Polymorphs, Hydrates, Cocrystals, and Cocrystal Hydrates

Article pubs.acs.org/crystal

Polymorphs, Hydrates, Cocrystals, and Cocrystal Hydrates: Thermodynamic Modeling of Theophylline Systems Linda Lange and Gabriele Sadowski* Department of Chemical and Biochemical Engineering, Laboratory of Thermodynamics, TU Dortmund University, Emil-Figge-Strasse 70, D-44227 Dortmund, Germany ABSTRACT: Polymorphic transitions and hydrate formation often occur in systems of cocrystal-forming components. To increase the efficiency of cocrystal formation and purification processes, the complex phase behavior of such systems was modeled using perturbed-chain statistical associating fluid theory (PC-SAFT). This is demonstrated for theophylline, a well-studied pharmaceutical, exhibiting polymorphs, as well as formation of a hydrate, cocrystals, and even cocrystal hydrates. The solubility of theophylline in water was modeled including hydrate formation (1:1) as well as polymorphic transitions of theophylline between the anhydrate forms IV, II, and I. The solubilities of theophylline(IV), the thermodynamically stable form at ambient conditions, and the theophylline/glutaric acid (1:1) cocrystal could be predicted without performing additional measurements. Moreover, the complex phase behavior of the theophylline/citric acid/water system could be correlated accounting for the formation of the theophylline hydrate (1:1), citric acid (1:1) hydrate, theophylline/citric cocrystal (1:1), and the corresponding cocrystal hydrate (1:1:1). By accounting for the thermodynamic nonideality of the components in the cocrystal system, PC-SAFT is able to model the solubility behavior of all above-mentioned components in good agreement with the experimental data.

1. INTRODUCTION Pharmaceutical cocrystals (CCs) represent an emerging class of solids1−6 consisting of an active pharmaceutical ingredient (API) and at least one coformer (CF) in a defined stoichiometry.7 Because of the weak interactions between API and CF in the CC, the latter decomposes into its components upon dissolution. In most cases, these interactions are executed via hydrogen bonds that can be also formed with water molecules.8−11 Thus, besides CCs often also hydrates of the API, CF, or even of the CC12,13 occur in aqueous solutions. Furthermore, polymorphism can potentially be found in any type of crystalline solid,11,14,15 including APIs,16−19 hydrates,20,21 and CCs.13,22−25 At the industrial scale, CCs are primarily generated by crystallization from solution.26−28 In many API/CF/solvent systems, polymorphic transitions and hydrate formations of API, CF, and/or the CC itself influence the design of CC formation and purification processes. Therefore, an effective CC formation requires knowledge of the thermodynamic phase diagram, exhibiting the concentration range in which the stable CC can be found.26,29−31 However, the reliable experimental determination of these diagrams is time- and cost-intensive,29,32−35 particularly if polymorphism occurs in the investigated API/CF/solvent system.36−39 Polymorphs are crystalline solids with the same chemical composition, but different crystal structures, resulting in different physicochemical properties.11,40−42 However, only one polymorphic form of a solute is thermodynamically stable at a given temperature and pressure.43 Besides, often metastable polymorphs are formed when the transition to the thermodynamically stable form is kinetically inhibited,36,43,44 © 2016 American Chemical Society

whereas the most stable form is less soluble than the metastable forms.18,36,45,46 For enantiotropic polymorphic systems, the transition between two polymorphic forms occurs at the socalled transition temperature Ttrs as one form is more stable below Ttrs and the other form is more stable above Ttrs.47−50 Various approaches in the literature illustrate the thermodynamic stability relationships of polymorphs.41,47,48,50,51 Among others, Burger and Ramberger developed thermodynamic rules for the existence of enantiotropism in polymorphic systems,47,48 while Yu et al. calculated transition temperatures Ttrs for these systems using melting properties of the polymorphs.49,50 In further studies, activity-coefficient models were applied to also predict the solubility of (pharmaceutical) polymorphs.52−54 The solubilities of hydrates and CCs are usually calculated using a corresponding solubility product Ks described in eq 1.30,31,55,56 Ks =

∏ ai ν

i

= (xAγA )νA ·(x BγB)νB

i

(1)

Using this approach, formation of hydrates or CCs is treated as a chemical reaction of component A and component B in a liquid solution, resulting in the respective solid complex. In the case of a hydrate, A refers to the API (or CF) and B refers to water, whereas for a CC, components A and B correspond to the API and CF. xi and γi are the mole fractions and activity Received: April 11, 2016 Revised: June 13, 2016 Published: June 17, 2016 4439

DOI: 10.1021/acs.cgd.6b00554 Cryst. Growth Des. 2016, 16, 4439−4449

Crystal Growth & Design

Article

coefficients of A and B at solubility, respectively. ai is the thermodynamic activity of these components, while the activity of the solid hydrate or CC is one. νA and νB represent the stoichiometric coefficients of A and B in the solid complex. It needs to be mentioned that Ks only varies with temperature and can be therefore calculated based on only one hydrate or CC solubility data point, regardless of solvent and concentration. However, in many approaches3,35,57−63 the activity coefficients in eq 1 are neglected, resulting in Kideal s : K sideal

