PHENOMENOLOGY OF CRYSTALLINE POLYMORPHISM: OVERALL MONOTROPIC BEHAVIOR OF THE CARDIOTONIC AGENT FK664 FORMS A AND B Ivo B. Rietveld*, René Céolin Caractérisation des Matériaux Moléculaires à Activité Thérapeutique (CAMMAT) Faculté de pharmacie, 4 avenue de l’observatoire, 75006 Paris *Corresponding author: Ivo B. Rietveld, e-‐mail: [email protected]
, tel. +33 1 53739675
ABSTRACT The stability hierarchy of crystalline polymorphs is often determined on the basis of limited calorimetric data even when other useful data such as specific volumes may be available. This may lead to wrong conclusions or an incomplete picture of the stability behavior of a compound and may be problematic in many applications, for example for pharmaceuticals. Therefore, the topological approach has been applied to the pharmaceutical FK664, which exhibits crystalline dimorphism, using heat and work related thermodynamic data to obtain a pressure – temperature phase diagram and elucidate its phase behavior. The approach leads to the conclusion that FK664 is overall monotropic with form B the most stable solid phase. In other words form A does not have a stable domain for any pressure-‐temperature coordinate. The case of FK664 demonstrates that making use of the full available thermodynamic data set in combination with statistical information of the phase behavior of small organic molecules and classical thermodynamics leads to a sound evaluation of the stability hierarchy for crystalline dimorphism. Keywords: monotropy, phase diagram, topological pressure-‐temperature diagram, crystalline polymorphism, phase relationship, stability, (E)-‐6-‐(3,4-‐dimethoxyphenyl)-‐1-‐ethyl-‐4-‐mesitylimino-‐3-‐methyl-‐3,4-‐dihydro-‐2(1H)-‐ pyrimidinone
INTRODUCTION In 1901, Bakhuis-‐Roozeboom published four pressure-‐temperature (P-‐T) phase diagrams representing the four possible topological layouts of the four phases (two solids, the liquid, and the vapor) present in the system in the case of crystalline dimorphism. These diagrams, whose original drawings have been reproduced recently , demonstrate the effect of pressure and make it possible to evaluate the phase behavior of two polymorphs over the entire pressure – temperature domain. Although Nagasako evoked the fact that the influence of pressure on phase equilibria needs to be taken into account , at present it is not the case when describing the stability hierarchy of pharmaceutical polymorphs. This may be due to the difficulty to measure the pressure (the vapor pressure and the hydrostatic pressure); However, this is not necessary, because only the inequality between the vapor pressures is required to establish a stability hierarchy between polymorphs, as shown in Figure 1. In Figure 1, the lines represent the pressure and temperature coordinates, where a condensed phase (form I, form II, or the melt (L = liquid)) is in equilibrium with its vapor phase. Point 1 is the intersection of the curves representing the equilibria between form I and the vapor and between the melt and the vapor. Thus, at the intersection, form I, the liquid, and the vapor (which has the same pressure for both phases at the temperature of intersection) are in equilibrium defining the intersection as a triple point with its coordinates being temperature T1 and the vapor pressure at the intersection. Thus, if a melting point is obtained while the condensed phases are in equilibrium with their vapor phase, which is the case under ordinary conditions1, it is equivalent to 1 “Ordinary conditions” signifies that the system is in thermal equilibrium with its surroundings and that a vapor phase is present which is in equilibrium with the condensed phase, thus ordinary in the sense of
the triple point and the temperature of triple point 1 is equivalent to the melting temperature T1. The same is true for triple point 2, at which form II, the liquid and the vapor are in equilibrium and for which T2 is the melting point under ordinary conditions. Finally, point 3 is the triple point where form I, form II, and the vapor are in equilibrium, but it also represents the solid-‐solid transition temperature T3 between forms I and II under ordinary conditions.
