The Design Method

These vacuum morphology prediction methods do not take into account the actual growth processes taking place on a molecular scale at the crystal surfaces. Our goal is to predict the effect of an impurity that affects these growth processes. Such a method should include calculations of the interaction energy between the impurity and the surface. This interaction energy affects the molecular growth processes taking place at the surface.

To make such a prediction, knowledge must first be obtained about the crystal surface structures present. Simple energy calculations determine the morphologically important surface structures. Then more sophisticated calculation methods such as molecular mechanics, molecular dynamics, Monte Carlo techniques or even quantum mechanical calculation methods are needed to calculate the interaction energies of the morphologically important crystal surface structures with the impurity. These interaction energies can be translated into growth rates of the surface structures by assuming a model which interrelates growth rate and interaction energy. The general method for the design of additives is shown in Figure 5.

A good way of determining surface structures and their morphological importance is by performing a periodic bond chain (PBC) analysis. A PBC is an uninterrupted chain of strong bonds between growth units in which periodicity is based on the unit cell parameters and symmetry and which is stoichiomet-ric with regard to the unit cell content. Two sets of intersecting PBCs make a connected net. The connected net can be considered a (hkl) growth slice or growth layer. If a surface does not contain a connected net, the surface cannot grow with a layer growth mechanism and is rough. This means that the relative growth rates of (hkl) surfaces that do not contain connected nets, are very large. These surfaces are not present on the crystal morphology, as a surface needs to contain a connected net to be present. If a surface contains more than one connected net (slices at different heights) more surface structures are possible for one crystal surface. Energy calculations should determine which of the surface structures actually occurs on the surface. However, it is important to remember that the morphologically important surface structure for a particular (hkl) surface may change because of interactions with impurities.

The morphologically important surface structures can be determined by calculating the attachment energy Ea of a surface slice containing a connected net from the bond energies in the slice (the slice energy Esl) and the lattice energy Ecr. The attachment energies of all the possible surface structures can be used to determine the relative growth rates of these surface structures:

This results in morphologically important crystal surface structures for crystals grown from systems without boundary layer influences such as solvent interactions with the surface structures.

Figure 3 The RDX (cyclotrimethylene trinitramine) unit cell (top left), and the (2 0 0) and (0 2 0) slices, indicating their surface structure. For the (2 0 0) slice the connected net is given on the right of the molecular view of the slice while for the (0 2 0) slice the connected net is given below the molecular view of the slice, as they occur in the unit cell. The surfaces are given perpendicular to the paper. The dotted line in the molecular view of the surfaces indicates the surface.

Figure 3 The RDX (cyclotrimethylene trinitramine) unit cell (top left), and the (2 0 0) and (0 2 0) slices, indicating their surface structure. For the (2 0 0) slice the connected net is given on the right of the molecular view of the slice while for the (0 2 0) slice the connected net is given below the molecular view of the slice, as they occur in the unit cell. The surfaces are given perpendicular to the paper. The dotted line in the molecular view of the surfaces indicates the surface.

Commercial software packages nowadays have several tools to calculate interaction energies of one or more impurity molecules on these surface structures.

Figure 4 The faces with the smallest growth rates determine the morphology. A growth rate decrease (R3 p R3) because of a face-specific interaction of the foreign compound causes a morphologically more important surface and thus a change in the morphology.

Once the interaction energies have been calculated, they should be related to the growth rates of the crystal surface structures. The models used for this relation are generally based on the assumption that interaction reduces the growth rate. First the impurity has to be removed from the surface before the face can grow. This takes energy and thus the growth rate decreases. An energy correction term Es for the vacuum attachment energy can be introduced, which is a function of the interaction of the impurity and the flat crystal surface structure:

In most cases only the interaction of a single ion, complex or molecule with the different crystal surfaces has to be considered. By determining the strongest interactions the corrected attachment energies can be calculated and changes in the morphology can be predicted.

By dynamically simulating a solvent layer on a surface structure (with either a molecular dynamics or a Monte Carlo procedure), the energy changes induced in the solvent layer by the presence of the

Figure 5 Method of calculating the effect of an impurity on the growth rates of the different crystal faces.

crystalline interface can be estimated, and thus the interaction energy Es can be calculated. The procedure has to be repeated for all important faces. Though this procedure is somewhat tedious, it yields important insights as to how a solvent can influence the morphology.

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