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Interplay between Thermodynamics and Kinetics on Polymorphic Appearance in the Solution Crystallization of an Enantiotropic System, Gestodene Liang Zhu,* Li-yu Wang, Zuo-liang Sha,* Yan-fei Wang, Li-bin Yang, Xiao-yu Zhao, and Wei Du Tianjin Key Laboratory of Marine Resources and Chemistry, College of Chemical Engineering and Materials Science, Tianjin University of Science & Technology, Tianjin 300457, China S Supporting Information *

ABSTRACT: The development of rational pharmaceutical polymorph control systems from crystallization requires the experimental manipulation of both thermodynamic and kinetic factors. Herein, we discuss the interplay between thermodynamics and kinetics on the formation mechanism responsible for concomitant polymorphs and their subsequent phase transformations. The polymorphic system studied is gestodene, which exhibits two enantiotropic polymorphs, I and II. The thermodynamic stability in ethanol is I > II above 18.5 °C and I < II below. At low supersaturation (1.09 to 1.25), plate-like crystals corresponding to form I become the dominant polymorph at T ≥ 19 °C, while at T ≤ 17 °C, needle-like solids corresponding to form II predominate. Solution crystallization at 5 ≤ T ≤ 25 °C and high supersaturation (1.36 to 1.81) results in concomitant polymorphs of forms I and II. The assessments of nucleation and growth kinetics indicate that at lower supersaturations, both nucleation and growth rates of the stable form are higher than that of the metastable one, while at higher supersaturations, the reverse occurs. It is therefore concluded that at lower supersaturations the stable form is favored by both thermodynamics and kinetics and at higher supersaturations concomitant polymorphism is the result of a balance between these competing driving forces. A semiempirical model that displays the influence of initial supersaturation and crystallization temperature on the relative nucleation rate of the two forms was derived and could be used to predict the polymorphic form resulting from nucleation with good accuracy. As the solvent-mediated polymorphic transformation kinetics between forms I and II is relatively fast at 5, 10, 30, and 35 °C, it can reasonably be expected that one can use a slurrying procedure to obtain the pure stable form when concomitant polymorphs appear at conditions of relatively high supersaturations.

1. INTRODUCTION Polymorphs are defined as a solid phase of a given compound resulting from the possibility of at least two different conformations in crystal structures. Due to the differences in arrangement of the molecules, polymorphs of pharmaceutical drugs usually exhibit different physiochemical properties and bioactivity in many cases, including solubility, dissolution rate, density, bioavailability, crystal habit, and intermolecular force.1,2 It is a common requirement in the pharmaceutical industry that a manufactured active pharmaceutical ingredient (API) must be in one, strictly defined polymorph.3,4 However, sometimes it is unfortunate to encounter problems, for example, the particular conditions of crystallization result in the production of multiple polymorphs.5 The simultaneous appearance of two distinct polymorphs of a compound in the same environment is named as “concomitant polymorphism”.6 Concomitant polymorphism is a very common phenomenon in the crystallization process.7−9 Since Wö hler and Liebig10 discovered the concomitant crystallization of the two forms of benzamide for the first time in 1832, concomitant polymorphism has attracted much attention from researchers,11−13 because the appearance of © 2017 American Chemical Society

concomitant polymorphs would affect the quality of the product.14,15 In the recent decade, concomitant polymorphism has been generally supposed to be dependent on the relative nucleation and growth kinetics16 of different polymorphs. Jiang determined the nucleation and growth rates of forms I and II of o-aminobenzoic acid (o-ABA) in an experiment of antisolvent crystallization performed at 298 K by rapidly mixing an ethanol solution of o-ABA with water as antisolvent. It was concluded that concomitant polymorphism of o-ABA was due to the competitive nucleation and growth rates of forms I and II. Teychené and Biscans investigated the nucleation kinetics of eflucimibe polymorphs in ethanol and n-heptane mixture at 35 °C and drew a conclusion that the relative nucleation rate of forms A and B determined the polymorph obtained at the end of the cooling crystallization. These works purported to address the crystallization of selective polymorphs from insight into kinetics at a defined crystallization temperature without relating Received: March 8, 2017 Revised: July 7, 2017 Published: August 21, 2017 4582

DOI: 10.1021/acs.cgd.7b00335 Cryst. Growth Des. 2017, 17, 4582−4595

Crystal Growth & Design

Article

to 35 °C) the solvent-mediated transition temperature (18.5 ± 0.5 °C) at predetermined initial supersaturations from ethanol solutions, it is anticipated that this enantiotropic system would allow us to investigate and report on the interplay between thermodynamics and kinetics that control the polymorph of the resulting solids. Moreover, The interconversions of the two concomitant polymorphs via solvent-mediated polymorphic transformations (SMPTs) were further studied.

to a general problem of the interaction between thermodynamics and kinetics. In 1897, Ostwald17 proposed that the solid first formed on crystallization of a melt or a solution would be the least stable polymorph. Many polymorphic systems in which crystalline forms are monotropically related, including L-glutamic acid,18 2,6-dihydroxybenzoic acid,19 benzamide,20 glycine,21 o- and maminobenzoic acids,22,23 and mannitol,24 offer continued support of Ostwald’s rule with initial crystallization of metastable forms over a wide range of operational conditions. Experimental data on p-aminobenzoic acid in aqueous solution25 showed this to be the case for the crystallization of the two enantiotropically related forms α and β at 15 °C and low supersaturation (1.30 to 1.45). However, at intermediate (1.48) and high supersaturation (1.50 to 1.60), direct crystallizations of concomitant polymorphism and the stable form α were possible. Apart from this, we are unaware of any systematic studies concerning the relationship of polymorphic outcome, crystallization temperature, and initial supersaturation in an enantiotropic system. Gestodene (13-ethyl-17β-hydroxy-18,19-dinor-17α-pregna4,15-dien-20-yn-3-one), a potent synthetic progestin, is available for clinical use through combination with low doses of ethinyl estradiol to provide low dose oral contraceptive preparations.26,27 The molecular structure of gestodene is shown in Figure 1. In the preliminary work,28 11 pure solvents

