Calculation of Attachment Energies and Relative Volume Growth

Apr 8, 2005 - ... Curtin University of Technology, P.O. Box U1987, Perth, Western Australia 6845, Australia, and Department of Chemistry, University C...
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Calculation of Attachment Energies and Relative Volume Growth Rates As an Aid to Polymorph Prediction Coombes,*,†

Catlow,†

David S. C. Richard A. Andrew L. Rohl,§ and Sarah L. Price⊥

Julian D.

Gale,‡

CRYSTAL GROWTH & DESIGN 2005 VOL. 5, NO. 3 879-885

Davy Faraday Research Laboratory, The Royal Institution of Great Britain, 21 Albemarle Street, London W1S 4BS, U.K., Nanochemistry Research Institute, Curtin University of Technology, P.O. Box U1987, Perth, Western Australia 6845, Australia, A.J. Parker CRC for Hydrometallurgy, Nanochemistry Research Institute, Curtin University of Technology, P.O. Box U1987, Perth, Western Australia 6845, Australia, and Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, U.K. Received August 20, 2004;

Revised Manuscript Received February 13, 2005

ABSTRACT: We calculate the morphologies of a number of the observed and hypothetical crystal structures of paracetamol, parabanic acid, and pyridine using the attachment energy model. We also estimate the relative growth volumes of the different polymorphs. This quantity is found to exhibit a large variation, which is generally well correlated with the attachment energy of the most dominant face of each polymorph, thus indicating how one face controls crystal growth. Such calculations suggest which thermodynamically feasible crystal structures could have a kinetic advantage in crystal growth. The application of the present results to polymorph prediction is discussed. Introduction The prediction of polymorphic structures of molecular crystals is a problem of fundamental scientific interest and of particular importance in the pharmaceutical industry. Changes in polymorphic form can affect properties such as bioavailability, solubility and morphology as well as being of significance in the area of patent protection.1 Over the past few years, polymorph prediction has been based largely on the relative lattice energies of different structures. It is commonly found that many unobserved hypothetical crystal structures are predicted close to the global minimum in lattice energy.2 In such cases, kinetic factors, such as ease of nucleation, barriers to transformation to more thermodynamically stable forms, and rates of crystal growth will influence whether metastable polymorphs are observed. The crystallization process, with the dependence on conditions including solvent, seeding and impurities is complex. Reliable polymorph prediction is clearly a major challenge.3,4 The morphology of a crystal can have a great impact on the properties of drugs at both the production and delivery stage. For example, in the production stage, there can be considerable problems if the morphology of the crystal is modified due to impurities or changes in solvent. At the delivery stage, the shape of a crystal can influence its dissolution rate, which is dependent on surface area. However, different morphologies do not imply different polymorphs, since a different morphol* To whom correspondence should be addressed. Mailing address: Davy Faraday Research Laboratory, Royal Institution of Great Britain, 21 Albemarle Street, London W1S 4BS, U.K. Telephone: +44 7409 2992. Fax: +44 7629 3569. E-mail: [email protected]. † Davy Faraday Research Laboratory, The Royal Institution of Great Britain. ‡ Nanochemistry Research Institute, Curtin University of Technology. § A.J. Parker CRC for Hydrometallurgy, Nanochemistry Research Institute, Curtin University of Technology. ⊥ Department of Chemistry, University College London.

