Predicting the Effect of pH on Stability and Solubility of Polymorphs

Article pubs.acs.org/crystal

Predicting the Effect of pH on Stability and Solubility of Polymorphs, Hydrates, and Cocrystals Linda Lange, Miko Schleinitz, and Gabriele Sadowski* Department of Chemical and Biochemical Engineering, Laboratory of Thermodynamics, TU Dortmund University, Emil-Figge-Strasse 70, D-44227 Dortmund, Germany S Supporting Information *

ABSTRACT: Cocrystal formation processes from aqueous solutions are often affected by pH-dependent dissociation, polymorphic transitions, and formation of hydrates and salts. To enhance the efficiency of those processes, the aqueous stability and solubility of pharmaceutical cocrystals were predicted in this study using the perturbed-chain statistical associating fluid theory (PC-SAFT). The solubilities in the binary systems caffeine/water and oxalic acid/water were modeled including hydrate formation and polymorphic transitions between the corresponding anhydrate forms I and II. Moreover, pH-dependent solubilities of these hydrate-forming components, their 2:1 cocrystal, and all appearing salts were measured and modeled at 298.15 K. It was found that the pH-dependent acid−base equilibria of caffeine and oxalic acid directly influence the stability and solubility of their cocrystal, their hydrates, and salts. In consideration of the thermodynamic nonideality of the components in the cocrystal system, PC-SAFT enables solubility predictions of the before-mentioned components as well as if any cocrystal is formed at given conditions of pH and temperature.

1. INTRODUCTION Pharmaceutical cocrystals (CCs) represent a promising class of solid forms as they can potentially improve the physicochemical properties of the solid active pharmaceutical ingredients (APIs) compared to their pure state.1−8 These CCs are stoichiometrically composed of the API and at least one coformer (CF) in the same crystal lattice,9 interacting via weak noncovalent interactions. Upon dissolution, CCs dissociate into their components therewith releasing the dissolved API in the same state as upon dissolution from the pure API crystal. At the industrial scale, the favored method for CC formation is the crystallization from solution.10−13 In aqueous solutions, however, the CC formation is commonly interfered by pHdependent dissociation as well as by salt formation of API and CF. Further, polymorphism and hydrate formation of the API, CF, or even of the CC are well-known phenomena occurring in API/CF/water systems. As demonstrated in previous studies,10,14−18 effective CC formation by crystallization from solution requires the knowledge of the thermodynamic phase diagram, particularly for systems exhibiting strongly nonideal phase behavior. For a given system of API, CF, and solvent (here water), this phase diagram depicts the concentration range in which the desired CC is formed, as was already earlier illustrated for systems exhibiting dissociation and salt formation16 and for systems including polymorphic transitions and hydrate formation.18 However, polymorphism,19,20 as well as formation of salts and hydrates, impede in turn the reliable and accurate measurement of these phase diagrams, resulting in a high experimental effort.10,12−15 © 2016 American Chemical Society

Several approaches were developed to increase the efficiency of CC formation processes, identifying the concentration range in which stable CCs are formed (CC stability).21−24 Most of these approaches explicitly consider the solubility of the CC using a so-called solubility product Ks:14,16−18,25 Ks =

∏ ai v

i

= (xAγA )vA ·(x BγB)vB

i

(1)

The solubility product in eq 1 can be adapted for solubility calculations of any solid complex, whether a CC, a salt, or a hydrate. In the case of a CC, A and B in eq 1 equal the neutral components API and CF, whereas for a hydrate, one of the components is specifically water. For a salt, A corresponds to the ionic API (or CF) species and B refers to the counterion. In eq 1, the mole fractions and activity coefficients of A and B of the saturated solution are represented by xi and γi, respectively, whereas ai is the thermodynamic activity of these components. The solubility product Ks depends on the temperature, but not on the solvent nor on concentration. Thus, it can be determined based on only one solubility data point of CC, salt, or hydrate, regardless of solvent, concentration, and pH. However, in almost all studies26−34 the solubility product Ks is calculated neglecting the activity coefficients. The resulting is described by eq 2: Kideal s Received: May 2, 2016 Revised: June 2, 2016 Published: June 14, 2016 4136

DOI: 10.1021/acs.cgd.6b00664 Cryst. Growth Des. 2016, 16, 4136−4147

Crystal Growth & Design K sideal

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

Article

∏ xi v = (xA)v (xB)v i

A

cannot be neglected and become increasingly important at very low solubilities. APIs and CFs usually exhibit functional groups that are ionizable in aqueous solutions upon pH change. During ionization of an acid AH, it dissociates into the ionized form A− and a proton-forming a hydronium ion (H3O+) in aqueous solution:

B

i

(2)

Kideal s

might be applicable for poorly soluble solutes when referring to the infinite-dilution reference state. However, this approach is not suitable for highly soluble components as investigated in this study. In contrast to these approaches, the thermodynamic nonideality of all components in aqueous CC solutions is factored into the solubility calculations of this work. Ks is determined applying PC-SAFT-predicted activity coefficients that constitute deviations from the pure-component reference state. PC-SAFT35 was already used in earlier studies to model and predict the activity coefficients of the neutral and ionic components involved in binary and ternary systems,35−39 including those with formation of CC,14,40,41 salts,42 and hydrates.16,18,40 In aqueous systems, PC-SAFT has been successfully applied for modeling pH-dependent solubilities42−47 of amino acids,43 as well as of APIs and their corresponding salts.42 On the basis of this, also CC solubilities in aqueous solutions could be predicted as a function of pH and temperature in excellent agreement with experimental data, accounting API dissociation, and salt formation.16 In a further study, the CC solubility in aqueous solutions was modeled for systems exhibiting polymorphism and hydrate formation.18 In extension to these studies, we now present the prediction of CC solubilities in aqueous systems including both pHdependent dissociation and salt formation as well as polymorphism and hydrate formation. Furthermore, we demonstrate that this approach is appropriate not only for prediction of solubility but also for stability predictions of CCs as a function of pH. The modeling results were validated by comparison with experimental data of caffeine (CAF) and oxalic acid (OA), both being polymorphic substances exhibiting formation of hydrates and CCs.