2. THEORETICAL BASIS 2.1. Nucleation. The classical approach for the estimation of the primary nucleation rate (J) comes from classical nucleation theory,30 which can be expressed by ⎛ ΔGC ⎞ ⎟ J = A exp⎜ − ⎝ kT ⎠

(1)

where A is the pre-exponential factor, ΔGC is the critical free energy barrier, k is the Boltzmann constant (1.38 × 10−23 J K−1), and T is the absolute temperature in kelvin. Generally, the pre-exponential factor (A) depends on the collision frequency between molecules, which is related to the solubility (C0), the interfacial tension (γ), and the energy barrier of diffusion of the molecules from the solution to the crystal (Dsl). The pre-exponential factor can be expressed as A=

⎛ γd ⎞1/2 3 Dsl dm 2(C0Na)7/3 ⎜ m ⎟ ⎝ kT ⎠ 2

(2)

where dm is the molecular diameter. According to classical nucleation theory, the change in Gibbs free energy (ΔG) between the crystalline phase and the surrounding mother liquor is described by ΔG = Figure 1. Molecular structure of gestodene.

4 3 πr ΔG V + 4πr 2γ 3

(3)

where ΔGV is the free energy change per unit volume, γ is the interfacial tension of the nucleus−solution system, and r is the radius of the nucleus (assumed to be spherical in shape for simplicity). The first term on the right-hand of eq 3 is a negative quantity, while the second term is a positive quantity. Thus, at the critical state, the free energy of nucleus formation obeys the condition of d(ΔG)/dr = 0. The critical size of the nucleus (rC) can be expressed as

and 26 binary mixtures of solvents were selected as the crystallization media to produce crystals. It was shown that gestodene crystallized into two crystallographic forms, which were identified as I and II. These two forms of gestodene were characterized by the techniques of X-ray powder diffraction (XRD), differential scanning calorimetry (DSC), and Raman spectroscopy. Accurate crystal structures of forms I and II were determined by single crystal X-ray diffraction (SXRD) method. Solid-to-solid phase transition of form II to I was confirmed preliminarily by thermal analysis of DSC and variable temperature X-ray powder diffraction (VT-XRD). Form II converted endothermically to form I (about 36 °C) on heating, which established the enantiotropic relationship between the two forms. Subsequent research29 focused on the study of the thermodynamic equilibrium, metastable zone widths, and nucleation behavior in the cooling crystallization of a gestodene−ethanol system. Experiments on the determination of metastable zone widths showed that in ethanol solutions, form I, form II, and a mixture of both forms could be crystallized, and concomitant polymorphism appeared over a wide temperature range. Therefore, gestodene crystallizing from ethanol solution has been chosen as a model system. The objective of this work is to make a full record of initial supersaturation and crystallization temperature effects on the crystallization behavior of gestodene. By performing solution crystallization experiments below (5 to 17 °C) and above (19

rC = −

2γ ΔG V

(4)

where ΔGV = −kT ln(S/v) = −kT ln[(C/C0)/v], S is the supersaturation ratio, C and C0 are the concentrations of the supersaturated and saturated solutions, respectively, and v is the specific volume of the solute molecule. The critical energy change (ΔGC) can be obtained by the combination of eqs 3 and 4 as follows: ΔGC =

4 2 πrC γ 3

(5)

On the substitution of rC in eq 5, we get ΔGC =

16πγ 3Vm 2 3(kT ln S)2

(6)

where Vm is the molecular volume. Therefore, the nucleation rate can be expressed from eq 1 as 4583



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⎛ 16πγ 3V 2 ⎞ J = A exp⎜ − 3 3 m 2 ⎟ ⎝ 3k T (ln S) ⎠