ogy can be observed by preferential crystal growth in one direction.5 One approach that is routinely used in predicting the morphology of crystals is the calculation of the attachment energy of different surfaces, i.e., the energy released when a stoichiometric growth layer of material is added to each surface. This model is based on the idea of layers attaching themselves to a growing crystal and is most appropriate for crystals grown from the vapor or low supersaturations.6 Attachment energy calculations have been used in a wide range of studies for morphology prediction of organic molecular crystals.7-13 Some previous crystal structure prediction studies (see, for example, ref 14) have used the growth morphology in order to assess whether any hypothetical lowenergy crystal structures would have a face that would present growth difficulties. Such crystal structures, although thermodynamically feasible, would be unlikely to be observed polymorphs. This approach was extended to consider the relative volume growth rates of the predicted low energy crystal structures of p-dichlorobenzene.15 In this case, the three known polymorphs were found as the 2nd, 3rd and 4th most stable in lattice energy but had a significantly greater relative volume growth rate than the hypothetical structure, which was 0.4 to 0.6 kJ/mol more stable. This observation suggests that the most stable predicted structure has not been observed, because closely related observed polymorphs grow more quickly. In this paper, we explore the variation in the attachment energies of the slowest growing face, and the relative volume growth rates, for sets of real and hypothetical crystals structures that are predicted to have very similar lattice energies. This analysis shows the likely strengths and limitations of using attachment energy calculations to predict which thermodynamically feasible crystal structures may be observed polymorphs.

10.1021/cg049707d CCC: $30.25 © 2005 American Chemical Society Published on Web 04/08/2005

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Figure 1. Molecules considered in the study: (a) paracetamol; (b) parabanic acid; (c) pyridine.

Methods We have calculated the attachment energies for the observed polymorphs and other hypothetical crystal structures of similar energetic stability for three contrasting organic molecules. All crystal structures are lattice energy minima, and for each molecule, all the crystal structures were optimized with the same model intermolecular potential and same rigid ab initio optimized molecular structure. Thus the comparison of calculations for different crystal structures of the same molecule gives an overall picture of how one slow growing face affects the predicted relative growth rate of the crystals, within the attachment energy model. In particular, we have calculated the growth morphologies for the observed and hypothetical crystal structures generated in computational searches for the global minimum in the lattice energy for paracetamol,14 parabanic acid16 and pyridine,17 whose molecular structures are shown in Figure 1. In this section, we first discuss the model used to calculate the attachment energies and then the programs and methods used. Attachment Energies. The attachment energy model6 for the growth morphology of a crystal is based on the assumption that the growth rate of a face is proportional to the absolute value of the attachment energy. The morphology is defined by a Wulff plot, in which the distance from the origin to the (h,k,l) face, Rhkl, is proportional to the magnitude of the attachment energy (which is negative), i.e., Rhkl ∝ |Eatt(h,k,l)|. In this paper, we make the assumption that the proportionality constant is the same for all faces of crystals composed of the same molecules. Thus we can compare both the growth rates of faces of both real and hypothetical crystal structures, and compare the volumes enclosed by the growth morphology crystal shape, as a relative crystal growth rate. This assumption is as limited as the attachment energy model in its validity for individual crystals, i.e., it is only a valid approximation for vapor grown crystals bounded by faces in which there are two-dimensional strong intermolecular interactions (F-faces) below their roughening temperature.18 Despite these theoretical restrictions, the attachment energy model is widely used as a routine modeling method for predicting the morphologies of the crystal structures of organic molecules.19 This is because faster growing faces are generally not observed in the growth morphology and so considerable errors in the modeling of their growth rate by the attachment energy model are often not apparent. The attachment energy model cannot reflect any effects of solvent20 or additives21 on crystal growth, and also cannot predict the differential growth rate of the (h,k,l) and (-h,-k,-l) faces in polar crystals such as urea.22 Nevertheless, a variety of studies show that the model readily provides a reasonable estimate of the dominant faces of the crystal habit for a range of organic molecules.19,23 It is undoubtedly

