CRYSTAL GROWTH & DESIGN
Molecular Modeling of Crystal-Crystal Interactions between the r- and β-Polymorphic Forms of L-Glutamic Acid Using Grid-Based Methods
2007 VOL. 7, NO. 5 875-884
R. B. Hammond, K. Pencheva, and K. J. Roberts* Institute of Particle Science and Engineering, School of Process, EnVironmental and Materials Engineering, UniVersity of Leeds, Leeds LS2 9JT, U.K. ReceiVed August 3, 2006; ReVised Manuscript ReceiVed January 16, 2007
ABSTRACT: Grid-based search methods, employing interatomic potential force fields and appropriate distance constraints, were applied to model interparticle interactions between faceted nanosized molecular clusters of L-glutamic acid having a polyhedral shape as predicted using molecularly based, morphological modeling. Calculations revealed that an interparticle central distance of ca. 30 Å was sufficient for satisfactory energy convergence, reflecting the dominance of short-range van der Waals interactions for this material. The modeling indicated that preferential binding takes place where the interacting crystal surfaces on two separate particles form a relatively large number of hydrogen bonds per unit area of the interacting surfaces. In particular, binding is preferred between the (111) and (101) faces of the R- and β-polymorphic forms, respectively, in excellent agreement with experimental data. Potential development of the methodology is reviewed. 1. Introduction A major challenge to advancing current, process-engineering design capabilities is the need to develop molecule-centered and system-specific modeling techniques to aid in the design, operation, and control of unit operations involved in processing materials in their solid form (notably crystalline products). Such materials acquire high added value through processing and are of significant commercial importance to the fine chemicals sector, in particular pharmaceutical materials. Interestingly, while a simplification, it is probably reasonably accurate to state that in the commodity chemical sector, where the value added by processing is much lower, the sophistication of process equipment control and optimization is much greater than in the speciality sector. This reflects, of course, the fact that often, for commodity materials, the manufacturing processes involve materials in gaseous and liquid forms where the fundamental understanding of thermodynamics, kinetics, and materials transfer is well-developed and enshrined in routine computer design tools. The modeling of properties of solid-state materials has advanced significantly in recent years, particularly for organic materials, and computational approaches to predict the physical properties of particulate solids, such as polymorphic structures, crystal shape, elastic properties, and surface properties, are now becoming much more routine. Challenges still remain, however, notably in the development of modeling approaches for predicting the kind of interparticle interactions involved in, for example, sedimentation, agglomeration, and compaction, which are central to the design and effective manipulation of unit operations such as crystallization, granulation, thickening, filtering, and drying. The description of the effects of molecular interactions between macroscopic bodies is important for the analysis of a variety of processes such as adhesion, sintering, colloidal interactions, tableting, granulation, and nucleation. The origin of the interaction force that operates between two bodies was first given a molecular perspective following London’s hypothesis regarding the physical mechanism that generates van der * Corresponding author: tel +44 (0)113 343 2408, fax +44(0)113 343 2405, e-mail
[email protected].
Waals forces.1,2 Qualitatively, the dispersion force originates from fluctuations in the electron density associated with the constituent atoms of the molecules of which a material is composed. These fluctuations lead to formation of temporary dipoles that in turn induce further dipoles and generate a net force. Later, Hamaker utilized London’s physical model together with an atomistic formulation in which it is assumed that an overall macroscopic resultant force can be evaluated quantitatively as the pairwise summation of interatomic or intermolecular central forces. Hence, if the pairwise interaction energy of two molecules as a function of separation distance D is denoted as w(D), then the interaction energy E(D) between two elemental volumes dV1 and dV2 of two bodies comprised of these molecules is given by
∫V ∫V
E(D) ) F1F2
2
1
w(D) dV1 dV2
(1)
where F1 and F2 are the respective number densities of molecules. The total force of interaction between two bodies of volume V1 and V2 can be calculated as the double volume integral given in eq 2 where vector quantities are denoted by a carat and ˆf(D) is the interaction force that is given by the spatial gradient of the interaction energy; thus
∫V ∫V ˆf(D) dV1dV2
Fˆ ) F1F2
1
where ˆf (D) ) -∇w(D) (2)
2
The analytical integration in determining the total force of interaction of macroscopic bodies is one of the basic problems of the approach. This was solved previously for the specific case of van der Waals interactions.3 However, because the approach assumes additivity, the total interaction energy can be calculated through a pairwise summation of the individual molecular energy contributions rather than an integration based on the volume of the particles, and eventually the interaction force between the two particles may be computed through calculating the gradient of the function E(D). Engkvist et al.4 have reported a computational approach that helps to explore the forces between the crystal faces of crystalline materials by cutting a fragment of the morphologically important habit faces
10.1021/cg0605234 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/03/2007
876 Crystal Growth & Design, Vol. 7, No. 5, 2007
Hammond et al. Table 1. Attachment Energies Calculated Using HABIT98 Software, Applying Atom-Atom Summation Method and Momany Force Fielda R-form
β-form
form
attachment energy
form
attachment energy
{001} {101} {111}
1.00 2.54 3.20
{101} {020} {021}
2.50 1.00 1.09
a Scaling factors of -5.89 and -5.54 kcal/mol have been used for Rand β-forms.
Figure 1. L-Glutamic acid is an amino acid with two carboxylic groups and one amino group (a). L-Glutamic acid has two polymorphic structures, metastable R-form (b) and stable β-form (c), differing in both solid-state crystal structure and molecular conformation (d).
and calculating the interaction energy between two fragments. The authors report that the interaction between two different crystal faces can be studied to predict forces between crystalline surfaces and their tendency to aggregate. The computational approach described in this work adopts a similar strategy but differs in that the interactions studied are between nanosized, polyhedrally shaped molecular clusters rather than between flat crystal surface fragments. It has also been demonstrated experimentally elsewhere5 that the existence and strength of any agglomeration interactions can be expected to vary considerably with the atomic structure of the crystal surface and its orientation. Hence, a systematic analysis of interparticle interaction energies as a function of the relative orientation between two interacting particles provides an approach for characterizing agglomeration at the nanoscale in terms of interparticle bonding. Recently, progress has been made in this area that facilitates the generation of molecular models of nanosized, faceted particles using a newly developed polyhedral molecular-packer program, POLYPACK.6 This program has been applied successfully to model the surface6 and solubility7 properties of particulate clusters together with their polymorphic stability as a function of particle size.8 The approach is attractive in that a molecular-scale model of particles, when combined with grid-based intermolecular search programs, for example, SYSTSEARCH,9,10 provides a meth-
odology with the potential to model interparticle interactions using an atom-atom modeling approach. In the application described here, despite the large number of pairwise interatomic interactions involved in definition of the interparticle binding between two nanosized particles, the calculation remains tractable because for organic solid systems, atom-atom interactions involved in the overall interparticle binding energy are dominated by short-range van der Waals forces with only the atom-atom interactions characterized by small (