Chemical Trade Co., Ltd. (Tianjin, China) and employed without further purification. 3.2. X-ray Powder Diffraction (XRD) Method. X-ray powder diffraction (XRD) was performed on gestodene polymorphs samples using an XD-3 automatic diffractometer system (Puxi, China). The radiation source was Cu Kα (λ = 0.15 nm); the tube voltage and current were set at 40 kV and 40 mA, respectively. The samples were scanned at the room temperature (25 ± 2 °C) from 5° to 75° (2Theta), at a step size of 0.02° and a scanning rate of 2° min−1. For XRD calibration, varying amounts of forms I and II of gestodene were weighed out to a total mass of 0.20 g. The mixtures were homogenized by shaking for 10 min using a reciprocating shaker (type HY-2A, Jintan Shunhua Instrument Co. Ltd., China). The samples were then placed on a zero background holder and scanned from 5° to 75° (2-Theta) with a scanning speed of 2° min−1. Analysis of the XRD data was completed using the software of Jade 6.0. Solid phase collected from the crystallization and solvent-mediated polymorphic transformation (SMPT) experiments were also analyzed in this way. The characteristic peak intensities of form I at 13.9° (2-Theta) and form II at 12.3° (2-Theta) were used to calculate the relative amounts of form I in a mixture of the two forms (as shown in section 4.1, Figure 3). In each case, calculation of the relative peak intensity of form I was achieved by dividing the peak intensity of form I by the total peak intensity of the forms I and II. The correlation curve was constructed from the relative peak intensity of form I measured by XRD and the weight fraction of form I in the mixture. 3.3. Hot-Stage Polarized-Light Video Microscope. Nucleation and crystal growth of gestodene polymorphs were investigated in a hot-stage (type KER3100-08S, Guangmi Instrument Co. Ltd. Shanghai, China) adapted to an optical microscope system (type GMR-213, Guangmi Instrument Co. Ltd. Shanghai, China), as shown in Supporting Information Figure S1. Morphology changes were recorded online and viewed using a 10× magnifying lens. Images were processed and analyzed using GM-2000M software (Guangmi Instrument Co. Ltd. Shanghai, China). 3.4. Focused Beam Reflectance Measurement (FBRM). A focused beam reflectance measurement (FBRM) probe (type D600LC22-K, Mettler Toledo, Swizerland) was used in this work to provide an in situ detection of nucleation. The FBRM probe was equipped with iC FBRM software for recording and analysis of the experimental data. For all the measurements, a frequency of 2 s was employed with a chord count of 5 # s−1 in the 1 to 1000 μm range taken to correspond to the point of nucleation in the solution. 3.5. Exploring of Crystallization Factors Affecting Crystal Forms. Ethanol was selected as the crystallization solvent in this study. The initial supersaturation and crystallization temperature were changed to assess the critical parameters affecting the crystal form. All the crystallizations were conducted using the Easy Max 102 system (Mettler Toledo, Switzerland), which focused on chemical process optimization at the volume of 25−100 mL. A 100 mL automatic reactor was used in this experiment. Solution crystallization of gestodene was performed in batch experiments over the supersaturation range from 1.09 to 1.81 by rapidly cooling an ethanol solution of gestodene. Crystallization temperatures both below (5 to 17 °C) and above (19 to 35 °C) the transition temperature (18.5 ± 0.5 °C) were chosen. Each solution was quickly cooled to the target temperature and maintained with stirring at 300 rpm. Immediately after the nucleation occurred, the crystals were collected by vacuum filtration and evaluated for polymorphic form by XRD and microscope. 3.6. Thermodynamic Stability Analysis. The thermodynamic stability of forms I and II was investigated in the ethanol solution at different temperatures. Slurrying was carried out in an automated lab reactor (Easy Max 102 system) with a volume of 25 mL at a stirring rate of 300 rpm. In the experiments, 0.20 g of forms I and II with a weight ratio of 1:1 was added to 2 mL of ethanol and suspended for 1, 5, and 72 h at 5, 10, 15, 18, 19, 20, 25, 30, and 35 °C. The mixture of forms I and II was weighed using an analytical balance (type AR2140, Ohaus Corp, Pine Brook, USA) and homogenized by shaking for 10

(7)

The induction period (tind) is defined as the time elapsed from the creation of supersaturation in solution to the detection of a new solid phase upon nucleation.31 In practice, the induction period is assumed to be inversely proportional to the nucleation rate (J). t ind ∝

1 J

(8)

By the combination of eqs 7 and 8, we get ln t ind = −ln B +

16πγ 3Vm 2 3k3T 3(ln S)2

(9) 2

where B is a constant. The plot of ln tind against 1/(ln S) is a straight line with a slope given by m=

16πγ 3Vm 2 3k3T 3

(10)

Therefore, the interfacial tension (γ) of the crystal−solution system can be calculated by ⎛ 3mk3T 3 ⎞1/3 ⎟ γ=⎜ 2 ⎝ 16πVm ⎠

(11)

To quantify the competition between nucleation of forms I and II, the relative nucleation rate (JI,II)32 is introduced and expressed as JI,II =

JI JI + JII

−1 ⎡ 3 ⎞⎤⎤ ⎡ 16πV 2 ⎛ γ 3 γ A II m II I = ⎢1 + − 2 ⎟⎟⎥⎥ exp⎢ − 3 3 ⎜⎜ 2 ⎢ ⎢⎣ 3k T ⎝ ln SII AI ln SI ⎠⎦⎥⎦⎥ ⎣

(12)

Equation 12 indicates there are three main variables governing the relative nucleation rate: temperature (T), interfacial tension (γ), and supersaturation ratio (S). As shown in eq 11, the interfacial tension can be assumed as a function of temperature, γ = f(T). Hence, at the predetermined supersaturation and crystallization temperature, the relative nucleation rate can be calculated by JI,II

−1 ⎡ ⎤⎤ ⎡ AII 16πVm 2 3 3 ⎥ ⎢ exp⎢ − 3 3 2 (fII (T ) − fI (T ))⎥ = 1+ ⎢⎣ AI ⎦⎥⎦ ⎣ 3k T ln S

(13)

2.2. Crystal Growth. The crystal growth is a two-step process involving diffusion of the solute molecules from solution to the crystal surface and integration of the crystal surface. The growth rate can be expressed as

G = k G(ln S)n

(14)

where kG is the growth rate constant and n is the growth order, which depends on the growth mechanisms.

3. EXPERIMENTAL SECTION 3.1. Materials. Gestodene (form I, mass fraction purity ≥99.5%) was provided by China Resources Zizhu Pharmaceutical Co. Ltd. (Beijing, China). Ethanol (analytical grade) was purchased from Bolt 4584



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min using a reciprocating shaker. The polymorphic composition of the resulting solids was quantified by XRD analysis. 3.7. Determination of the Induction Period. An Easy Max 102 system combined with a focused beam reflectance measurement (FBRM) probe was employed to determine the induction period of the gestodene−ethanol system. The experimental apparatus is shown in Supporting Information Figure S2. Based on the solubility data of form I, various quantities of gestodene form I and ethanol were transferred into the 100 mL automatic reactor and heated to ensure complete dissolution of the solute (5−10 °C higher than the saturation temperature), then allowed to equilibrate for 1 h. Subsequently, the solution was rapidly cooled at a constant rate of 5 °C min−1 to the required temperature. Rapid cooling rate is crucial to ensure the measured induction period reliable, as the time required to induce a certain level of supercooling must be significantly less than the induction period.33 The reactor was then held isothermally at the corresponding temperature at a constant agitation rate of 300 rpm until nucleation occurred. The time between reaching the required temperature and the detection of the first nucleation events was taken as the induction period. To ensure the accuracy of the experiments, three cycles were employed for each successive run. The trend of crystal counts versus time is shown in Figure 2. The solid forms of the obtained crystals were characterized by XRD and microscope.

