Experimental Determination and Computational Prediction of

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Experimental Determination and Computational Prediction of Androstenedione Solubility in Alcohol + Water Mixtures Weiwei Tang, Zhao Wang, Ying Feng, Chuang Xie, Jingkang Wang, Chengsheng Yang, and Junbo Gong Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 26 Jun 2014 Downloaded from http://pubs.acs.org on June 28, 2014

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Industrial & Engineering Chemistry Research

Experimental Determination and Computational Prediction of Androstenedione Solubility in Alcohol + Water Mixtures Weiwei Tang1, Zhao Wang1, Ying Feng3, Chuang Xie1, 2, Jingkang Wang1, Chengsheng Yang3, Junbo Gong*, 1, 2 1

State Key Laboratory of Chemical Engineering, Collaborative Innovation Center of Chemical

Science and Chemical Engineering (Tianjin), School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China 2

Tianjin Key Laboratory for Modern Drug Delivery and High Efficiency, Tianjin University, Tianjin 300072, China

3

School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China

* Corresponding author Tel.: 86-22-27405754. Fax: +86-22-27374971. E-mail: [email protected].

ACS Paragon Plus Environment

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TOC

Experimental

Determination

and

Computational

Prediction

of

Androstenedione Solubility in Alcohol + Water Mixtures Weiwei Tang, Zhao Wang, Ying Feng, Chuang Xie, Jingkang Wang, Chengsheng Yang, Junbo Gong

σ-Profiles and COSMO-cavities of studied compounds. Different colors denote various natures of surface screening charge, and blue, red and green represent negative, positive and almost neutral surface charges, respectively.


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ABSTRACT: This article evaluates the accuracy and applicability of three of the most common solubility models (i.e., Jouyban-Acree, NRTL-SAC, and COSMO-RS) in prediction of androstenedione (AD) solubility in binary mixtures of methanol + water and ethanol + water. The solubilities were measured from (275 to 325) K using medium-throughput experiments and then well represented mathematically by modified Apelblat and CNIBS/Redlich–Kister equations. The computational results show that AD solubility decreases monotonically with increasing water concentration in methanol + water mixtures, but it has a maximum at 0.15~0.30 mole fraction of water in the ethanol aqueous solution. Moreover, the performance of three solubility prediction models in this particular case was compared to identify the advantages and disadvantages of each model. The overall average relative deviation (ARD) for solubility prediction is 4.4% using Jouyban-Acree model, while it is 18.3% with NRTL-SAC model. Surprisingly, COSMO-RS model in combination with reference solubility achieves a good performance for solubility prediction in mixed solvents, including the prediction of synergistic effect of solvents, with overall ARD of only 4.9%.


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1. INTRODUCTION Androstenedione (AD, C19H26O2, molecular weight: 286.41 g·mol-1, Figure 1), with high additional value, is a significant intermediate for producing different kinds of steroid derivatives like prednisolone, progesterone, and testosterone.2 Since AD possesses many indispensable androgen properties, it has been the raw material in the manufacture of androgen and anabolic drug for a long time.3,

4

In industry, AD can be produced through fermentation, and then

separated and purified by extraction, decolourization, and crystallization. Solution crystallization, controlling the quality attributes of product (i.e., purity, yield, and crystal size distribution etc.), therefore is a critical step of purification process. But the development of crystallization process strongly depends on accurate Solid–Liquid Equilibrium (SLE) solubility data that vary with temperature and solvent composition. Recent work in our laboratory has shown that AD crystallization with lower alcohols, such as methanol and ethanol, can improve the product purity, but cooling crystallization is not an appropriate method due to the low final yield. To improve AD yield in crystallization with high purity, the anti-solvent crystallization method was proposed. AD is practically insoluble in water (57.2 mg·L-1 at 298.2 K), 5 and water is easily available with low cost. Thus water is chosen as an antisolvent for crystallization. In order to achieve a wellcontrolled crystallization process and hence high purity and yield, AD solubilities in methanol + water and ethanol + water are of great importance. However, up to now, these data are not available in the literature. In addition to the solubility determination from experiment, some mathematical and thermodynamic models of SLE have been presented to calculate the solubility in pure solvents and solvent mixtures.6-10 One of the most common used prediction models of drug or drug-like solubility in mixed solvent systems is Jouyban-Acree model, which fits a lot of experimental


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data using several effective model parameters for reducing the error in prediction. Chen and Song11,

12

introduced a new semi-empirical model, nonrandom two-liquid segment activity

coefficient (NRTL-SAC) model, for predicting drug solubility in pure or mixed solvents. The NRTL-SAC model was found promising for solubility prediction,13 but the results were not compared to much more models. Recently, Klamt and coworkers14 developed a more physically founded model, conductor-like screening model for real solvents (COSMO-RS) model, to predict thermodynamic equilibria of liquid mixtures according to the unimolecular quantum chemical calculations. This model does not require plenty of experimental data, and in some cases only the structural information of desired compounds are required to predict thermodynamic properties. Therefore, COSMO-RS model was achieved a remarkable interest in many areas,15 and it has been successfully applied to predict aqueous solubilities of drugs,8 cellulose solubilities in ionic liquids,16 and other properties such as excess enthalpies of ionic liquids with molecular solvents17 and Henry’s Law coefficients of CO2 in ionic liquids.18 COSMO-RS model has been previously employed by Ikeda et al.19 to predict the solubility of various drugs in four pure solvents and two binary solvent mixtures only at 298.15 K. Reasonable results for solubility prediction in pure solvents were obtained, but the model systematically underestimated the solubilities in mixed solvents and exhibited the great predicted errors in these systems.19 Albeit the performance of several predictive solubility models like UNIFAC, COSMO-SAC and NRTL-SAC has been compared in the literature,6, 20-22 little attention was paid to the comparison of Jouyban-Acree, NRTL-SAC, and COSMO-RS models, which were selected on the basis of their theoretical bases, predicted approaches, and predictive ability where solubility data were used or not. In this present, experimental solubility data of AD in methanol + water and ethanol + water at different temperatures were reported. Solubility data were represented mathematically using


