The phase transitions of 4-aminopyridine-based indolocarbazoles: twinning, local- and pseudo-symmetry

The phase transition behaviour and twinning of 4-aminopyridine-based indolocarbazoles are analyzed using the order–disorder theory and group–subgroup relationships.


Introduction
Symmetry relationships are crucial in understanding and describing phase transitions (Mü ller, 2013). In most cases of displacive phase transition (Tolé danoc et al., 2006), the symmetry of a high-temperature (HT) phase is a strict super group of the symmetry of the low-temperature (LT) phase (disregarding minor variations of cell parameters). Nevertheless, exceptions exist. For example, numerous incommensurate phases feature a lock-in phase transition to a periodic (and therefore higher-symmetry) LT structure on cooling (Cummins, 1990). In such a case, both phases are derived from a higher-symmetry prototype structure, which may exist at high temperatures or may be purely hypothetical.
Reconstructive phase transitions are generally not restricted by group-subgroup relationships because, as the name implies, a significant rearrangement of atoms or molecules takes place. There are intermediate cases of symmetry transformations, where modules (layers, rods) are preserved but are arranged differently. In such a case, an interpretation using local symmetry can be insightful.
In this context, we present the structural phase transitions of three 4-aminopyridine derivatives of indolo[3,2,1-jk]carbazole (ICz), whereby C atoms para to the N atom of ICz are replaced by an N atom. The IUPAC atom numberingscheme is given in Fig. 1(a). The molecules under investiga-tion, 5NICz, 2NICz and 2,5NICz feature substitution of C atoms by N at the respective positions ].
Crystals of 5NICz and 2NICz exist in distinct HT and LT polymorphs, which interconvert below room temperature. 2,5NICz exists in the solid state as three polymorphs. The bulk 2,5NICz-1 crystallizes in a structure unrelated to 5NICz and 2NICz. We could not find any evidence of a phase transition in the solid state for this polymorph. While attempting to obtain improved diffraction data, we found isolated crystals of a different polymorph, which is isostructural to 2NICz. These crystals featured an analogous phase transition (polymorphs designated 2,5NICz-2LT and 2,5NICz-2HT), though with a transition temperature above room temperature.
The observed phase transitions are analyzed with respect to symmetry relationships. Whereas the symmetries of 2NICz-LT and 2NICz-HT polymorphs (and the 2,5NICz-2LT and 2,5NICz-2HT polymorphs) can be described using classical group-subgroup relationships, the local symmetry has to be considered for 5NICz. For this purpose, we use the formalism developed in the framework of order-disorder (OD) theory (Dornberger-Schiff & Grell-Niemann, 1961;Ferraris et al., 2008). Despite being of the same name, this theory of polytypism is not related to order-disorder phase transitions. A summary of the phase transitions and the structural relationships between the seven polymorphs is schematized in Fig. 2.

Data collection and refinement
2.2.1. General. Intensity data were collected in a dry stream of nitrogen on a Bruker Kappa APEX II diffractometer system using graphite-monochromated Mo K radiation. Data were reduced to intensity values using SAINT (Bruker, 2017). Corrections for absorption and related effects were applied using SADABS (Bruker, 2017). The structures were solved with SHELXT (Sheldrick, 2015a) and refined with SHELXL (Sheldrick, 2015b). The atoms were labelled according to IUPAC rules (Fig. 1a). In the case of two crystallographically different molecules (Z 0 = 2), prime characters are added for the second molecule. For molecules located on twofold axes, atoms pairs that are equivalent by symmetry are assigned the lower out of the two possible numbers. More data collection  Schematics of (a) ICz with IUPAC numbering scheme and (b-d) the 4aminopyridine derivatives 5NICz, 2NICz and 2,5NICz.

