Supramolecular Architecture in a Disordered Perhydrotriphenylene

Supramolecular Architecture in a Disordered Perhydrotriphenylene Inclusion Compound from Diffuse X-ray Diffraction Data† H. B. Bu¨rgi* and J. Hauser

CRYSTAL GROWTH & DESIGN 2005 VOL. 5, NO. 6 2073-2083

Laboratory of Chemical and Mineralogical Crystallography, University of Bern, Freiestrasse 3, CH-3012 Bern, Switzerland

T. Weber Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zu¨ rich, Switzerland

R. B. Neder Institut fu¨ r Mineralogie, Am Hubland, 97074 Wu¨ rzburg, Germany Received May 13, 2005

ABSTRACT: The diffraction pattern from the disordered host-guest compound perhydrotriphenylene and 1-(4nitrophenyl)piperazine (PHTP5‚NPP) shows a complex variety of diffuse diffraction phenomena. A qualitative and semiquantitative interpretation of the diffuse intensities aided by group theoretical considerations provides detailed information concerning substitutional disorder of (R)- and (S)-PHTP molecules. On the basis of these results a Monte Carlo model for structure simulations has been developed, which comprises only five parameters. The parameters are determined with an evolutionary technique, testing 40 sets of parameters and improving them by mutation and recombination. It is found that the relevant structural information is available after relatively few generations from the average of all individuals rather than from the best individual within a population optimized through many generations. This realization accelerates the algorithm by at least an order of magnitude compared to its earlier implementations. R values between 0.125 and 0.13 indicate an agreement between experimental and simulated diffraction patterns that is pleasingly low for a disorder problem of this kind. The investigation of the structural properties of the host structure indicates that the most dominant structural features include sequences of 10-20 homochiral molecules along the c axis and a weak tendency of building heterochiral pairs in perpendicular directions. Strain associated with the difference in intermolecular contacts between homo- and heterochiral molecules is released by local shear deformations. 1. Ideal and Real Crystals Crystal structure analysis is based on the idealizing assumption that crystals are 3D periodic objects giving rise to sharp, pointlike Bragg scattering. The simplicity of this model has made it possible to develop highly efficient algorithms which solve and optimize crystal structures nearly automatically at the push of a button. However, every real crystal shows a multitude of defects: i.e., deviations from crystallographic symmetry. Defects may occur at different length scales. Large-scale defects, including grain boundaries and dislocations, are characteristic of the notion of a “real structure”. They do not change the individual unit cell substantially but distort the overall lattice. Small-scale defects such as point defects, rotational disorder, displacement of atoms or molecules, etc. locally change the structure of individual unit cells. They are often associated with shortrange order different from the long-range order of the average structure. This situation is referred to as “defect structure”. Bragg scattering provides only a time and space average picture of the molecular composition and of intermolecular geometry in a crystal. The more * To whom correspondence should be addressed. E-mail: [email protected]. Tel: +41 31 631 4282. Fax: +41 31 631 3996. † Dedicated to Professor Mike McBride, scholar and friend, on the occasion of his 65th birthday.

conspicuous the defects, the more information regarding nearest-neighbor relationships with respect to their chemical nature and supramolecular architecture is lost. Information on disorder is found as diffuse scattering between the Bragg peaks. Its interpretation is far from routine, however. Whereas there are only 230 space group symmetries, there are an infinite number of ways for structures to deviate from these symmetries. In this report we emphasize three aspects of deriving structural information from diffuse scattering: (1) defining a model of the disordered crystal by combining the average structure, especially its chemically unreasonable aspects, with a symmetry analysis and with distinctive features of the diffuse scattering, (2) simulating disordered crystal structures with Monte Carlo calculations and finding the optimal parameters for the model with a genetic algorithm called differential evolution,1 and (3) characterizing the disordered structure in terms of frequently occurring supramolecular building blocks extracted from the simulated structures. Novel methodologies are combined with known ones1,2 and are applied to the structure determination of a molecular cocrystal built from perhydrotriphenylene hosts and 1-(4-nitrophenyl)piperazine guests (PHTP5‚NPP). This material is of interest because its crystals are polar, a property which has been ascribed to the mechanism of crystal growth.3-6

10.1021/cg050211l CCC: $30.25 © 2005 American Chemical Society Published on Web 08/24/2005

2074

Crystal Growth & Design, Vol. 5, No. 6, 2005

Figure 1. Crystal structure of PHTP5‚NPP in a projection along c, showing the honeycomb arrangement of PHTP stacks with NPP filling the channels. Molecules with and without shadows are at different heights: z ) 0.5 and 0.0, respectively. Carbon atoms shown as filled black circles are above the mean molecular plane and those shown as filled gray circles are below. The unit cell is outlined.

The discussion starts with a summary of previous work on PHTP5‚NPP in section 2. Qualitative and semiquantitative interpretations of diffuse scattering are summarized in section 3. Monte Carlo simulations and parameter optimization are the topics of sections 4 and 5. Structural chemists interested primarily in a crystal chemical description of the disordered structure may want to skip sections 3.4 to 5 and proceed directly to section 6 and the summary in section 7. 2. Previous Work on PHTP5‚NPP 2.1. Average Crystal Structure. The average crystal structure of PHTP5‚NPP has been determined from the Bragg reflections measured at 100 K (a ) 15.023 Å, b ) 23.198 Å, c ) 4.73 Å).6 The PHTP molecules form stacks extending along c which are arranged in a honeycomb pattern. The intermolecular distance within stacks is 4.73 Å and corresponds to the translation distance c. Stacks with different orientations of the PHTP molecules are c/2 apart (Figure 1). The host structure built from chiral, D3 (32) symmetric PHTP molecules shows occupational disorder: on average each of the two sites is occupied by half of an R and half of an S molecule (Figure 2a). The NPP guests form hydrogen-bonded, polar chains along the tunnel axis with a translational period c′ 5 times as long as that of the PHTP host structure. Although these chains are parallel to each othersin agreement with the observed macroscopic polaritystheir lateral arrangement along a and b is also disordered: each chain randomly locks into one of five different positions relative to the host lattice. Again only a superposition of the five possibilities is seen in the average structure. The various types of disorder raise the question about the defect (as opposed to the average) structure of these crystals. Which host positions are taken by (R)-PHTP

Bu¨rgi et al.