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modified Apelblat equation and CNIBS/Redlich–Kister model, respectively. For reducing the predicted deviation, COSMO-RS in combination with reference solubility approach was introduced for solubility prediction. Moreover, solubility prediction was performed at different temperatures and in two different kinds of mixed solvent systems (i.e., the maximum solubility occurs or not), and thus the effects of temperature and non-ideal behaviour of mixed solvent systems on the prediction were also tested. Finally, the performance of Jouyban-Acree, NRTLSAC, and COSMO-RS models for AD solubility prediction was tested and compared. 2. SOLUBILITY MODELS 2.1. Jouyban-Acree Model. To interpret the cosolvency mechanisms and correlate or predict the drug solubility in cosolvent-water system, many cosolvency models were presented.7 The Jouyban-Acree model is one of the simplest but promising in such models, and it’s general form is23 log X 1 = f 2 log X 2 + f 3 log X 3 +

f 2 f3 2 J i ( f 2 − f 3 )i ∑ T i=0

(1)

where X1, X2 and X3 denote, respectively, the drug solubility in cosolvent + water mixtures, pure cosolvent and water; f3 and f2 represent the solute-free fractions of water and cosolvent, respectively, and Ji is the model constants regressed from experimental solubility data. To improve the solubility prediction ability, a series of modified versions of the model were developed to reduce the experimental data used in the regression. In a recent work, it has achieved a significant improvement in predictive ability by using partial solubility parameters. The drug solubility in binary solvent mixtures could be calculated by eq 2 using a combination of partial solubility parameters and Jouyban-Acree model.23


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log X 1 = f 2 log X 2 + f 3 log X 3 f 2 f3 A0δ d1 (δ d2 − δ d3 ) 2 + A1δ p1 (δ p2 − δ p3 ) 2 + A2δ h1 (δ h2 − δ h3 )2 } { T  f f ( f − f ) +  2 3 2 3  { A3δ d1 (δ d2 − δ d3 )2 + A4δ p1 (δ p2 − δ p3 ) 2 + A5δ h1 (δ h2 − δ h3 )2 } T  

+

(2)

 f 2 f 3 ( f 2 − f3 )2  +  { A6δ d1 (δ d2 − δ d3 ) 2 + A7δ p1 (δ p2 − δ p3 ) 2 + A8δ h1 (δ h2 − δ h3 ) 2 }   T   where A0 to A8 represent the model constants; δ is solubility parameter; subscripts h, p and d denote hydrogen-bonding, polar, and dispersion parts of solubility parameter, respectively. Combining the Jouyban-Acree model and modified Apelblat equation, eq 1 can also be further simplified and rearranged as eq 3, which describes solubility varying with solvent composition and temperature.24 x3 ( x3 ) 2 ( x3 )3 ( x3 )4 A2 ln x1 = A1 + + A3 ln T + A4 x3 + A5 + A6 + A7 + A8 + A9 x3 ln T T T T T T

(3)

in which x1 and x3 denote solubility (mole fraction) and the initial mole fraction of water in binary solvent mixtures, and A1–A9 are the adjustment parameters regressed from experimental data.

2.2. NRTL-SAC Model. Chen and Song introduced a new predictive model, NRTL-SAC model, which combines the original NRTL model with the segment contribution concept in the polymer NRTL model.11 The molecule is characterized by four predefined conceptual segments X, Y−, Y+, and Z, which represent hydrophobic, polar attractive and repulsive, and hydrophilic, respectively. All the interactions of each molecule are considered by those segments.11 The activity coefficient of a component in a solution is calculated by:12 ln γ I = ln γ IC + ln γ IR


(4)

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The combinatorial term is computed on the basis of Flory–Huggins approximation for mixing entropy:20 ln γ IC = ln

φI

+ 1 − rI ∑

xI

J

φJ rJ

rI = ∑ ri , I

(5)

(6)

i

rI xI ∑ rJ xJ

φI =

(7)

J

where φI and rI are, respectively, the segment mole fraction and the total segment numbers of component I. The residual part, γ IR , is given as:12 I  ln γ IR = ln γ Ilc = ∑ rm,I ln Γ lcm − ln Γ lc, m 

(8)

m

∑x G τ = ∑x G j

ln Γ

lc m

jm

jm

j

k

km

k

∑x G τ = ∑x G j,I

ln Γ

lc, I m

jm

j

k ,I

k

km

jm

 ∑j x j G jm′τ jm′  xm′Gmm′  +∑ τ mm′ − xk Gkm′  m ′ ∑ xk Gkm ′  ∑ k k    ∑j x j ,I G jm′τ jm′  xm′, Ι Gmm′  +∑ τ mm′ − xk , I Gkm′  m ′ ∑ xk , I Gkm ′  ∑ k k  

∑x r = ∑∑ x r

(9)

(10)

J j ,J

xj

J

(11)

I i ,I

I

x j,I =

i

rj , I

∑r

(12)

i ,I

i

G ji = exp(−α jiτ ji )


(13)

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in which I and J stand for the components; i, j, k, m, and m' represent segment species, respectively. The detailed parameter definitions in eqs 8−13 can be found in the literature.11 Based on the SLE relationship, the equilibrium solubility can be calculated as follow:9

ln xJ =

∆H fus  1 1   −  − ln γ J R  Tm T 

(14)

where xJ refers to the mole fraction solubility of solute J; ∆Hfus and Tm are, respectively, the fusion enthalpy and melting temperature of solute. For the calculation of solubility, ∆Hfus and Tm are determined experimentally using differential scanning calorimetry (DSC). The values of γ J can be calculated by NRTL-SAC model.