Figure 3. XRD patterns of gestodene forms I and II at room temperature (25 ± 2 °C).

Figure 4. Standard curve indicating the relationship between the weight fraction of form I and the calculated relative intensity of form I in a mixture of forms I and II, as measured by XRD.

Figure 2. Sample Easy Max program and experimental counts data at T = 5 °C and S = 1.24.

limit of detection is 2.81%. The root-mean-square deviation (RMSD) was calculated with the help of the following equation:34,35

3.8. Solvent-Mediated Polymorphic Transformations (SMPTs). The solvent-mediated polymorphic transformation (SMPT) experiments were carried out at temperatures of 5, 10, 15, 20, 25, 30, and 35 °C in an Easy Max 102 system. Slurrying procedure was conducted in a 100 mL reactor, where 2.00 g of the pure form was added to the saturated ethanol solution after the solvent temperature had been controlled to the predetermined temperature. The suspension was kept under a constant agitation rate of 300 rpm for 30−1200 min, and then a metal spoon was used to remove multiple samples of the crystals from the suspension. The collected crystals were filtered with a membrane (0.2 μm) and placed into a vacuum oven at 30 °C for 10 h. The composition of the solid phase was analyzed by XRD.

RMSD =

1 N

N

∑ (ωI − ωIcalcd)2 i=1

(15)

where ωI represents the experimental weight fraction of form I, ωIcalcd represents the calculated weight fraction of form I, and N equals the number of experimental data points. It is observed that the method is precise with RMSD of 1.71%. 4.2. Crystallization Conditions Affecting the Polymorphic Outcome of Gestodene Crystals. Figure 5 indicates the nature of the polymorph obtained by cooling crystallization from the ethanol solution. The polymorphs of the resulting solids are changed by two crystallization factors: initial supersaturation (or concentration) and crystallization temperature. Form I was predominantly crystallized at high crystallization temperature (27 °C ≤ T ≤ 35 °C). At intermediate crystallization temperature (19 °C ≤ T ≤ 25 °C) and low initial supersaturation (1.14 ≤ S ≤ 1.28), form I could also be obtained. The crystals were further observed with

4. EXPERIMENTAL RESULTS 4.1. Quantitative Phase Analysis. In this study, the polymorphic forms of gestodene, I and II, were identified by XRD at room temperature (25 ± 2 °C), as shown in Figure 3. The calibration curve for the actual weight fraction of form I (ωI) versus the calculated relative peak intensity of form I (RII) is shown in Figure 4. The correlation curve can be described by the linear equation of ωI = 1.1979RII − 0.2445 with a R2 value of 0.9971 illustrating a good fit of the experimental data. The 4585



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temperature was below 17 °C, form II was crystallized at lower initial supersaturations, and all the crystals showed a needle-like shape as shown in Figure 6b. Concomitant crystallization of forms I and II was observed at relatively high initial supersaturations and at the crystallization temperatures of 5 to 25 °C. Both plate-like form I and needle-like form II were observed by optical microscope as shown in Figure 6c. 4.3. Thermodynamic Stability in the Ethanol. Initially, solvent-mediated polymorphic transformation (SMPT) experiments for the mixture of forms I and II with a weight ratio of 1:1 were carried out to determine the thermodynamic stability of the two forms in ethanol. The polymorphic fraction of samples after slurrying at various temperatures are given in Figure 7. According to Ostwald’s rule of stages, the metastable

Figure 5. Impact of crystallization conditions on the polymorphic form of the resulting solids: (a) mole fraction concentration and crystallization temperature; (b) initial supersaturation ratio and crystallization temperature.

the help of an optical microscope, and all of them were platelike as shown in Figure 6a. When the crystallization

Figure 7. Polymorphic composition of gestodene in the samples after slurrying a 1:1 mixture of forms I and II for 1, 5, and 72 h in ethanol at the temperatures of (a) 5, 10, 15, and 18 °C and (b) 19, 20, 25, 30, and 35 °C.

form should move through each possible polymorphic structure until the thermodynamically stable form crystallizes. The results of slurrying experiments indicated that at the temperatures of 5, 10, 15, and 18 °C, form I converted into form II (Figure 7a), while the form II was converted into form I at the temperatures of 19, 20, 25, 30, and 35 °C (Figure 7b). Therefore, gestodene forms I and II established an enantiotropic relationship with a phase transition temperature of 18.5 ± 0.5 °C. Form II is the stable form below 18 °C, while form I is the stable form above 19 °C. It should be mentioned here, in our previous work, the solid-to-solid polymorphic transition of form II to I was investigated in DSC alumina pans under a flow of high-purity nitrogen at the heating rate of 10 K min−1. The temperature of solid-to-solid polymorphic transition was determined as 36 °C, which showed the difference between phase transition temperatures of “solid-to-solid phase transition” and “ethanol

Figure 6. Polymorphic morphology in polarizing microscope: (a) form I, (b) form II, and (c) concomitant crystallization of forms I and II. 4586



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Figure 8. Dependence of the ln tind on 1/(ln S)2 and the nature of polymorph that crystallized at (a) 5, (b) 10, (c) 15, (d) 20, (e) 25, (f) 30, and (g) 35 °C. The straight lines indicate the simulations according to eq 9.