Figure 1. The two possible cases for the inequality in the vapor pressures of two polymorphs (a) enantiotropy (b) monotropy (see text for an explanation of the figures) In Figure 1a, below T3, form II is more stable than form I. The most stable form has the lowest vapor pressure for a given temperature, which is clearly the case for form II below T3. Form I is the more stable form from temperature T3 up to temperature T1, above which the liquid is the more stable phase. Such stability behavior between two solid phases has been named enantiotropy by Lehmann [4, 5]. In contrast, in Figure 1b, form I has the lowest vapor pressure and is thus the more stable phase for the entire temperature domain below the melting temperature how a drug may be stored. It implies that the thermodynamic pressure of the system is equivalent to its partial vapor pressure, even if an inert gas is present. 2
T1, while form II is less stable. Lehmann called such behavior monotropy, after he observed cases for which form II transformed non-‐reversibly into form I on heating at ordinary pressure [4, 5]. “Non-‐ reversibly” indicates that form I does not transform back into form II on cooling. Lehmann neglected the influence of pressure on phase changes; however, by including pressure, Bakhuis-‐Roozeboom arrived at the general description of the four possible P-‐T diagrams for crystalline dimorphism. The dimorphism of FK664 is a further example in a series of theoretical and experimental studies on polymorphism through topological pressure – temperature phase diagrams [2, 6-‐23]. FK664 is a 2-‐ pyrimidone derivative (Figure 2) developed in 1988-‐ 1989 by Fujisawa Pharmaceutical Company as a drug against heart failure [24, 25]. The stability hierarchy between its two polymorphs is re-‐examined using heat-‐ and work-‐related data at ordinary pressure.
comparing the calculated densities (Dcal). The authors also calculated with results from DSC measurements that the heat of transition (ΔA→BH) of form A to form B is endothermic (+3.6 kJ mol-‐1 or +8.834 J g-‐1) and they concluded that “Form B is at least more stable than form A”, although no interconversion between the two crystal forms had experimentally been observed. Table 1. Crystallographic and calorimetric data at T = 295 K for forms A and B of FK664, C24H29N3O3 with M = 407.505 g mol-‐1 a Form A Form B Crystal system, Monoclinic, Monoclinic, space group P21/c P21/c a/ Å 13.504 8.067 b/ Å 6.733 15.128 c/ Å 24.910 18.657 96.55 102.34 β /° Vunit-‐cell/ Å3 2250.09 2224.252 Z 4 4 Dcal/g cm-‐3 1.20293 1.216909 vspec/ cm3 g-‐1 0.83130 0.821754 391.0 413.8 Ti→ L/ K Δi→ LH/ kJ mol-‐1 (J g-‐1) 33.2 (81.5) 36.8 (90.3) a
Crystallographic data based on values reported in the CIF files (Cambridge Structural Database codes VUSZEO and VUSZEO01), Dcal: calculated density from CIF data, vspec: specific volume from CIF data, Ti→L: melting point of phase i (= form A or form B), Δi→LH: heat of fusion of phase i.
RESULTS AND DISCUSSION
Figure 2. Chemical formula of FK664, (E)-6-(3,4dimethoxyphenyl)-1-ethyl-4-mesitylimino-3-methyl-3,4dihydro-2(1H)-pyrimidinone.
LITERATURE Structural studies and physicochemical characterization of the two forms of FK664 have been published in 1991 by Miyamae et al.  and the results are compiled in Table 1. The crystal structures were solved at the same temperature (295 K) and the authors reported that form B is the denser one after
HEAT-‐RELATED DATA Form B exhibits the higher melting temperature and the higher melting enthalpy, which differs by +22.8 K and +3.6 kJ mol-‐1 (8.8 J g-‐1) from those of form A, respectively. Neglecting the difference in the specific heats of the two forms, the enthalpy of transition from A to B is calculated to be −3.6 kJ mol-‐1 (−8.8 J g-‐ 1 ), according to the thermodynamic cycle shown in Figure 3. Thus ΔA→BH is negative, opposite to the conclusion reported by Miyamae et al , and this implies that the difference in Gibbs energy between the forms will decrease with increasing temperature and that form B should eventually spontaneously 3
transform into form A on heating following the Le Chatelier principle.
