Visualization of texture components using MTEX

Knowledge of the appearance of texture components and fibres in pole figures, in inverse pole figures and in Euler space is fundamental for texture analysis; an MTEX-based tool is presented to predict their appearance to enable subsequent quantitative texture analysis.


Introduction
Quantitative texture analysis is concerned with detailed analysis of the orientation distribution function (ODF), e.g. measuring weight fractions of texture components or texture fibres. These components in turn allow one to deduce, for example, whether an observed change in the microstructure during a thermo-mechanical processing step is due to deformation, recrystallization etc. as each of these processes develops typical texture components. For cubic materials, such as steels, these texture components are very well understood, and a large body of research exists defining the typical components as well as identifying the processes during which they develop. For example, Ray et al. (1994) and Hoelscher et al. (1991) provide reviews of texture components for steel. Hu (1974) provided a condensed review for polycrystalline metals and alloys. Typical pole figures (PFs), inverse pole figures (IPFs), and, more importantly, locations of components and fibres in ODF plots, i.e. orientation densities in Euler space, are provided in these papers. Matthies et al. (1987) compiled standard distributions for which the computed ODFs, IPFs and PFs for various components with varying half-width of the distribution are displayed. This book provides an atlas that aids texture analysis of cubic systems, as experimentally observed ODFs can be divided into their components for further quantitative analysis. Without the knowledge of these components, quantitative texture analysis is not possible and at best a qualitative comparison of PFs or ODF sections can occur.
For non-cubic systems, however, typical texture components are material specific and much less well documented. In their overview paper, Wang & Huang (2003) attempt to overcome this situation for hexagonal crystal systems. For other crystal symmetries, to the best of our knowledge, such reviews of typical texture components do not exist except for specific systems such as quartz (Law, 2014). This limitation prevents in many cases a quantitative texture analysis as well as a deeper assignment of observed texture components to thermomechanical processes in non-cubic materials. This has become particularly evident with the development of modern large-scale neutron and synchrotron user facilities in combination with sophisticated data analysis tools -especially Rietveld analysis tools -enabling texture analysis [such as MAUD (Lutterotti et al., 1997) or GSAS (Von Dreele, 1997)]. These resources allow measurement of textures of multi-phase systems, as well as low-crystal-symmetry materials in the bulk that are inaccessible or at least difficult to access by PF measurements with laboratory X-ray diffractometers or electron diffraction. But because of this limitation, the vast majority of applications of texture analysis for non-cubic materials at best report PFs. Only very rarely is a quantitative texture analysis presented, although this is almost standard for cubic materials.
The 'real-life' texture of a given material is typically a combination of several texture components. Knowledge of the individual texture components enables one to identify the components present in an experimental texture, thus allowing quantitative texture analysis, e.g. determining the weight or volume fraction of a component, to ultimately interpret how a given texture correlates with the material processing history.
In this paper, we present a tool that overcomes the limitation of needing to have a comprehensive atlas of texture components available for a given system by providing short scripts for the open-source toolbox for texture analysis MTEX (Bachmann et al., 2010). MTEX can produce similar plots of ODF sections, IPFs and PFs to those provided by Matthies et al. (1987). For the texture practitioner, this means that observed texture components of the material under investigation can be compiled from the literature or estimated via texture modelling to produce an atlas of how these components occur in ODF sections, IPFs and PFs. Once the components are identified visually, either their volume fraction can be directly refined with MAUD (using the 'Standard Functions' representation of the ODF in MAUD) or the volume fractions can be calculated (e.g. by using the 'volume' command in MTEX) to develop a deeper understanding of microstructural changes as a function of thermomechanical treatment of a material. While MTEX is also a texture analysis tool similar to popLA (Kallend et al., 1991), BEARTEX (Wenk et al., 1998) and LABOTEX (Pawlik & Ozga, 1999), utilizing the powerful MATLAB (The MathWorks Inc., Natick, MA, USA) graphing capabilities allows this kind of plot to be readily produced with MTEX. This paper provides a brief introduction to MTEX, describes the scripts used to produce the ODF, PF and IPF plots, and gives examples for cubic, hexagonal and orthorhombic unit cells.