2.3. COSMO-RS Model. Built on quantum chemical calculations of uni-molecule, COSMORS provides a novel and fast approach to predict thermodynamic equilibria of liquid systems. The theory of COSMO-RS has been introduced elsewhere by Klamt and coworkers.14 Briefly, the model combines ab initio calculations of solute and solvent molecules with statistical thermodynamics procedure to determine the interactions between molecules, and then chemical potential of arbitrary compound can be calculated to complete the prediction of thermodynamic properties in liquid mixtures. COSMO-RS calculations have two steps. Firstly, quantum chemical calculations (usually Density Functional Theory (DFT) method) for all investigated compounds are performed to produce a screening charge density σ (σ-profile). Then the interaction energy of pair-wise contacting surface segments is determined using the generated σ-profile. This interaction energy is computed from the sum of electrostatics, hydrogen bonding (HB), and van der Waals (vdW) interaction energies.14 The interactions of electrostatic and H-bonding in liquid mixture are defined as contact interactions of two molecular surfaces with polarization change densities σ


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and σ′ , and vdW or dispersive interactions are approximately treated with the specific dispersion coefficient for each element. Thus the chemical potential of a surface segment with polarization change density σ and average molecular contact area aeff in an ensemble S, µS (σ) , can be calculated using the normalized distribution function pS (σ′) given as follow:8

   1  µS (σ) = − RTln  ∫ pS (σ′)exp  µS (σ′) − E HB( σ, σ′) − E MF( σ, σ′) − E vdW( σ, σ′) )  dσ′ (15) (  RT    where EMF

,

EHB and EvdW represent the electrostatic, HB, and vdW interaction energies,

respectively. Then chemical potential for any molecule J in system S, µSJ , is computed from the integration of µS (σ) over the surface of J. COSMO-RS for solubility prediction is based on the calculation of µSJ of arbitrary solute J in any system S (i.e., pure solvent or solvent mixtures).7 Solubility describes the thermodynamic equilibria between the solute concentration in the solid state and it in a solution. Therefore, the solubility can be computed from the compounds chemical potential difference between the subcooled liquid state µ J(p) and the infinite dilution state in a given solvent or solvent mixture µ J(S) . It is noted that the solid solute is treated as a hypothetical subcooled liquid by introducing the concept of fusion Gibbs free energy ∆Gfus (T ) of solute. Thus the solubility of solute J in system S is calculated via a zeroth order approximation.25

log10 ( xJSOL ) =  µ J(p) − µ J(S) − max(0, ∆Gfus (T ))  / ( RT ln (10 ) )

(16)

The predictive accuracy can be improved by iterative refinement algorithm,25 as expressed in eq 17. log10 ( xJSOL( n +1) ) =  µ J(p) − µ J(S) ( xJSOL( n ) ) − max(0, ∆Gfus (T ))  / ( RT ln (10 ) )


(17)

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The fusion Gibbs free energy ( ∆Gfus (T ) ) of a solid solute at temperature T can be estimated via a QSPR (Quantitative Structure Property Relationship) approach and approximated from the following COSMOtherm descriptors using:26

−∆Gfus (T ) = c1µ HJ 2 O + c2 N JRing + c3VJ + c4

(18)

where c1 to c4 are the QSPR parameters; µ HJ 2O denotes the chemical potential of solute J in water;

N JRing and VJ are the number of ring atoms and molecular volume of solute J, respectively. It is noted that COSMO-RS treats the quantities size, rigidity, polarity, and number of hydrogen bonds as descriptors of potential significance for ∆Gfus (T ) . Molecular size and rigidity are, respectively, described by VJ and N JRing , and µ HJ 2O is a combined measure of polarity and Hbonding. All these descriptors can be available from the calculation of COSMO-RS. Moreover, QSPR parameters are derived from aqueous solubility data of 150 solid compounds.8 Therefore,

µ HJ O is of special importance for estimation of ∆Gfus (T ) and then calculation of solubility in 2

COSMO-RS.26 Alternatively, ∆Gfus (T ) can also be estimated by reference solubility method: if the experimental solubility of the studied compound in a given pure or mixed solvent at a given temperature is known, this reference solubility is used to determine ∆Gfus (T ) (or ∆H fus ) at these conditions by solving eq 16.27 In the current COSMOtherm program, ∆Gfus (T ) is calculated from both reference solubility and QSPR approaches to compare the accuracy of AD solubility prediction from these two different approaches. All the solubility calculations were performed using COSMOthermX_3.0 program (COSMOlogic GmbH & Co. KG, Leverkusen, Germany). The COSMO files of methanol, ethanol and water used in this study were taken directly from a database of DMol3-


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COSMO-PBE. The input file for AD excluded in this database was generated using the DMol3 package from Accelrys at the DFT level, applying the functional of PBE and DNP basis set.28, 29 For ensuring an efficient calculation, all stable conformations generated were taken into account and weighted by the Boltzmann distribution function,27 and solubility prediction was performed using the iterative algorithm.