mediated polymorphic transformation”. A similar phenomenon was reported by Sjölin36 and Wu et al..37 For the case of gestodene, the discrepancy of the transition temperatures has to do with the unary system (solid-to-solid, gestodene only) and binary system (SMPT, gestodene + ethanol). The results of solvent-mediated polymorphic experiments indicated that the process of polymorphic transformations need a period of time

to complete. Complete polymorphic transformations of the metastable form to stable form were observed at 5, 10, 30, and 35 °C after only 1 h of slurrying. The phase transitions were slower at 15, 18, 19, 20, and 25 °C, and complete phase transitions to the stable form were observed after 72 h of slurrying. 4587



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4.4. Nucleation Kinetics of Gestodene Polymorphs. The isothermal dependence of the induction period on supersaturation of the gestodene−ethanol system at seven different temperatures was measured. The tind−S experimental data allows the determination of the crystal−solution interfacial tensions of the gestodene polymorphs by plotting the evolution of the logarithm of the tind versus 1/(ln S)2 as shown in Figure 8. The polymorphic nature of the crystals obtained upon nucleation at the investigated temperatures are also presented in this figure. Figure 8a,b,c,d indicate two distinct regions with different slopes, which may be attributed to the nucleation of different polymorphs. At lower supersaturations, only a pure form nucleated (form I at 20 °C, form II at 5, 10, and 15 °C), while at higher supersaturations, concomitant nucleation of forms I and II occurred. At the crystallization temperatures of 25, 30, and 35 °C, only form I was obtained as shown in Figure 8e,f,g. Therefore, the distinct straight lines in Figure 8 actually represent the primary nucleation of different pure forms and concomitant polymorphism of gestodene. From the plot of ln tind versus 1/(ln S)2 and according to eq 11, the values of interfacial tensions were calculated from the slopes of the straight lines. The results obtained for the form and concomitant polymorphism of forms I and II are shown in

tension reflects the ability of the solute to nucleate from the solution: the higher its value the more difficult it is for the solute to nucleate.40 Using eq 4 and the estimated interfacial tension, the radius of critical nucleus (rC) can be calculated. The values of rC range from 0.60 to 7.61 Å. Because the interfacial tension of form II is lower than that of form I at the crystallization temperatures of 5, 10, and 15 °C, the radius of the critical nucleus of form II (rCII) is lower than that of form I (rCI). This is one of the reasons that form II seems to be favored at lower crystallization temperatures and lower supersaturations. At higher crystallization temperatures (20, 25, 30, and 35 °C) and lower supersaturations, rCI < rCII, form I was the preferred form. However, the radius difference between the forms I and II is not significant at higher supersaturations, which might lead to the concomitant crystallization of the two forms. By use of eqs 2 and 7, the primary nucleation rates of the two forms at the studied temperatures can be calculated from the interfacial tensions. The results of nucleation rates of the two forms are shown in Figure 9. From Figure 9a,b,c, at lower initial supersaturations, it is found that the nucleation rate of form II is higher than that of form I, while the reverse occurs at higher initial supersaturations. At the crystallization temperatures of 20, 25, 30, and 35 °C as shown in Figure 9d,e,f,g, at lower initial supersaturations, JI > JII, while at higher initial supersaturations, JI < JII. From the perspective of nucleation kinetics, the simultaneous nucleation of forms I and II is possible at the initial supersaturation corresponding to the crossover of the curves where the nucleation rates of the two forms are equal. Theoretically, the polymorphic composition of the formed crystals should follow the same trend as the relative nucleation rate of polymorphs in terms of kinetics. The blue points in Figure 10 show the effects of initial supersaturation ratio and crystallization temperature on the polymorphic content of gestodene crystals obtained after nucleation. It is found that at low crystallization temperature (5 °C ≤ T ≤ 17 °C), the weight fraction of form I in the resulting solids generally increases with rising initial supersaturation, while at intermediate crystallization temperature (19 °C ≤ T ≤ 25 °C), the weight fraction of form I decreases with increasing initial supersaturation. At high crystallization temperature (27 °C ≤ T ≤ 35 °C), the weight fraction of form I always equals 1 at the studied supersaturations. As shown in eq 12, the relative nucleation rate (JI,II) reflects the fraction of nucleation of form I relatively to the total number of nucleation events. Therefore, JI,II could be used to quantify the competition between nucleation of forms I and II. According to eqs 12 and 13, a semiempirical model relating to the relative nucleation rate is established.

Table 1. Results of the Correlation Analysis and the Related Parameters: Slope (m), Square of Correlation Coefficient (R2), and Interfacial Tension (γ) T (°C) 5 10 15 20 25 30 35

m

R2

γ (mJ m−2)

0.0796 0.2861 0.0868 0.2671 0.1308 0.2169 0.1326 0.1800 0.0711 0.0353 0.0233

0.9287 0.9313 0.9452 0.8341 0.8676 0.9695 0.9774 0.9426 0.8642 0.9629 0.8756

1.12 1.72 1.17 1.71 1.37 1.62 1.40 1.55 1.16 0.93 0.83

nucleation form II I+ II I+ II I+ I I+ I I I

II II II II

Table 1. The interfacial tensions (J m−2) of the two forms were correlated as a function of temperature (K) and expressed as γI = 1.4 × 10−6T 2 − 8.8062 × 10−4T + 0.1393

(16)

γII = 3.0 × 10−6T 2 − 0.0017T + 0.2346

(17)

The interfacial tensions of forms I and II at the temperatures of 5, 10, 15, 20, 25, 30, and 35 °C are listed in Table 2. The calculated interfacial tension ranges from 0.83 to 3.52 mJ m−2, which agrees well with other organic compounds.38,39 As shown in Table 2, it is observed that the interfacial tensions for different polymorphs change with temperature. The interfacial