∏i (ai)νi Ks = = = ∏i (γi)νi ∏i (γi)νi

(A), pure crystalline API hydrate (C), a mixture of both crystals, water and API hydrate, (B), pure crystalline API(I) (E), or pure crystalline API(II) (F). The temperature as well as the API/water ratio determine which of these solid phases is formed. Below the so-called eutectic temperature Teut and below an API/water ratio of 1:1, which corresponds to the stoichiometry in the API hydrate, a mixture of water crystals and API hydrate crystals is formed (B). Above Teut and for very high water and very low API concentrations (A), pure water crystals are in equilibrium with the liquid solution (L) and the respective solubility ends on the left axis in the melting temperature of water TSL water. For higher API concentrations but still below an API/water ratio of 1:1 (C), pure API hydrate crystals are formed. The solubility lines of water and API hydrate intersect at the eutectic point, where the solution (L) is in equilibrium with the heterogeneous mixture of water crystals and API hydrate crystals (B). The API hydrate crystals are stable below the transition temperature Ttrs I/hydrate. The same applies for a heterogeneous mixture of API hydrate crystals and API(I) crystals that are present at API/ water ratios higher than 1:1 (D). Ttrs I/hydrate marks the transition between the API hydrate and the API(I) anhydrate, since above Ttrs I/hydrate (E), pure API(I) crystals are thermodynamically stable. Pure API(II) crystals are formed above the second transition temperature Ttrs I/II (F). A hypothetical extension of the solubility curve of API(I) to temperatures above Ttrs I/II (gray dotted line in Figure 1) ends in the respective melting temperature TSL API(I). The same applies for the solubility of API(II), exhibiting a higher melting temperature TSL API(II) than API(I). The phase diagrams of the binary CF/water and API/water systems according to Figure 1 can be used to construct the ternary phase diagram of an API/CF/water system. This is illustrated in Figure 2 which schematically shows such a phase diagram of an aqueous solution in which API and CF form a 1:1 hydrate, as well as a 1:1 CC and a 1:1:1 CC hydrate at Tternary. The solid phases and their solubilities in the binary API/water system at a certain temperature correspond to the API/water axis of the ternary phase diagram at the same temperature. The same applies for the CF/water axis where the solid phases and corresponding solubilities equals that in the binary CF/water system at the same temperature. Thus, at high API concentrations (C), pure API hydrate crystals are in equilibrium with the aqueous solution (L), whereas at high CF concentrations (Q), pure CF hydrate crystals are formed. CCs are present in region (N) and CC hydrates in region (K), whereas the CC or CC hydrate solubility line is the one dividing regions (L) and (N) or (L) and (K), respectively. Moreover, six regions exist in which at least two solids crystallize at the same time: API hydrate and CC hydrate (G), API(I), API hydrate and CC hydrate (H), API(I), CC and CC hydrate (J), CC and CC hydrate (M), CC and CF (O), and CF and CF hydrate (P). In summary, this example shows the complex phase behavior in an aqueous API/API hydrate/CF/CF hydrate/CC/CC hydrate system that may include in total 12 different phases. Thus, it becomes obvious, that the knowledge of the thermodynamic phase diagram is required to determine the concentration range (N) in which only the CC is formed.

∏ xi ν = (xA)ν (xB)ν i

A

B

i

(2)

In contrast to the above-mentioned approaches,3,35,57−63 this work considers thermodynamic nonideality in API/CF/solvent systems. For this purpose, the perturbed-chain statistical associating fluid theory (PC-SAFT)64 was applied to calculate the activity coefficients in eq 1 that describe deviations from the pure-component reference state. In prior studies, PC-SAFT has already been successfully used to model solubilities in binary and ternary systems,64−68 including those with CC30,31,56,69,70 and hydrate31,69 formation. Furthermore, we consider polymorphism and enantiotropism in the investigated API/CF/ solvent systems. The modeling results were validated by comparison with experimental data of theophylline, a polymorphic model substance exhibiting formation of hydrate, CC, and CC hydrate.

2. PHASE DIAGRAMS OF SYSTEMS EXHIBITING POLYMORPHIC TRANSITIONS AND HYDRATE FORMATION Figure 1 schematically shows the solubility lines occurring in a binary API/water system, including formation of a 1:1 API

Figure 1. Schematic phase diagram of an active pharmaceutical ingredient (API) with formation of 1:1 hydrate and a polymorphic transition of the anhydrate forms I and II. Solid lines are the solubility (thick) and phase boundary (thin) curves of water, API hydrate, API(I), and API(II). The phases are denoted as follows: L, liquid mixture consisting of water and completely dissolved API; A, pure water crystals in equilibrium with liquid mixture L; B water crystals and API hydrate crystals in equilibrium with liquid mixture L; C pure API hydrate crystals in equilibrium with liquid mixture L; D API hydrate crystals and API(I) crystals in equilibrium with liquid mixture L; E pure API(I) crystals in equilibrium with liquid mixture L; F pure API(II) crystals in equilibrium with liquid mixture L.

hydrate as well as an enantiotropic transition between two anhydrate polymorphic forms API(I) and API(II). The aqueous solution L contains completely dissolved API. Along the solubility lines that separate region L from regions A, B, C, E, and F, the saturated solution is in equilibrium with the respective solid phase, which consist of pure crystalline water

3. THEORY 3.1. Solubility Calculations. As postulated by Prausnitz et al.,71 the solubility of an (anhydrate) API or CF is calculated 4440



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Ks =

∏ ai ν

i

= (xAPIγAPI)νAPI ·(xCFγCF)νCF ·(x H2OγH O)νH2O 2

i

(4)

All solubility products Ks used within this work depend on temperature according to the Gibbs−Helmholtz equation: ln K s = ln K sref +

a residual = a hard chain + adispersion + aassociation

(6)

