4.94 MPa K -1 , and with P II-III-liq = 117 MPa and T II-III-liq = 436 K, the linear equation for curve II-liq becomes: Page 17 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 18 P II-liq = -2040 + 4.94 T (11a) This equation gives T II-liq-vap = 413 K at P = 0 MPa. This value, slightly higher than T III-liq- vap = 412.34 K (see Table 3), is in close agreement with the observation of Kuhnert- Brandstätter et al. by optical microscopy that the melting of crystals assumed to be forms II and III occurred at 415.5 K and 413 K, respectively. 3 The triple point I-II-liq is the intersection of equilibrium curves I-II, I-liq, and II-liq. Setting eqs. 5c and 11a equal for curves I-liq and II-liq, respectively, the triple point coordinates T I-II-liq = 450 K and P I-II-liq = 182 MPa are obtained. A second combination of equations 5a (I-liq) and 8a (I-II) from PTA data only leads to the coordinates T I-II-liq = 459 K and P I-II-liq = 207 MPa, i.e. higher but still comparable to the foregoing results. Finally, using eqs. 8a and 11a (II-liq), T I-II-liq = 451 K and P I-II-liq = 186 MPa are found, which is also near the previous values. Moreover in all cases, the triple point coordinates are higher than those of triple point I-III-liq. 4.5 Construction of the complete P-T diagram A phase diagram involving three crystalline forms possesses ten triple points irrespective of their relative stabilities. 32 Three triple points, I-liq-vap, II-liq-vap, and III- liq-vap, are located on the liq-vap equilibrium, the saturating vapor pressure of the liquid. This equilibrium separates the phase diagram into a high-pressure and a lowpressure part. In the low-pressure domain, three more triple points are located, I-II-vap, I-III-vap, and II-III-vap, all part of the condensed-phase – vapor equilibria. In the highpressure domain, another four triple points, I-II-III, I-II-liq, I-III-liq, and II-III-liq are located. 4.5.1 Regions involving only condensed phases Using the results obtained or inferred from PTA and HP-DTA experiments, the diagram shown in Figure 5 represents the high-pressure domain with the condensed phases. It is important to realize that an equilibrium curve passing through a triple point must change its stability level. In fact, around a triple point, the stability ranking must alternate for each equilibrium line, as can be seen for triple point I-II-III, where line 4 below is stable, line 5 is metastable (broken line), line 6 is stable, continuing clockwise, equilibrium 4 is encountered again, which is now metastable etc. This fact, which is due Page 18 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 19 to the intersecting Gibbs energy surfaces of which the pressure-temperature phase diagram is a projection, is called the alternation rule. 33,34 The alternation of the twophase equilibria around the different triple points can be clearly observed in Figure 5a. Once the stability rank of one of the triple points has been determined, the ranks of the others are defined and therefore also those of the different two-phase equilibria. The result is presented in Figure 5. Figure 5a. Location of the triple points in the P-T region involving only condensed phases. Two-phase equilibrium curves: 1: I-liq, 2: II-liq, 3: III-liq, 4: I-II, 5: I-III, and 6: IIIII. Filled circles: stable triple points, half-filled circles: metastable triple points. Solid lines: stable equilibria, dashed lines: metastable equilibria, dotted lines: super – metastable equilibria. Stable phase regions: dark grey: liquid, white: form I, medium grey: form II, light grey: form III. Figure 5b. Detail of the P-T diagram with the three triple points along the liq-vap curve (= x-axis): solid circle: I-liq-vap, stable, half-filled circle: II-liq-vap, metastable, open Page 19 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 20 circle: III-liq-vap, super-metastable. The diagram shows the same two-phase equilibrium lines as in Figure 5a. Stable phase regions: see Figure 5a. The results presented in Figure 5 and obtained by high-pressure measurements and thermodynamic inferences can be compared to a phase diagram obtained by the topological method based on the Clapeyron equation and the assumption that all twophase equilibria can be represented by straight lines in a P-T diagram. 