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considerably more successful for organic crystal morphologies than the Bravais-Friedel-Donnay-Harker (BFDH) rule,24 which assumes that the growth rate is proportional to the interplanar spacing, dhkl, although the BFDH model is used to select the faces for consideration. We note that considerable progress is being made in developing more accurate models for the morphologies and relative growth rates of organic crystals,25,26 considering temperature27 and solvent effects28 and a variety of other factors.29 For example, Winn and Docherty have developed a method of predicting crystal morphology which can account for realistic processing conditions which uses knowledge of properties such as the crystal structure and internal energy of the solid and the pure component surface free energy of the solvent.30 This approach has been successfully applied to predicting the morphology of a number of amino acids31 as well as the morphology of urea grown from polar and nonpolar solvents.32 More recently, Cuppen et al.33 have performed Monte Carlo simulations using the program MONTY34 to investigate the effect of supersaturation on the predicted morphology of paracetamol. Here, it was found that the choice of force field and atomic point charge model used influenced the results to a large extent, which was attributed to the small differences in the step energies that determine the growth rates of the crystal faces used in this method. However, the attachment energy model is currently the most appropriate model that can be routinely applied to considering the relative crystal growth rates of sets of hypothetical and known crystal structures of the same organic molecule. The attachment energy Eatt is given by ∞

Eatt )

Ei(hkl) ∑ i)1

(1)

where Ei(hkl) is the interaction energy per molecule between a slice of thickness dhkl and the ith underlying slice. The relationship between the attachment energy and slice energy is given by

Ecrys ) Eatt + Eslice

(2)

where Ecrys is the energy of the crystal, Eatt is the attachment energy and Eslice is the energy of a slice of depth dhkl. The growth rate of a crystal face is proportional to the absolute value of the attachment energy i.e., faces with low absolute attachment energies grow most slowly and thus have the most morphological importance. Since Ecrys is constant for a given crystal structure, a large slice energy implies a low absolute attachment energy and so the face will be morphologically important.18 The growth volume was estimated by calculating the volume within the Wulff shape using a numerical integration technique. We now compare the growth volume (relative to a cube with Eatt ) 10 kJ mol-1 for each face) with the attachment energy of the most dominant face for each of the three systems studied. Procedure. In this paper we have used the low energy crystal structures generated in previously published crystal structure prediction studies, based on searches for the global minima in the lattice energy. Our calculations used the same molecular structure and the

Calculation of Attachment Energies

same repulsion-dispersion potential as in the original work. For paracetamol and pyridine, the potentials of Williams and Cox35,36 were used for interactions involving C, O, N and Hc (hydrogen bonded to carbon atoms), with the potential for Hp (hydrogen bonded to nitrogen or oxygen) obtained by empirical fitting to structures containing N-H‚‚‚N and N-H‚‚‚O-C hydrogen bonds37. The calculations for parabanic acid used the potential of Williams38, with the carbon repulsion parameter reduced by 25%.16 We employed the program GAUSSIAN9839 to calculate CHelpG charges40 using an MP2/ 6-31G** wave function. Potential derived charges were used to reproduce the electrostatic potential around the molecules as accurately as is possible within the atomic charge model.41 Morphologies predicted with different realistic intermolecular potentials have been shown not to be too sensitive to the choice of potential model.23 We also restrict our calculations to cases where there is one molecule in the asymmetric unit cell for hypothetical structures. The program GDIS42 was used to generate the surfaces to study, via a Bravais-Freidel-Donnay-Harker (BFDH) analysis.43 We then calculated the attachment energy for the surfaces using the GULP code.44 The surface simulation cell in GULP has planar 2D periodic boundary conditions and atoms are divided into two regions (1 and 2). The atoms in region 1, which are closest to the surface, can be relaxed until there is zero force on each atom, while those in region 2 are kept fixed to reproduce the potential of the bulk lattice on region 1. Thus the program could be used for more detailed investigations of surface phenomenon, such as relaxation of the surfaces, as discussed in detailed studies of the morphology of urea.22,45 However, for this study comparing many hypothetical crystal structures, we calculate the attachment energies for the unrelaxed surfaces that are generated by cleaving the crystal lattices produced in the original searches for low energy minima in the lattice energy. The structures considered are only those which are well within the energy range of possible polymorphism.1 However, since there is some variation in lattice energy, we also consider the energy above the global minimum (∆E ) Eo - Ei where Eo is the global minimum in the lattice energy found in the search and Ei is the lattice energy of the structure being considered). Results Since the molecules paracetamol, parabanic acid and pyridine, were chosen as having contrasting intermolecular forces and consequently crystallize under different conditions, the results are presented separately. For each, we compare the calculated relative volume growth rate with the relative growth rate of the most dominant face, i.e., that with the smallest magnitude attachment energy. We also demonstrate how the relative growth volumes and attachment energies of the most dominant face vary with the lattice energies of the polymorphs calculated using the repulsion-dispersion plus DMA electrostatic model. Finally, we compare the aspect ratios for each polymorph with the attachment energy of the most dominant face. Paracetamol. Paracetamol has two known polymorphs: form I (which is monoclinic) is used com-