As a detailed description of the different contributions is given in the literature,55,64−66,73,74 these are only briefly depicted here. In PC-SAFT, a nonassociating component is characterized by using three pure-component parameters, namely, the number of segments mseg i , the segment diameter σi and the dispersion-energy parameter ui/kB.64 For associating components, additionally the association-energy parameter εAiBi/kB and the association-volume parameter κAiBi are considered.65 In mixtures of substances i and j, the resulting segment diameter σij (eq 7) and the dispersion-energy parameter uij (eq 8) are estimated by the Berthelot-Lorentz combining rules.75

considering the thermodynamic equilibrium between the pure solid phase and the liquid solution: ⎡ Δh SL ⎛ ⎞ ΔcPSL,0i 1 ⎢ − 0i ⎜ 1 − T ⎟ − exp R ⎢⎣ RT ⎝ γi L T0SLi ⎠ ⎛ T SL T SL ⎞⎤ ⎜ 0i − 1 − ln 0i ⎟⎥ T ⎠⎥⎦ ⎝ T

(5)

In eq 5, Kref s refers to a reference solubility product at a reference temperature Tref. Δhref s equals the enthalpy of fusion of the complex at the reference temperature Tref, which can be also interpreted as the enthalpy of melting. If Δhref s was not available from the literature, it was derived in this work by an approach described by Folas et al.72 using at least two different solubility data points of the respective hydrate, CC, or CC hydrate at two different temperatures. 3.2. PC-SAFT. The activity coefficients γi in eqs 1, 3, and 4 were calculated in this work using the thermodynamic model PC-SAFT. The derivation of PC-SAFT was in detail described in the literature.64,65 It considers the residual Helmholtz energy aresidual as the sum of a contribution caused by the repulsion of the molecules (hard chain) and various contributions accounting for attractive forces (dispersion) as well as for hydrogen-bonding interactions (association):

Figure 2. Schematic phase diagram for an active pharmaceutical ingredient (API)/coformer (CF)/water system with hydrate, cocrystal (CC), and CC hydrate formation at Tternary marked in Figure 1. Solid lines are the solubility (thick) and phase boundary (thin) curves of API hydrate, CF hydrate, CC, and CC hydrate. The phases are denoted as follows: L, liquid mixture consisting of water with completely dissolved API and CF; C, pure API hydrate crystals in equilibrium with liquid mixture L; G, API hydrate crystals and CC hydrate crystals in equilibrium with liquid mixture L; H, API(I) crystals, API hydrate crystals, and CC hydrate crystals in equilibrium with liquid mixture L; J, API(I) crystals, CC crystals and CC hydrate crystals in equilibrium with liquid mixture L; K, pure CC hydrate crystals in equilibrium with liquid mixture L; M, CC and CC hydrate crystals in equilibrium with liquid mixture L; N, pure CC crystals in equilibrium with liquid mixture L; O, CC and CF crystals in equilibrium with liquid mixture L; P, CF and CF hydrate crystals in equilibrium with liquid mixture L; Q, pure CF hydrate crystals in equilibrium with liquid mixture L.

xiL =

Δhsref ⎛ 1 1⎞ ⎜ − ⎟ R ⎝ T ref T⎠

(3)

σij =

In eq 3, xLi refers to the solubility or rather the mole fraction of anhydrate component i (API or CF) in the liquid phase. T is the temperature of the system and R is the ideal gas constant. Further, the melting temperature TSL 0i and the heat of fusion of component i (API or CF) are considered in eq 3 as ΔhSL 0i well as the difference in the solid and liquid heat capacities of component i at its melting point ΔcSL P,0i. The activity coefficients γLi of API or CF describe the deviation from the purecomponent reference state. They were calculated in this work by PC-SAFT depending on temperature and on the concentration of all components in the liquid mixture. Thus, also the API solubilities in the presence of the CF and vice versa were calculated via eq 3. Hydrate formation or CC formation (commonly referred to as complex formation) was treated as a chemical reaction between its components, namely, water/API (or water/CF) and API/CF. The same applies for a CC hydrate consisting of API, CF, and water. The solubility of the latter was calculated using the solubility product Ks similar to that for hydrates and CCs in eq 1 as follows:

1 (σi + σj) 2

uij = (1 − kij) uiuj

(7) (8)

Further, the cross-dispersion energy of the two substances in mixtures is corrected by a binary interaction parameter kij. As shown in eq 9, kij is assumed to linearly depend on temperature. In this work, kij was adjusted simultaneously with the purecomponent parameters of the solutes to solubility data. kij(T ) = kij ,slopeT + kij ,int

(9)

The cross-associating interactions between two associating components i and j were defined by the association-energy parameter εAiBj/kB and the association-volume parameter κAiBj. εAiBj/kB was calculated via eq 10 and κAiBj was calculated via eq 11 using the combining rules derived by Wolbach and Sandler.76 ε Ai Bj = 4441

1 Ai Bi (ε + ε Aj Bj) 2

(10) DOI: 10.1021/acs.cgd.6b00554 Cryst. Growth Des. 2016, 16, 4439−4449


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Table 1. PC-SAFT Pure-Component Parameters for the Active Pharmaceutical Ingredient (API), Coformers (CFs), and Solvents Considered within This Work component

M [g/mol]

mi [−]

σi [Å]

ui [K]

εAiBi [K]

κAiBi [−]