Ka

AH + H 2O ↔ A− + H3O+

The dissociation equilibrium is characterized by the dissociation constant Ka that corresponds to the activity product of the involved species: γA−γH O+ x A−x H O+ a A−a H3O+ 3 3 Ka = = aAHa H2O γAHγH O xAHx H2O (5) 2 The relationship between the commonly used acid constants Kcacid and Ka is presented in eq 6 where Kγ is obtained from the activity coefficients of the respective species according to eq 5: c K acid =

K a,1

AH 2 + H 2O ←→ AH− + H3O+ K a,2

AH− + H 2O ←→ A2 − + H3O+

(6)

(7) (8)

In eq 7, Ka,1 describes the dissociation equilibrium of the first step, whereas Ka,2 in eq 8 describes that of the second step. The two dissociation constants Ka,1 and Ka,2 can be estimated analogous to Ka in eq 5. This approach is applicable to polytropic acids AHz that can dissociate in z dissociation steps each described by the respective equilibrium constant. The ionization of a base B is caused by protonation to BH+ as shown in eq 9. In correspondence to a polytropic acid, a polytropic base can be protonated z times to BHz+. Ka

BH+ + H 2O ↔ B + H3O+

(9)

This equilibrium can be described by dissociation constant Ka but this time considering the ionized form BH+ as an acid. γBγH O+ x Bx H O+ aBa H3O+ 3 3 Ka = = aBH+a H2O γBH+γH O x BH+x H2O (10)

⎡ Δh SL ⎛ ⎞ ΔcPSL,0i 1 ⎢ − 0i ⎜ 1 − T ⎟ − exp R ⎢⎣ RT ⎝ γi L T0SLi ⎠ ⎛ T SL T SL ⎞⎤ ⎜ 0i − 1 − ln 0i ⎟⎥ T ⎠⎥⎦ ⎝ T

Ka c− c H2O = c H3O+ A Kγ cAH

As given in eqs 7 and 8, a two-protic acid AH2 dissociates in two steps. The neutral form AH2 dissociates to the oncecharged ion AH−, which further dissociates to a twice-charged ion A2−, each of them generating one hydronium ion (H3O+).

2. THEORY 2.1. Solubility Calculations and Dissociation Equilibria. The solubility of the neutral API (or CF) species is determined accounting for a thermodynamic equilibrium between a pure solid phase and the liquid solution as postulated by Prausnitz et al.:48 xiL =

(4)

2

c Again, the corresponding acid constant Kacid can be calculated as described in eq 11.

(3)

c K acid =

In eq 3, xLi corresponds to the mole fraction of neutral component i (API or CF) in the liquid phase, also referred to as intrinsic solubility. T represents the temperature of the system and R is the ideal gas constant. Further, the melting temperature TSL 0i and heat of fusion of component i (API or CF) ΔhSL 0i as well as the difference in the solid and liquid heat capacities of component i at its melting point ΔcSL p,0i are considered. The activity coefficients referring to the purecomponent reference state γLi of API or CF were calculated in this work by PC-SAFT. As they depend on temperature and on the concentrations of all components in the liquid mixture, they

Ka c c H2O = c H3O+ B Kγ c BH+

(11)

Kcacid values are for many acids and bases given in the literature. However, Kcacid values are usually measured by titration methods49,50 and extrapolated to infinite dilution. Moreover, they solely describe the concentration product of the species involved in an acid−base equilibrium, but do not account for the influence of other species present in the aqueous solution. Consequently, they are not valid at high concentrations, particularly in the case of strongly nonideal interactions between the different species. The concentrations 4137


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of the highly soluble components considered in this work are far away from infinite solution. Thus, equilibrium constants Ka were used in this work that account for the thermodynamic nonideality in an aqueous system. The corresponding Ka-values are experimentally not available and were therefore adjusted to experimental solubility data. The total amount of an acid (or a base) in solution is summarized by its neutral and ionized species. In the case of an acid AHz that can dissociate z times, this value corresponds to nacid = nAHz + nAH−z−1 + ... + n Az−1 (12)

2.2. PC-SAFT. In this work, the activity coefficients γi in eqs 1, 3, and 15 are calculated via PC-SAFT, a thermodynamic model using the hard-chain as a reference system.35,36 As given in eq 17, the residual Helmholtz energy aresidual equals the sum of different independent contributions, namely, those accounting for repulsive interactions (hard chain), for attractive forces (dispersion), for hydrogen-bonding interactions (association), as well as for Coulomb interactions (ion):

Accordingly, the total amount of a base B, that can be protonated z times, amounts to nbase = nB + nBH+ + ... + nBHz (13)

In PC-SAFT, each molecule is characterized by three purecomponent parameters: the number of segments mseg i , the segment diameter σi, and the dispersion-energy parameter ui/ kB.35 For associating components, two additional parameters are required: the association-energy parameter εAiBi/kB and the association-volume parameter κAiBi.36 In mixtures of substances i and j, the segment diameter σij in eq 18, as well as the dispersion-energy parameter uij in eq 19 are determined applying the Berthelot−Lorentz combining rules.52

a residual = a hard chain + adispersion + aassociation + a ion

The amount of neutral and ionized species can be determined using the above-mentioned dissociation equilibria (eq 5 or 10), whereas the electroneutrality in a solution of N ionic species, each carrying a charge qi, is considered by eq 14. N

∑ qini = 0

σij =

(14)

i=1

The API (or CF) solubility in the presence of the CF (or API) was calculated using eq 3, as well as the dissociation equilibria (eq 5 or 10), and the total amount of an acid (or a base) (eq 12 or 13), regarding the overall electroneutrality (eq 14). Formation of salts, hydrates, or CC was calculated using the solubility product Ks in eq 1, again together with the corresponding dissociation equilibria (eq 5 or 10), the associated increase of solubilities (eq 12 or 13) due to pH, as well as the overall electroneutrality (eq 14). In aqueous CC systems, also hydrate formation of salts may occur. The solubility of the latter was calculated using the solubility product Ks similar to that for CCs, salts, and hydrates in eq 1 as follows: Ks =

∏ ai v

i

(19)

Additionally, a binary interaction parameter kij is introduced in order to correct the cross-dispersion energy between two substances. According to eq 20, kij was assumed to linearly depend on temperature: kij(T ) = kij ,slopeT + kij ,int

(20)

As postulated by Held et al.,53 dispersion interactions between noncharged species, between ions and noncharged species, as well as between anions and cations were considered, but whereas two anions or two cations they were not. Besides, cross-associating interactions between two associating components i and j were considered. Therefore, the combining rules derived by Wolbach and Sandler54 were required for calculation of the association-energy parameter εAiBi/kB in eq 21 and the association-volume parameter κAiBj in eq 22.