Figure 2
Phase transition temperatures (red arrows) and structural relationships (black arrows) relating the seven polymorphs described in this work. The meaning of 'same OD groupoid family' is explained at the end of x3.1.1. Polymorphs not connected by arrows are structurally unrelated. and refinement details are summarized in Tables 1 and 2, and described in the following sections. 2.2.2. Details for 5NICz. Crystals of 5NICz were small, yet of reasonable quality according to optical microscopy. Nevertheless, in preliminary scans at the routine temperature of 150 K, all plates featured mediocre reflection quality and diffracted only to small 2 angles. Such a bad diffraction quality for optically flawless crystals can be a sign of a reconstructive phase transition on cooling. Indeed, crystals cooled to 150 K showed clear signs of fracturing. Two data sets were, therefore, collected with long exposure times, one above the phase transition temperature at 270 K and one after slow cooling to 150 K. To our surprise, even at 270 K the reflection quality was not significantly improved.
For the 150 K data set a reasonable structure solution and refinement, considering the mediocre diffraction quality, was possible in the space group Pca2 1 .
The 270 K phase had apparent orthorhombic C-centred (oC) metrics. But, since a sensible structure solution was not possible in this setting and slight splitting of reflections indicated a lower metric symmetry, the data were reprocessed in the corresponding monoclinic primitive (mP) setting. Structure solutions and refinements were performed in the space group P2 1 =a under consideration of twinning by pseudomerohedry. The non-standard setting of the space group P2 1 =c was chosen to ease comparison with the LT polymorph.
The cell parameters of the LT-polymorph of 2NICz were apparently orthorhombic primitive (oP) and, therefore, data were at first processed assuming such a symmetry. A structure solution was successful in the space group Pccn. But all attempts at refinements resulted in excessively anisotropic atomic displacement parameters (ADPs) and mediocre residuals. Since, in analogy to 5NICz, reflections at higher diffraction angles were split, an attempt was made in the P2 1 =c space group under consideration of twinning by pseudomerohedry. The ADPs as well as the residuals improved significantly (R obs > 10% to $5.5%). For the HT phase, on the other hand, a refinement using Pccn symmetry was successful. In this case, reducing the symmetry to monoclinic did not improve reliability factors.
The cell parameters of 2,5NICz-2 L T suggested a structure isostructural to 2NICz. Refinements were, therefore, performed using starting models derived from the 2NICz model. Even at 300 K, refinements in the LT P2 1 =c model resulted in significantly improved residuals (R obs > 10% to $5.3%), even though the metrics are orthorhombic within the estimated standard errors. Only when heating to 380 K was the Pccn HT phase clearly observed. In 2,5NICz-2HT, the molecules are located on a twofold axis and the N5 atom is accordingly positionally disordered with the C11 atom in a 1:1 manner. In 2,5NICz-2LT, this position splits in two and both positions were refined as positionally disordered, by constraining the sum of the N-occupancies of both positions to 1. Ultimately, the N-occupancy of one position refined 0.58 (4) (the other accordingly being constrained to 0.42).
2.2.4. Details for 2,5NICz-1. The structure of 2,5NICz-1 was determined by routine refinement. The 2,5NICz molecule is located on a twofold axis and, therefore, the N5 and C11 atoms are positionally disordered in a 1:1 manner.

X-ray powder diffraction
Low-temperature X-ray powder diffraction (XRPD) experiments were performed on a Panalytical X'Pert Pro diffractometer equipped with an Oxford Cryosystems PheniX closed cycle cryostat in Bragg-Brentano geometry using Cu K 1,2 radiation ( = 1.540598, 1.544426 Å ) with an Ni filter and an X'celerator multi-channel detector. The ground bulk sample was placed on an Si single crystal cut along the ð711Þ plane. Scans were recorded in vacuum in the 2 ¼ 10-70 range in 10 K steps from 300 K to 100 K and back to 300 K with heating and cooling rates of 1 K min À1 and 5 min isothermals between scans.

