Sublimation of Molecular Crystals: Prediction of Sublimation Functions

Apr 21, 2010 - Studying the sublimation thermodynamics of ethionamide and pyridinecarbothioamide isomers by transpiration method. Svetlana Blokhina ...
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DOI: 10.1021/cg1001946

Sublimation of Molecular Crystals: Prediction of Sublimation Functions on the Basis of HYBOT Physicochemical Descriptors and Structural Clusterization

2010, Vol. 10 2707–2712

German L. Perlovich*,†,‡ and Oleg A. Raevsky‡ †

Institute of Solution Chemistry, Russian Academy of Sciences, 153045 Ivanovo, Russia, and Institute of Physiologically Active Compounds, Russian Academy of Sciences, 142432, Chernogolovka, Russia

‡

Received February 7, 2010; Revised Manuscript Received March 26, 2010

ABSTRACT: Quantitative structure-property relationship (QSPR) models on the basis of HYBOT physicochemical descriptors (molecular polarizability, the sum of all H-bond acceptor, and the sum of H-bond donor factors in a molecule) have been developed to predict sublimation enthalpies and Gibbs energies of molecular crystals. The experimental database analyzed includes 1766 values of sublimation enthalpies and 965 values of Gibbs energies. Fragmentation of the training sets for structurally similar groups/clusters both for enthalpies and Gibbs energies has been carried out using Tanimoto similarity coefficients Tc with the restriction of 0.5 e Tc e 1. According to this approach, any thermodynamic function, Y(i) (where Y= 298 298 298 , ΔHsub , ΔSsub ), has been calculated on the basis of experimental values of the same function of structurally closely related ΔGsub compounds (nearest neighbors, nn), Y(nn), and the difference in their relevant physicochemical descriptors, ΔY: Y(i) =Y(nn) þ ΔY. The value of unknown thermodynamic function has been calculated as a mean value from the values obtained for each of the elements in a cluster.

Introduction Thermodynamic characteristics of sublimation processes of molecular crystals are significant parameters determining the solubility of compounds and giving an opportunity to estimate molecular solvation properties experimentally. It is wellknown that the key issue in developing bioavailable drugs is to solve the problem of poor solubility and permeability of many drug candidate substances. Therefore, it is essential for the pharmaceutical industry and medicine field to understand the fundamental aspects of studying the solubility and membrane permeability processes. It should be noted that there are a lot of experimental data for aqueous solubility and this fact can be illustrated in detail by the example from Yalkowsky and He’s handbook.1 Moreover, there are a lot of models for prediction of aqueous solubility of a wide range of diverse compounds, which have been developed by various scientific groups: Yalkowsky and Valvani2 (general solubility equation), Myrdal et al.3 (group contribution methods: AQUAFAC approach), Huuskonen4 (molecular topology approach), Fredenslund et al.5 (group contribution methods: UNIFAC or ASOG), Abraham et al.6 (solvation energy relationship), Jorgensen and Duffy7 (QikProp method), Tetko et al.8 (electrotopological state indices method), Raevsky et al.9 (HYBOT descriptors), McFarland et al.,10 Livingstone et al.,11 Bruneau,12 Liu et al.,13 Taskinen and Norinder,14 Dearden,15 Bergstrom,16 Hansch et al.17 There are several sources summarizing sublimation enthalpies18-21 and saturated vapor pressure22 data. In recent decades, a lot of approaches for sublimation enthalpy estimation have been developed. These approaches can be conditionally divided into two groups. The first group includes various additive methods/schemes.23-25 The second one is based on

using pair potential functions and X-ray diffraction data from single crystals.26-28 This direction has been developed in recent decades due to an essential progress in X-ray diffraction techniques and creation of an X-ray database.29 Nevertheless, the approach has some disadvantages. First, for some groups of substances it is impossible to solve crystal structures as it is difficult to prepare single crystals. Second, for each class of the compounds it is necessary to normalize the pair potential functions. In this work, we would like to propose the QSAR approach for estimation of sublimation thermodynamic functions based on both the HYBOT descriptors30 and the conception of the nearest neighbors.31-33 The approach gives an opportunity to predict the above-mentioned thermodynamic functions of various molecular crystals only by their chemical structures. It should be noted that if the experimental data volume growth persists the proposed calculation algorithms will work more effectively. Experimental Section

*To whom correspondence should be addressed: Tel.: (þ7) 4932 533784; fax: (þ7) 4932 336237; e-mail: [email protected].