MTEX texture analysis toolbox
MTEX (Bachmann et al., 2010) is a free MATLAB toolbox for texture analysis and interpretation, which provides a variety of texture analysis tools. MTEX can be downloaded from http:// mtex-toolbox.github.io/download.html and allows users to produce different visualizations of a given texture component: PF and IPF as well as sections of Euler space showing the orientation density. A PF is a stereographic projection of the orientation distribution function showing the resulting pole densities of the plane normals of a single crystallographic lattice plane for all sample directions in the coordinate system of the sample. An IPF is a stereographic projection of the pole density of all crystal lattice plane normals for a single sample direction. Since both of these visualizations are projections of the ODF, information is lost, and a more complete visualization of the ODF is provided by sections through Euler space, showing the frequency or probability of all possible crystal orientations in the sample coordinate system (Suwas & Ray, 2014;Kocks et al., 2000;Randle & Engler, 2014).
In order to plot PFs, IPFs and ODFs in MTEX, crystal symmetry and specimen symmetry need to be defined. A texture component, e.g. a single orientation or a fibre, is defined in the specimen coordinate system. The mathematical description of texture components in MTEX is accomplished by kernel functions, and one of the available functions needs to be chosen. Finally, MTEX offers five types of texture components, two of which, the unimodal and fibre components, will be used here. In the following sections, more detail for each of the aforementioned parameters is provided to allow the reader to understand the examples, followed by a short example of the corresponding MTEX command. Complete descriptions of these commands are given in the MTEX documentation.

Crystal symmetry
Texture analysis requires solely the point group of the crystal symmetry. In MTEX, however, the crystal symmetry may be defined by the space group, point group or Laue group of the crystal. The crystal symmetry is stored in MTEX as a variable of the type crystalSymmetry. Command 1 shows an example of the definition for hexagonal -Ti. The parameters of the crystalSymmetry object in the form used below are point group, lattice parameters and alignment of the crystal axes with the Euclidean crystal coordinate system. The result of this command is assigned to the variable CS.

Specimen symmetry
The specimen symmetry can add symmetry due to thermomechanical processes, e.g. the fibre symmetry around the deformation axis from an extrusion process or uni-axial compression, or the mirror symmetry of a rolling process. Therefore, specimen-symmetry definition can affect the Euler space minimal dimension. For the general case, when the sample has no symmetry or is misaligned with respect to the symmetry of the texture, the full range of Euler space needs to be considered in order to represent all possible orientations. Any other sample symmetry will reduce the Euler space. For example, if a sample was deformed by rolling and is perfectly aligned with the instrument coordinate system, orthorhombic symmetry can be assumed and the Euler space will be reduced. Command 2 illustrates two different ways to define triclinic specimen symmetry (the general case). The result of this command is assigned to the variable SS.

Orientation
The weight, or probability, of all possible orientations describes the texture of the sample. Any orientation can be described by three rotation angles, the Euler angles, which may be defined using conventions given by Bunge (1982), Roe (1965) and Matthies et al. (1987). Furthermore, orientations can be given by pairs of Miller indices and crystal directions, {hkl}huvwi or {hkil}huvtwi for hexagonal crystal symmetry, indicating for example a set of lattice plane normals {hkil} parallel to the sample normal direction (ND) and a family of crystal directions parallel to the rolling direction (RD) of the sample. Command 3 shows orientations for which the {0001} plane normals are parallel to ND and the h1010i directions are parallel to RD. In this command ND and RD are determined by a sequence of lattice vectors.  The PDFs for all kernel functions in MTEX.

Figure 2
ODF maps for the Goss component with 7.5 half-width: (a) schematic representation after Matthies et al. (1987) and (b) produced by an MTEX script.

