LCDiXRay: a user-friendly program for powder diffraction indexing of columnar liquid crystals

The formulation of a standard computerized procedure for the indexing of powder X-ray diffraction (PXRD) patterns of columnar liquid crystals, with the determination of all structural information extracted from a properly indexed PXRD spectrum and the attribution of the columnar mesophase symmetry, is presented. In particular, the proposed program notably accelerates the identification of columnar mesophases together with the in situ determination of their structural parameters such as mesophase type, space group, cell parameters, cross-section area, intermolecular stacking distance between consecutive discoids and, in the case of ordered mesophases, the estimation of the number of molecules constituting each discoid.

1. Introduction

Liquid crystals (LCs) are often characterized by powder X-ray diffraction (PXRD) analysis since the molecular organization within the mesophase gives rise to well defined PXRD patterns. Surprisingly, to the best of our knowledge, no specific computerized tools have been developed for the resolution of liquid crystal PXRD patterns. To date, the analysis of PXRD patterns has been performed manually by scientists through semi `house-made' rules which are time consuming and often originate indexing mistakes, resulting in misleading determinations of the symmetries and/or cell parameters.

Columnar LCs (or discotics) can be divided according to their cell geometry into three different categories, columnar hexagonal (Col_h), columnar rectangular (Col_r) and columnar oblique (Col_o), which were first classified by Levelut in 1983 (Levelut, 1983). In addition, columnar tetrahedral structures (Col_t, also denoted Col_squ) have been observed for phthalocyanin liquid crystals, probably as a result of their peculiar structure (Ohta et al., 1991; Komatsu et al., 1994), and have been also encountered for nonconventional T-shaped polyphilic triblock molecules based on rod-like biphenyl core systems (Chen et al., 2005). However, a Col_t phase can be considered as a Col_r mesophase of equal lattice parameters; in this regard, a detailed discussion will be presented below. Moreover, a columnar organization showing characteristics of both columnar and smectic phases is known as a lamello-columnar phase (Col_L). However, the indexing of PXRD patterns related to smectic mesophases will not be considered in the present work. While discotics may also show low-ordered nematic mesophases [discotic nematic (N_D) and columar nematic (N_C)], these LC phases will not be discussed in this report because of the lack of singularity of their PXRD patterns, due to the high intrinsic disorder. A comprehensive description of these mesophases is given in the excellent review written by Laschat et al. (2007), to which interested readers are referred. Moreover, highly ordered columnar mesophases presenting three-dimensional ordered structures such as H phase or plastic phases (Chandrasekhar et al., 2002) will not be considered in this report.

The present article reports on standardized guidelines for the indexing of columnar liquid crystal PXRD patterns and the implementation of these guidelines in the program LCDiXRay. This protocol is based on necessary initial hypotheses coupled with mathematical expressions specific to each mesophase type. In this way the generation of various sets of possible indexing values is allowed. Comparing the obtained sets with experimental data, the identification of the most likely solution is achieved, reducing the probability of error. Furthermore, from the generated data set reproducing the experimental PXRD pattern, it is straightforward to access all the structural parameters of the mesophase, such as unit cell geometry and dimension, cross-section area, intermolecular stacking distances, and number of molecules within the discoid constituting the columns for ordered mesophases. All the classical columnar mesophases included in this report are summarized in Table 1 with related schemes and structural parameters.

Table 1
Two-dimensional space groups of columnar liquid crystals with corresponding cross-section area and number of discoids (disc or ellipsoid) Z_disc in a cross section

			Cross section
Type (notation)	Space group	Extinction rules	Schema	Area	Discoid number	Three-dimensional representation
Columnar hexagonal (Colh)	p6mm	No conditions		Sh = a^2h3^1/2/2	Zdisc = 1
Columnar tetragonal (Colt)	p4mm	No conditions		St = a^2t	Zdisc = 1
Columnar rectangular (Colr)	p2mm	No conditions		Sr = arbr	Zdisc = 1
	c2mm	hk: h + k = 2n, h0: h = 2n, 0k: k = 2n			Zdisc = 2
	p2gg	hk: no conditions, h0: h = 2n, 0k: k = 2n			Zdisc = 2
	p2mg	hk: no conditions, h0: no conditions, 0k: k = 2n			Zdisc = 2†
					Zdisc = 4†
Columnar oblique (Colo)	p1	No conditions			Zdisc = 2

†The two different schematic representations of p2mg are reported (Levelut, 1983; Laschat et al., 2007; Tschierske, 2007; Grolik et al., 2012).