Methods. Sublimation Enthalpy and Gibbs Energy Database. In order to analyze the experimental data, we used a database created by us on the basis of values published in the literature. The database includes the following information: the method used to measure the thermodynamic characteristics; the experimental temperature interval; the equation describing temperature dependence of saturated vapor pressure; the sublimation enthalpies both at the experimental T 298 temperature, ΔHsub (exp), and at 298.15 K, ΔHsub (exp), in kJ 3 mol-1; 298 the standard Gibbs energy values, (ΔHsub (exp), in kJ 3 mol-1; the fusion enthalpies and temperatures of molecular crystals and, finally, the refcodes29 of the compounds described by X-ray diffraction experiments. The database includes enthalpies and Gibbs energies obtained by various methods and at different experimental temperatures; therefore, we used a special algorithm to reduce the experimental values to comparable conditions. If the same substance had been earlier described in the literature by several methods, preference was given to the methods allowing us to obtain both sublimation enthalpy and Gibbs energy (saturated vapor

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pressure) data at the same time: transpiration, torsion effusion, mass effusion, isoteniscope, mercury manometer methods, etc. Moreover, with other conditions being identical, preference was given to the data received at the temperatures maximally close to the standard condition (298.15 K). The saturated vapor pressure at 298.15 K was calculated from by temperature dependence. It should be noted that the temperature dependencies obtained above 423.15 K were not taken for the analysis in order to avoid approximation errors. If the data was only taken for sublimation enthalpies, preference was given to the values obtained by the calorimetric method. In the final analysis, the database included 1766 values for sublimation enthalpies. Among these compounds, only 965 values of the Gibbs energies were known. So, it is evident that the number of the sublimation enthalpy values was 1.8 times bigger than the Gibbs energy ones. This relationship can be explained by the fact that the earlier works described a lot of calorimetric experiments which do not give an opportunity to estimate saturated vapor pressure. The experimental data for the sublimation enthalpies was derived by the following methods: 904 - mass effusion - knudsen effusion; 325 - transpiration; 305 - calorimetry; 71 - torsion; 20 manometer (mercury, piston); 10 - head space analysis; 5 - isoteniscope and 126 - others. On the other hand, the saturated vapor pressure values of 965 substances were measured by the following methods: 515 - mass effusion - knudsen effusion; 272 - transpiration; 47 - torsion; 16 - manometer (mercury, piston); 5 - isoteniscope and 110 - others methods. Computer Programs. All the descriptors were calculated by the program package HYBOT-PLUS (version of 2003) in Windows.30 The intervals of descriptor values for the selected 1766 compounds were equal to 3-60 for molecular polarizability (R), 0-15 for the sum of H-bond acceptor factors (ΣCa) and (-11) - (0) for the sum of H-bond donor factors (ΣCd). The intervals of experimental parameters were equal to 37.7-209.0 kJ 3 mol-1 for the sublimation enthalpies and 0.92-120.0 kJ 3 mol-1 for the Gibbs energy. Statistically, such wide intervals together with the distribution of parameter values close to the normal distribution increased the possibility of obtaining stable models.34 The structural similarity was estimated using Tanimoto similarity indices (Tc) obtained by means of the program MOLDIVS (MOLecular DIVersity & Similarity).35 T c ¼ N ðABÞ=½N ðAÞ þ N ðBÞ - N ðABÞ

ð1Þ

where N(A) is the number of fragments in molecule A, N(B) is the number of fragments in molecule B, N(A&B) is the number of common fragments in molecules A and B. In this program, molecular fragments are defined as atom centered concentric environments. The fragments consist of a central atom and neighboring atoms connected to it within a predefined sphere size (the number of the bonds between the central and edge atoms). For each atom in a fragment the information on the atom and bond type, charge, valency, cycle type and size is coded into fixed-length variables, which are subsequently used to define a pseudorandom hash value for this fragment. The program permits an estimation of similarity of each molecule in the database with all other molecules by sorting them according to the value of similarity with the initial molecule.