and which ones by (S)-PHTP molecules? Is the chirality of neighboring molecules correlated? Where in its channel is a specific NPP chain locked in? Is the distribution of chiral host molecules related to the polarity of the crystal structure? Bragg diffraction does not provide answers to these questions. They have to be searched for from among the multifarious 1D, 2D, and 3D features in the diffuse scattering pattern. 2.2. Experimental Determination and Description of Diffuse Scattering. The diffuse scattering of PHTP5‚NPP, which is relatively weak compared to Bragg diffraction, has been measured to a resolution of ∼1.7 Å at 100 K with synchrotron radiation at the Swiss-Norwegian Beamline at ESRF, Grenoble, France, using an image plate detector. Its various parts have been assigned with the help of molecular scattering factors to the occupational disorder of (R)- and (S)-PHTP molecules, to the positional disorder of the NPP chains, and to additional types of disorder (for details see ref 7). In the following we focus attention on the features in reciprocal space arising from disorder in the host structure. Two characteristic patterns, concentrated on and near the Bragg layers hk1, hk2, etc., are shown as examples in Figure 3, both of them due solely to the host molecules. They include a system of streaks extending along b* near h ) 11, k ) (9 and two symmetry-independent patches extending in the a* and b* directions near h ) 7, k ) 0 and h ) 3, k ) (9. The intensities of the streaks tend to decrease with increasing |l| and increase with increasing |h|. They reflect disorder resulting from a shear deformation of the host lattice, which does not depend on the chirality of the PHTP molecules and is thus independent of the occupational R,S disorder.7,8 The patches actually belong to a three-dimensional system of diffuse columns with pseudo-hexagonal symmetry running parallel to c*. The intensities along the columns reach maxima for integral l with half-widths ranging from about 0.05 to 0.08c*. The maxima are slightly asymmetric along c* with the broader tail toward high |l|. The broad, irregularly shaped patches shown in Figure 3 are cross-sections through these columns. Their maxima are highly asymmetric in the a* direction and are located close to and on opposite sides of the nearby Bragg reflections for k ) 0 and k ) 9, respectively. The patches cannot be seen in the hk0 layer, in contrast to the streaks mentioned above. 3. Qualitative and Semiquantitative Interpretations of Diffuse Scattering Data 3.1. Identification of Diffuse Scattering Due to R,S Disorder. To disentangle the average electron density with its superposition of (R)-PHTP and (S)PHTP half-molecules into a defect structure with fully occupied R or S molecules, it is necessary that one or the other of the two half-molecules must be removed and replaced by its enantiomer to obtain full occupation with a single enantiomer everywhere in the structure. In reciprocal space this modification of the average electron density is expressed by the difference of the form factors of R and S molecules. This difference, which may be calculated on the basis of the average structure, has its main maxima in the patches shown in Figure 3.

A Perhydrotriphenylene Inclusion Compound

Crystal Growth & Design, Vol. 5, No. 6, 2005 2075

Figure 2. (a) Substitutional disorder of the PHTP molecules: superposition of an R and an S molecule with population coefficients 0.5. (b) Intermolecular contact between heterochiral molecules “bumps onto bumps”. (c) Intermolecular contact between homochiral molecules “bumps into hollows”. Different chiralities are given in different colors.

Figure 3. Experimental (left) and simulated reciprocal layers (right): (a) hk1; (b) hk2; (c) hk3. The simulation is from model 3 in Table 1. The layers are mirror symmetric with respect to h ) 0 and k ) 0. Experimental data are shown for h < 0 and simulated data for h > 0.

It disappears in the hk0 plane, which represents the scattering associated with the projection of the crystal

structure onto the a,b plane. In this projection (R)- and (S)-PHTP molecules are indistinguishable (Figure 1), implying that the difference form factor is zero, in agreement with experiment. 3.2. Disorder within the Stacks. The concentration of diffuse intensities close to integral values of l implies extended sequences of equidistant, homochiral host molecules parallel to c (positive correlation) with occasional switches from one chirality to the other (negative correlation). In the following such sequences will be called c stacks or simply stacks. From the observed average half-width of the profiles along c* the average length of a homochiral sequence along c has been estimated to be ∼10 PHTP molecules.6 On steric grounds, the stacking distance is longer for RS than for RR or SS contacts, the former being of the “bumps onto bumps” type and the latter being “bumps into hollows” (Figure 2b,c).6 Thus, the substitutional disorder brings about displacive disorder as well; most likely the latter is responsible for the observed asymmetry of the diffuse profile.9 3.3. Relation between the Stacks. Given the occurrence of extended homochiral domains within stacks of PHTP molecules relationships between stacks are discussed assuming homochiral stacks of infinite length. The correlation between two stacks adjacent along b may be deduced from pseudo-extinctions in the diffuse intensities close to k ) 0: they are weak in the hk1 and hk3 layers, strong in hk2, and absent in hk0. This feature suggests a local c-glide operation in the real structure. Two stacks adjacent along b are therefore preferentially heterochiral, as symbolized in the abbreviations RS(b) and SR(b) (Figure 4c). In addition, most of the diffuse intensity in the hk1, hk2, and hk3 layers is found in the immediate vicinity of the Bragg peaks at h + k ) 2n, but not near the systematic absences at h + k ) 2n + 1, indicating a tendency for PHTP molecules related by (a + b)/2 or (a - b)/2 to have the same chirality. Taken together, the two observations suggest that a stack of PHTP molecule of a given chirality tends to be surrounded by three stacks of opposite chirality, but not always. The apparent space group of the average structure is Cmcm; stacks adjacent along a of opposite chirality (RS(a) and SR(a)) are related by an inversion and stacks of the same chirality (RR(a) and SS(a)) by a 21 operation. Similarly, stacks adjacent along b of opposite chirality (RS(b) and SR(b)) are related by a c-glide operation and stacks of the same chirality (RR(b) and SS(b)) by a 21 operation (from here on only one of the two equivalent enantiomeric configurations will be

2076


Figure 4. Interactions between PHTP molecules in neighboring stacks; only the C2H4 groups facing each other are shown (see Figure 1). The plane of projection is a(b,c for the RS(a) and RR(a) pairs in parts a and b and a,c for the RS(b) and RR(b) pairs in parts c and d. The arrow indicates a z shift of the molecule with a positive value of dRS. C2H4 groups from PHTP molecules with different chiralities are in different colors: lightly colored if behind the plane of projection and strongly colored if in front.