2.4. Regression Analysis. The performance of three different predictive models used in this study was compared and assessed with the root-mean-square error (RMSE), which is calculated by:

1 RMSE =  N

1

∑

N k =1

exp. 1,k

(x

−x

2 )  

calc. 2 1,k

(19)

exp. calc. where x1,k and x1,k are experimental and predicted mole fraction solubilities, respectively. N is the number of applied data points.

The average relative deviation (ARD) was also presented in this study to measure the predictive accuracy of various models due to the measured solubility values vary in several magnitudes. The ARD is defined as:

1 ARD = N

N

∑ k=1

calc. x1exp. ,k − x1,k x1exp. ,k

(20)

3. EXPERIMENTAL SECTION 3.1. Materials. The crystalline powder of AD, supplied by Shandong Dongyao Pharmaceutical Co., Ltd. (Shandong, China), was further purified via recrystallization with ethanol. The purity of final AD production determined by HPLC (Agilent 1100, Agilent Technologies, USA) is better than 99.0% (mass fraction). Methanol and ethanol (purchased from Tianjin Kewei Chemical Co., China) both with the purity of greater than 99.5% (mass fraction) were employed without further purification. Deionized water used was of HPLC grade.


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3.2. Solubility Experiments. AD solubility data in the methanol aqueous solution and ethanol aqueous solution were measured at different temperatures by a dynamic method described in a previous work.1 Briefly, the experiments were carried out in a medium-throughput multiple reactor setup (Crystalline, Avantium, Amsterdam). To prepare AD slurries with different concentrations, the predetermined amounts of crystalline material and about 3 mL of prepared solvent mixtures were added in the vials, and then the vials were placed in the setup at a stirring speed of 1200 rpm. The heating rate employed was 0.05 K/min, while 0.5 K/min cooling rate was applied. The clear point that the light transmission through a suspension obtains a maximum value at a certain temperature was taken as the saturation temperature. To reduce errors of measurement, the clear point was determined four times per sample via cycles of cooling and reheating, and then the average value was utilized. In all measurements, the chemical stability of AD samples in the studied systems was verified, and there was no degradation found throughout the experiments. The variation of system temperature measured was estimated within ±0.1 K. The predetermined sample was weighed by an analytical balance (type AB204, Mettler-Toledo, Switzerland) with an accuracy of ±0.0001 g. Based on repeat experiments and error analysis, the uncertainty of determined solubility data was estimated less than 2%.

3.3. Characterization Methods. 3.3.1. Differential Thermal Analysis. Differential Scanning Calorimetry (DSC) was performed on a Mettler DSC 1/500 under the protection of nitrogen atmosphere at 2 K/min heating rate. Samples (5-10 mg) were prepared within a 40 µL aluminum crucible, and then sealed with a lid, which has a hole to allow venting. The DSC experiments for AD were conducted three times, and then the melting temperature Tm and enthalpy of fusion ∆Hfus were evaluated using Mettler STARe software version 10.00 (build 2480). The enthalpy of fusion is determined by drawing a frame around the peak using a linear baseline and integrating


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Page 14 of 40

the area of the endothermic peak. Uncertainties of the measured melting temperature and enthalpy of fusion were ±0.30 K and less than 3%, respectively. 3.3.2. X-ray diffraction analysis. The crystallinity of AD sample was analyzed with Powder Xray Diffraction (PXRD) pattern, which was conducted on a D/MAX 2500 X-ray diffractometer using Cu Kα (1.54) radiation. Crystal samples were slightly ground and measured over a diffraction-angle (2θ) range of 2-50° with a step size of 0.02°, at 40 kV for voltage and 100 mA for current.

4. RESULTS AND DISCUSSION 4.1. Physical Properties. The results of XRD and DSC of AD are presented in Figure S1 (Supporting Information). The crystallinity of AD sample was identified from its Powder XRD pattern, which did not show any polymorphism throughout the experiments. The melting point Tm of AD measured by DSC was 444.54 K, and its molar fusion enthalpy ∆H fus was determined

to be 25.70 kJ/mol. Then the entropy of fusion ∆Sfus was calculated to be 57.81 J/ (mol·K).

4.2. Solubility Profiles. 4.2.1. Solubility data and correlation. AD solubility data (in terms of mole fraction) in methanol + water and ethanol + water with various water compositions and temperatures was summarized in Table S1 (Supporting Information). Figure 2(a) and 2(b) show that AD solubility varies with temperature at a fixed solvent composition. The solvent composition (x3) was expressed as the solute-free water mole fraction in the binary solvents. For a fixed water ratio, AD solubility increases with increasing temperature, and thus the dissolution process is endothermic. In addition, AD solubility in methanol is much higher than it in ethanol at a given temperature, but it is practically insoluble in water. Therefore, the solubility in methanol + water mixtures remarkably decreases with the addition of water into methanol, as