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Figure 2. Relative growth volume, attachment energy, Eatt, for the dominant face and the energy above the global minimum, ∆E, for crystal structures of paracetamol. The growth volume is plotted relative to a cube where each face has an attachment energy of 10 kJ mol-1. The lattice energies and the crystal structure identifiers were taken from ref 14. The / denotes crystals with polar faces.

mercially and is experimentally observed to be thermodynamically stable at room temperature with respect to the orthorhombic form II. Since form II has advantageous properties for formulation without binding agents, considerable efforts went into finding a method of crystallizing it from solution.46 The attachment energy model provides a good prediction of the morphology of form I grown from industrial methylated spirits, but a poorer prediction of form II grown from the same solvent,46 which can be attributed to the profound effect that solvent and degree of supersaturation has on the morphology of paracetamol crystals.46 Recently, a third highly metastable form of paracetamol has been found by crystallization from the melt.47 The powder pattern of this new experimental form has sufficient similarity to that of one (AK6) of the structures found in the polymorph prediction study,14 that AK6 was suggested as a possible structure for form III.47 However, this proposal is controversial, as the absence of an expected peak in the powder pattern47 had to be attributed to there being only short range order in the direction predicted to have growth problems because of its low attachment energy.14 The ready transformation of form III to form II suggests that the structures may be more closely related, and the low energy of a structure (CB9) with the same hydrogen bonding sheets as form II suggests that a variety of stackings of these sheets may be possible. The results of our attachment energy calculations reported in Figure 2 show that both the well-characterized forms of paracetamol grow relatively fast, with the metastable form II growing the fastest. There is a good correlation between growth volume and the attachment energy of the most dominant face, suggesting that a single face is indeed dominating the overall crystal growth. The aspect ratios are listed in Table 1, together with the lattice energy of the bulk crystal and the attachment energy of the most dominant face. We see that the majority of structures are fairly equant in that they have a small aspect ratio. Structures AK22, AK4 and AK6 all have a large aspect ratio and their morphologies are fairly platelike. In these cases, the most dominant face has an attachment energy whose absolute

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Table 1. Lattice Energies (from Ref 14), Attachment Energies (Eatt) of the Most Dominant Face, and Aspect Ratios for Computed Crystal Structures of Paracetamola

structure form I form II AI22 AY8b AQ6 AQ14 AK6 CB9 CC8 AK22 AI16 AM4 AK4

lattice energy, kJ mol-1 -110.1 -106.5 -105.6 -105.4 -102.8 -101.9 -101.9 -101.8 -101.0 -100.2 -100.2 -100.1 -100.0

Eatt, kJ mol-1 -46.7 -48.2 -33.6 -33.2 -42.6 -46.8 -9.8 -40.5 -30.8 -7.9 -30.8 -43.4 -5.9

Coombes et al. Table 2. Lattice Energies (Taken from Ref 16), Attachment Energies (Eatt) of the Most Dominant Face, and Aspect Ratios for Computed Crystal Structures of Parabanic Acida

aspect ratio 1.65 1.56 2.20 1.97 1.47 1.55 8.76 1.58 1.94 9.02 2.28 1.52 11.7

a

The crystal structures are identified using the same notation as that in the paper describing the search for minima in the lattice energy.14 b Crystals with polar faces, for which the attachment energy model is less likely to be realistic.