⎡ ⎡ k JI,II = ⎢1 + k1 exp⎢ 3 2 2 ((a1T 2 + a 2T + a3)3 ⎣ T ln S ⎣ −1 ⎤⎤ − (b1T 2 + b2T + b3)3 )⎥⎥ ⎦⎦

In eq 18, k1, k2, a1, a2, a3, b1, b2, and b3 represent the model surface-fit parameters calculated from a regression analysis using the measured polymorphic composition versus supersaturation and temperature. The model parameters and error analysis results are reported in Table 3. Values of JI,II will be between 0 and 1. If JI,II = 1.0, it is predicted that only form I will nucleate. JI,II = 0.5 indicates that there is an equal probability for the nucleation of forms I and II. Values of JI,II at different initial

Table 2. Interfacial Tensions of the Forms I and II at Different Temperatures γ (mJ m−2) polymorph

5 °C

10 °C

15 °C

20 °C

25 °C

30 °C

35 °C

I II

2.63 1.12

2.15 1.17

1.74 1.37

1.40 1.69

1.16 2.16

0.93 2.77

0.83 3.52

(18)

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Figure 9. Dependence of the nucleation rates of gestodene polymorphs as a function of initial supersaturation at (a) 5, (b) 10, (c) 15, (d) 20, (e) 25, (f) 30, and (g) 35 °C.

concomitant polymorphs of forms I and II will result. Thus, it is possible to predict polymorph of the formed crystals provided that values of relative nucleation rates are available. The values of relative nucleation rates calculated from eq 18 for various crystallization temperatures and initial supersaturations are listed in Table 4. Based on the calculated relative nucleation rate of forms I and II, the polymorphic form of gestodene was

supersaturation ratios and crystallization temperatures are also presented in Figure 10. The experimental results of relative nucleation kinetics of gestodene in the ethanol solution suggest that the difference in JI,II would possibly result in different polymorph. It is found when JI,II is smaller than 0.0399, pure form II would be generated, while JI,II ≥ 0.8718 gives rise to the pure form I. When JI,II is between 0.0399 and 0.8718, 4589



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Figure 10. Dependence of the relative nucleation rate and the weight fraction of form I as a function of the crystallization temperature and initial supersaturation ratio. The surface indicates the calculated relative nucleation rate. The blue point indicates the weight fraction of form I.

Table 3. Parameters of Eq 18 and Error Analysis Results parameter

value

R2

RMSD (%)

k1 k2 a1 a2 a3 b1 b2 b3

0.9938 0.002466 −0.3194 8.014 −5.860 0.2012 −1.302 2.479

0.7741

17.22

Table 4. Predicted Polymorphs of Gestodene in the Ethanol Solvent Based on Relative Nucleation Rate (JI,II) Together with Observed Polymorphs

predicted. The predictions along with the experimental findings are reported in Table 4. As can be seen the predictions are in good agreement with the experimental observations. 4.5. Growth Rate Measurement. Although it is tacitly assumed that the polymorph of the grown crystals was inherited directly from that of the nucleus, the polymorphic forms of the resulting solids determined by XRD were those of the grown crystals. Therefore, the effect of relative growth rates of the two forms on the polymorphic outcome should be considered. The growth rates of forms I and II were measured by hot-stage polarized-light video microscopy in an experiment of cooling crystallization starting at the concentration of 0.0071 (in mole fraction) and crystallizing at the temperatures of 15 and 20 °C, since concomitant crystallization was observed at these crystallization conditions. As shown in Figure 11, the solution was clear before the occurrence of nucleation (Figure 11a,e), while concomitant polymorphs of forms I and II were actually formed through cooling crystallization (Figure 11b,f). The appearance of forms I and II were observed at the same time, which indicated the occurrence of concomitant nucleation of the two forms. Figure 11c,d,g,h shows the crystals of both forms growing. From the sequential images, the surface areas of forms I and II were measured using the software of GM-2000M (Guangmi Instrument Co. Ltd. Shanghai, China) under equal conditions. The geometrical shape of the selected crystal was converted to the square with an equivalent surface area. The crystal size (L) is defined as the length of the diagonal of the equivalent square. Figure 12 shows the growth rate (G(t) = dL/ (2dt)) versus time. At the crystallization temperature of 15 °C as shown in Figure 12a, it is found that the growth rate of form

T (°C)

S

JI,II

5 5

1.22 1.43

0.0001 0.0578

5

1.66

0.2003

10 10

1.20 1.39

0.0042 0.1580

10

1.50

0.2424

15

1.19

0.2097

15

1.36

0.4002

15

1.50

0.4410

20

1.22

0.7861

20

1.32

0.6588

20

1.41

0.6098

25 25 25

1.15 1.28 1.47

0.9993 0.9084 0.7211

30 30 30 35 35 35

1.13 1.16 1.20 1.08 1.15 1.21

1.0000 1.0000 0.9998 1.0000 1.0000 1.0000

predicted polymorph II I (5.78%) + II (94.22%) I (20.03%) + II (79.97%) II I (15.80%) + II (84.20%) I (24.24%) + II (75.76%) I (20.97%) + II (79.03%) I (40.02%) + II (59.98%) I (44.10%) + II (55.90%) I (78.61%) + II (21.39%) I (65.88%) + II (34.12%) I (60.98%) + II (39.02%) I I I (72.11%) + II (27.89%) I I I I I I

observed polymorph II I (19.80%) + (80.20%) I (38.21%) + (61.79%) II I (19.16%) + (80.84%) I (39.01%) + (60.99%) II