Results and Discussion To create the training and test sets, we employed the following procedure. All the analyzed compounds of the initial database (sublimation enthalpies) were ranked from minimal to maximal molecular weights. After that we collected every third substance from the database by scanning the structures from minimal to maximal molecular weights. These compounds composed the test set. The rest of the substances were used as a training set. A similar algorithm was applied for the Gibbs energy values. Such database subdivided into test and training sets seems reasonable as the distribution functions of both sets are similar. To build correlation models describing sublimation thermodynamic functions, we used HYBOT physicochemical

descriptors.30 It should be noted that for searching appropriate correlations within the training sets for both thermodynamic functions, all HYBOT (32) descriptors were applied to compose one-parametric equations. After that descriptors with maximal values of pair correlation coefficients were selected. On the basis of the descriptors set, we tried to examine possible variations of three-parametric models. The procedure resulted in choosing the most suitable (with the maximal correlation parameters) variables: molecular polarizability (mainly the bulk effect descriptor), R, the sum of all H-bond acceptor factors in a molecule, ΣCa, and the sum of H-bond donor factors, ΣCd. The outlined descriptors display the main interaction types in molecular crystals and have a strict physical sense. We used the same descriptors both for the sublimation enthalpies and for the Gibbs energies, as this approach gives an opportunity to estimate the sublimation entropy term easily and, as a consequence, to predict all thermodynamic functions of the sublimation process. The choice of the same descriptors for the two functions was based on the following factors. On the one hand, ΣCa and ΣCd descriptors (free energy acceptor and donor factors) were taken from the correlation equations, built on H-bonding Gibbs’ energies analysis of various compounds.36 The physical sense of these descriptors was directly related to the Gibbs’ energies; therefore, it was assumed that they were to describe the sublimation process Gibbs energies adequately. On the other hand, ΣEa and ΣEd descriptors (enthalpy acceptor and donor factors), taken from the correlation equations, built on the basis of H-bonding enthalpy analysis of various compounds, could be ideal for analyzing sublimation enthalpies (according to their physical sense). However, experimental value libraries, taken as the ground for calculating these descriptors, were considerably smaller in volume as compared to similar libraries for Gibbs energies. As a result, the accuracy of ΣEa and ΣEd estimation was considerably lower than that for ΣCa and ΣCd. This assumption was verified by replacing ΣCa and ΣCd for ΣEa and ΣEd descriptors in a multiparameter equation for enthalpy description. It resulted in a decline of correlation parameters. However, there is another argument in favor of using/applying ΣCa and ΣCd to sublimation enthalpies. It is common knowledge that for a number of processes (sublimation process in this case) a linear compensation effect between enthalpy and entropy terms is observed. Therefore, it is this effect that may ensure a proper description of sublimation enthalpies using ΣCa and ΣCd descriptors. The correlation equations describing sublimation enthalpies and Gibbs energies on the chosen training sets can be presented as 298 ðcalÞ ¼ ð39:8 ( 1:4Þ þ ð2:03 ( 0:05ÞR ΔHsub

þ ð4:6 ( 0:2ÞΣCa - ð4:7 ( 0:2ÞΣCd

ð2Þ

rms ¼ 14:7; R ¼ 0:8141; n ¼ 1316; F ¼ 858:1 ΔG298 sub ðcalÞ ¼ ð- 0:5 ( 1:6Þ þ ð1:37 ( 0:06ÞR þ ð3:84 ( 0:25ÞΣCa - ð2:97 ( 0:26ÞΣCd

ð3Þ

rms ¼ 10:9; R ¼ 0:7761; n ¼ 686; F ¼ 344:4 The entropy sublimation term can be described by the equation: 298 298 ðcalÞ ¼ ΔHsub ðcalÞ - ΔG298 TΔSsub sub ðcalÞ ¼ 40:3 þ 0:66R þ 0:76ΣCa - 1:73ΣCd

ð4Þ

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298 298 Figure 3. Plot of ΔHsub (exp) vs ΔHsub (cal) predicted by the method of nearest neighbors.