Results and discussion
3.1. The OD polytypism of 5NICz 3.1.1. Local symmetry. The HT and LT phases of 5NICz are structurally closely related. They crystallize in the P12 1 =a1 and Pca2 1 symmetry, respectively and contain Z 0 = 2 5NICz molecules in the asymmetric unit. The structures can be considered as being composed of A n layers (n is a sequential integer) extending parallel to (001) (Fig. 3). These layers are made up of rods of molecules which connect via short C-HÁ Á ÁN contacts (Fig. 4). Whereas these rods are very similar in both structures (differences will be discussed below), their inclination with respect to the layer plane (001) differs significantly (Fig. 3). The angles of the least-squares planes of the molecules to the (001) plane are 67.5 and 67.6 versus 55.7 and 56.5 for the LT and HT phases, respectively. Thus, the two kinds of layers can be derived from each other, but they might not be considered as isostructural (Ká lmá n et al., 1993) in the strict sense.  (Bruker, 2017), SHELXL2014/7 (Sheldrick, 2015b). † Not determined owing to a lack of significant resonant scatterers.
Adjacent molecules in the rods described above are related by a 2 1 operation in the [010] direction. The operation is exact in the HT phase (one crystallographically unique molecule per rod) but only approximate in the LT phase (two molecules per rod). Adjacent rods are, in both polymorphs, related by an a operation in the [010] direction. Moreover, they are related by inversions, which is a space group operation in the HT and a local operation in the LT phase. Thus, the layers possess P12 1 =að1Þ actual (HT) or pseudo (LT) symmetry. Since we will perform an interpretation according to the OD theory, here we use the OD notation of layer symmetry, whereby parentheses indicate the direction lacking translational symmetry (Dornberger-Schiff & Grell-Niemann, 1961). In the LT polymorph, adjacent A n layers are related by actual 2 1 screw rotations in [001] and c glide reflections in [100] direction, whereas in the HT polymorph, these are only a pseudo-symmetry operations. In total, both polymorphs are made up of Z 0 = 2 crystallographically different molecules.
Recognizing the pseudo-symmetry of layers is the key to an OD interpretation. By assuming the pseudo-symmetry to be exact, both polymorphs can be described as members of OD families. The symmetries of OD families are classified into OD groupoid families, which correspond to space group types in   classical crystallography (Fichtner, 1979a). The symmetries of both polymorphs belong to the same OD groupoid family, which is described by P 1 2 1 =a ð1Þ f 2 rÀ1 =n sÀ1;2 À ð2 2 =n r;s Þ g according to the notation of Dornberger-Schiff & Grell-Niemann (1961). The metric parameter s adopts the value s = 1 in both cases, which can be expressed by P 1 2 1 =a ð1Þ f 2 rÀ1 =c 2 À ð2 2 =n r;1 Þ g : OD groupoids are made up of partial operations (POs), which relate layers but need not apply to the whole stacking sequence. The first line in these symbols gives the symmetry group of the layers [the -POs, here P12 1 =að1Þ]. The second line lists one possible set of operations relating adjacent layers (-POs). Since the relative intrinsic translations of the -POs are not restricted to those found in space groups, generalized Hermann-Mauguin symbols are used. For example, the n r;s glide reflection in the symbol above has the glide vector ra=2 þ sb=2. As can be seen in Fig. 3, the x-component of the glide vector is approximately 1 4 and thus r % 1 2 (x3.1.5). Intrinsic translation components in the stacking direction ½001 are given with respect to the vector c 0 , which is perpendicular to the layers and of the length of one layer width. Thus, the 2 2 operation in [001] direction has the screw vector c 0 (Fichtner, 1979b) since n m stands for an n-fold screw rotation with intrinsic translation of m n parts of the shortest lattice vector in the translation direction.
3.1.2. Stacking possibilities. The crucial aspect of OD structures is their ability of crystallizing in different polytypes, which are all locally equivalent (more precisely: pairs of adjacent layers are equivalent). If interactions beyond one layer width and deviations from the prototype layers are neglected, all polytypes can therefore be considered as energetically equivalent. The NFZ relationship (Ď urovič, 1997) is used to derive these stacking possibilities. For 5NICz, there are -POs that invert the orientation with respect to the stacking direction (--POs). But owing to r = 2 Z none of these is a reverse continuation, which would mean that it maps A n on A nþ1 and vice versa. In such a case, the NFZ relationship reads as Z ¼ 2N=F ¼ 2½G n : G n \ G nþ1 , where Z is the number of positions A nþ1 can adopt given A n and G n is the group of those A n operations that do not invert the orientation with respect to the stacking direction (--POs).
Since s = 1, the a glide planes of all A n overlap and G n ¼ G n \ G nþ1 ¼ P1að1Þ.