1.1. General definition and properties of columnar liquid crystals

Since their first identification by X-ray diffraction data in 1977 by Chandrasekhar et al. (1977), columnar (or discotic) liquid crystals have received much attention and have attracted growing interest, especially in the past decade, for their applicative role in optoelectronic devices such as light emitting diodes, photovoltaic cells, field effect transistors, liquid crystal displays and photoconductors (Schmidt-Mende et al., 2002; Hesse et al., 2010; Zheng et al., 2011; Shimizu et al., 2007; Kaafarani, 2011; Sergeyev et al., 2007). The high interest shown towards both organic and inorganic columnar liquid crystals is testified by the high number of dedicated peer reviews, regarding their structure, properties and applications, published over the past few years (Laschat et al., 2007; Kaafarani, 2011; Sergeyev et al., 2007; Kumar, 2006; Kato et al., 2006; Tschierske, 2007). The typical disc-like structure of discotic molecules comprises a flat and rigid aromatic core surrounded by numerous long flexible alkyl chains. The columnar organization is directed by π–π stacking interactions of the aromatic cores of the disc-shaped molecules (discoidal or ellipsoidal). However, discoids can be obtained by self-assembly of several molecules, held together via secondary interactions such as π–π stacking or hydrogen-bonding networks. Rod-like molecules have been also reported to self-assemble into ellipsoidal discs, originally observed through association of three single molecules (Guillon et al., 1987) and, as recently reported, association of 15–19 single molecules (Shimogaki et al., 2011). The presence of secondary interactions often contributes to increasing the order and stability of the resulting mesophase, widening the temperature range of the columnar organization. In conventional discotics, this can be achieved, for instance, by both increasing the dimension of the aromatic core and introducing, within the alkyl chains, some chemical functions favouring hydrogen bonds (Kumar, 2006; Gehringer et al., 2005).

The concept of order in discotics is often ambiguous. There is a difference between the intrinsic order of columnar mesophases (within/between columns) and induced order of the bulk material obtained through alignment. Indeed, discotics can be macroscopically aligned (in a homogenous manner) in two different ways: (i) in a homeotropic manner (i.e. all column directors are placed orthogonally to the substrate plane) or (ii) in a planar orientation (i.e. all column directors are parallel to the substrate). Alignment, required depending on the application sought, can be achieved by appropriate techniques or by using suitable substrates (Li et al., 2010). Aligned samples could be described as a highly ordered discotic mesophase. However, the terminology Col_ho, Col_ro, Col_oo, indicating ordered hexagonal, rectangular and oblique columnar phases, respectively, is not referred to such aligned mesophases but only used for intrinsic ordered phases, i.e. when the intracolumnar degree of order is high enough to observe in the PXRD pattern the reflection referred to as h₀, due to the diffraction generated by stacking of discoids. In the case of ordered columnar phases generated by stacking of molecules within the discoids, supplementary reflections (h_i, i > 0) can be observed in the wide-angle region. The difference between an ordered and a disordered columnar phase is illustrated in Fig. 1, together with their corresponding expected PXRD patterns. Accordingly, Col_hd, Col_rd and Col_od are referred to as disordered hexagonal, rectangular and oblique columnar phases, respectively.

Figure 1 Ordered (a) and disordered (b) columnar phases and the schematic representation of their PXRD patterns. Note that for clarity all eventual d00l peaks have been omitted.

2. Indexing of a PXRD pattern of columnar liquid crystals

For practical reasons, all the diffraction angles (2θ) will be converted into the interplanar spacing distances through the Bragg law,

where θ is the diffraction angle, λ is the wavelength of the monochromatic X-ray beam, h, k, l are the Miller indices of the associated reflection and n is an integer. Hereafter, all diffractions will be considered as their distance equivalents instead of their diffraction angles.

