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ISSN: 2052-5206

Synchrotron X-ray diffuse scattering from a stable polymorphic material: terephthalic acid, C8H6O4

aSchool of Physical, Environmental and Mathematical Sciences, The University of New South Wales, Canberra, ACT 2600, Australia, and bMaterials Science Division, Drug Product Science and Technology, Bristol-Myers Squibb, New Brunswick, NJ 08901, USA
*Correspondence e-mail: darren.goossens@gmail.com

Edited by E. V. Boldyreva, Russian Academy of Sciences, Russian Federation (Received 31 August 2016; accepted 23 November 2016; online 31 January 2017)

Terephthalic acid (TPA, C8H6O4) is an industrially important chemical, one that shows polymorphism and disorder. Three polymorphs are known, two triclinic [(I) and (II)] and one monoclinic (III). Of the two triclinic polymorphs, (II) has been shown to be more stable in ambient conditions. This paper presents models of the local order of polymorphs (I) and (II), and compares the single-crystal diffuse scattering (SCDS) computed from the models with that observed from real crystals. TPA shows relatively weak and less-structured diffuse scattering than some other polymorphic materials, but it does appear that the SCDS is less well modelled by a purely harmonic model in polymorph (I) than in polymorph (II), according to the idea that the diffuse scattering is sensitive to anharmonicity that presages a structural phase transition. The work here verifies that displacive correlations are strong along the molecular chains and weak laterally, and that it is not necessary to allow the —COOH groups to librate to successfully model the diffuse scattering – keeping in mind that the data are from X-ray diffraction and not directly sensitive to H atoms.

1. Introduction

The short-range order (SRO) in a molecular crystal is the result of intermolecular interactions. At the simplest level, molecules may bump into each other, resulting in correlated displacements. This may be termed displacive SRO. If the crystal contains more than one species (or more than one possible orientation of a species) then occupancy ordering can occur, in which species cluster or alternate.

Here the X-ray single-crystal diffuse scattering (SCDS) from triclinic polymorphs (I) and (II) of terephthalic acid, C8H6O4, is modelled using the Monte Carlo (MC) ZMC code (Goossens et al., 2010[Goossens, D. J., Heerdegen, A. P., Chan, E. J. & Welberry, T. R. (2010). Metall. Mater. Trans. A, 42, 23-31.]), combined with a suite of toolbox programs to aid in assembling the simulation  (Goossens, 2015[Goossens, D. J. (2015). Adv. Condens. Matter Phys. 2015, 1-7.]). A combination of MC modelling (Metropolis et al., 1953[Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. (1953). J. Chem. Phys. 21, 1087-1092.]; Welberry, 2004[Welberry, T. R. (2004). Diffuse X-ray Scattering and Models of Disorder. Oxford University Press.]) and visual inspection of the data leads to the development of a useful model very quickly compared with previous methods.

The SCDS from polymorph (I) is also examined, although the data are more limited. It is found that the SCDS from polymorph (II) – the more stable polymorph – is more accurately modelled by a purely harmonic model than the SCDS for polymorph (I), suggesting that the SCDS is sensitive to the different energetics of the two structures.

2. Terephthalic acid, C8H6O4

Terephthalic acid, C8H6O4 (benzene-1,4-dicarboxylic acid, Fig. 1[link]) is a widely used industrial chemical which crystallizes as at least three polymorphs, two triclinic and one monoclinic (Śledź et al., 2001[Śledź, M., Janczak, J. & Kubiak, R. (2001). J. Mol. Struct. 595, 77-82.]). The polymorphs explored in the present work are known as triclinic polymorphs (I) and (II). Triclinic (II) has lattice parameters a = 5.027 (6), b = 5.360 (6), c = 7.382 (6) Å, α = 115.72 (7), β = 101.06 (10) and γ = 92.91 (14)°, with Z = 1 (Domenicano et al., 1990[Domenicano, A., Schultz, G., Hargittai, I., Colapietro, M., Portalone, G., George, P. & Bock, C. (1990). Struct. Chem. 1, 107-122.]).

[Figure 1]
Figure 1
A molecule of TPA, C8H6O4.

Form (I) shows a = 7.730, b = 6.443, c = 3.749 Å, α = 92.75, β = 109.15 and γ = 95.95°, with Z = 1 (Bailey & Brown, 1967[Bailey, M. & Brown, C. J. (1967). Acta Cryst. 22, 387-391.]).

As it is a linear dicarboxylic acid, the molecules show a strong preference for forming chains through the crystal, and this is a feature of the structures of the polymorphs, and has implications for the diffuse scattering. In particular, it results in sheets of diffuse scattering perpendicular to the chains. See Appendix A[link]. These sheets arise out of the interactions in the model, however, and will not be imposed.