180.16

13.7041

3.0354

163.97

976.19

132.12 192.124

4.4371 7.17

2.7990 2.237

257.67 267.99

18.015 41.052 119.378 46.069 32.04 130.23 60.096

1.2047 2.3290 2.565 2.3827 1.5255 4.356 3.0929

a 3.1898 3.449 3.1771 3.23 3.715 3.2085

353.94 311.31 267.19 198.24 188.90 262.74 208.42

API theophylline CFs glutaric acid citric acid solvents water acetonitrileb chloroformb ethanol methanol 1-octanol 2-propanol

assoc scheme

ref

ARD [%]

0.02

1/1

this work

2.12

1762.5 513.79

0.02 0.02

2/2 4/4

30 this work

1.88

2424.67 0 0 2653.4 2899.5 2754.8 2253.9

0.045 0.01 0.01 0.03284 0.035176 0.0022 0.024675

1/1 1/1 1/1 1/1 1/1 1/1 1/1

94 95 96 65 65 65 65

a The expression σ = 2.7927 + 10.11 exp(−0.01775T) − 1.417 exp(−0.01146T) was used.94 bConsideration of induced association in mixtures with associating components.

Table 2. PC-SAFT Binary Interaction Parameters for the Binary Sub-Systems of the Active Pharmaceutical Ingredient (API) Theophylline (TP) (or the coformers (CFs) Citric Acid (CA) and Glutaric Acid (GA)) and Solvents Considered in This Work and the Average Relative Deviations (ARDs) of Calculated and Experimental Solubilitiesa binary parameter kij,slope [−] API/solvent TP/acetonitrile TP/chloroform TP/ethanol TP/methanol TP/1-octanol TP/water CF/solvent CA/acetonitrile CA/ethanol CA/water CA/2-propanol GA/acetonitrile GA/chloroform solvent/solvent chloroform/methanol API/CF TP/CA TP/GA a

kij,int [−]

temperature range of the experimental data [K]

1.95 × 10 −6.56 × 10−4 −1.04 × 10−3 −5.67 × 10−4 −2.40 × 10−4

3.52 × 10−2 3.21 × 10−4 2.39 × 10−1 1.07 × 10−1 2.59 × 10−1 −8.80 × 10−2

303.1 293.15−323.15 313.0−325.2 283.15−323.15 322.3−336.7 293.15−303.15

this this this this this this

4.20 × 10−4 1.65 × 10−4 1.25 × 10−4 −1.55 × 10−4 −5.99 × 10−5 −6.03 × 10−4

−1.64 × 10−1 −1.94 × 10−1 −9.48 × 10−2 −6.90 × 10−2 1.90 × 10−2 1.90 × 10−2

313.2−333.2 313.2−333.2 313.2−365.64 313.2−333.2 283.15−308.15 293.15−323.15

this this this this 30 this

−0.045

326.5−336.9

96

100

−0.158 −0.18

298.15 303.15

this work this work

101 91

−4

ref for parameters

ref for exp data

ARD [%]

work work work work work work

91 91 92 90 92 90

0.00 1.31 4.87 1.98 2.45 1.88

work work work work

97 97 98 97 99 91

1.17 2.48 1.32 2.55 1.39 4.72

work

12.5 0.37

Binary parameters should be used together with pure-component parameters from Tables 1 and 3.

κ

Ai Bj

=

κ

⎛ ⎞3 σσ i j ⎜ (1/2)(σ + σ ) ⎟⎟ ⎝ i j ⎠

namely, theophylline (TP)/water (1:1), citric acid (CA)/ water (1:1), TP/glutaric acid (GA) (1:1), TP/CA (1:1), and TP/CA/water (1:1:1). The solubilities of the pure API TP, the pure CFs GA and CA, as well as of all above-mentioned complexes were modeled using PC-SAFT for the calculation of the respective activity coefficients γLi in eq 3 or 1. The application of PC-SAFT in turn requires the pure-component parameters of the API, the CFs, and all considered solvents as well as the binary interaction parameters. Among the above-mentioned pure components and complexes, polymorphism is reported for TP. To date, four anhydrate forms (I−IV), one hydrate form (1:1), and a dimethyl sulfoxide solvate (1:1) of TP are known.79−85 Form II is the usually available commercial form since it can be easily prepared by crystallization from many organic solvents.80,84 Thus, form II was long believed to be the thermodynamically

Ai Bi Aj Bj ⎜

κ

(11)

Ai and Bi refer to the number of association sites of a component i defined on the basis of associating schemes postulated by Huang and Radosz. 77 Further, induced association of polar but non-self-associating components was assumed in mixtures with at least one self-associating component.78 In these cases, the association-energy parameter εAiBi of the non-self-associating component was set to zero, and the association-volume parameter κAiBi was set to 0.01.