Figure 3. Relative growth volume, attachment energy, Eatt, for the dominant face and the energy above the global minimum, ∆E, for crystal structures of parabanic acid. The growth volume is plotted relative to a cube where each face has an attachment energy of 10 kJ mol-1. The lattice energies and the crystal structure identifiers were taken from ref 16. The * denotes crystals with polar faces.

value is much lower than that for the other faces, and the relative volume growth rate is notably smaller. Hence, one face dominates the crystal growth. Of the main candidate structures for form III, CB9 has a faster predicted growth rate from vapor and a very similar lattice energy to AK6. Parabanic Acid. Parabanic acid has one known polymorph, which was found as the global minimum in the polymorph prediction study16; but there are a number of hypothetical structures that are only 2-6 kJ mol-1 less stable than the known form. Thus the lattice energies suggested that the possibility of other polymorphs cannot be excluded.16 Crystallization from a variety of solvent yielded this crystal structure, and no other polymorphs were found from a range of solvent based crystallization experiments.16 Thus, the experimental evidence, and the estimate that the relative energy stability would be increased at room temperature, led to the conclusion that “it is unlikely that additional polymorphs of parabanic acid will be readily found as persistent metastable forms”.16 From Figure 3 we see that the known polymorph has a relatively fast growth rate. However, the less stable structures AZ32 and AQ17 are predicted to grow up to 15% faster. Again there is good correlation between the lowest attachment energy for a face of a crystal and its

structure

lattice energy, kJ mol-1

Eatt, kJ mol-1

aspect ratio

expt FC5 AZ32 AI28 AF28b AZ17 DD11 DE5 AQ17 DE12 AI25

-111.4 -109.0 -108.5 -107.3 -106.8 -106.4 -106.1 -105.8 -105.3 -105.1 -105.1

-29.4 -18.5 -27.2 -12.7 -24.2 -22.9 -19.9 -14.2 -33.0 -19.4 -17.9

2.49 3.93 2.91 5.15 2.93 3.39 3.59 4.37 2.11 3.51 3.13

a The crystal structures are identified using the same notation as that in the paper describing the search for minima in the lattice energy.16 b Crystals with polar faces, for which the attachment energy model is less likely to be realistic.

relative volume growth rate, but no correlation with the small variations in lattice energy. From Table 2 we see that there is only moderate variation in the aspect ratios for the different polymorphs. As in the previous work,16 none of the structures were predicted to have very slow growing faces and the variations in the calculated aspect ratios were consequently found to be small. Pyridine. Pyridine melts at -40 °C, and crystallization usually results in the unusual crystal structure of form I, which has four molecules in the asymmetric unit cell (Z′ ) 4). A computational search for Z′ ) 1 crystal structures17 found many hypothetical crystal structures which had a more favorable lattice energy than form I, by up to nearly 6 kJ/mol. This computational study encouraged a further experimental search during which a new polymorph was found for perdeutero-pyridine17, when crystallized from a 1:1 solution in pentane at -85 °C. This structure (form II) was predicted17 to be metastable, with an energy similar to that of form I. Harmonic phonon estimates of the zero point energy made very little difference to the relative stability of form II. It is interesting to note that, using the PixelSCDS method48 of evaluating the lattice energies, Gavezzotti found that form II was the most stable of the pyridine crystal structures.49 Thus, there is a significant difference in the quantitative modeling of C-H‚‚‚N “hydrogen bonds”49 between the charge density and the model potential estimates of the lattice energies for pyridine. From Figure 4 we see that form II shows a much greater volume growth rate than any hypothetical Z′ ) 1 crystal structure. Form I shows a fairly high relative growth rate but it should be noted that since form I of pyridine contains more than one molecule in the asymmetric unit cell, the growth layer that is deduced by symmetry and used in the attachment energy model is less likely to reflect the actual growth mechanism. Since approximate symmetry elements imply faster growth of some faces than predicted by crystallographic symmetry,50,51 and form I may contain these symmetry elements, the attachment energies might be underestimated for some faces, and so the relative volume growth rate in Figure 4 is a lower bound.