298 298 Figure 1. Plot of ΔHsub (exp) vs ΔHsub (cal) predicted by eq 2.

298 298 Figure 2. Plot of ΔGsub (exp) vs ΔGsub (cal) predicted by eq 3.

At the next stage of our analysis, we estimated predicted reliability of the derived equations on the created test sets for the sublimation enthalpies (Figure 1): 298 298 ðexpÞ ¼ ð- 3:8 ( 3:7Þ þ ð1:03 ( 0:03ÞΔHsub ðcalÞ ΔHsub

ð5Þ rms ¼ 14:9; R ¼ 0:8176; n ¼ 450 for the sublimation Gibbs energies (Figure 2): 298 ΔG298 sub ðexpÞ ¼ ð- 3:97 ( 2:25Þ þ ð1:07 ( 0:05ÞΔGsub ðcalÞ

ð6Þ rms ¼ 11:2; R ¼ 0:8185; n ¼ 279 298 It was evident that there was a dispersion between ΔHsub 298 (exp) and ΔHsub (cal) values. However, if the compounds from the training set were selected by means of the algorithm, which takes into account structural similarities of substances (for example, benzoic acid derivatives, biphenyl derivatives, and others), the dispersion decreased essentially. These regularities can be connected to the fact that each class/group of the compounds has a special topological and geometrical molecular conjugation with the nearest molecules in the crystal lattice. Such sort of molecular ordering creates a molecular

packing architecture in crystals. In its turn, with the defined packing architecture there is a strict proportion between different types of interactions: specific (hydrogen bonds, donor-acceptor, coulomb, π-π, etc.) and nonspecific (van der Waals) ones. Moreover, molecules belonging to various classes have (a) different conformational strains in the crystals and (b) different hydrogen bond networks topology (one-, two-, or three-dimensional). The HYBOT descriptors give a proper description of the above-mentioned types of interactions between molecules in solutions. However, these descriptors cannot fully take into account the interaction peculiarities between the molecules in the crystal lattice. If the compounds are divided into structurally similar groups/clusters, then within each group there will be identical factors characterizing these crystal structures. The uniformity properties will improve statistical parameters of the correlation equations. Guided by the observations presented above, we chose the algorithm for creating fragmentation of the training set, which included groups/clusters with structurally similar compounds. For these aims we used Tanimoto similarity coefficients Tc (Tc=0: no similarity; Tc=1: identity).30 A criterion for ascribing substances to the same class was introduced. It can be formulated by the expression: 0.5 e Tc e 1. In order to calculate the unknown thermodynamic function within the clusters, we applied an approach based on similarity and physicochemical descriptors introduced by Raevsly et al.31 According to this approach, any thermodynamic function, 298 298 298 , ΔHsub , ΔSsub ), is calculated on the basis Y(i) (where Y=ΔGsub of experimental values of the same function of structurally closely related compounds (nearest neighbors, nn), Y(nn), and the difference in their relevant physicochemical descriptors, ΔY: ð7Þ YðiÞ ¼ YðnnÞ þ ΔY If we knew the thermodynamic function of the nearest neighbor and the correlation equation describing the relationship between the thermodynamic function and physicochemical descriptors obtained on the basis of the training set (for example, eq 2), then the value of the unknown thermodynamic function was estimated by the following equation: Y ðiÞ ¼ YðnnÞ þ A1 ðRðiÞ - RðnnÞ Þ þ A2 ðΣCaðiÞ - ΣCaðnnÞ Þ þ A3 ðΣCdðiÞ - ΣCðnnÞ Þ

ð8Þ

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298 298 Figure 4. Plot of ΔGsub (exp) vs ΔGsub (cal) predicted by the method of nearest neighbors.