4. RESULTS AND DISCUSSION 4.1. Estimation of PC-SAFT Parameters. Different complex-forming systems were investigated in this work, 4442



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stable form at ambient conditions.86−89 In contrast, Seton et al. found form IV, an at this time unknown form, that is at room temperature even more stable than form II.79,80,83,84 However, the conversion of form II to form IV is kinetically inhibited, taking up to 2 weeks in supersaturated solutions and even several years in dry samples.83 In contrast, form I is found to be stable at higher temperatures,86 while form III is highly metastable at any temperature.89 The different polymorphic forms exhibit different melting properties and therefore different solubilities, whereas the physicochemical properties of dissolved TP do not depend on the solid form which is in equilibrium with the liquid (compare eq 3). Thus, irrespective of the different polymorphic forms, one and the same pure-component parameters can be used to describe the properties of TP, particularly its activity coefficient in any solution. Since solubility data of TP are extensively available in the literature,90−92 its PC-SAFT parameters were fitted to solubility data of the kinetically stable form II in organic solvents, namely, acetonitrile, chloroform, ethanol, methanol, and 1-octanol. For this purpose, the parameter-fitting procedure described by Ruether and Sadowski was applied,55 whereas the binary interaction parameters were fitted simultaneously with the pure-component parameters to the solubility data. Fitting of four pure-component parameters (mi, σi, ui, εAiBi) plus a temperature-dependent binary interaction parameter (kij,slope, kij,int) required at least the same number of experimental solubility data points of the pure components (API or CF). As the parameters of less soluble components exhibit a higher sensitivity on modeling results, the corresponding solubility data used for the parameter-fitting procedure needs to be as accurate as possible. Therefore, solubility data in solvents with high solubility are preferred for this purpose as the accuracy of concentration analysis increases with increasing solubilities. The pure-component parameters of CA were also fitted to its solubility data in organic solvents (here acetonitrile, ethanol, and 2-propanol). The pure-component parameters of all other components considered in this work, namely the solvents and GA, were taken from literature. All pure-component parameters used in this work are given in Table 1, whereas the corresponding binary interaction parameters are listed in Table 2. Additionally, the solubility calculations via eq 3 required the pure-component melting properties of all considered solutes. Therefore, the melting properties of TP(I), (II), and (IV), as well as of GA and CA are summarized in Table 3. The heat capacity for liquid TP, GA, and CA are not available in the literature because of decomposition at high temperatures. Instead, the group contribution method of Kolska was applied93 for that purpose. Further, it was assumed that the heat capacities of the different anhydrate TP polymorphs were the same.

The quality of the modeling was quantitatively evaluated by the average relative deviation (ARD) between the calculated xcalc,i and experimental solubility xexp,i and nexp being the number of experimental data points: 1 ARD = 100 nexp

component

Δh0iSL [kJ/mol]

Δcp.0iSL [J/mol]

ref

TP(I) TP(II) TP(IV) GA CA

546.55 542.25 540.70 370.95 428.75

26.4 28.2 28.7 20.9 40.3

114.91 114.91 114.91 107.45 159.46

86, 87, 93 86, 87, 93 this work, 87, 93 93, 102, 103 93, 104, 105

∑

xcalc, i − xexp, i

i=1

xexp, i

(12)

In the ternary API/CF/solvent system, the deviation of experimental and calculated data refers to that of the API, in this study TP. The resulting ARDs between the correlated and experimental solubilities of pure components are listed in Tables 1 and 2. ARDs for the predicted solubilities and/or the correlated solubilities of hydrates or CCs can be found in Table 4. Table 4. Average Relative Deviation (ARD) of Calculated and Experimental Solubilities of Systems Including Theophylline (TP), Citric Acid (CA), and/or Glutaric Acid (GA) solid component TP hydrate TP(IV)

CC CA hydrate CC CC hydrate

system

temperature [K]

ARD [%]

TP/water TP/acetonitrile TP/chloroform TP/chloroform/methanol TP/GA/acetonitrile TP/GA/chloroform CA/water TP/CA/water TP/CA/water

278.15−338.85 303.15 303.15 283.15−323.15 303.15 303.15 283.38−303.57 298.15 298.15

7.93 13.3 1.58 8.94 0.40 33.6 2.27 1.80 8.47

The pure-component parameters for TP and the corresponding binary interaction parameters were adjusted to the solubility data of TP(II) in 1-octanol,92 ethanol,92 acetonitrile,91 chloroform,91 and a chloroform−methanol mixture90 (0.8/0.2, x/x). Figure 3 compares the resulting correlations with

Figure 3. Solubility of TP(II) in 1-octanol92 (diamonds), ethanol92 (triangles), acetonitrile91 (circle, dotted line), chloroform91 (hexagon), and chloroform−methanol mixture90 0.8/0.2 x/x (squares). Lines correspond to the PC-SAFT correlations; symbols represent the experimental data points.90−92

Table 3. Melting Properties of Theophylline (TP) (Forms I, II, and IV), Glutaric Acid (GA), and Citric Acid (CA) T0iSL [K]

nexp

experimentally determined data. The solubility of TP(II) is highest in the chloroform−methanol mixture (0.8/0.2, x/x), followed by chloroform, ethanol, and 1-octanol. The correlated solubility curve for acetonitrile exceeds that of ethanol and 1octanol for lower temperatures, whereas it falls below those for higher temperatures. The pure-component parameters as well as the binary interaction parameters for CA solutions were adjusted to 4443



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solubility data97 in ethanol, acetonitrile, and 2-propanol as illustrated in Figure 4. The solubility of CA is highest in ethanol

trs between forms (I) and (II) Ttrs I/II, (II) and (IV) TII/IV, (IV) and trs hydrate TIV/hydrate (Table 5), as well as the stoichiometry, Ks and Δhref of the TP hydrate (Table 6).