It has been shown that at room temperature and pressure, polymorph (II) is the more stable. On heating, it undergoes a phase transition to polymorph (I) at temperatures in the range 348–373 K  (Davey et al., 1994a[Davey, R. J., Maginn, S. J., Andrews, S. J., Black, S. N., Buckley, A. M., Cottier, D., Dempsey, P., Plowman, R., Rout, J. E., Stanley, D. R. & Taylor, A. (1994a). Faraday Trans. 90, 1003-1009.],b[Davey, R. J., Maginn, S. J., Andrews, S. J., Black, S. N., Buckley, A. M., Cottier, D., Dempsey, P., Plowman, R., Rout, J. E., Stanley, D. R. & Taylor, A. (1994b). Mol. Cryst. Liq. Cryst. 242, 79-90.]). While reversible, the transition appears to be hysteretic, and temperatures of 303 K and lower are needed to drive a polymorph (I) crystal back into polymorph (II). Thus, it is possible for both polymorphs to exist at room temperature, with polymorph (I) the less stable.

3. Experimental data

After crystal screening using a mar345 desktop beamline in the Research School of Chemistry at the Australian National University, detailed data for analysis were then collected at the 11-ID-B beamline at the Advanced Photon Source, Chicago.

These reciprocal space sections for both polymorphs were extracted using the same method. Cuts (also called sections) were assembled using Xcavate (Estermann & Steurer, 1998[Estermann, M. A. & Steurer, W. (1998). Phase Transitions, 67, 165-195.]; Scheidegger et al., 2000[Scheidegger, S., Estermann, M. A. & Steurer, W. (2000). J. Appl. Cryst. 33, 35-48.]). For the frames in each cut, intensity was normalized for beam fluctuations and any over-exposed pixels were excized (Welberry et al., 2005[Welberry, T. R., Goossens, D. J., Heerdegen, A. P. & Lee, P. L. (2005). Z. Kristallogr. 220, 1052-1058.]). The corrections required have been found to vary from experiment to experiment, detector to detector (image plate versus an electronic detector, for example) and even wavelength to wavelength. In this case intensities were corrected by beam monitor counts, and then scaled again based on the median of each frame because it was found that monitor counts alone could not always provide consistent backgrounds, and background should be a smoothly varying function. This was done because the samples used are large by synchrotron crystallography standards (to maximize the diffuse scattering from these molecules that contain only light elements) and were not fully bathed in the beam, and so the scattering volume varied with sample angle. In the experiments here, background was not subtracted, and gives the bright regions in the centres of plots in Fig. 2[link].

[Figure 2]
Figure 2
Observed data, measured at the Advanced Photon Source at 295 K, showing streaks (cuts through diffuse planes) due to molecular chains (two-ended arrows) and what looks like multiple crystalites with similar orientations (arrowed in a and c) for TPA triclinic polymorph (II). (a) 0kl, (b) h0l, (c) hk0. White circles show the extent of the calculations in Fig. 5[link].

For polymorph (II) twofold rotational averaging was applied to the data. This was not done for polymorph (I) as the data reconstruction was not as precise and the averaging did not cause related parts of the image to properly superimpose.

Overexposed regions on the image plate resulted in two possible effects; nearby pixel intensities were unreliable, and the same pixels in ensuing frames were not always reliable as if the strong exposure was not fully washed out by reading the plate (an effect referred to as ghosting). These were dealt with using empirical corrections, including discounting all pixels within some radius of any overexposed pixel, and discounting pixels for some number of subsequent images if they had been overexposed in a previous image. The radius and number of images, respectively, were determined interactively by looking for inconsistencies, like abrupt steps in intensity, in reconstructed data. Some aspects of data collection and correction are considered in more detail elsewhere (Welberry et al., 2005[Welberry, T. R., Goossens, D. J., Heerdegen, A. P. & Lee, P. L. (2005). Z. Kristallogr. 220, 1052-1058.]).

Fig. 2[link] shows sections of observed diffuse scattering from TPA triclinic polymorph (II), measured using the ID-11-B beamline at the Advanced Photon Source (T = 295 K). Fig. 3[link] shows the same cuts from polymorph (I); the slices are less comprehensive but a range of features can be seen, very much analogous to those seen for polymorph (II). In both polymorphs diffuse streaks are apparent. Although running in different directions, they are consistent with being formed by taking sections through the planes of scattering that come from the longitudinal correlations of molecular chains (see Appendix A[link]). Some Bragg spots are surrounded by large spots of thermal diffuse scattering. Again, no features centred at non-integer hkl are visible, and none of the diffuse shows complex shapes. A handful of features are highlighted by white boxes for later reference.