Kernel function
In polycrystalline materials, texture components do not appear as single orientations but have a certain variance. This is commonly modelled by bell-shaped density functions like the Gaussian. As there is no Gaussian in orientation space, several generalizations such as the Abel-Poisson kernel, the Gauss-Weierstrass kernel, the von Mises-Fisher kernel and the de la Vallé e Poussin kernel have been suggested in the literature for this purpose. In practice, all these kernel functions behave quite similarly and the choice of a specific kernel function is much less important than the choice of its halfwidth. In this work all standard components are modelled using the de la Vallé e Poussin kernel, which is the default choice in MTEX. Fig. 1 shows the probability distribution functions (PDFs) for all kernel functions existing in MTEX. The script that produces this figure is described in Appendix A and the MTEX documentation.

Visualization of components
The previously introduced concepts are applied to visualize texture components in Standard Distributions in Texture Analysis (Matthies et al., 1987) for cubic crystal symmetry. To establish the compatibility of the MTEX scripts presented here with this standard work, we choose the {011}h100i cubic component (the Goss component), which is typical of recrystallization textures in face-(f.c.c.) and body-centred cubic (b.c.c.) crystal systems and is shown on pages 278-323 of Volume 1 of Standard Distributions in Texture Analysis.
Having established the reproducibility for cubic systems (Section 3.1), we extend the concept to non-cubic crystals (Sections 3.2 and 3.3). In non-cubic crystals (e.g. hexagonal) the components strongly depend on the lattice parameter ratio (e.g. c/a) of a given material and therefore are not readily covered by the literature. Striving to review common texture components, we show the basal fibre component of the hexagonal phase in Ti-6Al-4V (see Section 3.2), which results from rolling (Philippe, 1994). We note in passing that a   splitting of the (0001) pole maxima can occur for basal fibres in hexagonal cubic packed (h.c.p.) materials depending on the c/a ratio in the presence of twinning (Wang & Huang, 2003). The second example in a hexagonal unit cell shows the {1013}h1210i component, also occurring in hexagonal materials, which is known as the recrystallization component in titanium alloys (Wagner et al., 2002). Finally, in order to illustrate the ability of MTEX to deal with all crystal symmetries, we chose to plot the {001}h010i component in orthorhombic olivine (space group Pbnm) (see Section 3.3).

Goss component in a cubic system
The script visualizing the Goss component {011}h100i, typical for f.c.c. and b.c.c. crystals and known as recrystallization texture (Hu, 1974), using the de la Vallee Poussin kernel (default function in MTEX) with a half-width of 7.5 is described in Appendix B.

Hexagonal unit cell
As an example of a non-orthogonal unit cell we show the basal fibre and recrystallization component for the Ti crystal structure.
3.2.1. Texture fibre in a hexagonal system.   script is displayed and explained in Appendix C. The results of the plots agree with Fig. 1(a) (PFs) and Fig. 5(a) (ODFs) of Wang & Huang (2003).

{1013}(1210) component in a hexagonal unit cell.
The following example shows Euler space sections, IPFs and PFs for the {1013}h1210i component, which is known as the recrystallization component in hexagonal systems (Wagner et al., 2002). The MTEX script to produce plots like Fig. 6 is provided in Appendix D.
The c/a ratio of h.c.p. materials can influence the ODFs, IPFs and PFs. Figs. 7(a)-7(c) show a comparison of the {1013}h1210i components for values of c/a above, equal to and below the ideal c/a ratio of 1.633 in the (1011) PF, respectively. Zn has lattice parameters a = 2.665 Å , c = 4.947 Å and c/a = 1.856, which is above the ideal ratio. Mg has lattice parameters a = 3.210 Å , c = 5.210 Å and c/a = 1.623, close to the ideal ratio. Ti has lattice parameters a = 2.951 Å , c = 4.683 Å and c/a = 1.5787, which is below the ideal ratio. The difference can be observed clearly in the PFs and to a lesser extent in the ODFs ]. Note that the c/a ratio has no influence on the (1010) and (0001) PFs (not shown).