Accordingly, there are Z ¼ 2½P1að1Þ : P1að1Þ ¼ 2 ways of placing A nþ1 given A n . These two possibilities are obtained by applying a 2 rÀ1 or a 2 1Àr -PO on A n , respectively.
In our experience, the overwhelming number of polytypes characterized by single-crystal diffraction is of the MDO kind. Other stacking arrangements may exist at domain interfaces. Indeed, the HT and LT polymorphs of 5NICz are precisely of the MDO 1 and MDO 2 type, respectively. Thus, even though the space groups of the two phases are not related by a groupsubgroup relationship, their groupoids belong to the same groupoid family with the same restrictions on the metric parameters, viz. s = 1. Their local symmetries are therefore, in a sense, isomorphic, which demonstrates the usefulness of such a symmetry description.  linear parts of all POs of a polytype. This group is mmm for the OD groupoid family of the 5NICz polymorphs.
Thus, MDO 1 (HT) can appear in [mmm:12/m1] = 2 orientations, which are related by the operations of the twin law f2 x ; m x ; 2 z ; m z g. This corresponds precisely to the observed twinning. MDO 2 (LT) can appear likewise in [mmm:mm2] = 2 orientations. In this case, the twin law is f1; 2 x ; 2 y ; m z g. Since the 5NICz molecules possess no significant resonant scatterers under Mo K radiation, this twinning by inversion could not be seen from the diffraction data. Its existence is nevertheless nearly certain. Besides being predicted by OD theory, it is also expected owing to the phase transition from the centrosymmetric MDO 1 (HT) phase. Point operations lost on phase transformation are typically retained as twin operations.
3.1.5. Desymmetrization and metric parameters. An important step in assessing an OD model is the quantification of the desymmetrization (Ď urovič, 1979) compared to the ideal model. Such a desymmetrization is expected (these geometrical differences may stabilize the individual polytypes) but should not be unreasonably large.
In the MDO 1 (HT) polytype, the symmetry of the actual A n layers is identical to those of the idealized description [P12 1 =að1Þ]. According to the P2 1 =a symmetry of the polytypes, the layers are partitioned into two equivalence classes, viz. the A 2n and the A 2nþ1 layers. To evaluate the desymmetrization, the A 1 layer was mapped onto the A 0 layer by translation of Àc=2 and reflection at the rÁa = 0 plane. The discrepancies between both layers are minute (max: C2/C2 0 , 0.157 Å ), proving the validity of the pseudo-symmetry analysis.
In the MDO 2 (LT) polytype all layers are related by the Pca2 1 space group symmetry, but the symmetry of the layers is reduced by an index of 2 to P1að1Þ. To assess the degree of desymmetrization, the location of the pseudo-2 1 screw axis was determined by averaging the x-and z-coordinates of the non-H atoms of the two crystallographically independent molecules. The screw rotation was then applied to a layer. Here, the desymmetrization is even less pronounced than in the HT phase (max: C11/C11 0 , 0.086 Å ).
The metric parameter r of the OD groupoids can be derived in the case of MDO 1 (HT) directly from the cell parameters as r ¼ c cos =a þ 1 ¼ 0:505. Owing to r % 1=2, the lattice symmetry of MDO 1 is pseudo-oC and the twinning is by pseudo-merohedry (the reflections of both domains are nearly coincident). More precisely, the twin obliquity calculates from the cell parameters as ! ¼ tan À1 ½aðr À 1 2 Þ=c= sin % 0:1 . It has to be noted though that the derivation of the cell parameters from single-crystal data is inexact in such a case because overlapping reflections are treated as single reflections during integration. The deviation from r = 1=2 might, therefore, be larger than estimated here.
For MDO 2 (LT), r is derived from the x-coordinate of the pseudo-2 1 operation (see above) as r = 4x = 0.449. Thus, in both cases, despite the distinctly different orientation of the molecules, the parameter r is approximately 1=2.
3.1.6. Structural changes on phase transition. Even though symmetry considerations are the main focus of this work, changes at the crystallo-chemical level must not be neglected. As has been noted above, the structures of both 5NICz polymorphs are controlled by non-classical C-HÁ Á ÁN hydrogen interactions, forming chains extending in the [010] direction (Fig. 4). Each molecule forms a pocket delimited by N8 and the H7 and H9 are in meta position to N8. These two H atoms are expected to be the most 'acidic' and indeed interact with the N5 lone pair of the adjacent molecule. The hydrogen bonding is distinctly asymmetric with one short (C7Á Á ÁN5) and one long (C9Á Á ÁN5) interaction ( Table 2).
The CÁ Á ÁN distances are slightly longer in the HT phase. In return, the C-HÁ Á ÁN angles are closer to linear, owing to near coplanarity of the connected molecules [ Fig. 3(d)]. Overall, the hydrogen bonding can be considered as close to equivalent in both polymorphs.  Table 2 Non-classical C-HÁ Á ÁN hydrogen bonding in both polymorphs of 5NICz.  (7) 172.1