24,26,35-38 The dP/dT slopes of the P-T melting curves can be calculated using the enthalpy and volume changes at ordinary pressure (P = 0 MPa) from DSC data (Table 3) and specific volume measurements (equations 1 to 4a). This results in dP/dT I-liq = 4.80 MPa K -1 and dP/dT III-liq = 3.89 MPa K -1 and leads to the following equations (P in MPa, T in K, the subscript T in the equation number indicates “topological”): P I-liq = -2037 + 4.80 T (5 T ) and P III-liq = -1604 + 3.89 T (6 T ) for lines I-liq and III-liq, which intersect at T I-III-liq = 477 K and P I-III-liq = 252 MPa. This is higher than the coordinates obtained through the high-pressure data points (section 4.3); however, it leads to the same topological picture of two converging equilibrium lines with increasing pressure. Because the III-I equilibrium curve also passes through triple point I-III-liq its equation (P and T in MPa and K respectively) P III-I = -315 + 1.187 T (7 T ), has been determined using the mean value ∆ III→I h = 27.5 J g -1 (Table 3) and ∆ III→I v = 0.0486 cm 3 g -1 at 477 K (eqs. 1 and 3). Eq. 7 T leads to the transition point I-III at ordinary pressure of T I-III-vap = 266 K at P = 0 MPa. The experimental melting point of the II-liq equilibrium reported in the literature has not convincingly been determined. Nevertheless, if T II  liq = 413 K is used together with Page 20 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 21 ∆ II→liq h = 211.5 J g -1 (the algebraic sum ∆ II→I h + ∆ I→liq h = 24.4 + 187.1) and the volume change on melting, 0.1184 cm 3 g -1 calculated with equations 2 and 4b (at 413 K), dP/dT II- liq becomes 4.32 MPa K -1 . This leads to the following relation for the melting curve of form II (P in MPa and T in K): P II-liq = - 1786 + 4.32 T (11 T ) Although this value of dP/dT II-liq is different from that reported in section 4.4, the topological method leads to the same inequality in the slopes of the melting equilibria (III-liq < II-liq < I-liq). The coordinates of the “topological“ triple points I-II-liq, I-III-liq and II-III-liq calculated with equations 5 T , 6 T and 11 T are compiled in Table 4 and compared with the values obtained or inferred from the high-pressure data. If the literature value of 415 K is used for T II-liq , equation 11 T becomes P II-liq = -1769 + 4.26 T (11 T ’) which leads in combination with equations 5 T and 6 T , to the same inequalities T I-II-liq > T I- III-liq and P I-II-liq > P I-III-liq (Table 4). Clearly the melting point of form II must be located in between those of forms I and III and it can thus be concluded that topological tools to construct a phase diagram can be a powerful means to verify and complete a phase diagram partially obtained by experimental data. Table 4. Triple point coordinates I-II-liq, I-III-liq and II-III-liq determined directly from high-pressure data and by the Clapeyron equation (topological approach) I - II - liq I - III - liq II - III - liq P/MPa T/K P/MPa T/K P/MPa T/K High-pressure data 182 450 153 443 117 436 Topological data with TII-liq = 413 K 501 52 9 2 50 47 7 2 7 4 19 Topological data with TII-liq = 415 K 361 500 250 477 120 443 Page 21 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 22 4.5.2 Triple points involving the vapor phase Lines 4, 5 and 6 cross respectively triple points I-II-vap, I-III-vap and II-III-vap where the vapor pressure is the saturating vapor pressure of the involved condensed phases at the respective triple points. Their pressure