As clearly seen from the schematized PXRD pattern of a columnar LC presented in Fig. 2, two different angle regions can be defined: the small-angle area (from 2θ ≃ 0 to ca 2θ = 12–15°) and the high-angle area (for 2θ > ca 12–15°), the latter characterized by the broad halo generated by the slow motion of the flexible molten alkyl chains. The centre of this broad peak is conventionally referred to as the h_ch reflection halo.

Figure 2 A general schematic PXRD pattern of a columnar liquid crystal. Note that several d00l might be observed with d00l > d001.

Within the small-angle area, only reflection peaks relative to the intercolumnar d_hk0 distances are observable. In the wide-angle region characterized by the broad halo h_ch, only intracolumnar d_00l reflection peaks (note d₀₀₁ = h₀), if any, can be present, together with all possible reflection peaks due to intradiscoidal stacking (h_i). The indexing of a columnar PXRD pattern will be performed by comparison between the experimental distances d_hk0 observed in the small-angle region of the spectrum and the calculated data sets of d_hk0 distances obtained through initial assumptions and mathematical expressions of the specific columnar mesophase relative to its two-dimensional lattice geometry. For columnar mesophases Col_h with hexagonal geometry,

For columnar mesophases Col_r with rectangular geometry,

For columnar mesophases Col_o with plane-parallel geometry

Here a_h, a_r and b_r, and a_o, b_o and γ are the cell parameters of the columnar hexagonal, rectangular and oblique mesophases, respectively (see Table 1 for cell drawings).

2.1. Indexing of the columnar hexagonal mesophase (Col_h)

As already reported in various studies (Laschat et al., 2007; Chandrasekhar et al., 2002), the indexing of a columnar hexagonal mesophase is rather straightforward, because of the high symmetry of the p6mm space group. The calculation of a data set of interplanar spacing distances, as showed by equation (2), requires the determination of the unique unknown cell parameter a_h, implying the need of just one initial hypothesis. Taking into consideration the first reflection peak of the PXRD pattern (d_h1k10; Fig. 2) and assuming its identity as d₁₀₀, the following relations are easily deduced from equation (2):

and

From the resulting equation (6), it is now evident that all interplanar spacing distances and therefore all the reflection peaks observed in the PXRD pattern will be in typical ratio values with respect to the first indexed peak. These ratio values calculated from equation (6) are reported in Table 2.

Table 2
Characteristic ratios between the observed peaks of a PXRD pattern of a Col_h mesophase

Peak	d100	d110	d200	d210	d300	d220	d310	…	dhk0	…
Ratio	1							…		…
Note: dhkl = dkhl.

However, not all the d_hk0 peaks will be necessarily observed, since the number and nature of the observed peaks depend highly on the intercolumnar degree of order. Note that the first observed peak d_h1k10, usually indexed as d₁₀₀, could be d₁₁₀ or even of higher Miller indices in the case of very large diameter discoids. In this case, new correlation ratios between peaks have to be determined.

2.2. Indexing of columnar rectangular mesophases (Col_r)

The indexing of the PXRD pattern of a Col_r mesophase is more challenging because of the absence of a mathematical sequence as previously shown for the Col_h mesophase. Equation (3), which rules out interplanar spacing distances of a rectangular lattice geometry, clearly shows the dependency on two unknown unit-cell constants, a_r and b_r. Therefore, in the case of a Col_r PXRD pattern, an initial hypothesis for the first two reflection peaks of the spectrum (d_h1k10 and d_h2k20; Fig. 2), which are often observed rather close to each other, has to be imposed. The reason of the splitting of the first peak when passing from a Col_h to a Col_r mesophase has already been described (Laschat et al., 2007; Chandrasekhar et al., 2002). For each hypothetical value attributed to the couple h₁k₁/h₂k₂ used for indexing the first two peaks, a corresponding set of d_hk0 values can be calculated and compared with the experimental d_hk0 values. The calculation of these sets is based on the determination of the unknown constants A (A = 1/a_r²) and B (B = 1/b_r²) derived from equation (3), which can be rewritten for the chosen couple h₁k₁/h₂k₂ as

By combining these two equations, both A and B can be expressed and calculated as a function of the hypothesized first two reflection peaks.