II II

II II

I (24.53%) + II (75.47%) I (35.58%) + II (64.42%) I I (69.03%) + II (30.97%) I (47.76%) + II (52.24%) I I I (74.21%) + II (25.79%) I I I I I I

I (GI) initially is higher than that of form II (GII). The growth rate of form I decreases more steeply until the point of intersection, after which GII > GI. The solute concentration is consumed by the growing crystals. Thus, the supersaturation of the solution decreases as time elapses. It is concluded that at the crystallization temperature of 15 °C and at higher 4590



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Figure 11. Images show the nucleation and growth of form I (plate-like shape) and form II (needle-like shape) crystals at the initial concentration of 0.0071 (in mole fraction) and the crystallization temperatures of (a, b, c, and d) 15 and (e, f, g, and h) 20 °C.

ments performed at 5, 10, 15, 20, 25, 30, and 35 °C presented a series of plots documenting the composition of the solid phase during phase transition, as determined by the XRD method. Figure 13 shows the polymorphic composition of the solid phase during SMPT. It is found that complete phase transition of form I to II was observed in the ethanol at 5, 10, and 15 °C after 45, 60, and 390 min, respectively, while phase transition of form II to I was completed at 20, 25, 30, and 35 °C after 1080, 930, 35, and 20 min, respectively. The driving force of a transition is determined by Gibbs free energy difference

supersaturations, GI > GII, while at lower supersaturations, GI < GII. In contrast, at the crystallization temperature of 20 °C as shown in Figure 12b, the growth rate of form I is lower than that of form II at higher supersaturations, while the reverse occurs at lower supersaturations. 4.6. Kinetics of Solvent-Mediated Polymorphic Transformation (SMPT). Solvent-mediated polymorphic transformation (SMPT) can be used to produce the stable form if crystallization from a solvent gives a metastable form or concomitant polymorphs. The results of the SMPT experi4591



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absolute values of ΔGI,II at relatively low and high temperatures indicate that the metastable form has a greater potential to convert to the stable form. Therefore, the transformation kinetics of gestodene was relatively fast at 5, 10, 30, and 35 °C, an order of magnitude shorter compared to the phase transitions at 15, 20, and 25 °C (as shown in Figure 13).

5. DISCUSSION According to the experimental results of thermodynamic stability and crystallization kinetics, the polymorphic appearance of gestodene can be explained. The interplay between thermodynamics and kinetics that control the nature of polymorph obtained from solution crystallization are summarized in Table 5 and Figure 15. At the crystallization temperature of 5 °C and low supersaturation (S ≤ 1.34), JII > JI and the stable form II is favored by both kinetics and thermodynamics. Therefore, only form II crystallized from the ethanol solution. At high supersaturation (1.46 ≤ S ≤ 1.79), JI > JII and the metastable form I is preferred by kinetic factors while the stable form II is favored by thermodynamics. Thus, concomitant crystallization of forms I and II is a result of the balance between thermodynamics and kinetics. Similar behaviors were observed over the studied temperature range from 5 to 17 °C. When the crystallization temperature is higher than the transition temperature (19 °C ≤ T ≤ 25 °C), at lower supersaturations, JI > JII and the stable form I appeared, while at higher supersaturations, JII > JI and both thermodynamics and kinetics play important roles and concomitant polymorphism of forms I and II was observed. It is clear that whether above or below the transition temperature, both the supersaturation and temperature play crucial part in determining the polymorphic outcome, with form I always appearing at lower supersaturations and T ≥ 19 °C, form II at lower supersaturation, and T ≤ 17 °C, and concomitant polymorphs at higher supersaturations and 5 °C ≤ T ≤ 25 °C. A strategic approach to the development of a crystallization process might reasonably use the Ostwald’s rule of stages as a guiding principle. It would predict that below the transition temperature, the metastable form I of gestodene would be the form to crystallize first and that above the transition temperature the metastable form II would be favored. At the transition temperature of 18.5 °C where the solubilities and supersaturations are identical for the two forms, concomitant polymorphs of I and II will be obtained. However, the report of current work gives considerable support in a negative way to the propositions advanced in this contribution as explanation of the crystallization behavior described by the Ostwald’s rule of stages. It is commented here that the crystallization of gestodene from ethanol solution at lower supersaturations gives the stable form rather than the metastable one predicted by Oswald’s rule. There are two reasons why Ostwald’s rule may not be relevant in this case, related to the thermodynamic and kinetic explanations presented above. First, classical thermodynamics determines the production of the stable form. If the thermodynamics dominates the crystallization process, it is reasonably probable to obtain the stable form. Second, if the nucleation rate of the stable form is considerable higher than that of the metastable one, the appearance probability of the stable form will be higher than that of the metastable form. Under such conditions, the stable form will nucleate. It is therefore concluded that Ostwald’s rule is not universal law, but it is only in the nature of a possible preferred tendency.

Figure 12. Crystal growth rates of forms I and II as a function of time at the initial concentrations of 0.0071 (in mole fraction) and crystallization temperatures of (a) 15 and (b) 20 °C. The supersaturation of the solution decreases with the time.