mentioned in the discussion). The symmetry of such pairs is given by their rod groups.10 A pair of disordered stacks adjacent along a has rod symmetry p(1 1)21/m (symbols in parentheses indicate the finite dimensions a and b). RS(a) pairs have rod symmetry p(1 1)1 h , and RR(a) pairs have rod symmetry p(1 1)21, both groups being subgroups of p(1 1)21/m. Similarly for a pair of disordered stacks that are neighbors along b, the rod symmetry is p(2/m 2/c)21/m, turning into the subgroup p(1 2/c)1 for an RS(b) pair and into the alternative subgroup p(2 2)21 for an RR(b) pair. When these possibilities are combined, the neighborhoods for an R stack along +a, -a, and b are RS(a)S(-a)S(b), RR(a)S(-a)S(b), RS(a)R(-a)S(b), RS(a)S(-a)R(b), RR(a)R(-a)S(b), RR(a)S(-a)R(b), RS(a)R(-a)R(b), and RR(a)R(-a)R(b). Repeating the local operations for successive pairs of stacks adjacent along a or b leads to maximally ordered structures with space groups C2221, P21/b2/c21/n, P2/b2/n21/n, and C12/c1, all of which are maximal subgroups of Cmcm. Note that in all four cases the molecules occupy special positions with site symmetry 2. The contact interface of an RS(a) pair of stacks along a differs from that of an RR(a) pair. Similarly, an RR(b) pair of stacks differs from an RS(b) pair of stacks (Figure 4). Because of their diastereomeric relationships, the relative molecular positions and orientations in the different pairs of stacks may differ and deviate

Bu¨rgi et al.

from those found in the average structure. For an RS(a) pair with p(1 1)1 h symmetry the S(a) molecule is no longer constrained to be exactly midway between its two R neighbors, as it is in the average structure, but may be displaced along c because its contacts to these neighbors are different (Figure 4a). It may also be displaced along a and b. For an RR(a) pair with p(1 1)21 symmetry the screw axis requires the R(a) molecule to be midway between the two R molecules in the neighboring stack, but displacements along a and b are still possible (Figure 4b). Analogous reasoning for an RS(b) and an RR(b) pair shows that displacements are possible only along b (Figure 4c,d). If we further require that the molecules in all four pairs of stacks have 2-fold rotational symmetry around b, as required by the maximally ordered structures, orientational disorder of the PHTP molecules may only arise from rotations about this direction. 3.4. Semiquantitative Modeling of Diffuse Scattering. Given the different relationships between neighboring molecules, the diffuse scattering in the layers perpendicular to c* was scrutinized for features indicating deviations from the average structure which are not only allowed by symmetry but are also of significant magnitude. In the hk1, hk2, and hk3 layers the main such features are the presence of distinct asymmetries along a* but not along b* and the reversal in the direction of asymmetry between k ) 0 and k ) 9 (Figure 3). Three hypotheses were tested qualitatively: longitudinal displacements of molecules in neighboring stacks along a and transversal displacements along b and c. In each case a linear sequence of randomly distributed (R)- and (S)-PHTP molecules juxtaposed along a was built and the distance vector between subsequent molecules assumed to depend only on the position and chirality of the preceding molecule. First, different nearest-neighbor distances along a were considered: 0.5 + ∆x for RR(a) and 0.5 - ∆x for RS(a). Given the site symmetry 2 of the PHTP molecules in all four maximally ordered structures, there is no need to distinguish between RS and SR distances on one hand and SS and RR distances on the other. The diffuse diffraction patterns calculated for hk1 and hk2 layers with ∆x ranging from -0.05 to 0.05 do show asymmetries with regard to the Bragg positions, but their direction does not depend on k, contrary to experiment. Second, displacements along b were considered. Because such displacements are not visible in a projection of the structure along b, any asymmetries in the diffuse scattering would disappear for k ) 0, contrary to experimental observation. Therefore, combinations of x and y displacements along a ( b, i.e., along the contact interfaces between neighboring molecules along a, were tested. With displacements (0.02,0.02,0) and (0.02,0.02,0), depending on the contact interface, the diffuse features near k ) 0 are similar to those calculated for displacements along a only. For k ) 9 the asymmetries relative to the Bragg positions are very weak and, not unexpectedly, do not reverse direction at k ) 0. Calculations with different displacement amplitudes lead to essentially the same unsatisfactory results. Finally, it was tested whether different contact distances between homo- and heterochiral b neighbors would have any influence on the asymmetries of the diffuse intensities.



Figure 5. Stereoscopic representation of a structural motif consisting of a central stack of four homochiral molecules surrounded by three stacks of three molecules of opposite chirality. Of all clusters of 13 molecules only 12.8% show this ordered arrangement. Different chiralities are given in different colors.

With shifts ∆y ranging from -0.03 to 0.03, no asymmetries along a* could be observed. Another striking disagreement between the simulations described in this paragraph and the experiment is found in the neighborhood of h ) 11 and k ) 9; the simulations show strong intensities where there are weak or no patchlike intensities in the experiment. As discussed above, transversal displacements parallel to c are allowed for RS(a) pairs of stacks, but not for RR(a) pairs. In the two pairs of molecules RS(a,+c/2) and RS(a,-c/2), the R and S molecules are related by independent centers of inversion (at a/2 ( c/4; Figure 4a). Small shifts along c of the a neighbors above and below the R molecule to new positions (a,+c/2+dRS) and (a,-c/2+dRS) away from the average positions (a,+c/2) and (a,-c/2) preserve the rod symmetry p(1 1)1h and are therefore allowed. They will occur if they improve the local intermolecular interaction energy. Note that in the enantiomeric situation the R molecules will be at SR(a,+c/2-dRS) and SR(a,-c/2-dRS). The influence of such displacements on diffuse scattering was tested again with a random distribution of (R)- and (S)-PHTP molecules along a. A value of dRS ) -0.02 reproduced the observed asymmetries along a* and the reversal of asymmetry along b* as well as the overall appearance of the diffuse intensities. Results for different values of dRS were qualitatively similar. Since the chirality of the molecules along a and b was taken to be uncorrelated, the diffuse features calculated in this preliminary test were broader than those found experimentally. Random sequences of R and S molecules along a were also tested for orientational disorder of the PHTP molecules centered on their average positions. From the symmetry considerations given above, such disorder was restricted to rotations about b. Librational displacements between -15 and 15° were tested, choosing rotation directions either the same or opposite for both homo- and heterochiral neighbors, but no asymmetries relative to the Bragg positions were discovered and the positions of the local maxima and minima of the diffuse intensities differed substantially from those found in the experiment. 3.5. Summary of Qualitative Interpretations of Diffuse Scattering. The diffuse scattering in the hk1, hk2, and hk3 layers was compared with the squared difference between the molecular form factors of R and S molecules in terms of position and distribution in reciprocal space. It could be assigned unequivocally to occupational R,S disorder and indicates stacks of ∼10 homochiral PHTP molecules along c and a weak tendency for chirality to alternate in directions perpen-