[Figure 3]
Figure 3
Observed data, measured at the Advanced Photon Source, for TPA triclinic polymorph (I) at 295 K. The yellow arrow on (a) indicates a (weak) streak due to molecular chains (direction indicated by white double arrow). Yellow arrows on (b) indicate artefacts due to the intense scattering near a large Bragg peak, and also a spot due to a second crystalite. (a) 0kl, (b) h0l, (c) hk0. Reconstructions proved difficult in this case. Black areas indicate a lack of data. The extent of data is similar to the extent of the calculation in Fig. 7[link].

4. Specifying the simulation

Before any modelling began, consideration was given to what was needed to go into the model. TPA does not show chemical ordering — only one molecular species is present, and in one orientation. However, a model should be able to allow the —COOH groups to rotate. This was an important early consideration as it influenced how the molecule was described. The discussion that follows focuses on polymorph (II); similar steps were taken when modelling polymorph (I).

4.1. The molecule

The molecule had to be described in a way which allowed for suitable internal degrees of freedom, in this case rotation of the —COOH groups. If a molecule possesses a centre of symmetry, it is reasonable to use this as the molecular origin. In this case, such considerations helped determine (1) the atomic ordering, (2) the connectivity within the molecule, and (3) the presence of a dummy atom at the centre.

The molecule was described using a z-matrix (Hehre et al., 1986[Hehre, W. J., Radom, L., Schleyer, P. v. R. & Pople, J. A. (1986). Ab Initio Molecular Orbital Theory. New York: Wiley.]), which was assembled using a toolbox program called zmat_maker. zmat_maker takes as input a .mol2 file (Mercury, 2015[CSD (2015). Mercury, http://www.ccdc.cam.ac.uk/products/mercury/.]), which contains the contents of the unit cell, both the atoms and the bonds. The name ZMC comes from a combination of MC and z-matrix. A walk-through of the construction of a model using ZMC and the associated toolbox has been given elsewhere, and that discussion includes deposited material to allow a user to reproduce a simulation  (Goossens, 2015[Goossens, D. J. (2015). Adv. Condens. Matter Phys. 2015, 1-7.]).

(1) A CIF was obtained (CSD database code TEPHTH12; CSD, 2016[CSD (2016). http://www.ccdc.cam.ac.uk/products/csd/.]; Domenicano et al., 1990[Domenicano, A., Schultz, G., Hargittai, I., Colapietro, M., Portalone, G., George, P. & Bock, C. (1990). Struct. Chem. 1, 107-122.]) and the atomic ordering, including the H-atom ordering, modified and any dummy atoms added – in this case at the centre of the phenyl ring.

(2) The CIF was imported into Mercury (Mercury, 2015[CSD (2015). Mercury, http://www.ccdc.cam.ac.uk/products/mercury/.]) and the unit cell was packed out. The resulting structure was exported as a .mol2 file

(3) The .mol2 file was edited to control the connectivity.

(4) zmat_maker was used to produce the z-matrix.

The final z-matrix for TPA [based on atomic coordinates from the polymorph (II) structure] is shown in Table 1[link].

Table 1
The z-matrix for TPA, C8H6O4, as used in the analysis

The atom prefixed `x' is a dummy atom used to aid in defining the local coordinate system. Bold dihedral angles were allowed to vary to allow the —COOH groups to rotate. All other values were fixed during the modelling. The numbers in the first column correspond to those in Fig. 4[link].

p Label l Distance from l (Å) m Angle with lm (°) n Dihedral angle of plm with lmn (°)
1 x1 0 0.000 0 0.000 0 0.000
2 C1 1 1.383 0 0.000 0 0.000
3 C2 2 1.391 1 60.222 0 0.000
4 C3a 3 1.388 2 119.798 1 0.006
5 C1a 4 1.390 3 119.633 1 −0.006
6 C2a 5 1.391 4 120.569 1 0.006
7 C3 6 1.388 5 119.798 1 −0.006
8 C4 2 1.483 7 119.892 3 −179.920
9 O1 8 1.252 2 119.227 7 177.394
10 O2 8 1.280 9 123.576 2 179.873
11 C4a 5 1.483 6 119.539 4 −179.920
12 O2a 11 1.280 5 117.197 6 −177.592
13 O1a 11 1.252 12 123.576 5 179.875
14 H2A 12 0.930 11 112.747 13 1.411
15 H4A 4 0.941 5 120.582 3 179.899
16 H3A 6 0.971 7 118.546 5 177.872
17 H2 10 0.930 8 112.747 9 −1.411
18 H3 3 0.971 4 118.546 2 −177.872
19 H4 7 0.941 6 119.785 2 180.000

4.2. The model crystal

A model crystal of size 48×48×48 unit cells was constructed. Each unit cell contained a single TPA molecule. Each molecule was described by the z-matrix plus a vector of variables, [{\bf v}]. [{\bf v}] contained three sub-vectors: a vector [{\bf x}] which gave the coordinates of the molecular origin (within the cell); [{\bf q}], which was a quaternion (a normalized four component vector) that gave the molecular orientation; and [{\bf i}] which contained the values of the internal degrees of freedom (the two φ values illustrated in Fig. 4[link]).