(ΔGI,II) between two phases, which reflects the ratio of escaping tendencies of the two phases. The escaping tendency is termed as fugacity ( f). The fugacity is proportional to the thermodynamic activity (a), while a is approximately proportional to the solubility (C0) in any given solvent provided that the laws of dilute solution apply. Therefore, ⎛f ⎞ ⎛ C0,II ⎞ ⎛a ⎞ ⎟⎟ ΔG I,II = RT ln⎜⎜ II ⎟⎟ = RT ln⎜ II ⎟ ≈ RT ln⎜⎜ ⎝ aI ⎠ ⎝ C0,I ⎠ ⎝ fI ⎠

(19)

where R is the universal gas constant and T is the absolute temperature (K). The ΔGI,II and its temperature dependence was calculated from the solubility data of gestodene. The ΔGI,II versus T diagram is shown in Figure 14. From this figure, it can be found that the Gibbs free energy difference between forms I and II increases as a function of temperature. At the temperatures of 5, 10, and 15 °C, ΔGI,II < 0, and the phase transition of form I to form II is a spontaneous process. At the temperatures of 20, 25, 30, and 35 °C, ΔGI,II > 0, and the phase transition of form I to II is a nonspontaneous process, while the transformation of form II to I is favored. Moreover, the higher 4592



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Figure 13. Solvent-mediated polymorphic transformations between forms I and II in ethanol at (a) 5, (b) 10, (c) 15, (d) 20, (e) 25, (f) 30, and (g) 35 °C.

6. CONCLUSIONS

In our previous work, the experimental results of SXRD data indicate that the crystal structures of gestodene forms I and II exhibit conformational differences. It is suspected that the ratio of different conformations in solution varies with crystallization conditions and affects the polymorphic composition of the nucleated crystals. The effect of conformations in solution on polymorphic outcomes of gestodene will be studied in the future.

In this study, the solution crystallization behavior of gestodene was investigated in detail. To fully understand the mechanism of the polymorphic outcomes, the cooling crystallization process, thermodynamic stability, nucleation and crystal growth kinetics, and polymorphic interconversions of the forms I and II of gestodene were discussed. 4593



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Figure 14. Diagram of Gibbs free energy difference of forms I and II (ΔGI,II) versus temperature (T).

Figure 15. Graphical representation of the interplay between thermodynamics and kinetics on the polymorphic appearance of gestodene.

The nature of the polymorph obtained from the cooling crystallization depends on the initial supersaturation and crystallization temperature. X-ray powder diffraction patterns and optical microscope images of forms I and II showed distinct differences that could be used to identify the obtained polymorphs. Cooling crystallization results in stable form crystals at lower supersaturations and concomitant polymorphs at higher supersaturations. In the polymorphic system of gestodene, directly crystallizing the stable form and concomitant polymorphs indicate that Ostwald’s rule of stages is not a general physical law. This can be explained by the interplay between thermodynamics and kinetics. The thermodynamic stability analysis in ethanol confirms that forms I and II are enantiotropically related with the transition temperature of 18.5 ± 0.5 °C. Form II is the thermodynamic stable form below 18 °C, while form I is the stable form above 19 °C. The determination of the induction time for the polymorphs allows us to calculate the crystal−solution interfacial energy and the nucleation rate. The growth rate of both forms was measured by hot-stage polarized-light video microscopy under equal crystallization conditions. It is found that at lower supersaturations, both the nucleation and growth rates of the stable form are higher than those of the metastable form, while at higher supersaturations, the reverse occurs. Therefore, it is concluded that at lower supersaturations, the stable form is both thermodynamically and kinetically favored. Hence, only the stable form appeared. For higher supersaturations, concomitant crystallization of forms I and II took place because of the nearly equivalent importance of thermodynamics and kinetics.

The evolution of the relative nucleation rate (JI,II) of forms I and II with initial supersaturation ratio and crystallization temperature allows the quantification of the competitive nucleation of gestodene polymorphs. A semiempirical model relating to the relative nucleation rate was established based on classical nucleation theory. Using the values of JI,II derived from the initial supersaturations and crystallization temperatures, the resulting polymorph of gestodene can be predicted. These results are almost consistent with experimental observations. The solvent-mediated polymorphic transformation from form I to II was observed below the transition temperature, while the reverse occurred above the transition temperature. The transformation rates of gestodene polymorphs were relatively fast at 5, 10, 30, and 35 °C due to the higher Gibbs free energy difference between the two forms, which was calculated by the solubility data of both forms. In the case of gestodene, although cooling crystallization under conditions of higher supersaturations results in the crystallization of concomitant polymorphs, the pure stable form can be readily produced by using a slurrying procedure at the end of the crystallization process.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.7b00335.

Table 5. Interplay between Thermodynamics and Kinetics on the Polymorphic Outcomes of Experiments Performed at 5, 15, and 25 °C from Ethanol Solutions T (°C)

S

5

1.34 1.46−1.79 1.12−1.22 1.29−1.68 1.24−1.29 1.36−1.58

15 25

a

kinetically favored form form form form form form

II (JII > JI) I (JI > JII) II (JII > JI) I (JI > JII) I (JI > JII) II (JII > JI)

thermodynamically favored stable stable stable stable stable stable

form form form form form form

II II II II I I

outcome

conclusion

form II concomitant polymorphism form II concomitant polymorphism form I concomitant polymorphism

a b a b a b

Theory grounded on experimental observations. bBalance of thermodynamics and kinetics. 4594



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Hot-stage polarized-light video-microscopy experimental set up used to observe nucleation and crystal growth in situ, schematic diagram of experimental apparatus for the measurement of induction periods, and possible phase diagram of gestodene-ethanol system (PDF)

AUTHOR INFORMATION

Corresponding Authors

*Tel.: +86 022 60601110. Fax: +86 022 60601110. E-mail address: [email protected] (Liang Zhu). *E-mail address: [email protected] (Zuo-liang Sha). ORCID

Liang Zhu: 0000-0002-9086-4202 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support of the Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 14JCZDJC40900), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20121208120001), the Training Program for Changjiang Scholars and Innovative Research Team in University (Grant No. [2013]373), the National Natural Science Foundation of China (Grant No. 21506162), and the Innovative Research Team of Tianjin Municipal Education Commission (Grant No. TD12-5004) are gratefully acknowledged.

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