Note that the most frequently encountered h₁k₁/h₂k₂ couples for Col_r mesophases are 11/20 and 20/11, but other combinations have also been reported, such as 11/02, 01/11, 11/31, 20/02 and 02/13 (Kilian et al., 2000; Morale et al., 2003; Pucci et al., 2005; Venkatesan et al., 2008; Maringa et al., 2008; Kaller et al., 2009; Camerel et al., 2006; Seo et al., 2007; Wuckert et al., 2009; Kaller et al., 2010; Amaranatha Reddy et al., 2005). Once the ideal calculated set has been identified, the unit-cell parameters can be easily extracted, being equal to d₁₀₀ (a_r) and d₀₁₀ (b_r), and the cross-section area can be calculated (see Table 1).

A direct comparison between the observed d_hk0 distances of the PXRD pattern and the calculated sets generated through these equations gives rise to the best possible indexing of the studied spectrum.

The final analysis step to be performed is the eventual determination of the space group belonging to the studied Col_r mesophase. Four different space groups (c2mm, p2mm, p2gg and p2mg) can be encountered and the accurate determination can be rather tricky. The space-group attribution is based on the presence and/or absence of reflection peaks, which implies the availability of a rather high number of reflections in the PXRD pattern, not so often occurring. Extinction rules relative to all lattice space groups and the corresponding illustrations are reported in Table 1.

2.3. Indexing of the columnar tetragonal mesophase (Col_t)

The Col_t mesophase can be considered as a Col_r mesophase with a unique cell parameter (a_r = b_r). Hence, equation (3) can be written as follows:

with a_t the lattice parameter of the Col_t mesophase (see Table 1).

As for the Col_h mesophase, adopting a similar analysis, the interplanar spacing distances are dependent on a sole unknown constant (a_t), and the resulting ratio values with respect to the first reflection, assumed to be d₁₀₀, can be extrapolated from equation (9); the results are collected in Table 3.

Table 3
Characteristic ratios between the observed peaks of a PXRD pattern of a Col_t mesophase

Peak	d100	d110	d200	d210	d220	d300	d310	d230	d140	…	dhk0	…
Ratio	1									…		…
Note: dhkl = dkhl.

Again, not all reflections will be necessarily present in the PXRD pattern, though for the space group p4mm of the tetragonal lattice, all reflections are theoretically allowed. Moreover, for very large discoids, the first observed peak d_h1k10 could differ from d₁₀₀, and hence new correlation ratios must be determined through equation (9). The Col_t mesophase is, however, less frequently encountered and often is observed in the case of highly specific shaped LCs (Ohta et al., 1991; Komatsu et al., 1994; Chen et al., 2005).

2.4. Indexing of the columnar oblique mesophase (Col_o)

The columnar oblique mesophase (Col_o) represents the most complicated case in the indexing procedure. The presence of three consecutive rather close first reflection peaks in the small-angle region of the PXRD pattern is, however, the first indication of a possible Col_o mesophase. As clearly shown by equation (4), the introduction of the three unknown unit-cell parameters (a_o, b_o and γ) within the rather complex mathematical expression characterizes the Col_o lattice. Fortunately, the Col_o mesophase is more rarely encountered because strong core–core interactions between mol­ecules are required to develop this phase (Laschat et al., 2007). Similarly to the case of Col_r mesophases previously described, initial hypotheses for the resolution of the Col_o PXRD patterns have to be formulated. In this case, the indexing of the first three peaks of the spectrum (d_h1k10, d_h2k220 and d_h3k30; Fig. 2) has to be hypothesized. The calculation of the interplanar spacing distance sets will be performed through the determination of three unknown constants A [A = 1/(a_o²sin²γ)], B [B = 1/(b_o²sin²γ)] and C (C = −2cosγ/a_ob_osin²γ) derived from equation (4).

The PXRD pattern of a Col_o mesophase will be indexed by comparison between the experimental data and the sets of d_hk0 values generated via the procedure summarized in the following scheme:

Note that the symmetry of the Col_o lattice corresponds to the p1 space group, allowing all the hk0 reflections to be present. As already mentioned, Col_o mesophases are rarely observed and therefore it is difficult to define which initial triad values h₁k₁, h₂k₂, h₃k₃ are more frequently encountered for the indexing of the first three peaks. Examples reported for initial attribution are 20/11/1, 10/11/2, 11/20/1 and 10/01/11 (Morale et al., 2003; Trzaska et al., 1999; Choi et al., 2011; Pucci et al., 2011).