If the cluster consisted of more than one compound, the value of the unknown thermodynamic function was calculated as an arithmetical mean of the values received from each element of the cluster. If the substance of interest had no nearest neighbor (the number of elements of the cluster was zero), then the value of the required function was calculated directly from the correlation equation obtained on the training set. Following the presented algorithm, we derived the results for the test sets of the sublimation enthalpies (Figure 3): 298 298 ðexpÞ ¼ ð0:2 ( 3:2Þ þ ð0:99 ( 0:03ÞΔHsub ðcalÞ ΔHsub

298 Figure 5. Dependence between experimental ((ΔHsub (exp)) and 298 calculated (ΔHsub (cal)) on the basis of HYBOT descriptors using eq 2 sublimation enthalpies for the structurally closely related to octanamide compounds (cluster with 0.5 e Tc e 1). Numbering corresponds to 1 - tetradecanamide; 2 - dodecanamide; 3 - N-methyl hexadecanamide; 4 - decanamide; 5 - nonamide; 6 - hexanamide; 7 pentanamide; 8 - butanamide.

ð9Þ

rms ¼ 13:9; R ¼ 0:8427; n ¼ 450 for the Gibbs energy (Figure 4): 298 ΔG298 sub ðexpÞ ¼ ð- 3:16 ( 1:75Þ þ ð1:06 ( 0:03ÞΔGsub ðcalÞ

ð10Þ rms ¼ 9:35; R ¼ 0:8764; n ¼ 279 The comparison of eqs 5 and 9 leads to a conclusion that the estimation of sublimation enthalpies by clusterizing the training set improves prediction of the experimental values. An analogous tendency is observed for the Gibbs energies (eqs 6 and 10). The proposed approach will give an essential improvement if the training set is increased (database filling) and if a maximal number of structural diverse substances is included in the training set. With an essential database filling in the future, it would be reasonable to make the procedure of compounds clusterization stricter; that is, the lower limit of Tanimoto similarity coefficient would be increased from 0.5 up to 0.75 and higher. If the cluster includes a lot of elements/ substances to carry out statistical estimation, the values can be smoothed by the least-squares method in coordinates 298 298 (exp); ΔHsub (cal)) and the equation received can be (ΔHsub used for prediction of the required thermodynamic function. As an example, let us consider substances structurally similar to octanamide with Tanimoto similarity coefficient lying within 0.5 e Tc e 1. The cluster for the mentioned compound includes eight substances. Moreover, seven of them are amide derivatives with different variations of the alkyl chain length and 0.62 e Tc e 1 (tetradecanamide, nonamide, dodecanamide, decanamide, hexanamide, pentanamide,

298 Figure 6. Dependence between experimental ((ΔHsub (exp)) and 298 calculated (ΔHsub (cal)) on the basis of HYBOT descriptors using eq 2 sublimation enthalpies for the structurally closely related to dibenzo[a,e]pyrene compounds (cluster with 0.5 e Tc e 1). Numbering corresponds to 1 - pyranthrene; 2 - phenanthro[1,10,9,8-opqra]perylene; 3 - picene; 4 - 1,2:6,7-dibenzophenanthrene (benzo[b]chrysene); 5 - 1,2:3,4-dibenzanthracene (benzo[b]triphenylene); 6 - benzo[ghi]perylene; 7 - perylene; 8 - benzo[e]pyrene; 9 - benzo[a]pyrene; 10 - chrysene.

butanamide) and one substance is N-methyl hexadecanamide 298 298 (exp); ΔHsub (Tc = 0.52). The results in coordinates (ΔHsub (cal)) are shown in Figure 5. It is clear that the experimental value with Tc = 0.52 is slightly deviated from the common trend line. So, if the correlation equation in the reduced coordinates is known, then the sublimation enthalpy of any required compound belonging to this cluster can be estimated. It should be noted that the correlation equation of the cluster includes all information on the molecules in crystals connected with peculiarities of packing architectures, hydrogen bond networks, and conformational strains.