dicular to c. The absolute magnitudes of the correlation coefficients decrease in the order |Cc| > |Cb| ≈ |Ca|. For steric reasons contact distances along c are longer for RS than for RR and SS pairs, i.e., the occupational R,S disorder induces displacive disorder (Figure 2b,c). Heterochiral nearest-neighbor stacks along a, RS(a) and SR(a), show a shearing deformation along c (Figure 4a). No indications of orientational disorder have been found. 4. Monte Carlo Model for Crystal Structure Simulations 4.1. General Considerations. The qualitative conclusions from the preceding section have been quantified in terms of a model of the intermolecular interactions, from which disordered crystals have been built with a Monte Carlo procedure. The search for parameters that optimally reproduce the experimentally observed diffuse scattering is described in Section 5. Model definition and crystal building is the result of a compromise. On one hand, highly simplified ad hoc models of intermolecular interactions with a minimal number of parameters must be invented, because building disordered crystals molecule by molecule, calculating their scattering, and refining the parameters of the model is so demanding in terms of CPU time11 as to prevent the use of established quantum-mechanical or empirical force field models. On the other hand, the models must be sufficiently flexible to allow matching experimental observations. Given the empirical nature of such models, the numerical parameters have very limited direct physical meaning and are usually not transferable to related problems. 4.2. Procedure To Build Disordered Crystals. The intermolecular interactions in PHTP5‚NPP are mostly of the van der Waals type, especially those between the PHTP molecules, suggesting that only nearest-neighbor interactions have to be considered. A given PHTP molecule is in contact with altogether eight neighboring host molecules: two within a stack at ∆z ≈ +1 and -1 (c-neighbors), and two with ∆z ≈ +0.5 and -0.5 in each of three neighboring stacks (a and b neighbors; Figure 5). Each PHTP molecule is also in contact with the polar chains of guest molecules in the three channels it is adjacent to. These interactions are neglected, because the diffuse scattering provides no obvious indications that the disordered arrangements of the host and guest molecules are correlated.7 An (R)-PHTP molecule, once it is contained in the disordered crystal, does not transform into an S molecule and vice versa; in other words, the spatial distri-

2078


Bu¨rgi et al.

bution of R and S molecules is determined during crystal growth. Disordered crystals are therefore simulated layer by layer. In the first and second layers, at z ) 0 and z ) 0.5, respectively, rigid R and S molecules are distributed randomly. The molecules in subsequent layers make a face-to-face contact to one neighbor immediately below at ∆z ≈ -1 and edge-to-edge contacts to three molecules in neighboring stacks at ∆z ≈ -0.5. They are placed at a distance of ∆z ) -0.99, if they are homochiral, RR(c) with contacts of the type “bumps into hollows”. They are placed at a ∆z ) -1.09, if they are heterochiral, RS(c) with contacts “bumps onto bumps”. The difference of ∼0.1c was chosen to reproduce the difference in van der Waals distances between RR(c) and RS(c) contacts and to obtain on average a unit translation along c given the estimated ratio of homoto heterochiral contacts of ∼10 to 1.6 From the symmetry considerations in section 3.3 it was concluded that ∆z is 0.5 for RR(a), SS(a), RR(b) and RS(b) pairs but 0.5 + dRS for RS(a) and 0.5 - dRS for SR(a) pairs, with dRS being a refinable parameter of the empirical model. Differences of intermolecular distances along c for certain contacts include the possibility of uneven growth between different stacks. To minimize the resulting local strain energy, the empirical relationship describing the attachment energy of a molecule includes two parts: Ising-like terms Ji describing the bulk of the attachment energy and strain relaxation terms. For an S molecule the total attachment energy E(S) was chosen as

E(S) )

∑[JiσSσi + ki{(z(S) - zi) - di}2]/2;

i ) 1, 2, 3, 4 (1)

The parameters Ji differ for a, b, and c neighbors; the attachment energies differ in sign for homochiral and heterochiral contacts (σi ) 1 for S and σi ) -1 for R neighbors) and are assumed to be independent of the exact position of the newly attached molecule. The position of this molecule along c is z(S), those of its four neighbors are zi, the strain-free or reference distances between the new molecule and its neighbors along c are di, and the ki values are force constants. The position z(S) is chosen so as to minimize the strain energy. The total number of variable parameters in this model is 10, including 3 Ji values, the distance quantity dRS, and 6 ki values, namely one for homochiral and one for heterochiral contacts to a , b, and c neighbors. Previous computational experience has shown that the available experimental data do not allow us to distinguish between the different force constants, which were therefore chosen to be equal.11 The reference distances di have been chosen as discussed in the preceding paragraph and section 3.3. With these considerations the total number of adjustable parameters in eq 1 becomes 5: Ja, Jb, Jc, k, and dRS. The energy for attaching an R molecule E(R) is analogous to E(S) and contains the same adjustable parameters. The choice between attaching an R or an S molecule is based on a Boltzmann distribution assuming kT ) 1:

p(S) ) [1 + exp{E(S) - E(R)}]-1

(2)

with p(S) + p(R) ) 1.11 4.3. Calculation of Diffuse Intensities. Initial model crystals spanned 50 × 50 unit cells along a and