−

2

i

(15)

As described for CCs, salts, and hydrates, also the salt hydrate solubility is calculated considering the pH-dependent dissociation. All considered solubility products of salts, salt hydrates, hydrates, or CCs depend on temperature. In accordance with the Gibbs−Helmholtz equation, the solubility product Ks at any temperature can be estimated using a reference solubility ref product Kref and a reference s at a reference temperature T enthalpy Δhref : s ln K s = ln K sref

(18)

uij = (1 − kij) uiuj

= (x A−γA−)v A ·(x BH+γBH+)vBH+ ·(x H2OγH O)vH2O

Δhsref ⎛ 1 1⎞ ⎜ ref − ⎟ + ⎝ R T T⎠

1 (σi + σj) 2

(17)

ε Ai Bj =

1 Ai Bi (ε + ε Aj Bj) 2

Ai Bj

Ai Bi Aj Bj ⎜

κ

=

κ

κ

(21)

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

(22)

The number of association sites Ai and Bi of a component i were chosen as proposed by the associating schemes defined by Huang and Radosz.55 Interactions between cations and anions were taken into account in eq 23 using a Debye−Hückel56 term as already stated by Cameretti et al.:57

(16)

In eq 16, the reference enthalpy Δhref s refers to the enthalpy of fusion of the complex at the reference temperature Tref. As Δhref s is usually not experimentally available, it was calculated in this work by using an approach described by Folas et al.51 As to be seen in eq 16, the logarithm of the solubility product of any complex Ks conforms to a linear function of the negative inverse temperature −1/T. Therefore, Δhref s can be estimated from the slope and axis intercept of this linear function on the basis of at least two solubility data points of the respective salt, salt hydrate, hydrate, or CC at two different temperatures.51

a ion κ =− kBT 12πkBTε

∑ xiqi2χi i

(23)

The parameter χi is characterized by eq 24 with κ being the socalled Debye screening length. 4138


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3 ⎡3 ⎢ + ln(1 + κqi) − 2(1 + κqi) (κqi)3 ⎣ 2 +

⎤ 1 (1 + κqi)2 ⎥ ⎦ 2

differential scanning calorimeter (DSC) to verify whether OA hydrate or OA anhydrate was present during the solubility experiments. The apparatus (Q100, TA Instruments GmbH, Eschborn, Germany) was calibrated by using pure indium. Samples of around 3.5 mg suspension were heated in aluminum pans (TA Instruments GmbH, Eschborn, Germany) that were capped with a lid including a hole allowing evaporation of the remaining organic solvent during heating. This analysis was performed once for each sample using TA Universal Analysis Software (TA Instruments GmbH, Eschborn, Germany). As an example, Figure 1 compares the heat flow of a suspension containing acetonitrile and OA anhydrate with a suspension containing

(24) ion

Describing the Helmholtz-energy contribution a by eqs 23 and 24 only requires the knowledge of the charge qi but no additional pure-component parameters.

3. MATERIALS AND EXPERIMENTAL METHODS 3.1. Materials. Caffeine was purchased as crystalline powder from Alfa Aesar (Germany) with a purity of ≥99.00%. Oxalic acid, also used as crystalline powder (98.00%), was obtained from Sigma-Aldrich (Germany). Acetonitrile (≥99.9%), butyl acetate (≥96.00%), ethyl acetate (≥99.80%), and silica gel orange were purchased from SigmaAldrich (Germany). Sodium hydroxide was supplied as pellets (≥98.00%) from Bernd Kraft GmbH (Germany), whereas potassium dihydrogen phosphate (≥99.50%), sodium dihydrogen phosphate dihydrate (≥99.00%), hydrochloric acid (2 mol/L and 37%), and CombiTitrant 5 (≥99.99%) were purchased from Merck KGaA (Germany). All substances were used without further purification steps as obtained from the manufacturer. Water was filtered, deionized, and distilled with a Millipore purification system. 3.2. Measurement of OA Solubility in Organic Solvents. Solubility measurements of OA anhydrate were performed in acetonitrile, ethyl acetate, butyl acetate, and an acetonitrile/ethyl acetate mixture (0.5/0.5 w/w) at 293.15 K, 298.15 K, 303.15 K, 310.15 K, and 318.15 K. The solubility was determined from a supersaturated solution adding an excess of OA anhydrate in each solvent (50 mL). The solution at a low ratio of solid to liquid phase was mixed using a magnetic stirrer to ensure sample equilibration. Further, the solution was tempered by a heating jacket, whereas the temperature of the solution was determined by a PT100 element (accuracy of ±0.1 K). Prior to the experiments, it was found that OA forms hydrates even in organic solvents, probably with water from the atmosphere. To avoid hydrate formation during equilibration, OA anhydrate, stored under a vacuum before, was filled into the glass vessel, and the whole equipment including periphery was purged afterward with nitrogen for at least 60 min. The equipment also includes a second vessel filled with silica gel (50 mL) which was connected to the solubility vessel in order to bind potentially remaining water. Finally, the solvent was injected into the glass vessel, again in countercurrent nitrogen flow which was applied during any filling or sampling procedure. However, to avoid solvent evaporation during equilibration, the nitrogen flow was turned off as soon as a filling or sampling procedure was finished and the solubility equipment was sealed. For each temperature and each solvent, the solution was stirred for at least 48 h, which was prior to the experiments found to be appropriate to reach equilibrium. After equilibration, samples were taken from the liquid phase through a semipermeable membrane using a syringe. The removed sample was filtered (pore size 0.2 μm), and the concentration of OA was estimated by UV−vis spectrometry (Eppendorf BioSpectrometer, Hamburg, Germany) at 260 nm. Calibration curves were prepared for every solvent (coefficient of determination ≥0.99995) using standard solutions of known OA concentrations. To ensure reproducibility, every photometric analysis was executed three times for every sample and the average value is reported. Further, the water content of every sample was determined by using Karl Fischer titration (915 KF Ti-Touch, Metrohm GmbH & Co. KG, Fliderstadt, Germany) and the reagent CombiTitrant 5. It was found that the water content was less than 0.33 wt % for all investigated samples. To analyze the corresponding solid phases, a suspension of solid and liquid phase was also sampled via a syringe through the semipermeable membrane. During the whole sampling process and the following analysis, the solid phase was covered by the liquid phase to avoid hydrate formation of OA anhydrate with atmospheric water. The latter was performed qualitatively using a linear temperature

Figure 1. DSC data (heat flow) of suspensions containing acetonitrile and oxalic acid anhydrate (form II) (black solid line) or oxalic acid hydrate (gray dashed line).