[Figure 4]
Figure 4
A molecule of TPA, C8H6O4, showing the connectivity and atom numbering used in the z-matrix in Table 1[link]. The two internal degrees of freedom are denoted [\phi _{i}] and the dummy atom at the centre of symmetry is number 1 (`x1'). The dotted line shows an example of an intramolecular non-bonded contact.

In conventional crystallography, the model is a single unit cell, or even asymmetric unit. The assumption is that if the scattering calculated from this model agrees with that observed in the real crystal, then the arrangement of atoms in the model reflects that in the real crystal, and can be explored to improve understanding of the real material. The model discussed here is exactly the same, except that to capture local ordering it must contain multiple unit cells. The model, once established, can be explored by looking at correlations amongst its constituents, by looking at what forms of disorder had to be imposed on it to give a calculated pattern that matches the observations, and so on.

4.3. Interactions

Interactions between non-bonded atoms were modelled using Hooke's law springs

[E_{{\rm{inter}}} = \sum _{{\rm{cv}}}F_{{i}}(d_{{i}}-d_{{0i}})^{2}, \eqno(1)]

where di is the length of vector i connecting the interacting atoms, d0i is its equilibrium length and Fi is its force constant. The sum is over all contact vectors (cv).

The Fi were determined using an empirical rule (Chan, Welberry, Goossens & Heerdegen, 2010b[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]; Hudspeth et al., 2014[Hudspeth, J. M., Goossens, D. J. & Welberry, T. R. (2014). J. Appl. Cryst. 47, 544-551.])

[F_{i} = A\exp\left[{-B}\left(d_{{0i}}-R_{i}\right)\right]+C \eqno(2)]

where i indexes the contact vector and Ri is the sum of the van der Waals radii for the two contacting atoms (Bondi, 1964[Bondi, A. (1964). J. Phys. Chem. 68, 441-451.]). A, B and C are constants (A = 11.00, B = -0.40, C = 8.00; Chan, Welberry, Goossens & Heerdegen, 2010[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]). These values have been used as starting values for a number of studies, and found to be useful for several different organic molecular crystals, particularly those in which the bonding is not dominated by very strong hydrogen bonding. The —COOH—COOH— bonds in TPA need to be treated separately (Chan, Welberry, Goossens & Heerdegen, 2010[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]; Hudspeth et al., 2014[Hudspeth, J. M., Goossens, D. J. & Welberry, T. R. (2014). J. Appl. Cryst. 47, 544-551.]; Goossens, 2015[Goossens, D. J. (2015). Adv. Condens. Matter Phys. 2015, 1-7.]; Chan & Goossens, 2012[Chan, E. J. & Goossens, D. J. (2012). Acta Cryst. B68, 80-88.]). Hence, Fi for the —COOH—COOH— bonds were typically an order of magnitude larger than for the other interactions of similar length  (Martsinovich & Troisi, 2010[Martsinovich, N. & Troisi, A. (2010). J. Phys. Chem. C, 114, 4376-4388.]).

The φ angles in Fig. 4[link] indicate internal, torsional degrees of freedom that can be added to the model. These torsional degrees of freedom were modelled using torsional potentials

[\eqalignno{E_{{\rm{torsion}}} =\, & G_{{1}}\left(\phi _{{1}}-\phi _{{01}}\right)^{2}+G_{{2}}\left(\phi _{{2}}-\phi _{{02}}\right)^{2}\cr & +G_{{12}}\left(\phi _{{1}}-\phi _{{01}}\right)\left(\phi _{{2}}-\phi _{{02}}\right), &(3)}]

where [\phi _{{i}}] is the current value of the angle, [\phi _{{0i}}] is the equilibrium value, G1 = G2 are the force constants and G12 is a force constant for a cross term that allows for interaction between the degrees of freedom, but in these studies, where the rotating groups are separated by an entire phenyl ring, this term was set to zero.