2.5. Indexing refinement

Once the most likely nature of the columnar mesophase has been identified, in order to obtain a data set of calculated d_hk0 values with a more accurate fitting with respect to all the observed experimental ones, i.e. reducing the discrepancy between observed and calculated data, it is necessary to proceed to a re-evaluation (or refinement) of the values of the initial peaks, taking into account the maximum number of data available. Following this calculation, all the d_hk0 peaks are redetermined using the refined values, obviously through the mathematical expression of the identified mesophase.

For the Col_h mesophase, the value of d₁₀₀ can be re-evaluated from equation (10) (Zelcer et al., 2007):

where N_hk0 is the number of hk0 observed reflections. For Col_r mesophases, the re-evaluation of the first two interplanar distances (d₁₀₀ and d₀₁₀) can be performed. In order to take into account the maximum number of available data, it is recommended to determine first the mean value of d₁₀₀ through equation (11) when the number of observed h00 reflections is higher than the number of 0k0 observed reflection peaks,

(where N_h00 is the number of h00 observed reflections), and use this re-evaluated value of d₁₀₀ in equation (12) for re-evaluating d₀₁₀:

(where N_hk0 is the number of hk0 observed reflections). In the opposite case (when N_0k0 > N_h00), the re-evaluation of d₀₁₀ should be performed first. This can be achieved by rewriting equation (11) for d₀₁₀. Then, d₁₀₀ will be obtained through the appropriate transcription of equation (12).

The Col_t refinement procedure can be performed through an analogous methodology. Since in this particular rectangular lattice d₁₀₀ is equal to d₀₁₀, only the re-evaluation of the first interplanar distance value is necessary. This can be achieved via

where N_hk0 is the number of hk0 observed reflections. Finally, the Col_o refinement procedure requires the redetermination of the two first interplanar distance values d₁₀₀ and d₀₁₀ as well as the re-evaluation of the γ angle, which can be achieved by the following equations:

where N_h00 is the number of h00 observed reflections,

where N_0k0 is the number of 0k0 observed reflections, and

where N_hk0 is the number of hk0 observed reflections.

2.6. Number of molecules in discoids

Once the indexing of a PXRD pattern of a columnar liquid crystal has been performed, the number of molecules within the discoid at the origin of the columnar stacking can be determined, but only for ordered phases (i.e. when the h₀ = d₀₀₁ reflection peak is present). The number of molecules per unit cell (or cross section) (z) can be estimated according to (Lehmann et al., 2006)

where ρ is the density of the liquid crystal phase, N_A is Avogadro's constant, S is the columnar cross-section area, h₀ is the height of the columnar slice and M is the molecular weight of the constitutive molecule. S can be easily calculated from the cell parameters, keeping in mind that its expression depends on the geometry of the cell (see Table 1). The liquid crystal density can be estimated unless measured experimentally. Density values ranging from 0.9 g cm⁻³ up to 1.2 g cm⁻³ have been reported depending on the nature of the studied liquid crystals (full organic molecules, metallomesogens, neutral or ionic mesogens) (Gunyakov et al., 2003; Kaller et al., 2009, 2010; Ionescu et al., 2012). The number of discoids present in the unit cell (Z_disc) is characteristic of the lattice geometry and the proposed model (see Table 1). Consequently, the number of molecules per discoid (N) is easily accessible by dividing the number of molecules in the cross section (z) by the corresponding Z_disc value. Therefore, a hypothesis on how mesogen molecules are eventually organized to form the discoidal shape can be formulated. This finding can be further supported by experimental data when intradiscoidal stacking gives rise to sufficiently intense reflections in the wide-angle region of the PXRD pattern (see Fig. 2). In the absence of such information, only theoretical modelling can shed light on the discoidal molecular assembly. However, it has to be mentioned that this calculation is based on the initial hypothesis that columns are organized in the mesophase with a straight vertical stacking order. In reality the intracolumnar stacking distance h might be higher or lower than h₀. This is the case for tilted (h > h₀) and undulating (h < h₀) columnar organizations (Weber et al., 1991; Donnio et al., 1997). Neither case will be considered by LCDiXRay in its current version, but they will be considered in future upgrades.