Table 5. Transition Temperatures between the Different Anhydrate Forms I, II, and IV and the Hydrate of Theophylline (TP), as well as between the Anhydrate Form and the Hydrate of Citric Acid (CA) Used in This Study component TP I/II II/IV IV/hydrate CA anydrate/hydrate

Figure 4. Solubility of citric acid in acetonitrile (circles), ethanol (triangles), and 2-propanol (pentagons, dotted line). Lines correspond to the PC-SAFT correlations; symbols represent the experimental data points.97

Ttrs [K]

ref

498 483 344

this work 84 86

307

98

The melting properties of TP(I) and (II) are available from the literature.86 The melting properties of TP(IV) were solely reported in the literature by Seton et al.,85 namely, being the same as of TP(II). However, Seton et al. also reported an enantiotropic transition between forms (IV) and (II) which requires a lower melting temperature and a higher melting enthalpy of form (IV) compared to form (II).47,48 Therefore, the melting temperature and enthalpy of TP(IV) were not taken from Seton et al. but adjusted to solubility data in this work. The adjustment was performed using the correlated solubility data point of TP hydrate at Ttrs IV/hydrate and that of TP(II) at Ttrs in eq 3, as at those temperatures these II/IV solubilities need to be equal to those of TP(IV) to fulfill the thermodynamic-equilibrium conditions of the corresponding phases at the transition points. trs The transition temperatures Ttrs IV/hydrate and TII/IV, listed in 84,86 trs Table 5, were adopted from the literature. TIV/hydrate = 344 K was reported by Suzuki et al.86 Khamar et al. observed the transition of the anhydrate form IV to form II over a temperature range of 483−513 K,84 instead at a discrete temperature. Thus, it is assumed that the transition point equals the lowest temperature of Ttrs II/IV = 483 K where first crystals of TP(II) appeared during heating. The transition temperature of form II to form I, Ttrs I/II, was determined in this work by solubility calculations of TP(I) and (II), since the solubility curves intersect at this transition point. Below Ttrs I/II, TP(II) is more stable than TP(I) which also implies that the solubility is lower than that of TP(I). Above Ttrs I/II, just the opposite applies as TP(I) is the more stable form. The resulting Ttrs I/II = 498.0 K is in good accordance with the estimated temperature ranges of Szterner et al.87 of Ttrs I/II = 499.8−504.8 K, as well as with the one of Griesser at al.106 (Ttrs I/II = 468.2−504.2 K) which was calculated based on melting properties published by Burger and Ramberger.48 The solubility of the TP hydrate was correlated using eqs 1 and 5. First, the hydrate solubility product Ks in Table 6 was calculated via eq 1 using only one hydrate solubility point at 298.15 K90 (marked as star in Figure 5). Afterward, the temperature-dependent hydrate solubility is modeled via eq 5 using Δhref = 10.6 kJ mol−1 from the literature.86 As can be seen in Figure 5, the complete binary phase diagram of TP/water can be modeled consistently over the whole considered temperature range (266−344 K) and in high accordance to the experimental data90,107,108 of the TP hydrate. Although only one single solubility data point was used for the determination of the TP hydrate solubility product Ks, the ARD

followed by that in 2-propanol. Analogous to TP, the solubility of CA in acetonitrile exhibits a smaller temperature-dependence compared to that in the other solvents, resulting in a higher solubility for lower temperatures and a lower solubility for higher temperatures. The almost quantitative solubility calculations are reflected by the ARDs < 5% in Table 2. Consequently, PC-SAFT allows for an accurate correlation of the solubility data of both TP(II) and CA in a variety of solvents. 4.2. Hydrate Formation and Polymorphic Transitions of TP in Water. In the second modeling step, the solubility of TP in water was investigated. In the binary system TP/water, the hydrate form as well as three anhydrate forms I, II, and IV of TP are thermodynamically stable depending on temperature.79,80,83,84 Figure 5 shows the phase diagram for the binary system TP/ water. The solubility of water in the presence of TP is modeled

Figure 5. Solubility of theophylline (TP) in the binary system TP/ water with formation of a 1:1 hydrate. Solid lines correspond to the PC-SAFT correlations of TP hydrate, TP(IV), TP(II), and TP(I) solubility (thick) and to the phase boundary (thin) curves; symbols refer to the experimental data points of TP hydrate (triangles,107 circles,108 diamonds90). The solid phases are in equilibrium with liquid mixture containing water and totally dissolved TP. The star is the hydrate solubility point90 used for the calculation of the solubility product Ks.

via eq 3. However, the concentration range of crystalline water in equilibrium with liquid solution is too small to be visible in Figure 5. To correlate the solubility of the above-mentioned TP forms, the following information was required: PC-SAFT purecomponents parameters of TP (Table 1), the binary interaction parameter between TP and water (Table 2), melting properties of TP(I), (II), and (IV) (Table 3), transition temperatures 4444



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Table 6. Solubility Products Ks and Reference Enthalpies Δhref of Hydrates, Cocrystals (CCs), and CC Hydrate Consisting of Theophylline (TP), Citric Acid (CA), and/or Glutaric Acid (GA), Calculated within This Worka complex A/B or A/B/C

a

Ks [−]

complex stoichiometry A:B or A:B:C

TP hydrate TP/H2O CA hydrate CA/H2O

1:1 1:1

CC TP/GA CC TP/CA CC hydrate TP/CA/H2O

1:1 1:1 1:1:1

7.60 2.60 3.60 1.76 6.52 5.77

× × × × × ×

−2 ( )

10 * 10−2 (*) 10−2 10−2 (*) 10−4 (*) 10−4 (*)

temperature [K]

Δhref [kJ mol−1]

source for Δhref

298.15 298.15 307.00 303.15 298.15 298.15

10.6 28.0

86 eq 5

All reference solubility products Kref s are marked with (*).