Figure 6
Pairs of 5NICz molecules connected byinteractions in the (a,b) LT and (c,d) HT polymorphs, projected on the molecular plane. Atom colours of the top molecules as in Fig. 4; bottom molecules in red for clarity. Ellipsoids are drawn at the 50% probability levels.
layer plane (Fig. 6). The C-HÁ Á Á contacts relating adjacent layers are, like the hydrogen bonding, very similar in both polymorphs. In summary, the dominant factor in the phase transition seems to be thestacking.
3.2. Phase transitions of 2NICz, and 2,5NICz-2LT and 2,5NICz-2HT polymorphs 3.2.1. Symmetry relationships. The 2LT and 2HT polymorphs of 2,5NICz are isostructural to the corresponding LT and HT 2NICz polymorphs, whereby the N5 and C11 atoms are positionally disordered. In contrast to 5NICz, the symmetries of the respective HT and LT polymorphs are related by a group-subgroup relationship. As is often observed in such a case, the symmetry of the HT phase (P2 1 =c 2 1 =c 2=n, Z = 4) is a strict super group (here minimal) of the symmetry of the LT phase (P12 1 =c1, Z = 4).
The structures are again built up of rods of 2NICz (2,5NICz) molecules connected by short C-HÁ Á ÁN interactions extending along [001] (Fig. 7). In the HT phase, the molecules are located on a twofold rotation axis and adjacent molecules are related by c [100] and c [010] glide reflections. The rods, therefore, possess pcc2 symmetry (Kopsky & Litvin, 2006). In the [100] direction, adjacent rods are generated by lattice translations. From a thus constructed layer, the final structure with Pccn symmetry is generated by 2 1 screw rotations in the [010] direction.
In the LT phase, the twofold rotation symmetry of the rods is lost. Of the two c-glide reflections, only the operation with plane parallel to (010) is retained. Thus, the symmetry of the rods is reduced by an index of 2 from pcc2 to p1c1. The rods are again related by translations forming layers parallel to (010) and the whole structure then generated by 2 1 screw rotations in [010] direction, resulting in an overall P2 1 =c symmetry.
3.2.2. Twinning. Whereas the HT polymorphs are not twinned, on cooling below the phase transition temperature, the lost point operations are retained as twin operations. The twin law is obtained as a coset of the coset decomposition of the LT in the HT point group. Thus, the LT twin consists of [mmm:2/m] = 2 domains, whose orientations are related by the operations f2 x ; m x ; 2 z ; m z g. The twinning is by pseudo-merohedry, since the orthorhombic metrics of the lattice are approximately retained. The twin obliquity is derived from the cell parameters as 1.5 (2NICz-LT) and 0.0 (2,5NICz-2LT). Indeed, for 2,5NICz-2LT no splitting of reflections was observed in single-crystal experiments, whereas for 2NICz-LT the twin obliquity is reflected in rows of diverging reflections.
3.2.3. Desymmetrization. The deviation of from ideal orthorhombic metrics is a measure of desymmetrization. For a finer evaluation of the desymmetrization, the atomic coordinates were transformed in an orthonormal coordinate system and the pseudo-rotation axis located at ð 1 4 ; 1 4 ; zÞ was applied to a molecule. The atoms in the original and the transformed molecule are separated by 0.52-0.68 Å (2NICz-LT) and 0.08-0.53 Å (2,5NICz-2LT). Whereas in 2NICz-LT the deviation is mostly due to a translation away from the rotation axis, in   Fig. 4. The 2LT and 2HT polymorphs of 2,5NICz are isostructural and not shown. Ellipsoids are drawn at the 50% probability levels. Crystallographic symmetry elements are indicated by the common graphical symbols (Hahn & Aroyo, 2016).