3. Software description and examples

3.1. Algorithm and software implementation

A user-friendly program has been implemented in a Java object-oriented framework in order to identify correctly the mesophase via indexing of PXRD patterns and, as a consequence, determine the optimal parameters best fitting the observed data. It is worth underlining the advances in terms of computational speed of the proposed approach. The framework has been designed on the basis of the previously described indexing procedures for Col_h, Col_r, Col_t and Col_o mesophases.

The core idea is to rewrite the experimental data d_i as a function of the peak indices h_i, k_i and unknown parameters { p_k}ⁿ_{k = 1} for all types of mesophase according to the following equation:

For all the discussed mesophase types, a relative interface has been initialized with appropriate methods as shown in the class diagram reported in Fig. 3.

Figure 3 Class diagram of the LCDiXRay algorithm.

The mathematical equations of the interplanar distances [equations (2)–(4)] can be described in the form of equation (18) with { p_k}ⁿ_{k = 1} representing the unknown structural parameters of the hypothesized mesophase. Given an n-length user-selected input data set { d_k}ⁿ_{k = 1}, coupled with n different indices ( h_i, k_i), the related n parameters are computed by solving the system formed by the n equations derived from equation (18). Once such parameters are obtained, an estimate d_gen of the entire experimental data set is generated by varying the indices h and k in an appropriate user-selected discrete grid. An optimization search is performed with the indexing couples ( h_i, k_i) as control parameters, by considering a root-mean-square difference (RMSD) optimization process:

The output of the above procedure represents the set of optimal parameters providing the minimum RMSD index value within the range of user-selected discrete indices, according to the chosen generation model.

The generation model, i.e. hexagonal, rectangular or oblique, and the data set extracted from the experimental PXRD pattern are needed as initial input. The Java program is characterized by dynamic panels that are updated according to the chosen generation model and provides a test on the consistency of the data set for the indexing problem. The class diagram of the designed user interface is depicted in Fig. 4.

Figure 4 Class diagram of the user interface developed for LCDiXRay.

The `InteractiveTableModel' provides useful tools to choose an appropriate number of observed peaks representing the initial hypothesis for the indexing algorithm from which all the lattice parameter(s) will be determined. It is also possible to select the Miller index range defining the admissible indexing values. A similar range definition is required in the generation process of the estimated data set. To this end, two Miller indices (h_max and k_max) represent the maximum admissible values to generate all the observed diffraction peaks. An appropriate choice of parameters h_max and k_max is required to avoid a set of d_hk0 indexing that does not have any realistic significance but which could mathematically result in a lower RMSD index.

Finally, once a convergent result has been obtained which suits the user's expectations, a pop-up menu allows them, as a first option, to proceed to the refinement of the indexing, following the methodologies previously described. A second option allowing the determination of the number of molecules within the discoid is also accessible only for ordered mesophases.

Ultimately, all the generated data, the indexing set and the optimal parameters can be exported for further manipulations.

3.2. Practical examples

3.2.1. Ordered hexagonal columnar mesophase

Although the indexing of a Col_h mesophase is rather straightforward to perform because of the characteristic ratios between the observed peaks of a PXRD pattern (see §2.1), the following example of a Col_ho discotic mesophase exhibited by discotic (I) (see Fig. 5) (McKenna et al., 2005), a poly(propylene imine) dendrimer based on a triphenylene, is a perfect example to illustrate at first glance the performance of the LCDiXRay program. The experimental and calculated literature data for (I) are presented in Fig. 6, together with the initial window frame relative to the loaded data.

Figure 5 Discotic (I), an example of a Colh mesophase (McKenna et al., 2005).

Figure 6 d spacings and lattice parameter for the ordered columnar hexagonal mesophase Colho of discotic (I), and the corresponding LCDiXRay program window.