Orthorhombic unit cell
The texture of low-symmetry crystal systems is rarely analyzed in terms of weight fractions of components, as is the case, in particular, for steel textures. As mentioned above, the relative simplicity of texture component reconstruction using MTEX, even for low-symmetry structures, could significantly help with such quantitative analysis. Olivine is an example of that. Wenk & Tomé (1999) applied a deformation-based model for dynamic recrystallization to the prediction of texture and microstructure development of olivine deformed in simple shear to large strains (Fig. 9). The script visualization of the {001}h010i component for the orthorhombic unit cell of olivine is shown in Appendix E. In this example we show the ODF, IPF and PF in MTEX without the tilt observed in Fig. 9. The orientation assigned to variable o is given in the Bunge Euler-angle convention. From Fig. 9 we can conclude that the Euler angles for this texture component are   Simulated PFs for olivine during deformation and recrystallization (Wenk & Tomé, 1999). ' 1 = 0, È = 0, ' 2 = 0 orientation. The resulting orientation in {hkl}huvwi is {001}h010i, which is somewhat obvious for the PFs presented in Fig. 9 but is much more complicated in the general case. Fig. 10 represents the MTEX result for (a) ODF sections, (b) IPFs and (c) PFs of the {001}h010i component in olivine.
Geoscience literature often describes olivine textures with types A to E, as described for example by Jung (2017). The script in Appendix F shows an example of visualization of the A to E texture component types of olivine in terms of Miller indices. While the olivine texture component of type D is represented by a fiber, the other component types are reproduced by components of the form {hkl} [uvw]. Fig. 11 shows the pole figures of A to E texture components (Jung, 2017) and can be directly compared with Fig. 6 therein.

Summary and conclusions
We have provided scripts for the MTEX toolbox that provide a fast and convenient way of plotting PFs, ODFs as sections in Euler space and IPFs for a given texture component or fibre to establish where pole densities or orientation densities for this component or fibre occur. The scripts produce plots that agree with equivalent plots from standard references (Matthies et al., 1987;Wang & Huang, 2003). Examples for hexagonal and orthorhombic crystal structures are given. Similar to the standard reference for cubic systems (Matthies et al., 1987), such plots can help to identify texture components present in experimental textures and, for example, to assign them to specific thermomechanical processes in experimental and theoretical studies of materials with non-cubic crystal structures. Contrary to texture identification in cubic materials, where a vast literature of components (e.g. or fibre, cube, Goss) and their origin (e.g. rolling, recrystallization) exists, identification of texture components in non-cubic systems is rarely reported. This tool, in combination with components reported in the literature, can help the texture researcher to overcome this limitation by producing an 'atlas' of the appearance of individual components for a given material. This allows determination of volume fractions in the next step, thus enabling truly quantitative texture analysis for non-cubic systems as demonstrated, for example, for Ti by Lonardelli et al. (2007).

APPENDIX A
The variables psi, assigned to all possible Kernel functions in MTEX are assigned to an array psi. The following lines produce plots of the probability distribution function of all defined Kernel functions. The array colour defines different colours for the subsequent plotting command. The A to E texture components of olivine displayed as pole figures.

APPENDIX B
The first command defines the crystal symmetry, lattice parameters and orientation of the crystal axes relative to the sample axes when the Euler angles are (0, 0, 0), and the sample's symmetry is defined in the second command. The preferred axis direction is defined by the third and fourth lines. The definition of the desired texture component is assigned to the variable o.
The plot commands produce PFs, ODF sections and IPFs, respectively. Axes of PFs can be labelled using the annotate command or by changing the mtex_settings file in the MTEX installation directory. In this example we chose to change the mtex_settings file.

APPENDIX C
The following script produces plots for the basal fibre {0002} parallel to the Z direction of the sample in hexagonal -Ti using MTEX. This script shows how font size can be defined. The hkil_fibre variable is assigned to fibre orientation, from which the fibre component ODF_fibre is calculated.

APPENDIX F
The following script produces the five olivine texture components A to E. The type-D texture component of olivine, which can be described as a girdle distribution of the [010] and [001] axes, can be visualized using the fibre component.