Figure 8
Overlay of molecules and their images by twofold rotation about the ( 1 4 ; 1 2,5NICz-2LT the molecules are tilted with respect to the rotation axis of the HT phase (Fig. 8).
3.2.4. Crystal chemistry. As in the case of 5NICz, the central crystallo-chemical feature are rods connected by nonclassical C-HÁ Á ÁN hydrogen bonding involving the two H7 and H9 positions. Here, the bonding is more symmetrical, with two equivalent (HT) or only slightly different (by ca 0.05 Å ; LT) bonds (Table 3). Enlarged ADPs of the N2 atom (Fig. 7c) indicate that the desymmetrization is dynamic, i.e. the orientations of the molecules oscillate between the two possible asymmetric states. Since the remaining structural changes are likewise minute, one can assume that the desymmetrization of the hydrogen-bonding is the decisive factor in the phase transition. Numerous reported solid-solid phase transitions are due to such a dynamic desymmetrization, a classical example being the KH 2 PO 4 (KDP) family of ferroelectrics (Peercy, 1975).

2,5NICz-1
The bulk polymorph 2,5NICz-1 features a crystallographically non-challenging structure with Pmn2 1 symmetry. In analogy to the other structures presented here, the 2,5NICz molecules are connected by hydrogen bonds to chains (Fig. 9). In contrast to the 2,5NICz-2LT and 2,5NICz-2HT polymorphs, the connected molecules are coplanar (related by a b + c lattice translation), demonstrating that the inclination is determined by packing effects.

Powder diffraction
To determine the stability ranges of the LT and HT polymorphs and to rule out additional phase transitions, powdered samples of 5NICz and 2NICz were subjected to lowtemperature powder diffraction (Fig. 10). In a bulk sample of 2,5NICz only the orthorhombic polymorph 1 could be seen by X-ray diffraction, which does not possess a phase transition in the solid state. Thus, in this case the exact phase transition temperature could not be determined. In both cases, 5NICz and 2NICz, the HT$LT transitions are clearly showed by appearance/vanishing of peaks and a distinct hysteresis of $20 K is observed [5NICz: transitions at 180-170 K (cooling) versus 200-210 K (heating); 2NICz: 210-200 K (cooling) versus 230-240 K (heating)]. No other phase transitions are apparent. The hysteresis suggests a phase transition of the first order. Even though neither powder diffraction nor DSC data for the 2,5NICz-2 polymorph could be acquired, experiments on the single crystal showed a smooth transition to the orthorhombic phase. This phase transition might be, therefore, of the second order.  Table 3 Non-classical C-HÁ Á ÁN hydrogen bonding in the HT and LT polymorphs of 2NICz and the 2HT and 2LT polymorphs of 2,5NICz.

Figure 10
Low-temperature XRPD scans of 5NICz (a,b) and 2NICz (c,d) over the