The hexagonal model has been selected on the LCDiXRay program window, as well as the first observed peak (d₁ = 58.1 Å), which will be taken into account to determine the lattice parameter. On the far right of the program window, only the first Miller index subset is active, corresponding to the user-selected discrete grid of indices allowed for the selected initial peak. Having chosen as a first subset the values 0 and 2 for h₁ and k₁ minimum and maximum, respectively, the only possible values to be considered for d₁ are d_hk0 = d₁₀₀, d₂₀₀, d₁₁₀, d₂₁₀ and d₂₂₀. Finally, on the bottom right of the window, the maximum values of h and k Miller indices that will be considered to index all observed peaks (d₂–d₉) have both been fixed equal to 5. Note that the 11th observed peak d₁₁ = 3.55 Å has been assigned as h₀ and the tenth peak d₁₀ = 4.5 Å has been assigned as h_CH; both values will therefore be removed from the fitting procedure. The active window obtained after a validation check is presented in Fig. 7, clearly showing the expected fitting for a hexagonal lattice with an RMSD index of 0.4770 for a lattice parameter of a_h = 67.09 Å. To reproduce exactly the literature data reported in Fig. 6 (McKenna et al., 2005), the initial selected peak (from which all calculations are performed) must be d₃ = 28.9 Å (see supplementary information¹), the initial choice made by the authors. In these conditions, an RMSD index of 0.3933 and a lattice parameter a_h = 66.74 Å are obtained.

Figure 7 Obtained LCDiXRay result.

An active pop-up window in LCDiXRay allows the user to proceed to the data refinement to reduce eventually the RMSD between experimental and calculated interplanar distances, according to the method previously described. The refined data reported in Fig. 8 and obtained from the optimization results based on the selected observed peak at 58.1 Å show the lowering of the RMSD index to 0.2324 with a final lattice parameter a_h = 66.65 Å. Note that this RMSD index is even lower than that obtained reproducing the literature indexing (i.e. considering d₃ = 28.9 Å as initial selected peak).

Figure 8 LCDiXRay results obtained after refinement.

Through this example, we can clearly see the genuine performance of LCDiXRay, allowing in a few clicks the generation of the optimized indexing despite the initial assumption made.

3.2.2. Ordered rectangular columnar mesophase with determination of the number of molecules within a discoid

A relevant example of the indexing of a Col_r mesophase is represented by the discotic mesophase exhibited by the cyclopalladated photoconductive Nile red complex (II) (Fig. 9) (Ionescu et al., 2012).

Figure 9 Discotic (II), an example of a Colr mesophase, and its PXRD pattern recorded at 398 K after cooling from 433 K (Ionescu et al., 2012).

The active LCDiXRay window of the loaded interplanar distances observed on the PXRD pattern of discotic (II) with its initial assignment, together with the LCDiXRay result window obtained after checking the hypothesis of a Col_r model and refinement of the obtained preliminary results, are shown in Fig. 10.

Figure 10 LCDiXRay results obtained for discotic (II) in a Colr model.

Starting from the loaded data (active window of LCDiXRay; Fig. 10a) a two-click procedure (check and refinement; supplementary Figs. S2 and S3 ) allows the visualization of the most probable indexing of the mesophase of (II) through a Col_r model (Fig. 10b). We obtain a final RMSD value of 0.1901 with lattice parameters a_r = 37.09 Å and b_r = 65.04 Å. In this case, the first two observed peaks were initially chosen to perform all calculations, with a selected discrete grid of Miller indices ranging from 0 to 2 (h_imin = k_imin = 0; h_imax = k_imax = 2, with i = 1 or 2).

A second activated window allows the determination of the number of molecules within the cross-section area of the unit cell, as illustrated in Fig. 11(a). Following equation (2), the number of molecules present within the cross section has been determined for four different standard density values. As expected for metallomesogens, a density of 1.2 g cm⁻³ can reasonably be attributed for discotic (II), resulting in four molecules being present in the cross section. According to the obtained indexing together with the extinction rules (see Table 1), the space group of the mesophase for (II) can be deduced as p2gg. Unequivocally, since in the p2gg space group two discoids are present in the unit cell (Z_disc = 2; Fig. 11b), each disc contains two molecules, forming a dimer-like discoid. Theoretical calculations have been performed confirming the existence of a hydrogen-bond network established between two side-by-side molecules placed in a head-to-tail arrangement as illustrated in Fig. 11(c) (Ionescu et al., 2012).

Figure 11 (a) LCDiXRay active window, (b) representation of a cross-section area of a p2gg lattice (c) and schematic representation of the dimer of (II) generating a discoid.

