Full three-dimensional diffuse scattering data have been recorded for both polymorphic forms [(I) and (II)] of aspirin and these data have been analysed using MC computer modelling. Although the modelling was focused mainly on the key sections
, once appropriate modelling parameters were obtained the calculated patterns for other sections were checked and also found to agree satisfactorily with the observed data.
The limited size of the computer models that it was possible to use imposed some limitations on how true a representation could be made of the aspirin II structure. Some aspects are better represented than others. The linear dimension of 48 unit cells imposes an upper limit on the lateral extent (in the
directions) of the planar defects. The width of the diffuse streaks is reciprocally related to this dimension. This width does appear qualitatively similar in the simulated patterns (Fig. 6
) to that of the observed streaks (Fig. 2
), so this aspect appears to be reasonably well modeled. Similarly the limit of 48 unit cells in the stacking direction (along a
) limits the statistical accuracy that can be achieved in the faulting parameters. This affects the distribution of intensity along the streaks. Again, the models presented have been able to reproduce these effects quite convincingly. What is rather less satisfactory is that it was not possible to use defect concentrations lower than ~ 10%, as for example in Fig. 6. This is significantly higher than the concentration envisioned to exist in the real crystals. This means that the diffuse streaking produced by the models, though having the same form, is too strong relative to the thermal diffuse scattering. The images showing the real-space distributions of components given in Fig. 6 should therefore be considered to be indicative only.
The observed scattering in form (I) is well reproduced by a simple harmonic model of thermally induced displacements. This model used the recently reported strategy for modelling diffuse scattering from molecular crystals (Chan et al.
) in which a simple empirical formula [see (4)
] is used to specify the spring constants of the inter-atomic springs used to represent intermolecular interactions and to provide an objective means for limiting their number.
The data for form (II) exhibited, in addition to thermal diffuse scattering similar to that in form (I), diffuse streaks originating from stacking fault-like defects as well as other effects attributable to strain induced by these defects. An MC model has been developed which, within the limitations dictated by the finite simulation size currently feasible, satisfactorily accounts for all these effects. This model for form (II) incorporated a description of the thermal displacements carried over directly from the form (I) analysis and used the same parameter values.
The modelling of the defects used a description for the structure in which each molecular site contained either an
molecular component (see Table 2). A crystal comprising all
components corresponds to a perfect form (II) crystal, while one consisting of all
components corresponds to the same form (II) structure reflected in the
plane and translated by
. For perfect
planes of molecules the form (I) structure consists of a sequence along
. Here the
pairs correspond to bilayers of strongly hydrogen-bonded molecular dimers. In the model described it is assumed that these molecular dimers remain intact throughout and faulting of the sequence can only occur at the interface between two such bilayers. In the model the extent of the order in the
planes is induced by imposing correlations between components in the
directions, but although these were moderately large given the finite simulation size, they can only be considered as an approximation to the much longer range of order in the real crystals.
With this model it has been demonstrated that in principle a continuous sequence of different disordered structures can be generated from the perfect form (I) crystal at one extreme to a perfect form (II) crystal at the other. Through this progression the diffuse streaks that occur on the
odd, reciprocal rows change continuously from a sharp peak falling on the form (I) lattice at one extreme, through an intermediate continuous streak showing no peaking, to a sharp peak falling on the form (II) reciprocal lattice at the other.
The intermediate structures in this sequence are consistent with the description given by Bond et al.