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corrected according to the peculiarities of molecular interactions in the crystal lattice. Conclusion

298 Figure 7. Dependence between experimental ((ΔHsub (exp)) and 298 calculated (ΔHsub (cal)) on the basis of HYBOT descriptors using eq 2 sublimation enthalpies for the structurally closely related to hexacosane compounds (cluster with 0.5 e Tc e 1). Numbering corresponds to 1 - tetracosane; 2 - tricosane; 3 - 6-heptadecanol; 4 9-heptadecanol; 5 - heneicosane; 6 - 7-heptadecanol; 7 - 1-pentadecanol; 8 - n-nonadecane; 9 - n-octadecane; 10 - 1-tetradecanol; 11 2-pentadecanone; 12 - 1-tridecanol; 13 - n-heptadecane; 14 2-tetradecanone; 15 - 1-dodecanol; 16 - 7-tridecanone; 17 - nhexadecane; 18 - n-pentadecane; 19 - n-tetradecane; 20 - decanol; 21 - n-tridecane.

At the next stage, let us consider the cluster for dibenzo[a,e]pyrene (Figure 6), that is, a group of compounds with very different structures in comparison with the previous case. The cluster includes 10 substances: benzo[e]pyrene (Tc = 0.71); benzo[a]pyrene (0.68); pyranthrene (0.61); benzo[ghi]perylene (0.6); perylene (0.52); phenanthro[1,10,9,8-opqra]perylene (0.52); 1,2:6,7-dibenzophenanthrene (benzo[b]chrysene) (0.5); 1,2:3,4-dibenzanthracene (benzo[b]triphenylene) (0.5); picene (0.5); chrysene (0.5). In spite of the fact that the dispersion of Tanimoto similarity coefficients is 0.5-0.71, all values can be appropriately approximated by the linear trend in coordinates 298 298 (exp); ΔHsub (cal)). The correlation coefficients of the (ΔHsub equation differ from the previous ones, and this fact can be explained by the peculiarities of crystal structures of compounds of various clusters. It should be mentioned that the compounds belonging to some clusters cannot be smoothed by the linear trend. As an example, the substances with structures similar to hexacosane are presented in Figure 7. The maximal deviation is observed for hexadecane; however, its Tanimoto similarity coefficient is equal to 1. In this case, it is difficult to explain the reasons for the deviation. They can be connected both with peculiarities of the crystal structures (although for the similar compounds this trend should be kept) and with artifacts of the experiments carried out: the values for the compounds in the cluster are often received by different authors and, moreover, with various methods. As the training set fills with new experimental data, the number of both compounds in the cluster and the clusters will increase and, as a consequence, the thermodynamic functions will be predicted with higher accuracy. However, every time it will be difficult to describe the new class of substances, which are not presented in the training set. Nevertheless, at “zero” approximation the required values can be predicted by HYBOT descriptors and later on (as the cluster fills) they can be

In the presented work, we have analyzed the database of sublimation thermodynamic functions of molecular crystals composed on the basis of the available literature materials. To predict the sublimation thermodynamic functions, we have applied the QSAR approach using HYBOT physicochemical descriptors: molecular polarizability (R), the sum of all H-bond acceptor (ΣCa), and the sum of H-bond donor (ΣCd) factors in a molecule. We used the same descriptors both for the sublimation enthalpies and for the Gibbs energies, as this approach gives an opportunity to estimate the sublimation entropy term easily and, as a consequence, to predict all thermodynamic functions of the sublimation process. The study has proved that the developed correlation models work better for structurally similar compounds. Therefore, an algorithm of the training set clusterization using Tanimoto similarity coefficients Tc with restriction 0.5 e Tc e 1 was proposed. If the cluster consisted of more than one compound, the value of the unknown thermodynamic function was calculated as an arithmetical mean from the values received from each element of the cluster. If the substance of interest had no nearest neighbor (the number of elements of the cluster was zero), then the value of the required function was calculated directly from the correlation equation obtained on the training set. It was noted that as the training set fills with new experimental data, the number of both compounds in the cluster and the clusters increases and, therefore, the thermodynamic functions can be predicted with higher accuracy. Acknowledgment. We acknowledge financial support from the International Science & Technology Centre (Project #888) and Russian Foundation of Basic Research (No. 09-0300057). Supporting Information Available: Table 1 SI: HYBOT Descriptors, Experimental and Calculated by eq 1 Sublimation Enthalpies for the Test Set. Table 2 SI: HYBOT Descriptors, Experimental and Calculated by eq 1 Sublimation Energy Gibbs for the Test Set. This material is available free of charge via the Internet at http://pubs.acs.org.

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