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Figure 4. Relative growth volume, attachment energy, Eatt, for the dominant face and the energy above the global minimum, ∆E, for crystal structures of pyridine. The growth volume is plotted relative to a cube where each face has an attachment energy of 10 kJ mol-1. The lattice energies and the crystal structure identifiers were taken from ref 17. The / denotes crystals with polar faces. Table 3. Lattice Energies (Taken from Ref 17), Attachment Energies (Eatt) of the Most Dominant Face, and Aspect Ratios for Computed Crystal Structures of Pyridinea

structure

lattice energy, kJ mol-1

Eatt, kJ mol-1

aspect ratio

FA37 AK11 AM50 CB38 FC21 DD31 AM20 AK23 CD49 AV32b CA21 AM43 AI36 CB39 DE20 CA28 DE40 CB47 AI18 AM5 form II CA43 AK7 AZ5 AK14 form Ib AK15

-56.5 -54.8 -54.8 -54.5 -53.7 -53.5 -53.5 -53.4 -53.4 -52.9 -52.8 -52.6 -52.6 -52.4 -52.4 -51.7 -51.7 -51.5 -51.5 -51.3 -51.3 -51.1 -51.1 -51.02 -50.9 -50.7 -50.7

-17.0 -19.4 -19.1 -15.2 -14.7 -15.5 -17.2 -16.9 -17.8 -14.6 -15.5 -13.0 -17.8 -15.1 -14.9 -13.9 -14.4 -16.1 -13.9 -17.0 -23.0 -14.0 -17.3 -16.0 -14.2 -15.9 -13.2

1.60 1.68 1.47 1.81 1.68 1.70 1.61 1.80 1.52 1.92 2.00 2.70 1.80 1.50 2.00 1.90 2.00 1.67 1.90 1.83 1.47 1.83 1.68 1.90 1.83 1.66 1.78

a The crystal structures are identified using the same notation as that in the paper describing the search for minima in the lattice energy.17 b Crystals with polar faces, for which the attachment energy model is less likely to be realistic.

There is little variation in the aspect ratios (Table 3) and the structures are thus fairly equant. There is still considerable correlation between the attachment energy of the dominant face and the relative volume growth rate, and no correlation of either with the lattice energy. This is despite the weaker intermolecular forces between pyridine molecules resulting in significantly lower lattice energies and attachment energies than calculated for paracetamol and parabanic acid.

Discussion Despite sets of crystal structures having very similar lattice energies, there is a significant variation in the lowest attachment energies and relative volume growth rates. The relative volume growth rate is strongly correlated with the lowest attachment energy, i.e., one slow growing face tends to limit the growth rate of the crystal. We would expect the degree of variation in growth volumes to be very dependent on the variation of crystal packings within the low energy range of the different polymorphs. For example, in the case of paracetamol, the polymorphs which contain chains of dimers all have low relative growth volumes compared with structures that have other hydrogen bonding motifs; a slice with strong interactions within the growth unit results in a low attachment energy according to eq 2, since the lattice energy is approximately the same for all polymorphs. Hence a smaller volume of crystal is grown in a given period of time. Thus, different packings of organic molecules can produce crystal structures, which are approximately equally favorable thermodynamically, and yet, have very different growth rates. Therefore, although the use of the attachment energy model for predicting the relative growth rates of crystal faces has many limitations, this general conclusion is most certainly valid. These attachment energy calculations of relative growth rates of different polymorphs cannot be directly validated experimentally, in that experimental measurements of relative growth rates of different polymorphs for the same crystallization conditions would be limited to concomitant polymorphs,52 and we are not aware of such experiments. The dependence of crystal growth rates on temperature, supersaturation, solvent and other crystallization conditions are likely to obscure the variation intrinsic to the differences in crystal structures revealed by these attachment energy calculations. Nevertheless, these calculations do suggest that considering the attachment energy predictions of relative growth rates may help to distinguish which energetically plausible crystal structures may be observed polymorphs. In the case of parabanic acid, both calcula-