In determining the empirical form for Fi, the same workers determined a useful generic value for Gi (Chan, Welberry, Goossens & Heerdegen, 2010[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]), as used in equation (3)[link] ([G_{i}\sim 20] rad−2).

A single MC step consisted of randomly choosing a molecule, calculating its energy using equations (1)[link] and (3)[link], then randomly altering [{\bf v}] and recalculating the energy. Reduction in energy caused the new configuration to be accepted; changes that increased energy were accepted or rejected according to a probability that depended on the temperature parameter, T. In each simulation, the isotropic atomic displacement parameter, averaged over all atoms (Bisoave), was set to be 4.8 Å2 for both polymorphs (Bailey & Brown, 1967[Bailey, M. & Brown, C. J. (1967). Acta Cryst. 22, 387-391.]). This served to globally scale the interactions (their values relative to each other were maintained).

This approach is geared toward inducing correlations between scatterers to allow for calculation of diffuse scattering. Compared to molecular dynamics (Chan, 2015[Chan, E. J. (2015). J. Appl. Cryst. 48, 1420-1428.]) or phonon dispersion calculations, it has the disadvantage that MC does not give a real time scale to explore the evolution of the system in times that can be compared with experiment. The main benefit of this approach is its flexibility. The forms of order and disorder that have been modelled include static disorder both in displacements and occupancies, correlated thermal motion which is rather like a phonon but of limited range in real space (and therefore diffuse in reciprocal space) and disorder in which a molecule shows different orientations. Systems can be quite large – hundreds of atoms per unit cell – because the z-matrix description allows a large molecule to be well described by just the centre of mass, a quaternion and a z-matrix with a few carefully chosen degrees of freedom – positioning N atoms (where [N\geq 30]) does not need 3N variables, but perhaps only ten or a dozen.

5. Results

5.1. Polymorph triclinic (II)

Various models were explored and their diffraction patterns evaluated qualitatively against the observations. Due to the multiple small crystalites apparent in Fig. 2[link](c), quantitative analysis proved troublesome. However, this does not greatly affect the ability to draw useful conclusions.

Inspection of the data suggested some conclusions which are discussed in Appendix A[link]. The TPA molecules form chains that run through the crystal along [[\bar{1}11]]. The chains act like one-dimensional crystals embedded in the three-dimensional crystal. A one-dimensional crystal will give rise to two-dimensional sheets of scattering perpendicular to the chain direction. These features are arrowed in Fig. 2[link]. Since there is no chemical disorder, the overall locus of intensity is modulated by the molecular structure factor and the correlation structure of the molecular displacements.

These diffuse planes were found to arise out of the model as long as the —COOH—COOH— bonds were significantly stronger than the remainder; a factor of ten was sufficient, and is in reasonable accordance with the energy scales of the different bond types.

The φ angles in Fig. 4[link] indicate internal, torsional degrees of freedom that were added to the model. Rotations were hindered in four different ways: (1) The molecules were held completely rigid (same as simply not adding the torsional degrees of freedom); (2) a torsional force constant, Gi was applied as discussed in §4.3[link]; (3) intramolecular contact vectors similar in length to the intermolecular vectors were added, with force constants calculated the same way [equation (2)[link]], and the Gi were set to zero (an example of such a contact vector is illustrated in Fig. 4[link]); (4) torsional force constants and intramolecular contacts were used.

In brief, it was found that none of the models in which the —COOH groups were allowed to rotate were any better than the rigid molecule model, and in the case where the groups were allowed free rotation (Gi = 0), the agreement was noticeably worse. Hence, it was concluded (for both polymorphs) that while it might initially seem reasonable to allow this form of motion, there was no valid reason to add more degrees of freedom to the model. Indeed, this is backed up by an examination of the atomic displacement parameters for the atoms in polymorph (II) (Bailey & Brown, 1967[Bailey, M. & Brown, C. J. (1967). Acta Cryst. 22, 387-391.]), which shows that the ADPs for the O atoms in the —COOH groups are not notably larger than for the C atoms in the acid group (or indeed, in the molecule as a whole), which suggests that molecular librations and acid group rotations are not as important to determining the ADPs as the overall molecular displacements in x, y and z.

Fig. 5[link] shows the SCDS from the three basal plans of the final model for polymorph (II). The agreement is sound. All the streaks (really sections through diffuse planes) noted in Fig. 2[link] are well reproduced, and the large blobs of thermal scattering are equally reasonable. Although the circled spot in the h0l cut is more circular than is desirable, the angularity of the shapes in 0kl is well reproduced. Note that the cuts are not of exactly the same reciprocal space extent as those in the observed data, and cover slightly less reciprocal space.