potential) of the condensed phase. The vapor pressure as a function of the temperature may be approximated by the following expression: Ln (Pβ) = −Δβ→vaporH /(RT) + Bβ→vapor
Figure 3. Hess’ law for the heats of transition between form A, form B, and the liquid; the algebraic sum of the heats of transition over the cycle must be equal to 0. It follows that the heat of transition from A to B must be equal to −3.6 kJ mol-1. Although the A-‐B equilibrium has never been observed, its temperature (TB→A) at ordinary pressure can be calculated using the procedure described by Yu  leading to the following equation:
TB→ A =
Δ A→ L H − Δ B→ L H Δ A→ L H Δ B→L H − TA→ L TB→ L
Using the values provided in Table 1, TB→A = 895.22 K is found, a temperature for the B→A transition far higher than the temperatures of fusion for both forms. As explained above (Figure 1), the transition temperature corresponds to the invariant equilibrium of the two solid forms and their vapor also called the triple point A-‐B-‐vapor (in equilibrium, the vapor pressures of the two solid phases must be equal). It implies that the A-‐B-‐vapor triple point is metastable, because it is located in a domain of the P-‐T diagram where the solids are not stable anymore, as the highest melting form melts at 413.8 K. As the triple point lies on the condensed phase-‐vapor equilibrium curves, the vapor pressure of the triple point can be calculated. The vapor pressure is a direct reflection of the Gibbs energy of the vapor phase. Because the vapor phase is in equilibrium with the condensed phase (in other words the chemical potentials are equal), the saturated vapor pressure is also a reflection of the Gibbs energy (or chemical
in which Pβ is the vapor pressure (in Pa) of the condensed phase β (forms A, B, or the liquid) in equilibrium with the vapor, Δ β→vaporH is the enthalpy (in J mol-‐1) of either vaporization (liquid) or sublimation (solid), R is the gas constant (8.31446 J mol-‐1K-‐1), T is the absolute temperature and Bβ→vapor is a constant. The boiling point at a vapor pressure of 760 Torr (1.0133×105 Pa), identical to the atmospheric pressure, and the enthalpy of evaporation of organic compounds was evaluated with ACD/Labs software . Using the boiling point, TL→vapor, of 558.5°C (831.65 K) and the vaporization enthalpy, ΔL→vaporH, of 84060 J mol-‐1, the constant BL→vapor for FK664 can be calculated and is found to be 23.68. Now, the vapor pressure (in Pa) of liquid FK664 as a function of temperature (in K) can be expressed by: Ln(PL) = −84060/(RT) + 23.68
Similar expressions for the vapor pressure of the two solid phases can be derived from eq. 3. At the melting temperature of form B, TB→L = 413.8 K, the vapor pressure of the liquid can be calculated with eq. 3 (PL = 0.4726 Pa) because the vapor pressure of form B equals that of the liquid at the melting point under ordinary pressure. The difference in enthalpy between form B and the liquid is the melting enthalpy. Therefore, the sublimation enthalpy of form B is given by ΔB→vaporH = ΔL→vaporH + ΔB→LH leading to 120860 J mol-‐1. As the vapor pressure is known at TB→L, constant BB→vapor can be calculated leading to 34.38. The vapor pressure (in Pa) of form B can now be expressed by: Ln(PB) = −120860/(RT) + 34.38
The dependence of the vapor pressure of form A with the temperature can be obtained in the same way: (i) 4
ΔB→vaporH is obtained by adding ΔL→vaporH and ΔB→LH and equals 117260 J mol-‐1. (ii) Because form A and the liquid are both in equilibrium with the vapor phase at the triple point A-‐L-‐vapor (i.e. at TA→L= 391 K), the pressure obtained with eq. 3 at this temperature (equal to 0.1137 Pa) can be used to calculate constant BA→vapor leading to 33.90. The vapor pressure (in Pa) of form A can now be given by: Ln(PA) = −117260/(RT) + 33.90
The two curves defined by eqs. 4 and 5 intersect at the temperature where both solid phases have the same vapor pressure, the triple point A-‐B-‐vapor. The temperature can be calculated by setting the pressure equal in eqs. 4 and 5 leading to 895.22 K. Because this approach is identical to eq. 1, any difference in the obtained transition temperature is due to rounding errors. In addition to the metastable transition temperature, this approach allows the calculation of the vapor pressures of the condensed phases as they are given by eqs. 3-‐5. At the A-‐B-‐vapor triple point the vapor pressure of the solid phases equals 76 MPa. Instead, at the same temperature, the liquid has a vapor pressure of 0.24 MPa, which is far smaller. It indicates that the liquid is more stable at this temperature and that the A-‐B-‐vapor triple point is metastable because it is located in a P-‐T domain, where the liquid has a lower vapor pressure. Nonetheless, if one were to work at 1 atm pressure, and the liquid would be present in a container freely expanding against the atmosphere, even the liquid would completely evaporate as its 0.24 MPa vapor pressure would only be counterbalanced by a mere 0.1 MPa external atmospheric pressure. Following the discussion of the two scenarios in Figure 1, the dimorphism of FK664 is clearly reflected by the case illustrated in Figure 1b. In principle, form B would transform into A on heating following the Le Chatelier principle; however this transition point is metastable. Nonetheless, using only heat-‐related data limits the stability analysis to the temperature domain and no inferences can be drawn concerning the phase behavior under increasing pressure. It may
be interesting to know whether the metastable phase at ordinary pressure (here FK664 form A) becomes stable on increasing the pressure. This is carried out using work-‐related data, as shown in the following section.