coordinates can be approximately determined with the experimental vapor pressure results for form II obtained by Picciochi et al. 4 These vapor pressures in the temperature range of 356.66 K to 374.44 K have been fitted with the Clausius-Clapeyron equation, leading to the following expression (P in pascal, T in kelvin): ln(P II-vap )= 36.7 - (14143/T) r 2 = 0.996 (12) From eq. 12, an average sublimation enthalpy is obtained Δ II →vap H = 14143 R (the gas constant R = 8.3145 J K -1 mol -1 ), which equals 117.6 kJ mol -1 (827.2 J g -1 ). At the temperature of the triple point I-II-vap, 373 K, the vapor pressure of form I equals that of form II. Thus, with Δ I→vap H = Δ II→vap H - Δ II→I H = 117592 – 3462 = 114130 J mol -1 , the vapor pressure of form I is given by (P/Pa, T/K): ln(P I-vap ) = 35.6 - (13727/T) (13) This leads to a vapor pressure of 26 Pa at the melting point of form I or triple point T I-liq-vap = 424.5 K. At the temperature of triple point II-III-vap, 316 K, the vapor pressure of form III equals that of form II. Thus, with Δ III→vap H = Δ II→vap H + Δ II→III H = 117592 + 441 = 118033 J mol -1 , the vapor pressure of form III is given by (P/Pa, T/K): ln(P III-vap ) = 36.9 - (14196/T) (14) The vapor pressure of form I at the temperature (T I-liq = 424.46 K) of triple point I-liq- vap is the same as the vapor pressure of the liquid phase. With Δ liq→vap H = Δ I→vap H - Δ I→liq H = 114130 – 26598 = 87532 J mol -1 , this leads to (P/Pa, T/K): ln(P liq-vap ) = 28.0 - (10528/T) (15) Page 22 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 23 Eq. 15 gives a vapor pressure of 12 Pa at T III-liq = 412.3 K. By setting eqs. (12) and (15) equal, the pressure and temperature of the melting triple point of form II can be found: P = 17.3 Pa and T = 418 K , i.e. greater than T III-liq and close to the values of 413 K (see above) and 415 K. 3 (If T II-liq = 413 K is used, the vapor pressure is found to be 11.7 Pa with equation 12). The last vapor pressure to be determined is the vapor pressure at triple point I-III-vap, whose temperature was found to be 362 K from PTA experiments. By setting equal equations 13 and 14, the temperature of this triple point (at which the vapor pressure of the two forms should be equal) is found to be 360.8 K (i.e. the same value as 362 K, the difference is due to rounding). Using either of equations 13 and 14, the vapor pressure is found to be ≈ 0.1 Pa, slightly greater than the vapor pressure of form II of 0.093 Pa found with equation 12 (at 362 K), thus indicating that, at 362 K, the I-III-vap triple point is metastable with respect to the II-vap equilibrium. The results can be summarized topologically by representing the exponential curves as straight lines; the relative positions of the triple points involving the vapor phase are depicted in Figure 6. Figure 6. Topological representation of the two-phase equilibria involving the vapor phase in the P – T diagram of piracetam: 7 = liquid-vapor equilibrium, 8 = I-vapor, 9 = II- vapor, 10 = III-vapor Page 23 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review 24 5 Concluding remarks Piracetam polymorphism is a case that clearly demonstrates how delicate the determination of the stability hierarchy of polymorphs can be, in particular if only transition temperature inequalities are used. Moreover, when a melting temperature is not known, the case becomes even more difficult. It has been demonstrated how the stability hierarchy of forms I, II, and III was inferred from experimental results that do not rely on the measured solid-solid transition temperatures by DSC; the latter are heating-rate dependent and randomly distributed in relation to each other. High-pressure measurements (isothermal pressure dependent, and isobaric temperature dependent) provided the I-II-III triple point and the two equilibria II-I and III-I with which the position of the third