acetonitrile and OA hydrate. Prior to these analyses, the OA anhydrate was stored under a vacuum, whereas the OA hydrate was obtained from a supersaturated aqueous solution to ensure that OA is either totally the anhydrate or the hydrate. The analyses were performed directly after preparation of the respective suspensions to avoid anhydrate/hydrate back formation. During the measurement, an ambient atmosphere (instead an inert atmosphere) was ensured to avoid abrupt dehydration of a potentially occurring hydrate prior to the transition temperature Ttrs II/hydrate between OA anhydrate (form II) and OA hydrate. The heating procedure started at 318.15 K with a linear heating rate of 2 K min−1. The first peaks just before a temperature of about 340 K correspond to the evaporation of acetonitrile, whereas the size of these peaks depends on the liquid-tosolid ratio in the suspension. For the suspension of acetonitrile with the OA hydrate, another significant peak is visible at a temperature of approximately 350 K referring to dehydration of OA hydrate. After 2 min holding at 378.15 K which is close to the transition temperature 58 (visible as slight kink in the heatflow lines in Ttrs II/hydrate = 373.15 K Figure 1), the heating continued with 2 K min−1 up to approximately 398.15 K. The peak at approximately Ttrs I/II = 393.20 K refers to the transition between the OA anhydrate forms I and II.59 The experimental data and the standard deviations are listed in the Supporting Information. 3.3. Measurement of Aqueous Solubilities As a Function of pH. Aqueous solubilities of CAF and OA in water and in solutions of both CAF and OA were measured at 298.15 K as a function of pH. The measurements were performed in a 50 mL glass vessel that was tempered using a heating jacked. Temperature was determined via a PT100 element, whereas the pH was controlled by applying a pH meter (Mettler Toledo InLab 413 SG/2m, Schwerzenbach, Switzerland). Initially, a solution was prepared exhibiting an excess of CAF or OA in water. Then, hydrochloric acid or sodium hydroxide was added to the resulting aqueous solutions to fit the desired pH-values. After every addition of CAF, OA, hydrochloric acid, or sodium hydroxide, the solution was mixed for at least 48 h using a magnetic stirrer. Equilibrium was regarded when no change in pH-values was observed within this time. 4139



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After equilibration, samples from liquid phase were taken, filtered (pore size 0.2 μm), and analyzed via UV−vis spectrometry (Eppendorf BioSpectrometer, Hamburg, Germany). Because of the pH-dependent absorbance of OA, the pH-value of these samples was adjusted to pH 7 prior to the analysis using a buffer system consisting of disodium hydrogen phosphate dehydrate and potassium dihydrogen phosphate. The concentration of CAF in solutions, which did not contain OA, was measured at 273 nm, whereas the concentration in solutions containing OA was quantified at 208 nm. However, CAF shows also a significant absorbance at 208 nm, and even OA at higher concentrations is measurable at 273 nm. Thus, the concentrations of CAF and OA in mixed solutions of both components were measured at BOTH wavelengths, 208 and 273 nm, whereas the absorbance of the mixture amounts to the sum of the single absorbance of CAF and OA, respectively. Absorbance/concentration calibration curves were collected prior to the measurements. Every photometric analysis was performed three times, whereas the average value is presented. Further, the solid phase of each solution was analyzed via X-ray diffraction (XRD, Miniflex, Rigaku, Japan). The experimental data points with the standard deviations as well as the corresponding solid phases are reported in the Supporting Information.

parameter set for CAF (or OA) can be used to describe their activity coefficient in any solution, irrespective of the polymorphic form. Additionally, one hydrate form of CAF and of OA occurs in aqueous solutions. The pure-component parameters of CAF, of all considered solvents, namely, water, acetonitrile, ethyl acetate, and butyl acetate, as well as those of the ions H3O+, OH−, Cl−, and Na+ were taken from the literature. The pure-component parameters of OA were fitted to solubility data of form II the thermodynamic stable form under ambient conditionsin the organic solvents acetonitrile, butyl acetate, and ethyl acetate as no dissociation takes place in these solvents. The parameterfitting procedure was performed as described in a prior study of Ruether and Sadowski.25 The pure-component melting properties of CAF and OA, required for the solubility calculations (eq 3), are summarized in Table 1. The heat capacities for liquid CAF and OA are not Table 1. Melting Properties of Caffeine (CAF, Forms I and II) and Oxalic Acid (OA, Forms I and II)

4. RESULTS AND DISCUSSION 4.1. Estimation of PC-SAFT Parameters. In this work, the phase behavior of the CAF/OA/water system was investigated including formation of CC, salts, hydrates, and salt hydrates. The solubilities of the amphoteric API CAF, the acidic CF OA, as well as of all above-mentioned solid complexes were modeled using PC-SAFT for the calculation of the respective activity coefficients γLi in eqs 1, 3, 5, 10, and 15. The application of PC-SAFT in turn requires the purecomponent parameters of the neutral and ionized species of CAF and OA, that of all considered solvents and small ions (H3O+, OH−, Cl−, and Na+) as well as the corresponding binary interaction parameters. The total amount of the monoprotic amphoteric CAF in aqueous solutions involves the neutral species, CH, as well as the ionized species, C− and CH2+: nCAF = nCH + nCH+2 + nC− (25)

component

T0iSL [K]

Δh0iSL [kJ/mol]

Δcp.0iSL [J/mol]

CAF(I) CAF(II) OA(I)

509.15 500.39 462.65

21.60 22.45 18.58

101.52 101.52 93.00

OA(II)

455.15

19.60

93.00

ref 61, 63, 64 [61, 64, this work] [61, 62, 65, 66, this work] 61, 62, 65, 66

available from the literature due to decomposition in the course of melting.60 Instead, these properties were estimated using the group-contribution method of Kolská et al.61 Further, it was assumed that the different polymorphs of one component have the same heat capacities. Additionally, the melting enthalpy of OA(I) was calculated via the group-contribution method of Marrero and Gani.62 All pure-component parameters used in this work are summarized in Table 2. Binary interaction parameters, which were used as unequal to zero, are given in Table 3. The accuracy of the modeling results was quantitatively evaluated using the average relative deviation (ARD) between the calculated xcalc,i and experimental solubility data xexp,i of component i, whereas nexp is the number of experimental solubility points:

The total amount of OA, a two-protic acid, in aqueous solutions is the sum of the neutral species, OAH2, the oncedissociated OAH−, and the twice-dissociated OA2−: nOA = nOAH2 + nOAH− + nOA2− (26) PC-SAFT parameters of the ionic species of CAF and OA were assumed to be equal to those of the respective neutral species. Only the molar mass M of the species was changed by the molar mass of hydrogen that was added to (CH2+) or eliminated from the neutral component (C−, OAH−, and OA2−). The charge of the ionized species was taken into account applying the Debye−Hückel term via eqs 23 and 24. Moreover, the loss of hydrogen ions was considered by correcting the number of association sites. Consequently, the amount of association sites was reduced by one for OAH− or C− (respectively resulting in a 1/1 association scheme) and by two for OA2− (resulting in no association). Polymorphism is reported for CAF and OA. Depending on temperature, two anhydrate forms (I and II) of both CAF and OA are thermodynamically stable. The different polymorphic forms of CAF (or OA) show different solubilities because of different melting properties, whereas the physicochemical properties of dissolved CAF (or OA) do not depend on the respective solid form which is in equilibrium with the liquid (compare eq 3). Thus, one and the same pure-component

ARD = 100

1 nexp

nexp

∑ i=1

xcalc, i − xexp, i xexp, i

(27)

In the ternary CAF/OA/water system, the deviation of experimentally determined and computed data corresponds to that of CAF. The resulting ARDs for the solubilities in water and organic solvents are listed in Table 3, whereas those for the aqueous solubilities as a function of pH are listed in Table 4. The pure-component parameters for (the neutral species of) OA(II) and the corresponding binary interaction parameters were adjusted to the solubility data in acetonitrile, ethyl acetate, butyl acetate, and in an ethyl acetate/acetonitrile mixture (0.5/ 0.5 w/w). Figure 2 compares the PC-SAFT correlations with experimentally determined data. The solubility is highest in ethyl acetate, followed by that in the ethyl acetate/acetonitrile mixture, that in butyl acetate, and finally that in acetonitrile. The almost quantitative solubility calculations, reflected by the ARDs < 4% in Table 3, demonstrates that PC-SAFT allows for 4140



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Table 2. PC-SAFT Pure-Component Parameters for Neutral Caffeine (CH), Ionized Caffeine (C− and CH+2 ), Neutral Oxalic Acid (OAH2), Ionized Oxalic Acid (OAH− and OA2−), Small Ions (H3O+, OH−, Cl−, and Na+), and Solvents Considered within This Work component

M [g/mol]

mi [−]

σi [Å]

ui [K]

εAiBi [K]

κAiBi [−]

assoc scheme

charge [−]

ref

CH C− CH2+ OAH2 OAH− OA2− H3O+ OH− Cl− Na+ water acetonitrileb ethyl acetateb butyl acetateb

194.19 193.19 195.19 90.04 89.04 88.04 19.023 17.008 35.453 22.990 18.015 41.052 88.105 116.16

9.5552 9.5552 9.5552 3.3585 3.3585 3.3585 1.0000 1.0000 1.0000 1.0000 1.2047 2.3290 3.5375 3.9808

2.9590 2.9590 2.9590 2.7486 2.7486 2.7486 3.4654 2.0177 2.7560 2.8232 a 3.1898 3.3079 3.5427

428.51 428.51 428.51 180.14 180.14 180.14 500.00 650.00 170.00 230.00 353.94 311.31 230.80 242.52

827.7 827.7 827.7 1654.78 1654.78 1654.78 0 0 0 0 2424.67 0 0 0

0.02 0.02 0.02 0.02 0.02 0.02 0 0 0 0 0.045 0.01 0.01 0.01

2/2 1/1 2/2 2/2 1/1 0/0

0 −1 +1 0 −1 −2 +1 −1 −1 +1

14 14 14 this work this work this work 53 53 53 53 43 67 35 35

1/1

The expression σ = 2.7927 + 10.11 exp(−0.01775T) − 1.417 exp(−0.01146T) was used.43 bConsideration of induced association in mixtures with associating components. a

Table 3. PC-SAFT Binary Interaction Parameters Considered in This Work and Average Relative Deviations (ARDs) of Calculated and Experimental Solubilitiesa binary parameters kij,slope [−] API/solvent CAF/water CF/solvent OA/acetonitrile OA/butyl acetate OA/ethyl acetate OA/water ion/water OAH−/water C−/water CH2+/water H3O+/water OH−/water Na+/water Cl−/water ion/ion H3O+/Cl− Na+/Cl− solvent/solvent acetonitrile/ethyl acetate a

kij,int [−]

temperature range of the experimental data [K]

ref. for parameters

ref. for exp. data

ARD [%]

−2.79 × 10−4

6.51 × 10−2

293.15−495.13

this work

68, 69

9.29

2.01 × 10−4 6.46 × 10−4 3.79 × 10−4 0

−7.77 × 10−2 −2.29 × 10−1 −1.67 × 10−1 −0.0256

293.15−318.15 293.15−318.15 293.15−318.15 278.15−338.15

this this this this

work work work work

this work this work this work 70

3.77 1.47 3.30 15.31

0 0 0 0 0 −0.007981 0

0.05 −0.06 0.02 0.25 −0.25 2.37999 −0.25

298.15 298.15 298.15

this work this work this work 53 53 53 53

this work this work this work

4.72 4.63 5.53

71 71

0 0

0.654 0.317

53 53

71 71

0

0.002

72

73

273.15−298.15 273.15−298.15

313.15

All other binary parameters were set to zero.

liquid solution is too small to be visible in Figures 3 and 4. To correlate the solubility of the above-mentioned CAF (or OA) polymorphs, the following information was required: PC-SAFT pure-components parameters of CAF (or OA) (Table 2), the binary interaction parameter between CAF (or OA) and water (Table 3), melting properties of CAF (or OA) (I) and (II) (Table 1), transition temperatures between forms I and II Ttrs I/II and form II and hydrate Ttrs II/hydrate (Table 5), as well as the stoichiometry, Ks and Δhref of the CAF (or OA) hydrate (Table 6). The melting temperatures of CAF(I), as well as of OA(I) and OA(II), required for the anhydrate solubility calculations (eq 3), were adopted from the literature. The same applies for the

an accurate correlation of the solubility data in a variety of solvents. 4.2. Hydrate Formation and Polymorphic Transitions in the CAF/Water and OA/Water Systems. In the second modeling step, the solubilities in the CAF/water and OA/water systems were investigated. In both systems, one hydrate form as well as two anhydrate forms I and II are thermodynamically stable depending on temperature.58,59,68 Figure 3 illustrates the phase diagram for the binary system CAF/water, whereas Figure 4 depicts that for the binary system OA/water. The melting point of water in the presence of CAF in Figure 3 or OA in Figure 4 is correlated using eq 3. However, the concentration range of crystalline water in equilibrium with 4141