WORK-‐RELATED DATA The finite change ΔU of the internal energy of a one-‐ component system that undergoes a change of state is the algebraic sum of the heat received, Q, and the work done, W, during the change. This is written as ΔU = Q + W and, according to Gibbs, leads to the fundamental equation of (classical) thermodynamics ΔU = TΔS – PΔV. P and T are the pressure and the temperature of the system at which the change occurs, which is accompanied by changes in entropy (ΔS) and volume (ΔV). The first principle states that ΔS = Sfinal − Sinitial and ΔV = Vfinal − Vinitial, and the second principle states that ΔS = Q/T if the change occurs “reversibly“, i.e. at an infinitely slow rate. In particular, because the parameters P and T are more accessible, and because they are intensive, the fundamental equation of classical thermodynamics is rewritten through the Legendre transformation in the form of a variation of the Gibbs energy ΔG = −SΔT + VΔP. The pressure – temperature phase diagrams are a direct reflection, or one should say projection, of the use of the Gibbs energy to determine phase stability. Unfortunately, work-‐related data, i.e. P and v, are virtually never taken into account, because the pressure is seldom measured and work is assumed to be negligibly small, thus thermodynamics is often reduced to calorimetry. However, it is nowadays fairly easy to determine the unit cell volumes of crystal structures. In addition, in the Helmholtz function, which is also obtained by the Legendre transformation of the internal energy, F = U − TS and ΔF = −SΔT − PΔV, V and T are the independent variables instead of S and V (or P and T in the case of the Gibbs energy). v-‐T diagrams can be of practical interest in the study of polymorphism, because temperature and specific volumes are more easily obtained than pressure. v-‐T diagrams are projections of the function F(v,T) on the v-‐T plane and 5
the specific volume v is used, rendering an extensive variable intensive by dividing by the quantity of material present in the system. For a relevant use of the variables T and v, the system needs to be in equilibrium with its vapor phase. This implies that thermal expansion measurements have to be carried out under saturating vapor pressure. If the solid for which the volume is being measured is not placed in a closed container with a fixed volume, the surroundings that have to be saturated by the vapor pressure are virtually infinite and the saturation condition does not seem to be fulfilled. Nevertheless, because the saturating vapor pressure of pharmaceuticals is in most cases very small, the assumption that vapor pressure locally saturates the surroundings of the specimen is generally correct. In the case of FK664, the specific volumes obtained from the two crystal structures solved at the same temperature have enough precision to judge that vA is greater than vB. From the inequality ΔA→Bv = vB – vA < 0, it can be inferred that form A should eventually transform into form B on increasing the pressure, following the Le Chatelier principle. In combination with the previous conclusion involving the calorimetric data, it can be inferred that the slope dP/dT of the equilibrium curve A-‐B should be positive. To confirm the inference, the triple point A-‐ B-‐L will need to be placed in the P-‐T diagram. It is located at the crossing point of the P-‐T melting curves A-‐L and B-‐L. The melting curves are approximated by straight lines, which is the simplest case of a monotonous curve . As the slopes of the equilibrium curves (dP/dT) are given by the Clapeyron equation,
dP Δs Δh = = dT Δv T Δv
the inequalities in the specific entropy (Δs = Δh/T; the equality is valid at equilibrium, h is the specific enthalpy (in J g-‐1)) and in the specific volume (Δv) between the phases in equilibrium with each other need to be established. In the case that they are not available experimentally, they can be determined
based on the following generally accepted approximations: (1) the inequality in the specific volumes between two solid forms is virtually independent of the temperature. (2) Gavezzotti has reported that molecular solids with expansivities (or volumetric thermal expansion coefficients αv) of about 2 × 10-‐4 K-‐1 expand by about 5 to 6 % from 0 K to their melting temperature (, p 277). This value is close to the mean value found for several drugs, as shown in Table 2. (3) A reasonable estimate of the volume change on melting can be found in several references. Gavezzotti reported (, p 24) values from Ubbelohde for volume changes on the melting of organic compounds [31, 32]. The mean value in the increase of the volume that accompanies melting is 11.5 % with only 2 values out of the 7 – 16 % range. This is close to the value of 12 % reported by Goodman et al. . It is also near the 10 % value found for several molecular pharmaceuticals (see compilation in Table 2). Thus an increase of 11 % will be used in the following for the estimate on the volume change on melting.