solid-solid equilibrium III-II could be determined. The inequalities between the slopes of the solid-solid equilibrium curves involving forms I, II and III were experimentally obtained and obey the sequence dP/dT I-II > dP/dT I-III > dP/dT II-III . These inequalities can also be inferred from calorimetric and volumetric data obtained at “ordinary“ pressure, thus without explicit high-pressure measurements. Independent of the temperature coordinate, the inequality in the slopes can be calculated using the Clapeyron equation with the enthalpies of transition in J g -1 from Table 3 (∆ II→I h = 24.4, ∆ III→I h = 27.5 and inferred ∆ III→II h = 3.1) and with the differences in the specific volumes in cm 3 g -1 at for instance 100 K, 300 K and 500 K using equations 1, 2 and 3. In each case, one finds T(dP/dT I-II ) > T (dP/dT I-III ) > T (dP/dT II-III ) (see calculations in Table S11 in the supporting information). These three slopes will intersect at a single temperature, the triple point I-II-III, which leads to the unambiguous conclusion that the inequality in the temperatures of triple points I-II-vap, I-III-vap and II-III-vap (i.e. at P = 0 MPa) must be T I-II-vap > T I-III-vap > T II-III-vap , without resorting to the temperatures found by DSC for transitions II→I and III→I. To conclude, the P-T diagram for the phase relationships involving the three solid forms (I, II, and III) of piracetam can be topologically drawn as presented in figure 7a, in which all two-phase equilibria are present, or in figure 7b in which only stable equilibria and subsequent stable phase regions are present. On increasing the pressure, forms I and II both transform to form III, which is the stable phase at low temperature, including room temperature (<316 K), and high pressure. Thus, the conclusion from previous studies Page 24 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 1. Sublimationcondensation of piracetam crystals in a silica tube subjected to a temperature gradient for 6 days. The initial sample (form III) was placed in the small container on the left-hand side and maintained at 414 K. 28x5mm (600 x 600 DPI) Page 31 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 2. Onset temperatures of transitions II→I (solid squares), III→I (open circles) and melting of form I (solid diamonds) (A) of samples obtained by different methods and heated at 5 or 10 K min-1 (see for details Table S2 to S5) and (B) as a function of the heating rate. 221x615mm (600 x 600 DPI) Page 32 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 3. Specific volumes of piracetam forms I (open circles, eq. 1), II (solid diamonds, eq. 2) and III (solid circles, eq. 3) as a function of temperature from high-resolution X-ray powder diffraction and comparison with volumes from single crystal structures (a: ref 8, b and c: ref 11). The experimental (eq. 4a, open diamonds) and calculated (eq. 4b, dashed line, see text) specific volumes of liquid piracetam are also provided. 73x66mm (600 x 600 DPI) Page 33 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 4. Pressuretemperature phase diagram of the phase transitions of piracetam with experimental data points only. Fusion of form I: squares, fusion of form III: diamonds, III→I: plus sign, II→IPTA: circles, and II→IHP-DTA: triangles. 79x79mm (600 x 600 DPI) Page 34 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 5a. Location of the triple points in the P-T region involving only condensed phases. Two-phase equilibrium curves: 1: I-liq, 2: II-liq, 3: III-liq, 4: I-II, 5: I-III, and 6: II-III. Filled circles: stable triple points, half-filled circles: metastable triple points. Solid lines: stable equilibria, dashed lines: metastable equilibria, dotted lines: super – metastable equilibria. Stable phase regions: dark grey: liquid, white: form I, medium grey: form II, light grey: form III. 84x90mm (600 x 600 DPI) Page 35 