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tions and experiment indicate that the known form is the thermodynamically most stable structure, and its predicted growth rate is among the highest. This study gives no reason to expect that another polymorph is favored by the kinetics of crystal growth. For paracetamol, the room temperature metastable structure form II is predicted to have a slight growth advantage over form I; thus both well-characterized forms are thermodynamically favored and have surfaces whose growth rates provide a kinetic advantage. The most dramatic example is pyridine, where the relative volume growth rate clearly favors form II, and probably form I, above more stable hypothetical crystal structures. Since pyridine crystallizes at low temperatures, any metastable crystals formed are less likely to be able to transform to any thermodynamically more stable structures.53 Hence, these calculations demonstrate that the crystal structure of form II has a kinetic advantage, which may account for its appearance. Understanding how the crystallization of an organic molecule under different conditions is controlled by thermodynamics and the kinetics of nucleation, crystal growth and possible transformations is a major challenge that has to be met if we are to be able to predict polymorphism. That such a crude model for the kinetics of crystal growth often favors (and certainly does not eliminate) the known and long-lived metastable forms of these well-studied systems, suggests that it is worth pursuing for estimating one aspect of the kinetic factors that lead to polymorphism. Conclusions Attachment energy calculations of the relative growth rates of real and hypothetical crystal structures, which are within the energy range of possible polymorphs, can show considerable variation, as has been demonstrated for three disparate systems, paracetamol, parabanic acid and pyridine, two of which are polymorphic, and all of which have a variety of different crystal packing motifs of comparable lattice energy. This variation correlates well with the crystal structure having at least one face with a low attachment energy, suggesting difficulties in crystal growth on that face (at least by the mechanisms for which the attachment energy model is appropriate). Thus the variations in the crystal packing will have a major effect on the kinetics of growth of the crystal structures, should they nucleate. The application of attachment energy estimates of relative crystal growth rates in predicting whether a hypothetical crystal structure has such a kinetic advantage in growth, that it may be observed as a metastable polymorph, is speculative. It could only be applied in conjunction with predictions that such a crystal structure would nucleate and not easily transform to the thermodynamically stable form. However, the results in this paper show that it is worth calculating these simple measures of relative crystal growth rates in the development of an understanding of polymorphism. Acknowledgment. We would like to thank Dr. Graeme Day for useful discussions. Funding was provided by EPSRC under the project “e-Science Technologies in the Simulation of Complex Materials”. S.L.P.