CONSTRUCTION DIAGRAM dP/dT
OF THE PRESSURE-‐TEMPERATURE
SLOPES OF THE
CURVES B ETWEEN T HE C ONDENSED P HASES
One step in the construction of a topological P-‐T phase diagram involving two polymorphs is to determine the slopes dP/dT of the melting curves and to infer whether they cross at high pressure or at negative pressure, i.e. whether they diverge or converge with increasing pressure. In either case, the intersection is the triple point A-‐B-‐L. If it occurs at elevated pressure, the triple point is stable; at negative pressure, the triple point will be metastable as the system is in an expanded state. For the slopes of the melting equilibria, the volume difference between the solid form and the melt has to be determined at the temperature of fusion. A value of 2 × 10-‐4 K-‐1 for the volume thermal expansion 6
αv for each polymorph is taken and a linear change in the specific volume as a function of temperature is assumed: v(T) = v0 (1 + αv T),
with v0 at T = 0 K and v(T) in cm g , αv in K and T in K (cf. Figure 4). The values of the specific volumes at the melting temperatures of forms A and B are found to be 0.846372 cm3 g-‐1 at 391 K and 0.8401914 cm3 g-‐1 at 413.8 K, respectively, using the values obtained at 295 K and reported in Table 1 (v0 is 0.784986 cm3 g-‐1 and 0.7759720 cm3 g-‐1 for respectively form A and form B). It is generally accepted that the thermal expansion of organic molecular liquids is larger than that of their related molecular solids, and the expansivity αV of their melts on average is found to be approximately 1.0 × 10-‐3 K-‐1 . In the following, a value of 1.25 × 10-‐3 K-‐1 will be used, as it is close to the average of the values obtained from the pharmaceuticals in Table 2.
an expansivity of 1.25 × 10-‐3 K-‐1 for the melt, the specific volume of liquid, vL /cm3g-‐1, as a function of temperature (K) can be approximated with the following linear equation: vL = 0.619195 + 7.73993 ×10-‐4 T
It results in vL = 0.921826 cm3 g-‐1 at TA→L = 391 K. It follows that ΔB→Lv = 0.099281 cm3 g-‐1 at 413.8 K and ΔA→Lv= 0.075454 cm3 g-‐1 at 391 K. Schematically this is shown in Figure 4. It can be seen that the specific volume of the metastable melt is larger than the crystalline solid A at the temperature of the observed glass transition, Tg = 327 K . Along the same line, the Kauzmann temperature at which the curves of the specific volume of the melt and of form A cross, can be found below the experimental glass transition temperature. Thus even though the relationships of the specific volume with the temperature have been obtained from statistical information, they coincide with the available additional experimental and theoretical information.