14

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seen in Figure 2(c). However, the solubility in ethanol–water system shows different rules. The maximum solubility effect is observed when the mole fraction of water increases (Figure 2(d)). To describe the temperature dependence of solubility in binary solvent mixtures, experimental data were correlated by the modified Apelblat equation with the assumption that the enthalpy of solution is linear temperature dependence.30 The effect of temperature on solubility can be empirically described as follow:31 ln x1 = a +

b + c ln(T ) T

(21)

in which a, b, c are adjustable parameters and optimized by multivariable least square method. The values of R-squared in regression are all higher than 0.99 and the optimized parameters together with RMSE and ARD were listed in Table S2 (Supporting Information). The plot and the RMSE values indicate that the solubility data were well represented with the overall ARD of 1.90%. 4.2.2. Synergistic effect of the solvents on solubility. As reported in the previous experiments,21, 30, 32

the solubility of a solute may have a maximum in a binary solvent mixture. The occurrence

of such synergistic effect of mixed solvents on solubility relies on the properties of solute and solvents and is often observed when their polarities are closest to each other.21 Figure 2(d) shows the synergistic effect of ethanol–water system on AD solubility at various temperatures. One thing to note is that the solubility data in mixed solvents at exactly the same temperature were estimated with the modified Apelblat equation (eq 21) because those data were hardly measured in this work. The maximum solubility effect exhibited in ethanol + water may be attributed to a strong intermolecular association of solute molecule with solvent mixtures.30, 32-34 Similar results were also found from the solubility measurements of 2-benzoyl-1-naphthol in hexane + 1butanol mixtures and intermolecular association was verified with the study of UV–vis


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absorption spectra.30,

32

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Surprisingly, synergistic effect was not observed in methanol–water

system, indicating that the nature of solvents may play an important role in the formation of intermolecular hydrogen bond association. For quantitatively modeling the maximum of AD solubility, Combined Nearly Ideal Binary Solvent (CNIBS)/Redlich–Kister model was used to fit experimental data.31,

35

n

ln x1 = x2 ln( x1 ) 2 + x3 ln( x1 )3 + x2 x3 ∑ Si ( x2 − x3 )i

(22)

i =0

Substitution of ( 1 − x3 ) for x2 in eq 22 with n = 2 results in the following equation: 31 ln x1 = B0 + B1 x3 + B2 x32 + B3 x33 + B4 x34

(23)

where B0, B1, B2, B3, and B4 are adjustable parameters and optimized from experimental data using non-linear least-squares. The regressed parameter values together with calculated RMSE and ARD were presented in Table S3 (Supporting Information). The overall ARD in methanol + water and ethanol + water at different temperatures was lower than 6%. Figure 2(c) and 2(d) visually show the smoothed solubility curve, which shows that CNIBS/Redlich–Kister model has well reproduced the experimental data and successfully described the positive synergistic effect of mixed solvents on AD solubility. The system of ethanol–water exhibited the maximum solubility for x3 = 0.15 ~ 0.3 at all temperatures studied. Besides, the maxima slightly shift to pure ethanol with increasing temperature.

4.3. Solubility predictions. 4.3.1. Jouyban-Acree Model. To describe AD solubility varying with both temperature and solvent composition, Jouyban-Acree model, a semi-empirical model,21 was used to fit experimental solubility data in methanol + water and ethanol + water. The model parameters of eq 3 are optimized from experimental data using the linear least squares method, and their regression values together with RMSE and ARD were reported in Table 1. The ARDs


16

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for the simplified Jouyban-Acree model in the methanol aqueous solution and ethanol aqueous solution are 6.1% and 2.6%, respectively. Notably Jouyban-Acree model can also be applied to predict solubility with eq 2, but here we used a simplified Jouyban-Acree model (i.e., eq 3) to correlate the solubility data for attaining better fitting result. 4.3.2. NRTL-SAC Model. For solubility prediction with NRTL-SAC model, four segment molecular parameters (i.e., X, Y−, Y+ and Z) of AD molecule need to be identified from its solubility data, and then these generated parameters were used to estimate solubility in the remaining mixtures of solvents. To test the model sensitivity to the solvent selection and obtain a reliable regression, four molecular parameters for AD molecule were optimized in three cases:

Method I: parameters were estimated from two pure solvents and 14 solvent mixtures, and then solubility prediction was carried out in all the remaining solvent mixtures.

Method II: parameters were estimated from two pure solvents and two solvent mixtures, and then solubility prediction was performed in the remaining solvent mixtures.

Method III: parameters were estimated from two pure solvents, and then solubilities in the other remaining solvent mixtures were predicted. For the first estimation (method I), two pure solvents (i.e., methanol and ethanol) and 14 compositions in methanol + water and ethanol + water binary mixtures at the mean harmonic temperature Thm [calculated as: Thm = n / ∑ i =11 / T ] were selected to optimize AD molecular n

parameters. This selection considered the effects of the solvent composition and temperature on the parameters estimation. The identified molecular parameters were then applied to estimate AD solubility in the remaining solvent mixtures. The RMSE and ARD for AD solubility prediction in the remaining solvent mixtures were listed in Table 2. Figure 3 displays the predicted solubility data varying with temperature in methanol + water and ethanol + water binary systems