b with periodic boundary conditions. These numbers were increased to 70 × 70 and 90 × 90 in more advanced stages of the analysis. Along c the crystal was grown over 300 unit cells, corresponding to 600 layers of molecules. To ascertain a model crystal devoid of artifacts originating from the random distribution of R and S molecules in the two initial layers, its Fourier transform as well as all its statistical properties were calculated from the last 200 layers only. Test calculations showed that a steady growth regime under the control of the adjustable parameters was reached after only a few tens of layers. From the model crystals lots comprising 10 × 10 × 50 unit cells were selected at random and their Fourier transforms were calculated, squared, and averaged (incoherently), the number of such lots depending on the size of the model crystal (Table 1).11 One could argue that the molecular positions and orientations assumed during growth might relax as a given molecule is overgrown. Given the computational effort required to account for such effects, relaxation in the crystal bulk was not considered and the molecules were left at the positions they assume on the surface of the growing crystal. This simplification was justified a posteriori by the agreement between observed and calculated diffuse scattering. 5. Parameter Optimization using Differential Evolution 5.1. General Considerations. Methods to find the numerical parameters needed in eq 1 include trial and error12 and least-squares refinement.13 The former has the potential disadvantage of being very time-consuming, even for modest numbers of parameters; the latter has the well-known limitation of all least-squares procedures, namely that the starting values of the parameters must be sufficiently close to those of the global least-squares minimum. We have recently described a genetic algorithm for parameter refinement called differential evolution.1,11 A large number of parameter sets covering a wide range of parameter space is tested simultaneously with the help of a scheme of distributed computing. Parameter sets that lead to poor agreement between model and experiment are eliminated and replaced in subsequent generations by new ones obtained from recombining and mutating the more successful sets. This procedure eliminates the tediousness of the trial and error approach and is not as restricted as the least-squares approach with respect to initial parameter values. 5.2. Acceleration of the Calculations. Our early application of the method of differential evolution in 2001 turned out to be very time-consuming. Optimization of 7 parameters with 10 processors took ∼30 days.11 Since then the technique has been improved in several ways. (1) The most important gain comes from a much more efficient use of the information produced by differential evolution. (2) Several aspects of the computations have been optimized: the Fourier transform of the lots of point lattices representing the distribution of R and S molecules, the subsequent multiplication with the complex-valued molecular form factors, and the final incoherent summation have been combined into a single program, whichsas far as possibleskeeps all



Table 1. Substitutional Disorder between (R)- and (S)-PHTP Molecules: Models, Their Parameters, and Statistical Descriptors model from ref 11 no. of params no. of param sets model crystals per param set no. of generations size of model crystala no. of lots Ja Jb Jc dRS (in fractional units) ka or k kb kc R(best individual)b R(pop. av)b Ca Cb Cc RS(a)RS(a)..., SR(a)SR(a)... RR(a)RR(a)..., SS(a)SS(a)... RS(b)RS(b)..., SR(b)SR(b)... RR(b)RR(b)..., SS(b)SS(b)... RR(c)R(c)..., SS(c)S(c)...

from ref 11 (best individual)

1 (pop. av)

2 (pop. av)

3 (pop. av)

4 (best individual)

6 (best individual)

5 (pop. av)

7 40 1

7 1 1

7 40 1

7 40 1

5 40 1

5 1 40

5 40 1

5 1 40

220 90 × 90 150 0.082(4) 0.156(6) -1.19(3) -0.032(2) 18(5) 30(9) 17(4)

220 90 × 90 150 0.0822 0.1579 -1.1723 -0.0348 20.81 26.017 24.955 14.1

63 50 × 50 20 0.092(8) 0.162(26) -1.19(11) -0.030(3) 43(14) 73(27) 37(21) 22.3 13.2

108 90 × 90 150 0.089(4) 0.153(11) -1.16(6) -0.0289(7) 33(7) 38(10) 34(12) 14.2 12.5

31 50 × 50 20 0.084(23) 0.180(39) -1.22(15) -0.033(6) 18(38)

31 50 × 50 20 0.066 0.2032 -1.31 -0.0383 -6.798

83 90 × 90 150 0.081(9) 0.169(10) -1.23(6) -0.031(2) 17(15)

83 90 × 90 150 0.0768 0.1751 -1.261 -0.0313 9.372

22.3 12.9

22.3 14.4

14.5 12.4

14.5 13.0

Nearest-Neighbor Correlations -0.14(2) -0.14(1) -0.14(4) -0.30(4) -0.28(2) -0.32(6) 0.87(2) 0.86(1) 0.87(3)

-0.131(3) -0.359(3) 0.8711(4)

-0.14(1) -0.30(2) 0.87(1)

-0.135(2) -0.315(2) 0.8703(2)

Pattern Lengths along cc 8(1)/14(2) 7.7(6)/13(1) 8(1)/13(3) 6(1)/11(2) 5.9(5)/10(1) 6(1)/10(2) 9(1)/15(2) 8.5(7)/14(1) 9(2)/15(3) 5(1)/9(2) 5.0(4)/8(1) 5(1)/8(2) 14(2)/22(3) 12.6(9)/21(2) 13(2)/21(3)

8.01(4)/13.3 (3) 6.30(2)/10.49(8) 9.46(5)/15.51(8) 4.74(3)/7.5 (0) 13.25(5)/22 (0)

8.0(8)/13(1) 6.2(6)/10(1) 9.0(9)/15(1) 5.1(5)/8(1) 13 (1)/22(2)

8.00(2)/13.0(1) 6.24(2)/10.1(2) 9.14(2)/15 (0) 5.02(1)/8 (0) 13.15(2)/22.0(2)

-0.15 -0.28 0.859

a In unit cells along a and b. b R ) 100(∑∆I2/∑I2)1/2, summation over all pixels included in parameter determination.11 of unit cells, mean/median (with standard deviations in parentheses).

information in memory.14 Dead time between subsequent steps of the refinement has been reduced by introducing criteria for automatic switching to larger and larger model crystals. The first switch occurs if a new generation does not contain a new parameter set; subsequent switches occur after 15 generations at the latest or earlier, if a new generation does not contain updated parameters. (3) The number of parameters to be optimized has been reduced with no obvious loss in the performance of the model. (4) Processors have become faster during the 4 years after our first use of the differential evolution technique. The various changes accelerated the computations by 1-2 orders of magnitude (see below) and made it possible to double-check the seven parameters of the initial model. The mean values of Ji and dRS and their distributions are reproduced well, whereas the ki values are poorly determined and consistently cover a broad range of values amounting to 40-80% of their mean (Table 1, columns 2, 4, and 5). For small model crystals the best individual shows relatively poor agreement with experiment (R ) 22.3%), whereas for the larger model crystals the agreement is much better (R ) 14.2%) and agrees with our earlier result (R ) 14.1%). This and several analogous calculations provide strong evidence that the parameter optimization for this problem is not plagued by subsidiary minima in the agreement function. Given the poor reproducibility of the parameters ki, the model was simplified to a single strain constant k. However, the parameter k still covers a wide range of values, and sometimes it even becomes negative (Table 1, columns 6 and 7)! This seemingly unphysical result may be understood from the fact that with a single strain constant in eq 1 the equilibrium position of an

c

In numbers

attaching S molecule, the quantity affecting the appearance of the diffuse scattering directly, becomes independent of k:

z(S) )