Table 2. Thermal expansion of a number of active pharmaceutical ingredients (API) from the literature.; α V,L: isobaric thermal expansion coefficient (or expansivity) of the liquid defined by equation (7), α V,S: expansivity of the solid, and
v(liq)/v(solid):the ratio between the specific volumes of the liquid and the solid at the melting point. API
α V,L 3 -‐1 ×10 (K ) 1.26 1.49
biclotymol progesterone a (I/II) Rimonabant 1.38 a (I/II) lidocaine 1.30 prilocaine 1.01 ternidazole 0.81 Mean/median 1.21/1.28 a crystalline polymorphs
α V,S 4 -‐1 ×10 (K ) 1.72 1.71/1.63
v(liq)/v(solid) at Tfus 1.13 1.09/1.10
3.99 2.18 1.80 2.12/1.76
1.06 1.13 1.11 1.10/1.10
  
The specific volume of the liquid can be estimated, based on the statistical difference mentioned above of 11% and the specific volume of the highest melting form at the melting point. This leads to 1.11 × vB(413.8 K) = 0.939473 cm3 g-‐1 at TB→L. Then, assuming
Figure 4. Schematic thermal expansion of FK 664 form A (double trace line), form B (black line), and the liquid (black line turning into broken and subsequently dotted line). Tg: glass transition (ca. 327 K), Tk: Kauzmann temperature.
To determine the slope of the melting equilibria, the Clapeyron equation (6) is used. The results on the specific volume and the calorimetric data on the fusion in Table 1 can now be used to find the following slopes dP/dTA-‐L = 2.76 MPa K-‐1 and dP/dTB-‐L = 2.20 MPa K-‐1. Because form B has the higher melting point in combination with the more gradual slope and hence form A has a lower melting point with a steeper slope for the equilibrium, the equilibrium curves must intersect at negative pressure and thus the triple point A-‐B-‐L is metastable. The melting curves can be approximated by straight lines that pass at ordinary pressure through the triple points A-‐L-‐vapor and B-‐L-‐vapor, which are equal to the respective melting points. The pressure at these triple points is given by eqs 5 and 4 and equals 0.11 Pa and 0.47 Pa respectively. Because the slopes of the solid – liquid equilibria are in the order of 2 to 3 MPa K-‐1, the pressure coordinate of the triple points can be taken as 0 MPa. This leads to the following expressions using slope and triple point for the pressure (in MPa) of the two solid – liquid equilibrium lines as a function of the temperature (in K): A-‐L:
PA-‐L = 2.76 T – 1080
PB-‐L = 2.20 T – 910
With these equations, the coordinates of metastable triple point A-‐B-‐L are found to be P, T = -‐246 MPa, 302 K. Although the temperature can be found close to room temperature, the pressure is clearly negative. The last two-‐phase equilibrium curve that remains to be located is the one where A and B are in equilibrium with each other. It passes through triple points A-‐B-‐vapor (T, P = 895 K, 76 MPa, see the section on heat related data) and A-‐B-‐L whose P-‐T coordinates have been determined just above. Solid-‐ solid equilibrium curves are usually straight lines, at least up to a few hundred megapascals; thus, using the triple point coordinates, the equation is obtained in a straightforward way: A-‐B:
PA-‐B = 0.54 T – 409
It can now be concluded that the slope dP/dT for the A-‐B equilibrium curve is smaller (ca. 0.54 MPa K-‐1 from eq. 11) than those of the two melting curves. The purpose of these calculations is not to obtain accurate values for the expressions of the two-‐phase equilibria and of the coordinates of the triple points but to study with a high level of precision their inequalities and relative positions to each other in the phase diagram. To demonstrate the robustness of the topological placement of the phases, even though the accurate position of the phase equilibria is not known, a sensitivity parameter analysis has been carried out, which can be found in the supplementary information.
P-‐T DIMORPHISM O F FK664 TOPOLOGICAL
Because the coordinates of the four triple points that exist in the case of dimorphism have been determined (Table 3), the P-‐T diagram can be drawn by using straight lines for the two-‐phase equilibria passing through the triple points, as shown in Figure 5a. Table 3. Coordinates of the four triple points for the dimorphism of FK664 Triple point P /Pa T /K O1: B-‐L-‐vapor 0.47 413.8 6 O2: A-‐B-‐vapor 76 ×10 895 O3: A-‐L-‐vapor 0.11 391 O4: A-‐B-‐L -‐246 ×106 302 However, the resulting phase regions are not of equal stability ranking, because the P-‐T representation of phase equilibria in a one-‐component system is the projection of the intersections of monotonous Gibbs energy surfaces, each of which represents the Gibbs energy minimum as a function of P and T for a given phase. To determine which of these surfaces has the deepest minimum and to apply a stability ranking to the different phases, the alternation rule can be used. This rule imposes the alternation of stable equilibrium lines and metastable lines around a triple point, which is a direct result of the Gibbs energy surfaces intersecting and replacing one minimum 8
surface (or stable phase) by another . Metastability simply means that the Gibbs energy surface of the phase is not the lowest one. One metastable surface can intersect another and therefore the stability ranking projected on the P-‐T
plane can also be extended into metastable triple points surrounded by alternating metastable and supermetastable equilibrium lines (which translates into even higher placed Gibbs energy surfaces in relation to the stable and metastable phases).