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acknowledges funding from the Basic Technology Grant project “Control and Prediction of the Organic Solid State”. J.D.G. acknowledges the support of the Government of Western Australia through the Premier’s Research Fellowship programme. A.L.R. acknowledges the support of the Australian Government’s Cooperative Research Centre (CRC) Program, through the AJ Parker CRC for Hydrometallurgy. References (1) Bernstein, J. Polymorphism in molecular crystals; Oxford University Press: Oxford, U.K., 2002. (2) Beyer, T.; Lewis, T.; Price, S. L. CrystEngComm 2001, 3, 178-212. (3) Dunitz, J. D. Chem. Commun. 2003, 545-548. (4) Gavezzotti, A. Acc. Chem. Res. 1994, 27, 309-314. (5) Giron, D. Thermochim. Acta 1995, 248, 1-59. (6) Berkovitch-Yellin, Z. J. Am. Chem. Soc. 1985, 107, 82398253. (7) Coombes, D. S.; Catlow, C. R. A.; Gale, J. D.; Hardy, M. J.; Saunders, M. R. J. Pharm. Sci. 2002, 91, 1652-1658. (8) Clydesdale, G.; Roberts, K. J.; Telfer, G. B.; Grant, D. J. W. J. Pharm. Sci. 1997, 86, 135-141. (9) Docherty, R.; Roberts, K. J. J. Cryst. Growth 1988, 88, 159168. (10) Givand, J. C.; Rousseau, R. W.; Ludovice, P. J. J. Cryst. Growth 1998, 194, 228-238. (11) Pfefer, G.; Boistelle, R. J. Cryst. Growth 2000, 208, 615622. (12) Vogels, L. J. P.; Bennema, P.; Hottenhuis, M. H. J.; Elwenspoek, M. C. J. Cryst. Growth 1991, 108, 733-743. (13) Vogels, L. J. P.; Grimbergen, R. F. P.; Strom, C. S.; Bennema, P.; Roberts, S. A.; Blanks, R. F. Chem. Phys. 1996, 203, 69-80. (14) Beyer, T.; Day, G. M.; Price, S. L. J. Am. Chem. Soc. 2001, 123, 5086-5094. (15) Day, G. M.; Price, S. L. J. Am. Chem. Soc. 2003, 125, 1643416443. (16) Lewis, T. C.; Tocher, D. A.; Day, G. M.; Price, S. L. CrystEngComm 2003, 3-9. (17) Anghel, A. T.; Day, G. M.; Price, S. L. CrystEngComm 2002, 348-355. (18) Hartman, P.; Bennema, P. J. Cryst. Growth 1980, 49, 145. (19) Clydesdale, G.; Roberts, K. J.; Walker, E. M. The Crystal Habit of Molecular Materials: A Structural Perspective. In Theoretical aspects and computer modeling of the molecular solid state; Gavezzotti, A., Ed.; John Wiley & Sons: Chichester, 1997; pp 203-232. (20) Liu, X. Y.; Boek, E. S.; Briels, W. J.; Bennema, P. J. Chem. Phys. 1995, 103, 3747-3754. (21) Clydesdale, G.; Roberts, K. J.; Lewtas, K.; Docherty, R. J. Cryst. Growth 1994, 141, 443-450. (22) Engkvist, O.; Price, S. L.; Stone, A. J. Phys. Chem. Chem. Phys. 2000, 2, 3017-3027. (23) Brunsteiner, M.; Price, S. L. Cryst. Growth Des. 2001, 1, 447-453. (24) Donnay, J. D. H.; Harker, D. Am. Miner. 1937, 22, 446. (25) Liu, X. Y.; Bennema, P. Phys. Rev. B: Condens. Matter 1996, 53, 2314-2325. (26) Liu, X. Y.; Bennema, P. J. Cryst. Growth 1996, 166, 117123. (27) Grimbergen, R. F. P.; Bennema, P.; Meekes, H. Acta Crystallogr., Sect. A 1999, 55, 84-94. (28) Boerrigter, S. X. M.; Hollander, F. F. A.; van de Streek, J.; Bennema, P.; Meekes, H. Cryst. Growth Des. 2002, 2, 5154. (29) Roberts, K. J.; Docherty, R.; Bennema, P.; Jetten, L. J. Phys. D: Appl. Phys. 1993, 26, B7-B21. (30) Winn, D.; Doherty, M. F. AIChE J. 1998, 44, 2501-2514. (31) Bisker-Leib, V.; Doherty, M. F. Cryst. Growth Des. 2003, 3, 221-237. (32) Bisker-Leib, V.; Doherty, M. F. Cryst. Growth Des. 2001, 1, 455-461. (33) Cuppen, H. M.; Day, G. M.; Verwer, P.; Meekes, H. Cryst. Growth Des. 2004, 4, 1341-1349.

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