∑(zi + di)/4; i ) 1, 2, 3, 4

(3)

In addition, the strain energy term in eq 1 tends to be small compared to the attachment energies, Jc in particular: too small to significantly influence the Boltzmann statistics governing the R,S distribution. For larger model crystals the mean value of k does not change, but the range is now restricted to positive values (Table 1, columns 8 and 9). With hindsight the parameter k appears unnecessary; taking into account z(S) according to eq 3 seems sufficient. The most significant reduction in computing effort results from an alternative way of looking at the information produced by differential evolution. In our initial application of the technique we had compared observed diffuse intensities with those calculated for one crystal from one parameter set. When mutation and recombination no longer improved parameter sets, the size of the model crystal was increased in the a and b directions as described above. Agreement with experiment improved, because the parameters improved somewhat, butsequally importantsalso because the noise in the calculated intensities was reduced due to the larger, more representative size of the simulated crystal (Table 1, R(best individual)). In the modified procedure the parameters were also optimized to convergence. However, as a last step, the calculated intensities for an entire population of 40 individuals, all from different parameter sets, are averaged and compared to the observed diffuse scattering. The corresponding agreement factor R(population average) is significantly lower

2080


than that for the intensity calculated with the best parameter set in the population (Table 1). To distinguish improved statistics due to averaging over 40 patternss all calculated with different parameterssfrom effects due to an improved model, 40 model crystals were simulated with the best set of parameters and their calculated intensities averaged. Agreement with experiment also improves substantially, but not as much as for the population average. We conclude that model crystals built with a distribution of parameter values tend to give a better account of experimental data than those from best parameter values, at least in the early stages of the modeling process. Different model crystals calculated with the same set of parameters cover a range of R values due to the probabilistic nature of Monte Carlo procedures. Assessment of the sets of parameters is thus biased to some extent, the “best” set being the one for which the single model crystal calculated to gauge its quality happens to show a better than average R factor. This bias is reduced if the average is taken over individuals obtained from different parameter sets. At present we do not fully understand the statistical background behind these observations. In any case, making use of population averages in the early stages of the modeling has helped to reduce the computational effort from ∼30 to ∼1/2 days. This reduction implies no loss of structural information, at least not for the resolution of the data available in this study (∼1.7 Å). This will be shown in the next section, where the actual crystal structures simulated with different models and different implementations of the differential evolution procedure are compared, rather than just their R factors and parameter values. 6. Description of the Disordered Structure 6.1. Comparison of the Crystal Structures from Different Models and Different Implementations of the Differential Evolution Procedure. In all models the ratio of (R)- to (S)-PHTP molecules is 1 to within 0.1%. The mean distances along a, b, and c are 0.499, 0.499, and 0.999, respectively. The disordered arrangements of R and S molecules of the model crystals are best characterized by averages and other statistical quantities derived from the top 200 layers of a crystal 600 layers thick. Nearest-neighbor correlations C are the same irrespective of the number of parameters in the model, of the number of generations in the differential evolution procedure, and of the size of the model crystals. The mean and median lengths of homochiral stacks and pairs of such stacks are also independent of the model, the number of generations, and the size of the model crystal (Table 1). More detailed comparisons of other structural descriptors have shown similar results. It is concluded that most of the structural information is available after a few generations of differential evolution from population averages based on small model crystals and that the main gain from higher numbers of generations and larger model crystals is a somewhat narrower definition of the parameter ranges and the statistical descriptors. 6.2. Structural Motifs at the Nanometer Scale. Given the lack of crystallographic symmetry in a disordered crystal, the choice of relevant structural motifs is somewhat arbitrary. We have chosen to

Bu¨rgi et al.

consider the lengths of homochiral domains along c, pairs of stacks with the same or opposite chiralities, and nanometer-sized clusters consisting of 13 molecules in which a central c stack of four molecules is surrounded by three stacks of three molecules each (Figure 5 shows an example). In these clusters the emphasis is on the central contact in the middle stack. If it is between two homochiral molecules, we call the contact “regular”. If it is between two heterochiral ones, we call it a “fault”. The probabilities p of neighboring molecules along a, b, or c to be homochiral is obtained from nearestneighbor correlations C according to p ) (1 + C)/2. Quantitative results given below refer to model 3 (Table 1). 6.2.1. Stacks, Double Stacks, and the Polarity of the Host-Guest Compound. Figure 6 shows sections from the model crystal perpendicular to the a, b, and c axes. The dominant features in the a,c layer are long, homochiral sequences of PHTP molecules along c. They illustrate the large positive correlation coefficient Cc. Along a and the diagonal of the unit cell a+b heterochiral sequences are more frequent than homochiral ones, in agreement with the slightly negative correlation coefficients Cb and Ca. Overall the a,c and a+b,c layers of the PHTP host structure show chiral (monochrome) and nonchiral (dichromic) nanopatches. Whereas the packing of the racemic mixture of apolar host molecules shows locally chiral and achiral domains, the packing of the achiral but polar guest molecules is found to be polar at short and long ranges. This observation confirms the earlier hypothesis, according to which the parallel arrangement of the polar guest molecules may be explained with a Markov growth process that is largely independent of the structure of the host lattice.3-6 The probability for molecules of one chirality to be surrounded by three molecules of the opposite chirality is nonnegligible, as expected from Ca and Cb, both of which are negative, and is in agreement with the conclusion from section 3.3. The distribution of (R)- and (S)-PHTP molecules within the a,b layer indicates that molecules related by the vectors a and (a+b)/2 tend to be homochiral (Figure 6). The correlation coefficient Cc of 0.87 implies that about 93% of the contacts along c are homochiral, i.e., the average length of a homochiral sequence is 13 molecules, in fair agreement with the earlier semiquantitative estimate of 10 molecules (section 3.2 and ref 6). With increasing distance along the stack the correlation decreases about exponentially. For the 10th and 20th neighbors, for example, the correlation coefficients are 0.275 and 0.079, respectively, close to the values of 0.8710 ) 0.248 and 0.8720 ) 0.062, which are expected if the distributions of (R)- and (S)-PHTP molecules within a single stack are governed by nearest-neighbor interactions only. The length distribution of homochiral stacks is strongly skewed, the median length being almost twice as long as the mean and most frequent lengths. Thus, homochiral domains in a single stack may be several tens of nanometers long (Table 1 and Figure 7). The correlation Ca is -0.15 for nearest neighbors. The correlation coefficients between nextnearest neighbors and neighbors further apart along a do not exceed the range -0.04 to 0.02 but are somewhat larger along (a+b)/2. The average length along c of pairs