17


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versus experimental results. This model clearly captures the temperature dependence of solubility, but it over-predicts the solubilities in the methanol aqueous solution and fails to predict the solubilities in ethanol + water mixtures. The predicted AD solubility using NRTLSAC model (method I) varying with solvent composition versus experimental data was shown in Figure 4. The results show that the model nicely predicts the solubility trend in methanol + water and the solubility maxima in the ethanol aqueous solution. However, poor results were found in the accuracy of solubility prediction, especially in ethanol + water. To reduce the number of experimental data used and to achieve a good predicted result, the second possible method of estimation (method II) can be performed with two pure solvents (i.e., methanol and ethanol) and two solvent mixtures. The solubility data in methanol, ethanol, methanol + water mixtures with x3 = 0.6407 , and ethanol + water mixtures with x3 = 0.7188 at temperature Thm were employed to generate the model parameters. Then AD solubilities in the other remaining solvent mixtures were predicted using those generated parameters. The final way of estimation (method III) uses AD solubilities in methanol and ethanol at various temperatures. The error between the experimental and calculated solubility data was minimized with the experimental data in methanol and ethanol at different temperatures. For all three cases, RMSE and ARD were calculated and compared. The resulting molecular parameters for AD and summary of these methods are given in Table 2. The hydrophobic and polar attractive segments optimized in method I are very close to those in method II, whereas this behavior was not found in the other two segments. Compared with segment values optimized from method I and II, method III obtains a very near value for hydrophobic segment but entirely different values for other segments. Similar results were also seen in the literature.9 Those differences can be attributed to the number and the nature of solvents used in parameters estimation.20 Moreover,


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the polar attractive segments estimated in method I and method II are zero, which indicates that AD molecule possesses a repulsive nature against the studied alcohol + water mixtures. Due to the main part of AD molecular volume is made up of alkyl group, and thus the electron-rich part and the hydrophobic effect of the steroid ring result in a repulsive interaction with polar solvent mixtures. As seen in Figure 5 and Table 2, we can concluded that method I is the best for predicting and describing AD solubility data in methanol + water and ethanol + water, but it fits all compositions in studied systems at temperature Thm and hence the best predicted results. Method II is the next best, and it is on the basis of the regression of only two pure solvents and two solvent mixtures at temperature Thm. Thus it is an effective approach for predicting AD solubility in mixed solvents and has a comparable ARD with method I. Besides, method III is the worst one for prediction because of its parameters regression only from pure solvents without considering the effect of water on solubility. In fact, the model predictive accuracy strongly depends on the used regression values of four molecular parameters, and thus the selection of the parameter values is a key step for this model. In order to choose the appropriate representative solvents and thus obtain the best predicted results, Bouillot et al.20 suggested that solvents should be selected based on the following relationship:

∑ X = ∑Y i

i

i

i

+

= ∑ Yi − = ∑ Z i i

(24)

i

However, that balance is hardly achieved especially when the number and the diversity of solvents are limited.10 Despite of those drawbacks, NRTL-SAC is useful to create an estimate for solubility in various solvents.20 4.3.3. COSMO-RS Model. For the solubility prediction of solid compounds with this model,

∆Gfus (T ) of the solute has to be carefully considered. As in general ∆Gfus (T ) is one of the


19


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largest error sources for the solubility prediction of solids,27, 36 the reference solubility and QSPR were used to determine ∆Gfus (T ) to reduce this error. Thus AD solubility predictions in methanol + water and ethanol + water were performed by two following approaches:

∆Gfus (T ) was calculated based on the reference solubility: using the experimental solubility of AD in a certain pure or mixed solvent at a given temperature, ∆Gfus (T ) (or ∆H fus ) of AD at these conditions can be solved by eq 16.27 Then solubility predictions were performed for all the solvent mixtures. Since this method combines COSMO-RS model with the reference solubility approach, COSMO-RS + Ref was used to denote it in this study. It is worth mentioning that ∆H fus determined by DSC was not used for the estimation of ∆Gfus (T ) due to its large extrapolation error and thus for AD solubility prediction.

∆Gfus (T ) can also be estimated by a QSPR approach and then solubility prediction for all the solvent mixtures. This approach combining the solubility prediction of COSMO-RS and QSPR for the determination of ∆Gfus (T ) is denoted as COSMO-RS + QSPR in this study. Figure 6 illustrates the comparison of AD solubility predictions from COSMO-RS + QSPR and from COSMO-RS + Ref using the experimental data in each composition at temperature Thm. As displayed in Figure 6(b) and 6(d), the predicted solubility temperature dependences using COSMO-RS + QSPR have a good agreement with experimental results in pure methanol and ethanol, but this method over-predicts the solubilities in methanol + water and fails to predict the solubilities in aqueous solution of ethanol. But the correction through using the reference solubility dramatically improves AD solubility prediction accuracy in solvent mixtures. As seen


20

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in Figure 6(a) and 6(c), COSMO-RS + Ref successfully predicted temperature effect on solubility in mixed solvents. Figure 7 shows the experimental data versus predicted solubilities using COSMO-RS + QSPR and COSMO-RS + Ref varying with solvent composition at different temperatures. As presented in Figure 7 (b) and 7 (d), COSMO-RS + QSPR encounters poor accuracy for AD solubility prediction in studied mixed solvent systems, while this method can capture the solubility trend varying with solvent composition. By contrast, COSMO-RS + Ref achieves satisfactory results for solubility prediction in all cases. As presented in Figure 7 (a), the predicted results of AD solubility using COSMO-RS + Ref in methanol + water mixtures were well agreed with experimental data, and thus COSMO-RS + Ref successfully predicted the solubility that decreases monotonically with increasing water. Figure 7 (c) shows that experimental solubility data in the ethanol aqueous solution were well reproduced by COSMO-RS + Ref, and thus the synergistic effect of mixed solvents was also accurately described. To conclude on COSMO-RS model, a global summary of the predicted results using COSMORS + QSPR and COSMO-RS + Ref is presented in Figure 8. AD solubility prediction from COSMO-RS + QSPR gives poor accuracy in mixed solvents, but has a good agreement with experimental data in pure solvents. This method also provides a clear solubility trend in aqueous mixtures of methanol without a solubility peak, but it fails to accurately predict the synergistic effect of ethanol–water system on solubility. By contrast, COSMO-RS + Ref achieves good prediction performances on both pure solvents and solvent mixtures, especially in the prediction of the maximum point of solubility, but it requires several experimental data for the estimation of

∆Gfus (T ) .