Figure 5. Construction of the topological phase diagram of the dimorphism of FK664; P and T are not to scale (a) Placing the triple point coordinates (numbers can be found in Table 3) and intersecting each set of two triple points by a straight line representing one of the six two-phase equilibria (line intersecting O1-O2: B-vapor, O1-O3: liquid-vapor, O1-O4: B-liquid, O2O3: A-vapor, O2-O4: A-B, and O3-O4: A-liquid) (b) The lines and triple points have been ranked according to their stability hierarchy (c) Stable domain of form B (shaded region) (d) Metastable domain of form A (shaded region), where it can occur as a solid even if it will eventually turn into form B. Because the highest melting solid is the most stable phase just below its melting point, it is clear that the triple point B-‐L-‐vapor is stable. Three stable two-‐ phase equilibria must meet in this point and turn into metastable on the opposite site of the triple point. These are the liquid-‐vapor curve (down from the critical point), the sublimation curve of form B, and the melting curve of form B. This can be seen around
triple point O1 in Figure 5b. It can also be seen that the metastable extensions of these two-‐phase equilibria curves intersect metastable triple points. The metastable extension of the B-‐vapor curve intersects O2, which is the triple point between form A, form B, and the vapor at a location where the liquid is stable. This means that the triple point itself is metastable (as has already been discussed in 9
previous sections) and therefore any metastable equilibrium intersecting it must turn supermetastable. The same is true for triple point O3, the melting point of form A, which occurs in the region where B is stable (Figure 5c), and triple point O4, where the two solids and the liquid are in equilibrium. The latter triple point is located in the region where the system is expanded and the only possible phase that can exist in such a state is the vapor. Thus O1 is the only stable triple point and the intersection of the only three stable two-‐phase equilibria (Figure 5b).
Figure 6. Gibbs energy-temperature isobaric sections of the P-T diagram for an overall monotropic behavior (form A less stable than form B for any P and T). Left-hand side: section at pressure P1, right-hand side: section at pressure P2. The inset shows how form A becomes less metastable than form B as T increases, at constant pressure. However at point 5, the liquid has the lowest Gibbs energy. Points 1 to 6 are sections of two-phase equilibrium curves; 1: A-liquid, 2: A-vapor, 3: B-liquid, 4: B-vapor, 5: A-B, 6: liquid-vapor. The phase diagram possesses the same topology as Bakhuis-‐Roozeboom’s case 4, for which Bakhuis-‐ Roozeboom gave no example. Two other cases of dimorphism among APIs have been found to exhibit this so-‐called “overall monotropic behavior”: biclotymol  and rimonabant . In the case of overall monotropic behavior, the Gibbs energy of the less stable polymorph remains larger than that of the other phases at any P and T. This is
illustrated in Figure 6 by depicting two G-‐T isobaric sections of the G(P,T) diagram.
CONCLUDING REMARKS The preceding analysis demonstrates that Miyamae et al. were correct when they concluded that form B was the more stable one of the two known solid forms. It also places this conclusion on firm thermodynamic footing. Finally, it is clear from the analysis that form A does not have a stable domain under any pressure and temperature coordinates in relation to form B, in other words that the system is overall monotropic. In 1901, when Bakhuis-‐Roozeboom published the four possible cases of P-‐T diagrams for crystalline dimorphism, no example was provided of the fourth case, overall monotropic behavior. It may have been the lack of an example that caused this case to be all but forgotten, as citations have been scarce in the literature. However, systems with overall monotropic behavior do exist as compounds with such phase behavior can be found in the literature: trichloroacetic acid , hydrazine monohydrate , sulfanilamide [8, 10], ferrocene , and more recently two pharmaceuticals: biclotymol  and rimonabant . Supplementary information: Sensitivity parameter analysis of the inequality in the dP/dT slopes of the melting curves
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