Figure 6. Schematic representation of the distribution of R and S molecules in sections through the a,b, a,c, and b,c planes. The b,c plane passes through x ) 0 (Figure 1); the empty columns represent the channels. Molecules with different chiralities are represented by dots of different colors. The simulation is from model 3 in Table 1.

Figure 7. Length distribution of homochiral sequences along c (in unit cells, 1 unit cell ) 1 molecule). Insets: (top) length distributions of homo- and heterochiral zigzag sequences in pairs of stacks along a; (bottom) length distributions of homoand heterochiral zigzag sequences in pairs of stacks juxtaposed along b (in unit cells, 1 unit cell ) 2 molecules, one in each of the two stacks). The simulation is from model 3 in Table 1.

of heterochiral stacks is still considerable at eight to nine unit cells. Again, the distribution of their lengths is skewed and pairs of heterochiral stacks are generally longer than pairs of homochiral stacks. They are shorter, however, than the homochiral domains in single stacks. The difference represents the combined effects of strong positive correlation along c and weaker negative correlations along a and b. 6.2.2. Environment of Homo- and Heterochiral Contacts. Regular contacts RR(c) and SS(c) account for 93% of all contacts along c and faults RS(c) and SR(c) for 7%. We have analyzed the immediate environment of these contacts by classifying the clusters of 13 molecules described in section 6.2 above (Figure 5). For 53.4% of the clusters the central stacks are RRRR or SSSS; these 53.4% may be further subdivided: 12.8% are surrounded by stacks of molecules of opposite

chirality, i.e., RS(a)S(-a)S(b) or SR(a)R(-a)R(b), 17.9% are RS(a)R(-a)S(b), SR(a)S(-a)R(b), etc., 11.8% are RS(a)R(-a)R(b), SR(a)S(-a)S(b), etc., 8.1% are RR(a)S(-a)R(b), SS(a)R(-a)S(b), etc., and 2.6% are RR(a)R(-a)R(b) or SS(a)S(-a)S(b). In these clusters, which are free of faults, no local strain is generated, since all intermolecular distances within the stacks are the same. For another 39.6% of the clusters the central contact is also regular, but its environment shows at least one fault. About half of the remaining clusters (7%) shows a central RRSS or SSRR stack and is surrounded by three regular stacks. These numbers illustrate the high degree of disorder in the host structure: the last two groups, i.e., 46.6% of all clusters of 13 molecules, show at least one fault. The influence of these faults on the local intermolecular geometry may be gauged from the distribution of intermolecular distances along c. 6.3. Faults, Stacking Distances and Local Strain. The distributions of distances between homo- and heterochiral molecules in c stacks are shown in Figure 8a. Intermolecular contacts between homochiral molecules are generally shorter becausesas mentioned abovesthey are “bumps into hollows”, whereas contacts between heterochiral molecules are “bumps onto bumps”. The distribution of the former is trimodal with a maximum and mean of 0.997c (4.72(3) Å), and that of the latter is monomodal with a maximum and mean of 1.022c (4.83(3) Å) and with very long tails. Note that the difference of 0.025c corresponding to 0.11 Å is smaller than the 0.42 Å proposed from a consideration of van der Waals contacts.6 The resulting H‚‚‚H distances between the axial hydrogen atoms in a fault are relatively small but still reasonable at ∼2.2 Å (calculated with a C-H internuclear distances of 1.09 Å). Compressing the long equilibrium H‚‚‚H distances of the relatively rare heterochiral contacts (∼7%) seems to be

2082


Bu¨rgi et al.

cadence with the surrounding stacks and to keep strain in the next but one growth step small. Of the distances labeled homo that are longer than 1.0075c, 84.2% are associated with the second homochiral contact in central RR(c)R(2c) or SS(c)S(2c) stacks, if the two b neighbors, or the two a neighbors or the two -a neighbors, are heterochiral. The long intermolecular contact in the faulted side stacks thus pushes the homochiral molecules in the central stack further apart than expected from the strain-free equilibrium distance. The average contact distance along c between heterochiral RS(a) and SR(a) pairs is affected by a shear deformation, as mentioned in section 3.4. The respective most frequent and mean distances are 0.487c and 0.512c (Figure 8b). It is this small deviation from 0.5 that is responsible for the strong asymmetries of diffuse intensities along a* seen in the hk1, hk2, and hk3 layers (Figure 3)! The deviation of RR(a) and SS(a) contacts from ∆z ) 0.5 is negligibly small. There is no difference in the average contact lengths along c for RS(b) and SR(b) pairs; these lengths are very close to ∆z ) 0.5 (Figure 8c). The variance of the contacts along a and b is about the same, both distributions being much broader than those of the four resolved groups in Figure 8a. 7. Summary and Conclusions

Figure 8. Distributions of the c component of intermolecular distances between homo- and heterochiral PHTP molecules (a) between c neighbors, (b) between a neighbors, and (c) between b neighbors. The simulation is with parameters from column 6 in Table 1.

the most favorable way of minimizing nonbonded strain around a fault. This hypothesis is confirmed by an analysis of the subsidiary maxima in Figure 8a. Of the distances labeled homo that are shorter than 0.9875c, 88.2% are associated with the RR(c) or SS(c) contact in SR(c)R(2c) and RS(c)S(2c) fragments surrounded by three stacks of two homochiral molecules each. In other words, the long distance associated with the SR(c) or RS(c) contact tends to be compensated by a shortened distance in the subsequent formation of an RR(c) or SS(c) contact. This shortening helps to get the central stack back into