21


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4.4. Performance of the Different Predictive Models. Table 3 shows the performance comparison of three different models on AD solubility prediction. The RMSE and overall ARD of each model are used to compare predictive accuracy. Moreover, the number of regression parameters together with the number of experimental data used is also included to evaluate the applicability of each model. For simplified Jouyban-Acree model, nine empirical parameters were optimized from all the experimental data, with overall ARD of 4.4%, and thus it performs remarkably well as expected. In the case of NRTL-SAC model, the molecular parameters for AD were estimated in three cases, but method I gets the best prediction with overall ARD of 18.3%. This behavior is caused by the selection of representative solvent mixtures for parameters’ regression, because the effects of all the solvent compositions on the estimation of parameters were considered by method I. Nevertheless, this model employs only four molecular parameters to fit small amounts of experimental data, which maybe lead to its relative good accuracy on AD solubility prediction and the description of the maximum point of solubility. For COSMO-RS model, the Gibbs free energy of fusion ∆Gfus (T ) was estimated from both the reference solubility and QSPR approaches, and thus solubility prediction with COSMO-RS model was performed in two ways. It's not surprising that the performance of COSMO-RS + QSPR is quite poor for mixed solvents because no parameter needs to be optimized from experimental data, though this method has a good prediction in pure solvents (i.e., methanol and ethanol). In this approach, the predictive accuracy of solubility largely depends on the accuracy of estimated ∆Gfus (T ) and thus used QSPR parameters, whereas those parameters were derived only from aqueous solubility data without consideration of the effects of other solvents or mixed solvents on solubility, which maybe result in its extremely poor accuracy on AD solubility


22

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prediction in mixed solvents. By contrast, COSMO-RS + Ref has a nearly predictive accuracy in pure solvents but achieves better predicted results in mixed solvents, including the prediction of the maximum point of solubility, with overall ARD of only 4.9%. This may be because the reference solubility approach considers the effects of both solvent composition and temperature on ∆Gfus (T ) and thus improves its estimated accuracy. The weakness of this method is the requirement of the accurate reference solubility to determine ∆Gfus (T ) of the solute.

5. CONCLUSION The solubilities of AD in the methanol aqueous solution and ethanol aqueous solution from (275 to 325) K have been investigated. For pure solvents, AD solubility in ethanol is lower than it in methanol. The solubility in methanol + water dramatically decreases with increasing water in methanol; whereas the synergistic effect of mixed solvents was observed in ethanol–water system, and the maximum point of solubility occurs for x3 = 0.15 ~ 0.3 at studied temperatures. Furthermore, the modified Apelblat and CNIBS/Redlich–Kister models were used to describe successfully the effects of temperature and solvent composition on the solubility, respectively. Three most significant models (i.e., Jouyban-Acree model, NRTL-SAC model, and COSMORS models) for AD solubility prediction were studied. Among these predicted models, JouybanAcree model fits the experimental data best with the lowest overall ARD of 4.4%, but its applicability for solubility prediction is the lowest. Using some modifications for optimizing four segment molecular parameters in the NRTL-SAC model, a good predicted result with overall ARD of 18.3% and good applicability were achieved. But it can only give the relative accuracy of solubility prediction and just obtain a clear description of the maximum point of solubility in this study. The COSMO-RS model, when combined with QSPR for the estimation of fusion Gibbs free energy ∆Gfus (T ) , fails to predict AD solubility in mixed solvents; whereas the


23


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Page 24 of 40

combination of COSMO-RS model with reference solubility for estimating ∆Gfus (T ) achieves a better overall ARD of only 4.9% with good applicability. Moreover, the positive synergistic effect was accurately described by this approach.

AUTHOR INFORMATION Corresponding Author *Tel.: 86-22-27405754. Fax: 86-22-27374971. E-mail: [email protected].

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors are grateful to the financial support of National Natural Science Foundation of China (NNSFC 21176173), State Key Laboratory of Chemical Engineering of China (SKL-CHE11B02) and National High Technology Research and Development Program (863 program) (2012AA021202).

ASSOCIATED CONTENT Supporting Information Available Detailed numerical values of experimental solubility of AD in methanol + water and ethanol + water at temperatures from (275 to 325) K are listed in Table S1. Optimized parameters for the modified Apelblat model and CNIBS/Redlich–Kister model are given in Tables S2 and S3, respectively. The results of DSC and XRD of AD are presented in Figure S1. This information is available free of charge via the Internet at http://pubs.acs.org.