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Table 4. Average Relative Deviation (ARD) of Calculated and Experimental Solubilities in Aqueous Systems Containing the pH-Modifying Agent Hydrochloric Acid (Rather H+ and Cl−) or Sodium Hydroxide (Rather Na+ and OH−) component caffeine hydrate oxalic acid hydrate sodium hydrogen oxalate monohydrate disodium oxalate CC at pH 1 CC at pH 2 CC at pH 2.3 sodium hydrogen oxalate monohydrate at pH 2 sodium hydrogen oxalate monohydrate at pH 4 caffeine hydrate at pH 2.9 caffeine hydrate at pH 4

components in aqueous solution

temperature [K]

ARD [%]

298.15 298.15 298.15

3.93 0.97 7.96

acid acid acid acid

298.15 298.15 298.15 298.15 298.15

1.48 2.26 1.45 1.22 6.26

caffeine, oxalic acid

298.15

2.50

caffeine, oxalic acid caffeine, oxalic acid

298.15 298.15

1.17 4.95

caffeine oxalic acid oxalic acid oxalic acid caffeine, oxalic caffeine, oxalic caffeine, oxalic caffeine, oxalic

Figure 4. Solubility of oxalic acid (OA) in the binary system OA/water with formation of a 1:2 hydrate. Solid lines correspond to the PCSAFT correlations of OA hydrate, OA(II), and OA(I) solubility (thick) and to the phase boundary (thin) curves; symbols refer to the experimental data points of OA hydrate (light gray circles70). The solid phases are in equilibrium with liquid mixture containing water and totally dissolved OA. The star is the hydrate solubility point70 used for the calculation of the solubility product Ks.

Table 5. Transition Temperatures between the Anhydrate Form I, Anhydrate Form II and the Hydrate of Caffeine and Oxalic Acid Used in This Study component oxalic acid II/hydrate I/II caffeine II/hydrate I/II

Ttrs [K]

ref

373.15 393.20

58 59

353.15 414.15

68 68

heat of fusion of CAF(I), whereas that of OA(I) was calculated via the group-contribution method of Marrero and Gani.62 In contrast, the melting temperature and the heat of fusion of CAF(II), as well as the heat of fusion of OA(II), are not available from the literature. According to the thermodynamic rules of Burger and Ramberger,74,75 an enantiotropic transition between forms I and II requires a lower melting temperature and a higher melting enthalpy of form II compared to form I. These data were adjusted in this work using the modeled solubility data of CAF (or OA) hydrate at Ttrs II/hydrate and those of CAF(I) (or OA(I)) at Ttrs (eq 3), as at that temperatures these I/II solubilities need to be equal to those of CAF(II) (or OA(II)) to fulfill the thermodynamic-equilibrium conditions of the corresponding phases at the transition points. The transition trs temperatures Ttrs II/hydrate and TI/II, listed in Table 5, are available for both systems from the literature. The temperature-dependent solubility of the CAF (or OA) hydrate was calculated via eqs 1 and 16, using the hydrate solubility product Ks from Table 6 at 298.15 K as reference (marked as stars in Figures 3 and 4). The reference enthalpy Δhref for CAF hydrate was adopted from the literature,68 whereas that for OA hydrate was determined via eq 16 using an additional hydrate solubility point at 303.15 K. The reference enthalpies considered within this work are listed in Table 6. As shown in Figure 3, the complete binary phase diagram of CAF/water can be modeled consistently. Although the whole considered temperature range (293.15−495.13 K) is larger than 200 K, the resulting ARD of the obtained solubility curves exhibits only 9.29% (Table 3). Also the solubility lines in the

Figure 2. Solubility of oxalic acid in acetonitrile (stars), butyl acetate (circles, dotted line), ethyl acetate (diamonds), and an ethyl acetate/ acetonitrile mixture 0.5/0.5 w/w (triangles). Lines correspond to PCSAFT correlations; symbols represent experimental data points.

Figure 3. Solubility of caffeine (CAF) in the binary system CAF/water with formation of a 5:4 hydrate. Solid lines correspond to the PCSAFT correlations of CAF hydrate, CAF(II), and CAF(I) solubility (thick) and to the phase boundary (thin) curves; symbols refer to the experimental data points of CAF hydrate (triangles69 and light gray circles68), CAF(II) (white circles68), and CAF(I) (black circles68). The solid phases are in equilibrium with liquid mixture containing water and totally dissolved CAF. The star is the hydrate solubility point69 used for the calculation of the solubility product Ks. 4142



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Table 6. Solubility Products Ks of Hydrates and the Cocrystal, Consisting of Caffeine (CAF) and Oxalic Acid (OA), as Well As That of Salts (or Salt Hydrates) Consisting of OAH− or OA2− and a Respective Counter Ion, Na+, Calculated within This Work complex stoichiometry A:B or A:B:H2O

complex A/B or A/B/H2O

temperature [K]

Δhref [kJ mol−1]

source for Δhref

10−4 10−2 10−1 10−1

298.15 298.15 303.15 298.15

10.7 12.0

68 eq 16

298.15 298.15

Ks [−] × × × ×

CAF hydrate: CAF/H2O oxalic acid hydrate: OA/H2O

5:4 1:2

sodium hydrogen oxalate monohydrate: OAH−/Na+/ H2O disodium oxalate: OA2−/Na+ cocrystal: CAF/OA