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 5b. Detail of the P-T diagram with the three triple points along the liq-vap curve (= x-axis): solid circle: I-liq-vap, stable, half-filled circle: II-liq-vap, metastable, open circle: III-liq-vap, super-metastable. The diagram shows the same two-phase equilibrium lines as in Figure 5a. Stable phase regions: see Figure 5a. 48x28mm (600 x 600 DPI) Page 36 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 6. Topological representation of the two-phase equilibria involving the vapor phase in the P – T diagram of piracetam: 7 = liquid-vapor equilibrium, 8 = I-vapor, 9 = II-vapor, 10 = III-vapor 70x62mm (600 x 600 DPI) Page 37 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 7a. Complete topological P - T diagram for the phase relationships involving forms I, II and III of piracetam. Two-phase equilibrium curves: same numbering as in Figures 5 and 6. Stability ranking: same as in figure 6. 68x59mm (600 x 600 DPI) Page 38 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure 7b. Topological representation of the stable P-T diagram. The stable phase regions are shown and delimited by the equilibrium curves connecting the five stable triple points (from ten in total). Form III is stable under ambient conditions. 63x49mm (600 x 600 DPI) Page 39 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Tables Table 1. Specific volumes (v) from the literature of the three solid forms of piracetam obtained under ambient conditions Form System, space group v /cm 3 g -1 T /K Ref. I Monoclinic P2 1 /n 0.7661 0.7518 R.T. 150 9 7 II Triclinic P - 1 0.7383 0.7094 R.T. 100 10 11 III Monoclinic P2 1 /n 0.7216 0.7321 0.7245 0.7251 0.7058 R.T. R.T. R.T. 290 100 12 10 13 14 11 Table 2. Calorimetric data for piracetam forms I, II, and III from the literature T II-I /K ∆ II→I h /J g -1 T III-I /K ∆ III→I h /J g -1 T I-liq /K ∆ I→liq h /J g -1 T III-liq /K ∆ III→liq h /J g -1 Ref. 382 382 23.71 23.92 393 392 26.45 26.73 - 423 - 179.38 - - - - 14 5 348 or 377 21 390 26.7 426 180.7 413 206 3 399 24 392 28 426 180 412 210 6 382.7 22.72 394.4 26.24 425.9 176.6 or 182.9 4 Page 40 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S5 cont. Fourth series 4c - Other measurements with ground forms II and III Form II Heating rate/ K min -1 Mass/ mg T trans /K (onset) 25 2.68 397.59 22 2.32 397.20 50 2.80 395.80 1 3 . 10 383 . 43 Form III 25 3.14 385.71 22 2.71 386.09 1 2.75 387.84 50 2.67 407.89 Series 5. A fifth series of experiments has been performed with a Mettler-Toledo 822e thermal analyzer. A specimen of Form III (2.00 mg) showed a transition to form I (T III→I =393.39 K (onset), ∆ III→I H = 26.80 J g -1 ) on heating at a 10 K min -1 rate followed by melting of form I at T fus,I = 425.13K (onset), ∆ fus,I H = 189.80 J g -1 . A specimen of Form II (9.77 mg) heated at the same rate showed first a transition to form I at T II→I = 402.82 K (onset), ∆ II→I H = 23.96 J g -1 , then a small endothermic peak at 412.46 K (onset), ∆H = 0.40 J g -1 , followed by melting of form I at T fus,I = 425.07 K (onset), ∆ fus,I H = 189.31 J g -1 . On cooling, crystallization started at 360.44 K, ∆ cryst H = −145 J g -1 . On reheating at a 10 K min -1 rate, only the melting of form I was observed (T fus,I = 423.52 K (onset), ∆ fus,I H = 183.50 J g -1 ) thus indicating that the crystallization enthalpy previously observed should be ascribed to form I. Page 47 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S6a. Unit-cell parameters (with standard deviations in parentheses) and specific volumes for piracetam forms I, II, and III as a function of temperature (measurement series 1). The X-ray powder diffraction data of form I were collected upon cooling, except for three of them (the parameters derived from these three datasets are marked with an asterisk). Form II triclinic T/K a/Å b/Å c/Å α /° β /° γ /° Vcell(II)/Å 3 v(II)/cm 3 g - 1 298 6.395(2) 6.613(2) 8.531(4) 79.77(4) 102.38(3) 91.14(3) 346.7(2) 0.73432 313 6.406(2) 6.621(3) 8.554(5) 79.79(4) 102.48(7) 91.21(4) 348.6(3) 0.73835 333 6.400(3) 6.629(2) 8.568(5) 79.68(3) 102.56(8) 91.20(5) 349.0(3) 0.73919 353 6.407(2) 6.637(2) 8.593(2) 79.64(2) 102.70(2) 91.18(3) 350.5(2) 0.74237 373 6.401(2) 6.636(3) 8.612(3) 79.55(3) 102.79(3) 91.20(3) 350.7(3) 0.74280 383 6.404(3) 6.654(3) 8.630(3) 79.56(2) 102.94(2) 91.09(2) 352.4(3) 0.74640 Form I monoclinic (from Form II) T/K a/Å b/Å c/Å β /° Vcell (I)/Å 3 v(I)/ cm 3 g - 1 298 6.759(2) 13.426(4) 8.102(2) 98.95(3) 726.2(4) 0.76906 313 6.763(1) 13.454(2) 8.115(1) 99.06(1) 729.2(2) 0.77224 323 6.769(1) 13.464(2) 8.120(1) 99.12(2) 730.6(2) 0.77372 333 6.772(2) 13.477(3) 8.124(2) 99.10(3) 73 2.1(3) 0.77531 353 6.776(2) 13.504(3) 8.136(2) 99.21(2) 734.8(3) 0.77817 373 6.779(2) 13.526(2) 8.150(1) 99.31(2) 737.5(3) 0.78103 383 6.780(1) 13.543(2) 8.158(1) 99.35(2) 739.1(2) 0.78272 393 6.787(2)* 13.563(2)* 8.172(2)* 99.38(2)* 742.2(3)* 0.78600* 398 6.793(2) 13.564(3) 8.170(2) 99.41(3) 742.6(3) 0.78643 403 6.791(1)* 13.577(2)* 8.180(1)* 99.43(2)* 744.0(2)* 0.78791* 413 6.797(2)* 13.588(2)* 8.190(2)* 99.48(2)* 746.1(3)* 0.79013* Page 48 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Form III monoclinic T/K a/Å b/Å c/Å β /° Vcell (III)/Å 3 v(III)/ cm 3 g - 1 298 6.513(3) 6.426(2) 16.419(3) 92.09(2) 686.7(3) 0.72723 313 6.517(3) 6.435(3) 16.434(4) 92.13(2) 688.9(4) 0 . 72956 333 6.529(2) 6.443(2) 16.475(2) 92.15(2) 692.6(3) 0.73348 353 6.526(2) 6.449(2) 16.505(3) 92.17(2) 694.2(3) 0.73517 373 6.533(2) 6.460(2) 16.518(5) 92.22(2) 696.6(3) 0.73771 Form I monoclinic (from Form III) T/K a/Å b/Å c/Å β /° Vcell (I)/Å 3 v(I)/ cm 3 g - 1 353 6.781(3) 13.502(3) 8.147(2) 99.12(4) 736.5(3) 0.77997 383 6.784(2) 13.539(3) 8.166(2) 99.29(3) 740.2(3) 0.78389 393 6.789(2) 13.562(2) 8.173(1) 99.39(2) 742.4(3) 0.78622 Page 49 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S6b. Lattice parameters and specific volumes of forms I, II and III as a function of temperature (measurement series 2) Form II T/K a/Å b/Å c/Å α/° β/° γ/° v/cm 3 g -1 100 6.358 6.535 8.376 80.26 101.79 90.93 0.71104 130 6.362 6.546 8.393 80.21 101.85 90.96 0 . 71387 160 6.368 6.556 8.415 80.16 101.93 90.98 0.71719 190 6.373 6.566 8.439 80.10 101.99 91.01 0.72061 220 6.380 6.577 8.467 80.02 102.10 91.03 0.72452 250 6.385 6.590 8.496 79.95 102.21 91.03 0 . 72855 280 6.392 6.601 8.525 79.89 102.28 91.06 0.73273 310 6.399 6.616 8.558 79.79 102.46 91.08 0.73729 340 6.405 6.629 8.591 79.72 102.60 91.10 0.74172 355 6.407 6.636 8.606 79.67 102.69 91.13 0 . 74365 370 6.412 6.646 8.624 79.63 102.76 91.14 0.74660 380 6.412 6.650 8.635 79.59 102.83 91.16 0.74770 Form I from Form II 400 6.789 13.556 8.174 90 99.35 90 0 . 78608 410 6.792 13.567 8.181 90 99.37 90 0.78769 420 6.796 13.580 8.188 90 99.40 90 0.78952 Form III 100 6.457 6.387 16.186 90 92.05 90 0 . 70647 130 6.466 6.391 16.217 90 92.05 90 0 . 70925 160 6.474 6.394 16.249 90 92.04 90 0.71187 190 6.483 6.400 16.285 90 92.05 90 0.71510 220 6.490 6.406 16.324 90 92.06 90 0.71826 250 6.498 6.412 16.364 90 92.08 90 0.72157 280 6.505 6.420 16.405 90 92.11 90 0.72505 290 6.505 6.422 16.412 90 92.11 90 0 . 72558 310 6.512 6.430 16.447 90 92.14 90 0.72881 340 6.518 6.441 16.492 90 92.19 90 0.73270 370 6.526 6.452 16.540 90 92.23 90 0 . 73697 380 6.526 6.457 16.552 90 92.26 90 0 . 73806 Form I from Form III 390 6.789 13.556 8.136 90 99.63 90 0.78179 400 6.789 13.562 8.158 90 99.61 90 0 . 78429 410 6.791 13.571 8.172 90 99.62 90 0.78637 420 6.792 13.583 8.176 90 99.664 90 0.78747 Page 50 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S7. Specific volume of molten piracetam as a function of temperature a T/K Top of the melt /cm Bottom of the melt /cm Height of the melt /cm Volume of the melt /cm 3 Specific volume of the melt /cm 3 g -1 427.44 42.684 25.554 17.130 8.9408 0.88328 427.44 42.684 25.530 17.154 8.9533 0.88451 420 . 