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ABBREVIATIONS aeff = average molecular contact area

a = empirical constant in the modified Apelblat model A0–A8 = model constants for Jouyban-Acree model in eq 2 A1–A9 = regressed model constants for simplified Jouyban-Acree model in eq 3 ARD = average relative deviation b = empirical constant in the modified Apelblat model B0–B4 = model constants for the CNIBS/Redlich–Kister model c1–c4 = QSPR parameters

c = empirical constant in the modified Apelblat model f = volume fraction in Jouyban-Acree model G ji = interaction energy between segments i and j in NRTL-SAC model

N JRing = number of ring atoms in compound J ri , I = contribution of segment i to molecule I in NRTL-SAC model

rI = total segment number of component I in NRTL-SAC model R = gas constant, 8.3145 J/ (mol·K)

RMSE = root-mean-square error T = temperature (K) Thm = the mean harmonic temperature VJ = molecular volume of the solute J

x = mole fraction in the solution x1 = mole fraction of AD in the solution


25


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Page 26 of 40

x2 = the solute-free mole fraction of cosolvent in binary mixed solvents x3 = the solute-free mole fraction of water in binary mixed solvents

X = hydrophobic segments in NRTL-SAC model Y− = polar attractive segments in NRTL-SAC model Y+ = polar repulsive segments in NRTL-SAC model Z = hydrophilic segments in NRTL-SAC model

Greek Letters

α ji = nonrandomness parameter between segments j and i γ = activity coefficient δ = solubility parameter

µ J(p) = chemical potentials of the solute J in pure solute µ J(S) = chemical potentials of the solute J in system S J µ H2O = chemical potential of solute J in water

Γ lcm = activity coefficient of segment m in NRTL-SAC model I Γlc, = activity coefficient of segment m in component I in NRTL-SAC model m

∆Gfus (T ) = fusion Gibbs free energy at temperature T (J/mol) ∆H fus = enthalpy of fusion at the melting point (J/mol)

∆Sfus = entropy of fusion at the melting point (J/ (mol·K))

τ ji = binary interaction parameter between segments j and i in NRTL-SAC model

φI = fraction of the total contribution of segments from compound I in NRTL-SAC model Superscripts


26

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C = combinatorial calc. = calculated data exp. = experimental data R = residual

Subscripts d = dispersion, for solubility parameters h = hydrogen-bonding, for solubility parameters i, j, k, m, and m' = segment-based species indices I, J = component

m = melting p = polar, for solubility parameters

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(27) Eckert, F.; Klamt, A. COSMOtherm, Version C3.0, Release 13.01, COSMOlogic GmbH & Co. KG, Leverkusen, Germany, 2013. (28) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865. (29) Klamt, A.; Jonas, V.; Bürger, T.; Lohrenz, J. C. Refinement and Parametrization of COSMO-RS. J. Phys. Chem. A 1998, 102, 5074-5085. (30) Ding, Z.; Zhang, R.; Long, B.; Liu, L.; Tu, H. Solubilities of m-Phthalic Acid in Petroleum Ether and its Binary Solvent Mixture of Alcohol + Petroleum Ether. Fluid Phase Equilib. 2010, 292, 96-103.

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Table 1. Optimized Parameters and Deviations for Prediction of AD Solubility in Methanol + Water and Ethanol + Water Binary Mixtures by the Simplified Jouyban-Acree Model value

value

parameter

methanol + water

ethanol + water

parameter

methanol + water

ethanol + water

A1

23.905

-45.624

A7

18642

13029

A2

-3977.9

-1517.4

A8

-18003

-9553.0

A3

-2.732

7.939

A9

72.040

37.189

A4

-481.46

-249.44

A5

21622

13335

104RMSE

2.357

2.223

A6

-7898.5

-8217.9

ARD

0.061

0.026

Table 2. NRTL-SAC Model Optimized Molecular Parameters for AD Molecule and Deviations for Solubility Predictions from Three Regression Cases

method

X

Y−

Y+

Z

Na

103RMSE

ARD

I

0.765

0.000

1.792

0.427

16

1.244

0.183

II

0.735

0.000

1.578

0.274

4

1.868

0.254

III

0.758

0.635

3.013

0.000

20

1.293

0.341

Na = number of experimental data used.


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Table 3. RMSE and ARD Values for Different Models

model

103RMSE

overall ARD

na

Nb

Jouyban-Acree

0.229

0.044

18

145

NRTL-SAC (method I)

1.244

0.183

4

16

COSMO-RS+Ref

0.318

0.049

16

16

COSMO-RS+QSPR

3.010

1.912

0

0

na = number of parameters optimized. Nb = number of experimental data used.

Figure 1. Chemical structure (1) and 3D COSMO-surface screening charge densities (2) of AD. Reprinted from ref. 1, Copyright (2014), with permission from Elsevier.


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Figure 2. Solubility profiles of AD in different binary mixtures correlated by modified Apelblat equation ((a), (b)) and CNIBS/Redlich–Kister model ((c), (d)), respectively.


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Figure 3. Experimental solubility data of AD in different binary mixtures and their estimation curve as a function of temperature using NRTL-SAC model (method I).


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Figure 4. Experimental points of AD solubility and their estimation curve as a function of solvent composition using NRTL-SAC model (method I) at various temperatures.

Figure 5. Predicted AD solubilities versus experimental data in various binary mixtures at different temperatures using three different segment quadruplets of NRTL-SAC model.


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Figure 6. Experimental solubilities of AD and their predicted values with COSMO-RS + Ref ((a), (c)) and COSMO-RS + QSPR ((b), (d)) in mixed solvents as a function of temperature.


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Figure 7. Experimental solubilities of AD and their predicted data with COSMO-RS + Ref ((a), (c)) and COSMO-RS + QSPR ((b), (d)) in mixed solvents as a function of the solvent composition.


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Figure 8. Comparison of COSMO-RS + Ref and COSMO-RS + QSPR methods for AD solubility prediction.


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Experimental Determination and Computational Prediction of

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