Determination of the disordered crystal structure of the polar material PHTP5‚NPP was begun with a careful interpretation of the Bragg reflections, i.e., an accurate determination of the average crystal structure.6 This structure was scrutinized for chemically impossible and unreasonable features such as the superposition of half an (R)-PHTP on half an (S)-PHTP molecule and unusual interatomic distances and anisotropic displacement parameters.7 Maximally ordered variants of the average structure have been derived from the subgroups of the space group of the average structure to serve as models of more or less ordered domains in the disordered crystal. In the second step the diffuse scattering was classified qualitatively as streaks, planes, or 3D features in order to determine the nature of the disorder as 1D, 2D, or 3D. In the analysis presented here we have concentrated on reciprocal planes of diffuse scattering with a thickness exceeding the diameter of Bragg reflections. From the beginning these features indicated strong disorder parallel to the planes and lesser disorder perpendicular to them. In a third step the information on disorder from direct and reciprocal space was combined. On the basis of the information from direct space, models of disorder were formulated and translated into reciprocal space by making use of the molecular form factors of (R)- and (S)-PHTP molecules. Agreement of the difference between these form factors with experimentally observed features led to identification of those parts of the diffuse scattering, which are due to occupational R,S disorder. The subsequent quantitative analysis of the diffuse scattering started with the formulation of an empirical, probabilistic model describing the distribution of R and S molecules in the disordered crystal with a minimum number of parameters, but still allowing for local molecular reorientation and displacement in order to


minimize unfavorable intermolecular contacts in the disordered crystal. The parameters of the model are three attachment energies, a reference intermolecular distance, and a force constant governing the minimization of local strain arising from packing faults. They were optimized with a genetic algorithm called differential evolution that was implemented on a cluster comprising between 10 and 20 PC’s. Optimization converged within a half to a full day. The last step was the chemical interpretation of the results. The dominant motifs in the crystal are domains of ∼10-20 homochiral molecules extending along c. Pairs of neighboring domains are slightly biased toward being heterochiral, especially along b. Along the diagonals of the unit cell, and to a lesser extent along a, nanodomains of chains with alternating chirality outnumber those with the same chirality. Neighboring chains along a of opposite chirality show a small shear deformation of ∼0.15 Å with shear direction c. There is no indication of a correlation between the disordered distribution of the racemic mixture of PHTP host molecules on one hand and the polarity of the arrangements of the NPP guest molecules on the other. The simulated crystal structures suggest that relatively rare stacking faults induce strain, which is due to the difference in intermolecular distances between regular RR(c) or SS(c) contacts on one hand and faulted RS(c) contacts on the other. The strain appears to be local and is released by compression or extension of the nearby regular contacts, at least as far as we can tell from the limited resolution of the diffuse scattering data. The successful analysis of the disordered structure of PHTP5‚NPP suggests that the analysis of diffuse scattering from disordered crystals is a powerful technique for the investigation of supramolecular building principles, even when the driving force for an ordered structure of the bulk is too small. The analysis presented here demonstrates that diffuse scattering is capable of detecting disorder phenomena with characteristic length scales differing by several orders of magnitude: detailed information about local displacive disorder with amplitudes 100 Å have been obtained. The simulated crystal structures may be viewed not only as a catalog of probable, nanometersized supramolecular building units but also as static pictures of the variety of intermolecular geometries, which is usually hidden in the atomic displacement


parameters. Furthermore, they may be considered protocols of the crystal growth process. Complementary techniques for the investigation of real structures, such as the reconstruction of electron densities from coherent scattering,15 still lack atomic or even molecular resolution and are presently restricted to the study of nanoparticles. Atomic force microscopy and to a large extent also electron microscopy look at the surface of samples or at projections of structures. The investigation of disordered structures is still a demanding and time-consuming task. With the availability of more general and automated optimization techniques as well as faster computers, it may be expected, however, that investigations of diffuse scattering will become more accessible, provide a deeper insight into the structure of disordered materials, and may even provide hints concerning the molecular aspects of the growth of crystals, a phenomenon that is still far from being understood. References (1) Storn, R.; Price, K. J. Global Optim. 1997, 11, 341-359. (2) Proffen, Th.; Welberry, T. R. Phase Transitions 1998, 67, 373-397. (3) Hulliger, J.; Ko¨nig, O.; Hoss, R. Adv. Mater. 1995, 7, 719721. (4) Roth, S. W.; Langley, P. J.; Quintel, A.; Wubbenhorst, M.; Rechsteiner, P.; Rogin, P.; Ko¨nig, O.; Hulliger, J. Adv. Mater. 1998, 10, 1543-1546. (5) Hulliger, J.; Roth, S. W.; Quintel, A.; Bebie, H. J. Solid State Chem. 2000, 152, 49-56. (6) Ko¨nig, O.; Bu¨rgi, H. B.; Armbruster, T.; Hulliger, J.; Weber, T. J. Am. Chem. Soc. 1997, 119, 10632-10640. (7) Weber, T.; Estermann, M. A.; Bu¨rgi, H. B. Acta Crystallogr., Sect. B 2001, 57, 579-590. (8) Mayo, S. C.; Proffen, Th.; Bown, M.; Welberry, T. R. J. Appl. Crystallogr. 1999, 32, 464-471. (9) Welberry, T. R. J. Appl. Crystallogr. 1986, 19, 382-389. (10) International Tables for Crystallography; Kopsky, V., Litvin, D. B. Eds.; Kluwer Academic: Dordrecht, The Netherlands, 2002; Vol. E. (11) Weber, T.; Bu¨rgi, H. B. Acta Crystallogr., Sect. A 2002, 58, 526-540. (12) Welberry, T. R.; Butler, B. D. J. Appl. Crystallogr. 1994, 27, 205-231. (13) Welberry, T. R.; Proffen, Th.; Bown, M. Acta Crystallogr., Sect. A 1998, 54, 661-674. (14) The program code for the calculation of the Fourier transform was taken from the program DISCUS: Proffen, Th.; Neder, R. B. J. Appl. Crystallogr. 1997, 30, 171-175. (15) Miao, J.; Sayre, D. Acta Crystallogr., Sect. A 2000, 56, 596-605.

CG050211L

Supramolecular Architecture in a Disordered Perhydrotriphenylene

Recommend Documents