3.2.3. Disordered oblique columnar mesophase

Finally, to illustrate the performance and the swiftness of the LCDiXRay program, the identification of a columnar oblique mesophase is presented through the indexing of the PXRD pattern recorded at 393 K for the cobalt complex (III) (see Fig. 12) (Morale et al., 2003).

Figure 12 Discotic (III), an example of a Colo mesophase (Morale et al., 2003).

As shown in Fig. 13(a) for the Col_o model, all subsets of the LCDiXRay program window are now activated, since three interplanar distances have to be initially selected to index as a Col_o lattice. The literature-reported interplanar distances have been loaded and, again, in only one validation click, the results window allows us to check the feasibility of the Col_o hypothesis. The results window (Fig. 13b) gives the indexing with the smallest RMSD between experimental and calculated distances, based as always on the user-selected choice of the initial Miller indices. The obtained results are in agreement with reported literature data.

Figure 13 LCDiXRay results obtained for discotic (III) in a Colo model.

Note that in this example the refinement of the data performed following the previously described procedure will lead to a slight increase of the RMSD value, which varies from 0.2038 (unrefined) to 0.2218 (refined) (supplementary Figs. S4 and S5 ). This is due to the small number of observed reflection peaks on which obviously all the calculations are based. In particular, the lack of 0k0 reflection peaks means that the d₀₁₀ reflection value has to be fixed, excluding it from the refinement. Since for Col_o mesophases three initial peaks are required, the refinement tool can only achieve realistic results when a large number of reflection peaks are observed, which is unfortunately rarely encountered for this type of mesophase.

3.3. Experimental errors

LCDiXRay performs the refinement procedure from the loaded interplanar distances values, although the scattering angles 2θ are the experimental data collected over an X-ray powder diffraction analysis. The 2θ experimental error is highly dependent on the experimental technique used (reflection versus transmission). Furthermore, the interplanar distance accuracy is intrinsically generally much lower for large d values than for low d values. Not surprisingly, throughout the literature on PXRD of liquid crystal mesophases, data accuracy is most of the time left undiscussed and experimental data are reported in d values in ångström with one (in most cases) or at most two digits. In some cases, to take into account such error in the accurate determination of distances, several indexings of possible Miller indices are proposed. For all these reasons, the current version of LCDiXRay will not take into account this unpredictable experimental error. Consequently, the RMSD index given by LCDiXRay is mostly indicative and only relevant for comparisons, and the accuracy of the lattice constant values obtained through the program must be critically evaluated by the user. Similarly, in its current version only one set of data is proposed by the software, which corresponds to the lowest RMSD value obtained from the loaded values and initial restrictions. The proposal of several solutions, with restricted deviations reflecting the experimental errors obtained during the recording of the PXRD pattern, will be implemented in the near future.

4. Conclusions

LCDiXRay is a user-friendly program for powder diffraction indexing of columnar liquid crystals. The main objective of this powerful tool is to accelerate the determination of the exact nature of the mesophase presented by columnar LCs. Furthermore, the program contains all the mathematical expressions required for data refinement and determination of structural parameters of the identified mesophases (unit-cell geometry and dimensions, cross-area section, number of molecules within the discoid for ordered phases). The determination of columnar mesophases, in particular Col_r and Col_o, is often performed manually; LCDiXRay is able in a few minutes to perform all the time-consuming operations and provide the most likely indexing of a recorded PXRD pattern. Further implementations of LCDiXray are planned in due course and will concern also the determination of the most likely space group for Col_r mesophases on the basis of the extinction rules characterizing them, as well as the possibility of performing a refinement of Col_r data as a Col_t mesophase when the cell parameters are compatible (a_r ≃ b_r). Finally, it is expected that a similar procedure will be introduced for the indexing of calamitic mesophases, allowing the inclusion in LCDiXray of lamellar columnar and non-classical mesophases, such as plastic columnar phases or the three-dimensionally highly ordered mesophases (H phase).

The LCDIXRay program is freely available from the authors on request. We would like to suggest that users send feedback for future improvements. A request form is available in the supporting information .

Footnotes

¹Supporting information for this article is available from the IUCr electronic archives (Reference: NB5093 ).