of the obtained solubility curve to solubility data from three different literature sources is less than 8% (compare Table 4). 4.3. TP/GA CC Systems. In this section, the solubilities of the TP/GA (1:1) CC are correlated at 303.15 K in acetonitrile and predicted in chloroform. Besides, the solubilities of TP(IV) are predicted as this is a thermodynamically stable form at this temperature and in the considered solvents. The solubility of TP(IV) is predicted using the purecomponent parameters (Table 1) adjusted to solubility data of TP(II) as described in section 4.1, and the melting properties (Table 3) adjusted to the corresponding transition points of the TP/water system as described in section 4.2. Figure 6 compares the predicted and experimentally determined solubility data91 for TP(IV). Zhang and RasmuFigure 7. Solubilities for the theophylline (TP)/glutaric acid (GA)/ acetonitrile system at 303.15 K with shortened axes in mole fractions. Solid lines correspond to PC-SAFT calculations, whereas the TP(IV) solubility line is fully predicted; symbols refer to the experimental data points91 of GA (black), the cocrystal (gray), and TP(IV) (white). The star is the CC solubility point used for the solubility product Ks.

properties from Table 3. The same applies for the solubility of GA (in the presence of totally dissolved TP). The CC solubility was correlated using eq 1 and the CC solubility product Ks from Table 6 that was previously determined on the basis of one CC solubility data point (marked as a star in Figure 7) in acetonitrile at 303.15 K. The resulting CC solubility is in almost quantitative agreement with the experimental data91 (ARD of 0.40%). Using the same Ks from Table 6, the CC solubility was afterward predicted for the TP/GA/chloroform system. As can be seen from Figure 8, the predicted CC solubility is again in good agreement with the experimentally determined data,91 except for the eutectic point of GA and CC resulting in an ARD of 33.6%. The particularly high distance of this eutectic point from the remaining solubility points is also visible in the publication of Zhang and Rasmuson.91 However, the measurement of eutectic points is particularly difficult due to the higher number of adjacent phases that in turn impede equilibration. Thus, we assume that the deviation of the calculated and experimental eutectic point is probably caused by a kinetically inhibited system, being not yet in thermodynamic equilibrium. 4.4. TP/CA CC and CC Hydrate System. In this section, the modeling approach was finally used to correlate the solubilities of the TP/CA/water system at 298.15 K forming a CC (1:1) and a CC hydrate (1:1:1). Both TP and CA form a hydrate (1:1) in aqueous solution. Thus, depending on temperature the hydrate or the anhydrate of CA is thermodynamically stable in aqueous CA solutions as illustrated in Figure 9.

Figure 6. Solubility of theophylline(IV) in acetonitrile91 (circle, dotted line), chloroform91 (hexagon), and chloroform−methanol mixture90 0.8/0.2 x/x (squares). Lines correspond to the PC-SAFT predictions; symbols represent the experimental data points.90,91

son91 assumed TP(I) is the thermodynamically stable form during their solubility measurements. However, they characterized their solid phase with a reference X-ray pattern which equals that of form IV published by Khamar et al.109 Thus, we assume that actually the solubility of TP(IV) was measured by Zhang and Rasmuson.91 Although neither the pure-component parameters nor the melting properties were adjusted to solubility data of TP(IV), the predicted solubility lines are in very good agreement with the experimental data (compare Table 4).90,91 This in turn confirms our assumption that TP(IV) was present during the measurements of Zhang and Rasmuson.90,91 The highest ARD of 13.3% is observed in acetonitrile exhibiting the lowest solubility of TP(IV), where already small experimental uncertainties result in a high ARDs. Figure 7 illustrates the correlated solubilities in the TP/GA/ acetonitrile system at 303.15 K. The solubility of TP(IV) on the TP/acetonitrile axis in Figure 7 equals that of TP(IV) in acetonitrile in Figure 6 at 303.15 K. In accordance to the upper section, also the solubility of TP(IV) in the presence of totally dissolved GA was predicted via eq 3 using the respective purecomponent parameters from Table 1 and the melting 4445



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Finally, the ternary phase diagram of the TP/CA/water system was constructed using the information on the binary systems TP/water and CA/water. It becomes obvious from Figures 5 and 9 that both TP and CA are thermodynamically stable as the respective hydrate form at 298.15 K. Consequently, the solubilities of the hydrates in the presence of the respectivetotally dissolvedother component were correlated via eqs 1 and 5 using the corresponding reference solubility products Ks and reference enthalpies Δhref s from Table 6. Afterward, the CC and CC hydrate solubility products Ks were determined based only on one solubility data point, representing the eutectic point of both solid phases (marked as star in Figure 10). Figure 8. Solubilities for the theophylline (TP)/glutaric acid (GA)/ chloroform system at 303.15 K with shortened axes in mole fractions. Solid lines correspond to PC-SAFT predictions; the cocrystal (CC) and TP(IV) solubility lines are fully predicted; symbols refer to the experimental data points91 of GA (black), the CC (gray), and TP(IV) (white).

Figure 9. Solubility of citric acid (CA) in the binary system CA/water with formation of a 1:1 hydrate. Solid lines correspond to the PCSAFT correlations of CA hydrate and CA; symbols refer to the experimental data points98 of CA. The solid phases are in equilibrium with liquid mixture containing water and totally dissolved CA. The star is the hydrate solubility point101 used for the calculation of the solubility product Ks.