) that ‘each aspirin crystal is an integral whole in which the domains are intimately connected with each other, with possibly many turnovers of domain within a single crystal’. However, we have shown that when the frequencies with which the two components
occur are equal there are no Bragg peaks on the
odd, reciprocal lattice rows except at the limiting extremes (when the streaking also disappears). Moreover, for intermediate disordered models the interference of the scattering from the intimately intermingled
domains gives strong diffuse peaks in the
section and these are clearly not present in the observed scattering (see Fig. 7
These considerations led us to the conclusion that a crystal of form (II) is essentially comprised of a single domain of one component (
, say) with isolated planar defects comprised of bilayers of the other component (
). These defects may be essentially randomly occurring, in which case the diffuse streaks are continuous and show no peaking or they may cluster to form small regions of the form (I) structure as in Fig. 6(b
). In this case a diffuse peak appears at the position of the form (I) reciprocal lattice on the
odd, reciprocal lattice rows. For this model, sharp form (II) Bragg peaks are present in the
odd, rows throughout. Such sharp Bragg peaks are clearly present in the data (Fig. 2
), and moreover the diffuse peaks in the
section are now much reduced in intensity and are in much better agreement with the observed data (see Fig. 7
Finally it has been shown that various diffuse scattering features that occur in the
section of the form (II) data that are absent from the form (I) data (see Fig. 2
) are due to strain induced by the presence of the defects. The results of modelling these effects have shown that they arise from the mismatch of the packing of the
components at the edges of the planar defects in the
direction. The sign of the size-effect distortion indicates that the spacing of an
interface is increased relative to the normal
spacing. No evidence for similar mismatching in the
direction was found. With the current limitation on the size of the model system that we can use it is not possible to represent the domain structure completely realistically. Consequently, it is difficult to be sure of the lateral extent of the planar faults and of the overall volume fraction of the faulted material (i.e.
the concentration of component
). However, from the present study it is clear that strain effects are highly significant and since they only occur at the
edges of the faulted domains this implies that these must occur reasonably frequently in the real material.
Ouvrard & Price (2004
noted that aspirin form (II) should have ‘a low shear elastic constant, implying that it is so readily deformed that there may be problems in its growth’. This same point was picked up by Bond et al.
). The sample used for the present diffuse scattering study was a good quality prismatic crystal with well formed facets and edges with no outward signs of any problems in its growth. It was also significantly bigger than samples normally used for Bragg structure analysis. We consider that what the present study has revealed is that the kind of planar defects described are intrinsic to the form (II) stucture. It may well be that these arise as a result of ‘a low shear elastic constant’, but there is clear evidence that their inclusion results in significant strain to the form (II) lattice. This implies that there must be a balance. The defects may be easy to form but the strain that is induced results in a significant energy penalty that limits their number.
Bond et al.
) have also been concerned with trying to assess the composition of the crystals in terms of the relative fractions of forms (I) and (II). This is not completely straightforward. The example shown in Fig. 6(b
), which we consider to be qualitativly a good representation of what occurs in the real structure, has 10% of
bilayer defects. Since what comprises the form (I) structure is an alternating sequence of
it might be considered that 10% of
bilayer defects would correspond to 20% of form (I). However, a good fraction of the defects are composed of single isolated
bilayers. For these the neighbouring
bilayers are still part of the form (II) structure so cannot be counted as part of the form (I) structure.
The present results are based on data derived from a single aspirin (II) crystal grown via
super-cooling a 0.6–0.7 M
solution of aspirin in acetonitrile. Although no other crystals were examined in the same amount of detail, all of the samples that were surveyed showed similar streaking along (
odd) rows, with the form (II) peaks prominent and with a relatively minor presence of more diffuse form (I) peaks. Whether form (II) crystals with different degrees of faulting can be made by controlling the growth conditions remains open to conjecture.
In summary, the present study has provided strong evidence that the aspirin form (II) structure is a true polymorph with a structure quite distinct from that of form (I). The diffuse scattering evidence we have presented shows that crystals are essentially composed of large single domains of the form (II) lattice with a relatively small volume fraction of intrinsic planar defects or faults comprising misoriented bilayers of molecular dimers. There is evidence of some local aggregation of these defect bilayers to form small included regions of the form (I) structure. Evidence has also been presented that shows that strain effects arise from the mismatch of molecular packing between the defect region and the surrounding form (II) lattice. This occurs at the edges of the planar defects only in the