64 42 . 560 25 . 530 17 . 030 8 . 8886 0 . 87812 420.64 42.550 25.496 17.054 8.9011 0.87936 415.79 42.492 25.486 17.006 8.8761 0.87688 410.93 42.432 25.500 16.932 8.8374 0.87307 430 . 35 42 . 666 25 . 502 17 . 164 8 . 9585 0 . 88503 430.35 42.678 25.500 17.178 8.9658 0.88575 430.35 42.640 25.496 17.144 8.9481 0.88400 435.21 42.710 25.486 17.224 8.9898 0.88812 440.06 42.814 25.518 17.296 9.0274 0.89184 444.92 42.848 25.486 17.362 9.0619 0.89524 432.30 42.678 25.480 17.198 8.9763 0.88678 422 . 58 42 . 539 25 . 480 17 . 059 8 . 9037 0 . 87962 417.73 42.504 25.472 17.032 8.8896 0.87822 415.79 42.482 25.472 17.010 8.8781 0.87709 410 . 93 42 . 426 25 . 474 16 . 952 8 . 8479 0 . 87410 408.02 42.378 25.474 16.904 8.8228 0.87162 404.13 42.331 25.468 16.863 8.8014 0.86951 401.22 42.293 25.468 16.825 8.7816 0.86755 427 . 44 42 . 658 25 . 516 17 . 142 8 . 9470 0 . 88389 418.70 42.538 25.512 17.026 8.8865 0.87791 430.35 42.643 25.498 17.145 8.9486 0.88405 440 . 06 42 . 787 25 . 498 17 . 289 9 . 0238 0 . 89147 444.92 42.827 25.502 17.325 9.0426 0.89333 449.78 42.884 25.502 17.382 9.0723 0.89627 454.63 42.942 25.509 17.433 9.0989 0.89890 454 . 63 42 . 987 25 . 512 17 . 475 9 . 1209 0 . 90107 459.49 43.008 25.530 17.478 9.1224 0.90122 469.20 43.180 25.536 17.644 9.2091 0.90978 410 . 93 42 . 432 25 . 494 16 . 938 8 . 8406 0 . 87338 408.99 42.390 25.500 16.890 8.8155 0.87090 407.05 42.414 25.500 16.914 8.8280 0.87214 405.10 42.404 25.500 16.904 8.8228 0.87162 401.22 42.350 25.498 16.852 8.7957 0.86894 396.36 42.302 25.516 16.786 8.7612 0.86554 396.36 42.296 25.510 16.786 8.7612 0.86554 a Inner diameter of the cylindrical silica tube containing the melt measured at various heights from the bottom of the tube: 8.161, 8.157, 8.154, 8.149, 8 .140, 8.151 mm, Mean diameter: 8.152 mm (SD = 0.0073, SEM = 0.0030), Mass of piracetam: 10.1223 g Page 51 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S8. Piezothermal analysis: experimental melting pressure of forms I and III as a function of temperature, the experimental enthalpy changes, and the specific volume changes a on melting Melting of form I T /K P /MPa ∆ h /J g -1 ∆ s /J g -1 K -1 ∆ v a /cm 3 g -1 424.46 b 0 187.10 b 0.44080 b 0.072700 428.1 20.0 186.41 0.43542 0.071816 432.9 47.0 185.00 0.42769 0.070484 435.4 64.2 187.82 0.43191 0.071145 438.1 81.5 186.41 0.42487 0.070177 Melting of form III 412.34 b 0 211.50 b 0.51293 b 0.11262 412.60 1.4200 - - - 417.90 31.700 - - - 422.50 44.800 213.14 0.50506 0.11089 425.60 59.300 208.22 0.48959 0.10749 428.60 71.600 195.55 0.45653 0.10023 432.50 92.100 211.03 0.48748 0.10703 435.40 103.40 208.92 0.48044 0.10548 438.40 121.80 209.62 0.47763 0.10487 442.30 138.70 210.33 0.47552 0.10440 a Volume changes have been calculated with the Clapeyron equation: ∆v= ∆s dP dT ( ) and dP/dT = 5.96 from equation 5a for the fusion of form I and 4.55 from equation 6a for the fusion of form III b Mean values at ordinary pressure The values for the entropy change of form III in Table S8 lead to the expressions (r 2 = 0.93): ∆s/J g -1 K -1 = 1.09 – 0.00140*T/K = 0.513 – 0.000302*P/MPa (S8a) The value ∆s = 0.45653 J g -1 K -1 has been ignored. The values for the specific volume change of form III in Table S8 lead to the expressions (r 2 = 0.94): ∆v/cm 3 g -1 = 0.233 − 0.000292*T/K = 0.113 – 6.26*10 -5 *P/MPa (S8b) The value Δv = 0.10023 cm 3 g -1 has been ignored. Page 52 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S9. Piezothermal analysis: experimental pressure for transitions III→ →→ →I and II→ →→ →I as a function of temperature, experimental enthalpy changes, and volume changes a on transition T/K P/MPa ∆ h/J g -1 ∆ s/J K -1 g -1 ∆ v a /cm 3 g -1 transition III → →→ → I b 378 . 00 33 . 800 30 . 248 0 . 080020 0 . 042431 378.20 26.400 27.434 0.072538 0.038463 381.90 42.000 31.654 0.082887 0.043951 382.00 35.400 30.248 0.079182 0.041986 382 . 