Figure 10. Solubilities for the theophylline (TP)/citric acid (CA)/ water system at 298.15 K with shortened axes in mole fractions. Solid lines correspond to PC-SAFT correlations); symbols refer to the experimental data points101 of CA hydrate (black circle), the eutectic point CA hydrate/cocrystal (CC) (black and light gray semicircles), the eutectic point CC/CC hydrate (light gray/gray star), the eutectic point CC hydrate/TP hydrate (white and gray semicircles), and TP hydrate (white circle). The star is the solubility point101 used for the calculation of the solubility products Ks for the CC and the CC hydrate.

The solubility of the anhydrate form in water was correlated using eq 3, simultaneously with the adjustment of the binary interaction parameter between CA and water (Table 2). The solubility of the hydrate was calculated using a hydrate solubility product Ks (eq 1) based on the hydrate solubility point101 at 298.15 K as reference (marked as star in Figure 9). The reference melting enthalpy of the CA hydrate Δhref was calculated via eq 5 using an additional hydrate solubility data point. For this purpose, the solubility data point of CA at 98 Ttrs was used where the solubility of the anhydrate/hydrate = 307 K hydrate equals that of the anhydrate which in turn was correlated. Thus, no additional solubility point of the CA hydrate was needed. The solubility curve of water in CA was estimated using eq 3. Figure 9 compares the finally correlated and experimentally determined solubility data for water, CA, and CA hydrate.98 In contrast to TP (Figure 5), the solid water phase in equilibrium with liquid solution can be depicted. Further, solubility data of the anhydrate form are available due to the relatively low transition temperature Ttrs anhydrate/hydrate = 307 K (compare Table 5). The transition between hydrate and anhydrate form at 307 K is visible as a slight kink in the solubility line. The correlated solubilities show ARDs of less than 2.27% (compare Tables 2 and 4) and are in high accordance with the experimental data.98

Figure 10 shows the correlated solubilities of the TP/CA/ water system. The solubility of the TP hydrate is much lower than that of the CA hydrate. Thus, also the CC solubility is higher than that of the CC hydrate since the concentration range in which pure CCs are formed is close to that of CA hydrate. Although only few solubility data are available in the literature, the phase behavior of this complex system can be correlated in good agreement with the experimentally determined eutectic solubility points. As listed in Table 4, the ARD for the CC hydrate refers to 8.47%, whereas the ARD of the CC is even much lower (1.80%). This means that using one and the same pure-component parameters of TP, the solubility of six different solutes can be correctly correlated, namely, TP hydrate, TP(IV), TP(II), TP(I), TP/CA CC, as well as the corresponding CC hydrate (Figures 5 and 10). This in turn demonstrates the ability of our thermodynamic approach that allows us to reliably determine the concentration range in which CCs form. 4446



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5. CONCLUSIONS This study presents a thermodynamic approach accounting for the complex relationships in CC systems, including polymorphic transitions and hydrate formation. The solubilities of the kinetically stable and prevalent polymorph TP(II) as well as of the TP hydrate were modeled accounting for thermodynamic nonidealities. Using this information, the melting properties and therewith the solubility of the thermodynamically stable TP(IV) could be precisely predicted. Subsequently, the complex phase behavior of the binary TP/water system, including seven different crystalline phases, could be constructed with a minimum of experimental data. Further, CC solubility was predicted for the TP/GA/ chloroform system in almost quantitative agreement with experimental data using the information on only one CC solubility point in acetonitrile. Finally, the model could also reliably determine the complex phase behavior for the aqueous TP/CA system exhibiting 10 different crystalline phases including formation of TP hydrate, CA hydrate, TP/CA cocrystal, and the corresponding CC hydrate. The proposed approach allows modeling solubilities of polymorphs, hydrates, CCs, and CC hydrates in complex systems. In contrast to existing approaches, it accounts for thermodynamic nonidealities influencing solubilities and therewith the phase behavior of these complex systems that in turn provides a more reliable estimation of the concentration region in which CCs form as well as the prediction of CC solubilities in other solvents.

■

T TSL i Tref νi xi

temperature [K] melting temperature of component i [K] reference temperature [K] stoichiometric coefficient of component i [−] mole fraction of component i [−]

Abbreviations

API CC CF ARD TP CA GA PC-SAFT

active pharmaceutical ingredient cocrystal coformer average relative deviation theophylline citric acid glutaric acid perturbed-chain statistical associating fluid theory

Greek symbols

γi ui/kB εAiBi/kB κAiBi σ

activity coefficient of component i [−] dispersion energy parameter [K] association energy parameter [K] association volume parameter [−] segment diameter [Å]

Subscripts

i, j component i, component j

■

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: ++49 (0)231 755 2635. Fax: ++49 (0)231 755 2572. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the CLIB-Graduate Cluster Industrial Biotechnology. NOTATION a molar Helmholtz energy [J mol−1] ai activity of component i [−] ΔcSL difference of the heat capacity of the solid and the liquid P,i component i at its melting point [kJ K−1 kg−1] ci concentration of component i [mol L−1] ref Δhs reference enthalpy of complex formation [kJ kg−1] ΔhSL heat of fusion of component i [kJ kg−1] i kij(T) binary interaction parameter [−] kij,slope slope of the temperature-dependent binary interaction parameter [K−1] kij,int intercept of the temperature-dependent binary interaction parameter [K] Kγ product of the activity coefficients [−] solubility product [−] Ks mseg number of segments of component i [−] i nexp number of experimental data points [−] ni mole number of component i [mol] qi charge of component i [−] R gas constant [J mol−1 K−1] 4447



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