[Figure 5]
Figure 5
The basal plane cuts from the final model for polymorph (II), which uses the Fi from an empirical formula (Chan, Welberry, Goossens & Heerdegen, 2010[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]), a rigid molecule, and an overall average B-factor (non-H atoms) of 3.3 Å2. (a) 0kl, (b) h0l, (c) hk0.

5.2. Polymorph triclinic (I)

The data for polymorph (I) are of lower quality, but it was nevertheless possible to implement a model, using the same approach as for polymorph (II). Force constants were established using the same methods, with the model different only in that the crystal geometry is different.

The hydrogen bonds that were set to have strong interaction constants are illustrated in Fig. 6[link], which shows the ab plane of polymorph (I); the chains running along [{\bf a}+{\bf b}] are very apparent.

[Figure 6]
Figure 6
The chains of hydrogen bonds (thick black lines) running along [{\bf a}+{\bf b}] in polymorph (I) of TPA. Dummy atoms, added to simplify simulation but not included in the Fourier transform, are shown (in green) in the centres of the phenyl rings.

Fig. 7[link] shows the SCDS from the three basal plans of the final model for polymorph (I). The agreement with the observed data in Fig. 3[link] is reasonable but not as close as for polymorph (II). In the observed data some features are quite rounded while others are stretched out along the streak direction [this is particularly noticeable in the features in the inner box (`A') in Fig. 7[link]c]; in the calculated pattern the features are all similar in shape where in the observed pattern one spot is stretched out along the streak direction compared with the others.

[Figure 7]
Figure 7
The basal plane cuts from the harmonic springs model for polymorph (I), which uses the Fi from an empirical formula (Chan, Welberry, Goossens & Heerdegen, 2010[Chan, E. J., Welberry, T. R., Goossens, D. J. & Heerdegen, A. P. (2010). J. Appl. Cryst. 43, 913-915.]), a rigid molecule, and an overall average B-factor (non-H atoms) of 3.3 Å2. (a) 0kl, (b) h0l, (c) hk0.

Calculating the diffuse scattering using only a subset of the atoms in the molecule (for example, only the atoms lying on the molecular long axis, or only the —COOH groups, or only the phenyl ring) shows that the greatest contribution to the intensity of the rightmost spot in box A comes from the O atoms, suggesting that the different aspect ratio of this spot compared with the others is a result of some correlations/motions of the —COOH groups.

Based on this, numerous models allowing —COOH rotation were explored. However, the agreement with experiment was not improved by allowing torsional motions, or by any other harmonic motion of the molecule or its parts. It is therefore suggested, following previous work (Chan et al., 2009[Chan, E. J., Welberry, T. R., Goossens, D. J., Heerdegen, A. P., Beasley, A. G. & Chupas, P. J. (2009). Acta Cryst. B65, 382-392.]; Chan & Welberry, 2010[Chan, E. J. & Welberry, T. R. (2010). Acta Cryst. B66, 260-270.]; Chan, Welberry, Heerdegen & Goossens, 2010[Chan, E. J., Welberry, T. R., Heerdegen, A. P. & Goossens, D. J. (2010). Acta Cryst. B66, 696-707. ]; Chan, 2015[Chan, E. J. (2015). J. Appl. Cryst. 48, 1420-1428.]), that this disagreement may be an indicator of anharmonicity, or at least of more complex dynamics in polymorph (I) than (II), associated with it being less stable under the experimental conditions. The other layers show no such effects, leading to the suggestion that the relevant motion lies in the ab plane, but is not along [{\bf a}] or [{\bf b}]. The key directions in the ab plane are (1) along the chain directions and (2) perpendicular to the chains, where the —COOH group H atoms are closest to the phenyl H atoms on the molecules in adjacent chains (see Fig. 6[link]), so this is chemically reasonable.

It can be concluded that the diffuse scattering in TPA originates from correlations between displacements and to a lesser extent the global orientation of nearby molecules. Predominantly, these consist of longitudinal correlations, which is to say that the correlation is strong in the direction of the displacement, which is a mechanical effect of molecules shunting each other along in chains. X-rays cannot distinguish static from dynamic. The diffuse scattering is calculated from a static array of atoms, and cannot determine whether these are frozen or not. Only an energy-sensitive technique (like a spectroscopy or neutron diffraction where the energies of the neutrons can be analysed; Welberry et al., 2003[Welberry, T. R., Goossens, D. J., David, W. I. F., Gutmann, M. J., Bull, M. J. & Heerdegen, A. P. (2003). J. Appl. Cryst. 36, 1440-1447.]) can separate dynamic from static. In a room-temperature experiment on an organic material like TPA, where the melting temperature is not very much larger than 273 K, it is not unreasonable to consider the system as dynamic unless shown otherwise. A more comprehensive, modelling method, like molecular dynamics with realistic potentials, or DFT, may give energetics that can then be compared to experiment, but this is difficult on a system of sufficient size to capture the local order.