90 47 . 400 30 . 951 0 . 080833 0 . 042862 388.20 51.600 30.248 0.077918 0.041316 393.00 61.700 31.654 0.080546 0.042710 398 . 10 68 . 500 30 . 951 0 . 077747 0 . 041225 402.80 82.300 32.358 0.080332 0.042596 402.90 70.600 30.248 0.075075 0.039809 407.90 83.300 31.654 0.077604 0.041150 408 . 20 89 . 600 32 . 358 0 . 079270 0 . 042033 410.30 89.300 31.654 0.077150 0.040909 413.10 102.00 32.358 0.078329 0.041534 413 . 10 103 . 00 33 . 765 0 . 081735 0 . 043340 415.50 99.700 30.951 0.074491 0.039499 418.00 105.20 33.061 0.079094 0.041940 423.10 117.50 30.951 0.073153 0.038789 432.90 136.80 33.765 0.077997 0.041358 435.40 139.80 32.358 0.074318 0.039407 Transition II → →→ → I c T/K P/MPa ∆ H/J g -1 ∆ s/J K -1 g -1 ∆ v/ cm 3 g -1 375 . 70 7 . 5000 383.00 23.700 26.027 0.067956 0.028214 387.90 34.800 23.917 0.061657 0.025599 390.40 38.400 26.027 0.066668 0.027679 393 . 90 47 . 200 26 . 027 0 . 066075 0 . 027433 397.90 56.900 28.137 0.070715 0.029359 400.30 65.500 28.841 0.072048 0.029913 405 . 90 81 . 400 28 . 841 0 . 071054 0 . 029500 a Calculated with the Clapeyron equation (see Table S8) and dP/dT = 1.89 MPa K -1 from eq. 7a for III→I and dP/dT = 2.41 MPa K -1 from eq. 8a for II→I b Mean ∆s = 0.078 J K -1 g -1 (σ = 0.003), mean ∆v = 0.0385 cm 3 g -1 (σ = 0.0015) c Mean ∆s = 0.068 J K -1 g -1 (σ = 0.004), mean ∆v = 0.0282 cm 3 g -1 (σ = 0.0015) Page 53 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Table S10. Melting pressures and temperatures of form I and form III from HPDTA measurements obtained with 2 K min -1 Melting of form I Melting of form III Transition III → I Transition II → I T/K P/MPa T/K P/MPa T/K P/MPa T/K P/MPa 425.25 0 409.60 0 402.90 62.340 387.90 0 433.15 55.390 415.08 22.462 408.30 75.470 392.00 10.190 428.75 21.180 429.70 84.150 410.00 87.980 397.00 19.900 429 . 45 31 . 080 435 . 00 112 . 92 415 . 10 101 . 97 396 . 70 30 . 060 431 . 45 49 . 530 440 . 30 135 . 40 431 . 70 123 . 31 401 . 60 48 . 110 432.35 63.420 449.60 176.27 438.70 144.68 402.90 62.340 434.45 76.110 453.40 211.29 408.30 75.470 435 . 75 89 . 010 448 . 75 184 . 31 410 . 00 87 . 980 436.55 102.04 453.15 205.36 404.40 59.260 426.95 16.640 453.55 214.90 431.65 60.740 Table S11. Inequalities in the dP/dT slopes of the equilibria I-II, I-III and II-III The temperature of triple point I-II-III does not need to be known, but the slopes of the two-phase equilibria can be calculated at a given temperature, for instance at 100 K, 300 K or 500 K, assuming that the enthalpies and the specific volumes change little with pressure. Specific volumes as a function of temperature: v(I)/cm 3 g -1 = 0.7219 +(0.00016026 * T/K) equation (1) v(II)/cm 3 g -1 = 0.69661 +(0.00013037 * T/K) equation (2) v(III)/cm 3 g -1 = 0.69341 +(0.00011618 * T/K) equation (3) T (K) 100 K 300 K 500 K v I 0.73793 0.76998 0.80203 v II 0.70965 0.73572 0.76179 v III 0.70503 0.72826 0.75150 v I - v III 0.0329 0.04172 0.05053 v I -v II 0.02828 0,03426 0.04024 v II -v III 0.00462 0.00646 0.01029 With ∆ III→I h = 27.53 J/g and ∆ II→I h = 24.42 J/g, ∆ III→II h = 3.11 J/g (from calculation) Slopes (MPa/K) 100 K 300 K 500 K dP/dT(I-III) 8.37 2.20 1.09 dP/dT (I-II) 8.64 2.38 1.21 dP/dT(II-III) 6.73 1.39 0.60 Conclusion: dP/dT I-II > dP/dT I-III > dP/dT II-III Page 54 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Figure S1. Examples of curves obtained with the piezo-thermal analyzer on decreasing the pressure (arrows along the abscissa). Curves A and B: thermal effects associated to transitions II→I and III→I at 388 K, respectively. Curve C: thermal effects recorded using a solvent-free specimen of form III. Curve D: thermal effect recorded using a specimen of form III within which about 1% of the crystallization solvent was kept, resulting in the melting of form III. Page 55 of 55 John Wiley & Sons, Inc. Journal of Pharmaceutical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60