1:1:1

3.44 9.93 1.08 1.67

1:2 2:1

1.03 1.83 × 10−5

Figure 5 shows the pH-dependent solubilities of CAF hydrate in aqueous solutions containing sodium hydroxide or

binary system OA/water can be described consistently, as depicted in Figure 4, although the resulting ARD is a bit higher than for the CAF/water system (15.31%, Table 3). However, the experimentally determined solubilities of OA are quite small compared to that of CAF, which probably causes the relatively high ARD. 4.3. Solubility of CAF and OA as a Function of pH. In this section, the pH-dependent solubilities of CAF and OA at 298.15 K were determined. As can be seen from Figures 3 and 4, both components exist as hydrates at this temperature when no pH-modifying agent is added to solution. The solubilities of the CAF and OA hydrates as well as of potentially occurring salts (or salt hydrates) were correlated using eq 1 (or 15), whereas the solubilities at different pH-values were calculated considering the dissociation equilibria (eq 5 or 10). The dissociation constants Ka used in this work account for the thermodynamic nonideality by using activity coefficients. The corresponding Ka-values were fitted to pH-dependent solubilities using the equations for the hydrate or salt (or salt hydrate) solubility via eq 1 (or eq 15), the dissociation equilibria (eq 5 or 10), the related increase of solubilities (eq 12 or 13) due to pH, as well as the overall electroneutrality (eq 14). The fitting procedure was performed in pH ranges in which solubilities are significantly influenced by pH also accounting for the presence of the ions of the fully dissociated pH-modifying agents, namely, HCl (rather H+ and Cl−) or NaOH (rather Na+ and OH−). For CAF, Ka,1, describing the dissociation equilibrium between CH and CH2+ (eq 10), was adjusted to solubilities in the pH range of 0.5−2.5 and Ka,2, describing the dissociation equilibrium between CH and C− (eq 5), was independently fitted to solubilities in the pH range of 10−12.5. The dissociation constants of oxalic acid, Ka,1 and Ka,2, were adjusted in the pH range of 0−5 simultaneously with the salt (hydrate) solubility products Ks of sodium hydrogen oxalate monohydrate and disodium oxalate shown in Table 6. The Ks-values of the latter are in turn determined on the basis of one corresponding salt (hydrate) solubility point (marked as stars in Figure 6). The obtained Ka-values of CAF and OA are listed in Table 7.

Figure 5. pH-dependent solubilities of caffeine (CAF) in aqueous solutions containing hydrochloric acid or sodium hydroxide as a pHmodifying agent at 298.15 K. The line corresponds to the PC-SAFT correlation; symbols represent the experimental data points69 for CAF hydrate.

hydrochloric acid at 298.15 K. Over the whole considered pH range, CAF hydrate is the thermodynamically stable form. For pH-values lower than 2 the overall solubility of CAF hydrate increases with decreasing pH, according to the protonation of CAF as described by eq 10. The same applies for pH-values higher than 10.5 where dissociation of CAF takes place (eq 5). Figure 6 illustrates the solubilities of OA hydrate, sodium hydrogen oxalate monohydrate, and disodium oxalate in aqueous solutions including sodium hydroxide as a function of pH at 298.15 K. The different solid phases are characterized by different symbols. In accordance to eq 5, the solubility of OA increases with pH until a first solubility maximum is reached at pH 0.6. However, a further increase of pH results in a solubility decrease with a solubility minimum. This was followed again by a solubility increase up to a second solubility maximum at pH 4.1. In between the two solubility maxima, sodium hydrogen oxalate monohydrate is the thermodynamically most-stable form, whereas for higher pH-values disodium oxalate is formed. In comparison to CAF, the pH-dependent solubility of OA additionally shows the formation of a salt and a salt hydrate in the investigated pH range. As can be seen from Table 4, the modeling results show a high accordance with the experimental data. The ARDs for OA and its salts are below 8%, whereas the pH-dependent solubility calculation of CAF hydrate even exhibits an ARD of only 3.93%. 4.4. Solubility Predictions in the CAF/OA/Water System As a Function of pH. The solubility of CAF hydrate

Table 7. Acid Constants Kcacid from the Literature, As Well As Dissociation Constants Ka and Reference Enthalpies Δhref Fitted within This Work for Caffeine and Oxalic Acid at 298.15 K component caffeine oxalic acid

dissociation step 1 2 1 2

Kcacid [−] 1.00 1.00 5.89 6.46

× × × ×

−176

10 10−1476 10−277 10−577

Ka [−] 1.02 4.35 7.00 2.00

× × × ×

10−5 10−16 10−1 10−4 4143



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Figure 6. pH-dependent solubilities of oxalic acid in aqueous solutions containing sodium hydroxide at 298.15 K. Lines correspond to the PCSAFT correlations; symbols represent the experimental data points for oxalic acid dehydrate (light gray circle), sodium hydrogen oxalate monohydrate (white triangles), disodium oxalate (dark gray diamonds). The stars are the salt (or salt hydrate) solubility point used for the calculation of the solubility product Ks. Figure 7. Solubilities for the caffeine/oxalic acid/water system at pH 1 (circles, black dotted lines), pH 2 (triangles, black solid lines), and pH 2.3 (squares, gray solid lines) at 298.15 K in mole fractions. Caffeine hydrate and sodium hydrogen oxalate monohydrate are the thermodynamically stable forms, respectively. Water also contains the pH-modifying agent hydrochloric acid (HCl) or sodium hydroxide (NaOH). Lines correspond to the PC-SAFT calculations, whereas the solubility lines at pH 1 (black dotted lines) and pH 2.3 (gray solid lines) are fully predicted using PC-SAFT for calculating the activity coefficients. Symbols refer to the experimental data points of sodium hydrogen oxalate monohydrate (black), the CC (gray), and the caffeine hydrate (white). The star is the CC solubility point used for the calculation of the solubility product Ks.

in the presence of OA as well as the solubility of OA hydrate (or sodium hydrogen oxalate monohydrate/disodium oxalate) in the presence of CAF was calculated via eq 1 using the hydrate (or salt) solubility product Ks (compare Table 6). Moreover, the solubility of the CC formed by the neutral species of CAF and OA (2:1) was determined using eq 1, whereas the corresponding CC solubility product Ks was correlated on the basis of only one CC solubility data point (marked as star in Figure 7) at pH 2. The amount of neutral species at this pH was determined using the dissociation constants Ka from Table 7 in eqs 5 and 10. The resulting Ksvalue is also listed in Table 6. Figure 7 shows the modeled solubilities in the CAF/OA/ water system at pH 1, 2, and 2.3 using Ks from Table 6 and the dissociation constants Ka from Table 7 at 298.15 K. The CC solubilities at pH 1 and 2.3 were fully predicted using PC-SAFT for calculating the activity coefficients in eq 1. The solubilities of CAF hydrate and sodium hydrogen oxalate monohydrate in the absence of the respective other component were adopted from Figures 5 and 6 of the previous section. However, the corresponding solubilities in the presence of the respective other component were fully predicted as no binary interaction parameter between neutral or ionized species of CAF and OA was used. As can be seen from Figure 5, the solubility of CAF hydrate is slightly higher at pH 1 than at pH 2 and 2.3. According to Figure 6, the solubility of sodium hydrogen oxalate monohydrate is even twice as high at pH 1 as at the other pH-values. Further, the concentration ranges at pH 1, in which only CAF hydrate or sodium hydrogen oxalate monohydrate are formed, are smaller than for pH 2 and pH 2.3 (Figure 7). As the concentration range in which pure CCs can be formed is therefore bigger for pH 1 than for pH 2 and pH 2.3, this pH-value is more appropriate for successful CC formation. However, with increasing pH the concentration range in which the stable CCs form decreases. The predicted CC solubilities in Figure 7 are in almost quantitative agreement (ARDs lower than 2.26%) with the experimental data for pH 1 and 2.3. The same applies for the predicted solubilities of sodium hydrogen oxalate monohydrate in the presence of CAF at pH 2, showing an ARD of 6.26%. 4.5. Cocrystal-Stability Predictions in the CAF/OA/ Water System As a Function of pH. Finally, the