5.3. Analysis of the models

Fig. 8[link] shows one way of representing some of the conclusions from the modelling. It shows the displacive correlations amongst molecules in the molecular chains, as a function of the direction of the displacement component being considered. The correlations are strongest when considering displacement components parallel with the chain directions (up to around 0.6), which is not remarkable. Considering the molecules in adjacent chains, no displacive correlations larger than 0.4 occur for either polymorph, showing that the chain interactions are dominant. Correlations are similar between a chain and its nearest and next-nearest chains, showing that there is no tendency for the structure to break up into sheets and then chains within sheets, but that it really can be considered as an array of chains, where the intra-chain correlations are relatively isotropic in the two directions perpendicular to the chain direction.

[Figure 8]
Figure 8
Displacive correlations along the molecular chains in TPA. (a) Plots of the correlations amongst components of displacements in the ab plane of polymorph (I), in a polar plot in which the angle, θ, gives the direction of the displacement component to the a axis and the radius gives the strength of the correlation, so both axes give correlation magnitudes. This plot shows that correlations are strong in the [{\bf a}+{\bf b}] direction, and weak transverse to this. (c) Shows a schematic picture of real space for the polymorph, showing the direction of the chains, with the [{\bf a}] and [{\bf b}] axes oriented the same as in (a). (b) and (d) give the same results for polymorph (II); the magnitudes are similar, although the shapes slightly different.

This is reinforced by looking at the correlation diagram analogous to that in Fig. 8[link](a) but for next-nearest neighbours (nearest non-chain neighbour). Such a diagram is shown in Fig.  9[link], and while the correlations are significantly different from zero, they are considerably smaller than the largest values within the chains.

[Figure 9]
Figure 9
Displacive correlations along the molecular chains in TPA, showing the correlations amongst components of displacements in the ab plane of polymorph (I), in a polar plot in which the angle, θ, gives the direction of the displacement component to the a axis and the radius gives the strength of the correlation. The view is the same as in Fig. 8[link](a), but the molecules in question are second nearest-neighbours, and the correlations are weak.

For polymorphs (I) and (II), the x, y and z coordinates of molecules adjacent along the chains are, as may be expected, strongly correlated ([\gtrsim] 0.4), but there is virtually no correlation amongst quaternion components on adjacent molecules ([\lesssim 0.1]). In fact, the molecular orientations vary relatively little – hence the relatively small extra components in the ADPs of the outer atoms; were the molecules librating strongly, the outer atoms would show much larger ADPs, and they do not.

6. Conclusions

The key features in the single-crystal diffuse scattering from TPA triclinic polymorphs (I) and (II) can be modelled using a set of harmonic springs to model intermolecular interactions, although the model is less satisfactory for polymorph (I). This is in accordance with the fact that, although TPA is polymorphic, at the measurement temperature 295 K, and polymorphs are stable, polymorph (I) will transform into (II) with heating. This implies there is an anharmonicity that increases with T until significant enough to trigger the phase transition, but is present in some small degree at 295 K.

Polymorphic benzocaine [polymorph (II)] and aspirin [polymorph (II)] both showed non-harmonic aspects to the scattering due to an incipient phase transition and to correlated defects, respectively (Chan, 2015[Chan, E. J. (2015). J. Appl. Cryst. 48, 1420-1428.]; Chan, Welberry, Heerdegen & Goossens, 2010[Chan, E. J., Welberry, T. R., Heerdegen, A. P. & Goossens, D. J. (2010). Acta Cryst. B66, 696-707. ]; Chan et al., 2009[Chan, E. J., Welberry, T. R., Goossens, D. J., Heerdegen, A. P., Beasley, A. G. & Chupas, P. J. (2009). Acta Cryst. B65, 382-392.]; Chan & Welberry, 2010[Chan, E. J. & Welberry, T. R. (2010). Acta Cryst. B66, 260-270.]), neither of these mechanisms is apparent in either triclinic polymorph of TPA at room temperature. In this it is similar to paracetamol, which also showed scattering that could be modelled harmonically (Chan & Goossens, 2012[Chan, E. J. & Goossens, D. J. (2012). Acta Cryst. B68, 80-88.]). Because we can model harmonic effects readily, we can use diffuse scattering to gauge when the anharmonic effects become significant, giving insight into the evolution and onset of the phase transition.

TPA undergoes a polymorphic interconversion transition at ∼ 348 K, so the work here lays some groundwork for a qualitative study of the SCDS as a function of temperature as the phase transition temperature is approached. Such studies can be undertaken now that the modelling of the thermal diffuse is relatively routine. This is a significant step forward in the use of SCDS to explore phase transitions, and in parametric studies in general.

As expected, it is found that strong hydrogen bonding between —COOH groups cause the molecules to form long chains, and these are relatively weakly correlated laterally. The proton disorder on the —COOH is known to be strong at room temperature (Meier & Ernst, 1986[Meier, B. & Ernst, R. (1986). J. Solid State Chem. 61, 126-129.]), but the data used here, being measured using X-rays, are not directly sensitive to it.

APPENDIX A

Diffuse planes and chains of molecules

A molecular crystal will often show chains of molecules running through the crystal, where interactions within the chains are much stronger than those without. Thus, the motions of the molecules along the direction of the chain (longitudinal motions) are highly correlated, causing the chain to act as a one-dimensional crystal embedded in the three-dimensional crystal. The real space direction of a chain can be expressed as a vector, [{\bf p}]

[{\bf p} = p_{a}{\bf a}+p_{b}{\bf b}+p_{c}{\bf c}. \eqno(4)]

The scattering from the chain will be distributed in planes perpendicular to [{\bf p}]. A plane can be specified by giving two non-collinear vectors that lie in it, say [{\bf q}] and [{\bf r}] such that

[{\bf p}\cdot{\bf q} = {\bf p}\cdot{\bf r} = 0 \eqno(5)]

where

[{\bf q} = q_{{a^{\ast}}}{\bf a^{\ast}}+q_{{b^{\ast}}}{\bf b^{\ast}}+q_{{c^{\ast}}}{\bf c^{\ast}}. \eqno(6)]

Recalling the definitions of [{\bf a^{\ast}}], [{\bf b^{\ast}}] and [{\bf c^{\ast}}] and ignoring factors of [2\pi]

[{\bf q} = q_{{a^{\ast}}}({\bf b}\times{\bf c})+q_{{b^{\ast}}}({\bf c}\times{\bf a})+q_{{c^{\ast}}}({\bf a}\times{\bf b}). \eqno(7)]

To satisfy equation (5)[link] it is possible to arbitrarily set one of [q_{{a^{\ast}}}], [q_{{b^{\ast}}}] or [q_{{c^{\ast}}}] to zero, giving three solutions, [{\bf q}], [{\bf r}] and [{\bf s}], most conveniently expressed as

[{\bf q} = p_{b}{\bf a^{\ast}}-p_{a}{\bf b^{\ast}}\,\,{\rm and}\,\,{\bf r} = p_{c}{\bf a^{\ast}}-p_{a}{\bf c^{\ast}}\,\,{\rm and}\,\,{\bf s} = p_{c}{\bf b^{\ast}}-p_{b}{\bf c^{\ast}}.\eqno(8)]

In terephthalic acid (TPA), triclinic polymorph (II), each molecule has a —COOH group at each end. These H atoms bond strongly with the groups on neighbouring molecules giving correlated chains of molecules running along the [[\bar{1}11]] (direct space) direction. This then suggests that the planes of scattering perpendicular to these chains will extend in the directions

[{\bf q} = {\bf a^{\ast}}+{\bf b^{\ast}}\,\,{\rm and}\,\,{\bf r} = {\bf a^{\ast}}+{\bf c^{\ast}}\,\,{\rm and}\,\,{\bf s} = {\bf b^{\ast}}-{\bf c^{\ast}} \eqno(9)]

or

[{\bf q} = (110)\,\,{\rm and}\,\,{\bf r} = (101)\,\,{\rm and}\,\,{\bf s} = (01\bar{1}). \eqno(10)]

Fig. 2[link] shows the reciprocal axes and cuts through diffuse planes. These agree with the calculation above. This is a relatively trivial result, but it means that these features can be ascribed to correlations in the displacements of the TPA molecules linked by the —COOH groups.

For triclinic polymorph (I), the chains run along [110], so the diffuse plane sections (which appear as streaks) run along [(1\bar{1}0)] and [(00\bar{1})] (see Fig. 3[link]).

Acknowledgements

Thanks to Dr A. P. Heerdegen, Professor T. R. Welberry, Dr Marek Paściak, Dr Jessica Hudspeth and Dr Ross Whitfield for their advice and assistance, but they take no responsibilities for any inaccuracies or opinions expressed herein. Use of the Advanced Photon Source, an Office of Science User Facility operated for the US Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the US DOE under Contract No. DE-AC02-06CH11357. Some aspects of this research were undertaken on the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government.

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