thermodynamic model was used to predict the pH-dependent CC stability at 298.15 K. The predictions were performed using the previously determined solubility products of the CC, the CAF hydrate, and the sodium hydrogen oxalate monohydrate from Table 6. Once the CC solubility exceeds that of CAF hydrate and sodium hydrogen oxalate monohydrate, the CC is less stable than the latter. Figure 8 compares predicted and experimentally determined solubilities for the investigated CAF/OA/water system at pH 2.3 and 2.9. For better visibility, only the predicted solubilities of CAF hydrate, sodium hydrogen oxalate monohydrate, and the CC at pH 2.3 are shown in Figure 8 but not the experimentally determined solubility data (illustrated in Figure 7). The solubilities of CAF hydrate and sodium hydrogen oxalate monohydrate in the absence of the respective other component were adopted again from Figures 5 and 6 of the previous section. As can be seen from Figure 5, the solubilities of CAF hydrate at pH 2.3 and 2.9 are almost the same. However, the solubility of sodium hydrogen oxalate monohydrate at pH 2.3 is a bit higher than at pH 2.9. In contrast to pH 2.3, the experimentally investigated solid phases at pH 2.9 did not contain CC but only CAF hydrate. This is in agreement with the solubility predictions at pH 2.9 as the predicted CC solubilities exceed those of CAF hydrate and sodium hydrogen oxalate monohydrate. Figure 9 shows schematically an API/CF/water system for three different cases (a), (b), and (c), which only differ in pH. In cases (a) and (b) CC formation occurs, whereas the concentration range in which stable CCs form decreases from 4144



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same applies for the complex phase behavior of the binary CAF/water system, also including six different solid phases. In addition, the previously unknown melting properties of CAF(II) as well as the hypothetically heat of fusion of OA(II) could be determined. Besides, the dissociation equilibria of the amphoteric CAF and the acidic OA were modeled using dissociation constants accounting for thermodynamic nonidealities via activity coefficients of all considered neutral and ionic species. Simultaneously, the hydrate and salt formation of CAF and OA as well as the formation of their CC were accounted for in the modeling using the respective solubility products. Every solubility product was determined using only the information on one single experimental solubility point at any pH. On the basis of this approach, the solubilities of all abovementioned components could be modeled in high accordance with the experimental data over the whole considered pH range. Furthermore, it was found that besides the solubility the stability of the investigated CC depends on pH. However, the model could also precisely predict the pH up to which stable CCs can form and at which pH they do not form at all. The proposed approach allows predicting solubilities of polymorphs, hydrates, salts, salt hydrates, and CCs in aqueous solutions over a wide range of pH, using PC-SAFT predicted activity coefficients and the above-mentioned dissociation constants and solubility products. In contrast to former approaches, it therefore considers thermodynamic nonidealities affecting solubilities, dissociation reactions, as well as salt and CC formations that in turn offers a more reliable estimation of the pH at which CCs are formed.

Figure 8. Solubilities for the caffeine/oxalic acid/water system at pH 2.3 (gray solid lines) and pH 2.9 (diamonds, black dashed lines) at 298.15 K in mole fractions. Caffeine hydrate and sodium hydrogen oxalate monohydrate are the thermodynamically stable forms, respectively. Water also contains the pH-modifying agent hydrochloric acid (HCl) or sodium hydroxide (NaOH). Lines correspond to the PC-SAFT predictions; symbols refer to the experimental solubility of caffeine hydrate at pH 2.9.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.6b00664. Experimental solubility data, the respective standard deviations and corresponding XRD-patterns of the solid phase (PDF)

Figure 9. Schematic ternary phase diagrams for an active pharmaceutical ingredient (API)/coformer (CF)/water as a function of pH. (a) System with cocrystal (CC) formation, (b) concentration range of CC formation decreased compared to (a), and (c) no CC formation.

■

case (a) to case (b). As can be seen from Figure 7, case (a) compares to the caffeine/oxalic acid/water system at pH 1, whereas case (b) refers to the same system at pH 2 or 2.3. At even higher pH (case c)) CC formation disappears completely. For the caffeine/oxalic acid/water system, this pH was predicted in this work to be 2.9, which is in very good agreement with experimental data as shown in Figure 8. This means that the model allows for reliable predictions not only of the solubility but also of the stability of CCs.

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.

5. CONCLUSIONS This study presented a thermodynamic approach accounting for the complex relationships between dissociation equilibria and the solubilities using the example system of an aqueous solution of CAF and OA, exhibiting polymorphs, as well as formation of hydrates and a 2:1 CC. The solubilities of the (very hygroscopic) OA anhydrate were measured and modeled in organic solvents. Using this information, the complex phase behavior of the binary OA/ water system, including six different solid phases, could be constructed with a minimum amount of experimental data. The

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] concentration of component i [mol L−1] ci Δhref reference enthalpy of dissociation [kJ kg−1] a 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 [−] 4145



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kij,slope slope of the temperature-dependent binary interaction parameter [K−1] kij,int intercept of the temperature-dependent binary interaction parameter [K] Ka dissociation constant [−] Kcacid acid constant [−] Kγ product of the activity coefficients [−] Ks solubility product [−] Kideal solubility product in an ideal solution [−] s 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] T temperature [K] TSL melting temperature of component i [K] i Tref reference temperature [K] vi stoichiometric coefficient of component i [−] xi mole fraction of component i [−] z number of protons that an acid can donate Abbreviations

API CC CF ARD CAF CH CH2+ C− OA OAH2 OAH− OA2− PC-SAFT

active pharmaceutical ingredient cocrystal coformer average relative deviation caffeine neutral species of caffeine protonated species of nicotinamide dissociated form of caffeine oxalic acid neutral species of oxalic acid once dissociated form of oxalic acid twice dissociated form of oxalic 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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