Texture and Anisotropy

Home , Texture (crystalline)

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS Rep. Prog. Phys. 67 (2004) 1367–1428 PII: S0034-4885(04)25222-8

Texture and anisotropy

H-R Wenk1 and P Van Houtte2

1 Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA 2 Department of MTM, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium

E-mail: [email protected]

Received 17 February 2004 Published 5 July 2004 Online at stacks.iop.org/RoPP/67/1367 doi:10.1088/0034-4885/67/8/R02

Abstract

A large number of polycrystalline materials, both manmade and natural, display preferred orientation of crystallites. Such alignment has a profound effect on anisotropy of physical properties. Preferred orientation or texture forms during growth or deformation and is modified during recrystallization or phase transformations and theories exist to predict its origin. Different methods are applied to characterize orientation patterns and determine the orientation distribution, most of them relying on diffraction. Conventionally x-ray pole- figure goniometers are used. More recently single orientation measurements are performed with electron microscopes, both SEM and TEM. For special applications, particularly texture analysis at non-ambient conditions, neutron diffraction and synchrotron x-rays have distinct advantages. The review emphasizes such new possibilities. A second section surveys important texture types in a variety of materials with emphasis on technologically important systems and in rocks that contribute to anisotropy in the earth. In the former group are metals, structural ceramics and thin films. Seismic anisotropy is present in the crust (mainly due to phyllosilicate alignment), the upper mantle (olivine), the lower mantle (perovskite and magnesiowuestite) and the inner core (ε-iron) and due to alignment by plastic deformation. There is new interest in the texturing of biological materials such as bones and shells. Preferred orientation is not restricted to inorganic substances but is also present in polymers that are not discussed in this review.

Contents

Page 1. Introduction 1370 2. Measurements of textures 1370 2.1. Overview 1370 2.2. X-ray pole-figure goniometer 1371 2.3. Synchrotron x-rays 1371 2.4. Neutron diffraction 1374 2.5. Transmission electron microscope 1374 2.6. Scanning electron microscope 1375 2.7. Comparison of methods 1375 3. Data analysis 1376 3.1. Orientation distributions and texture representations 1376 3.2. From pole figures to ODF 1378 3.3. Use of diffraction spectra 1378 3.4. Statistical considerations of single orientation measurements 1379 3.5. From textures to elastic anisotropy 1380 4. Polycrystal plasticity simulations 1380 4.1. General comments 1380 4.2. Deformation 1381 4.3. Recrystallization 1383 5. Important texture types in metals 1385 5.1. Fcc metals 1385 5.2. Bcc metals 1389 5.3. Hcp metals 1390 5.4. Phase transformations 1391 6. Ceramic textures 1394 6.1. Bulk ceramics 1394 6.1.1. α-alumina (Al2O3) 1394 6.1.2. Silicon nitride (Si3N4) 1394 6.1.3. Zirconia (ZrO2) 1394 6.1.4. Ceramic matrix composites 1395 6.1.5. Bulk high-temperature superconductors 1395 6.2. Thin films and coatings 1396 6.2.1. Silicon and diamond 1396 6.2.2. Nitride, carbide and oxide coatings 1397 6.2.3. Epitaxial films 1398 7. Textures in minerals and rocks 1399 7.1. Calcite (CaCO3) 1400 7.2. Quartz (SiO2) 1402 7.3. Olivine (Mg2SiO4) 1404 7.4. Sheet silicates 1406 Texture and anisotropy 1369

7.5. Ice (H2O) 1407 7.6. Halite (NaCl) and periclase (MgO) 1408 7.7. Polymineralic rocks 1409 7.8. Cement minerals 1411 7.9. Earth structure 1412 7.10. Textures as indicators of strain history 1413 7.11. Anisotropy in the deep earth 1415 8. Textures in mineralized biological materials 1419 8.1. Nacre of mollusc shells (aragonite) 1419 8.2. Bones (apatite) 1419 9. Conclusions 1421 References 1422 1370 H-R Wenk and P Van Houtte

1. Introduction

Preferred orientation of crystallites (or texture) is an intrinsic feature of metals, ceramics, polymers and rocks and has an influence on physical properties such as strength, electrical conductivity, piezoelectricity, magnetic susceptibility, light refraction and wave propagation, particularly in the anisotropy of these properties. The directional characteristics of many polycrystalline materials were first recognized not in metals but in rocks and were described as ‘texture’ (d’Halloy 1833). In the 20th century texture research was largely pursued by metallurgists but recently it has gained importance in ceramics (e.g. high temperature superconductors), polymers, and regained interest in the earth sciences. The reason for the latter is that seismologists have discovered anisotropic wave propagation in large sectors of the earth’s interior and a likely cause is preferred orientation of crystals that developed by deformation during the earth’s long history. This review will highlight some aspects of textures with focus on new approaches and methods, as well as relevant problems. Metallurgists and ceramicists are engaged in texture research to develop materials with favourable properties. In contrast, geologists and geophysicists are using textures to interpret the past. The rationale is thus reversed. In metallurgy specimens are readily available for analysis, and theories can be tested with experiments. Deep-earth materials do not occur on the surface and many are unstable at ambient conditions. Also, many geological conditions are outside the realm of experiments, particularly the slow strain rates and highly heterogeneous nature of rock formations. Yet, in spite of these differences, methods and approaches are remarkably similar, even though the objects of interest vary greatly in dimension. This review is intended to provide a brief introduction for physical scientists, not for texture experts. Some of the important issues are highlighted with examples, and we refer new researchers in the field of texture and anisotropy to important publications. Since the classic books on metallurgy (e.g. Wassermann and Grewen (1962), Dillamore and Roberts (1965), Hatherly and Hutchinson (1979)) and geology (e.g. Sander (1950), Turner and Weiss (1963)), there have been newer books (e.g. Bunge (1982), Wenk (1985), Kocks et al (2000)), numerous journal articles and particularly research papers in the tri-annual proceedings of the International Conferences of Textures of Materials (ICOTOM). These publications need to be consulted for details. Textures in polymers, though important, are only mentioned peripherally (see, e.g. G’Sell et al (1999)). While preparing this review we noticed that almost half of the references are in physics journals and fifteen in Nature and Science, illustrating that texture and anisotropy are subjects of core physics as well as of general interest. We try to give a balanced account of recent progress in texture research; however, in this broad field it was hard to avoid some emphasis on our own specialities, particularly in the selection of examples which were more readily available. We do not suggest that these are in any way more important.

2. Measurements of textures

2.1. Overview Interpretation of textures has to rely on a quantitative description of orientation characteristics. Two types of preferred orientations are distinguished: the lattice preferred orientation (LPO) or ‘texture’ (also ‘preferred crystallographic orientation’) and the shape preferred orientation (or ‘preferred morphological orientation’). Both can be correlated, such as in sheet silicates with a flaky morphology in schists, or fibres in fibre-reinforced ceramics. In many cases they Texture and anisotropy 1371 are not. In a rolled cubic metal the grain shape depends on the deformation rather than on the crystallography. Many methods have been used to determine preferred orientation. Optical methods have been extensively applied by geologists, using the petrographic microscope equipped with a universal stage to measure the orientation of morphological and optical directions in individual grains (e.g. Phillips (1971)). Metallurgists have used a reflected light microscope to determine the orientation of cleavages and etch pits (e.g. Nauer-Gerhardt and Bunge (1986)). With advances in image analysis, shape preferred orientation can be determined quantitatively and automatically with stereological techniques. Optical methods of LPO measurements of some minerals have also been automated (Heilbronner and Pauli 1993). Today diffraction techniques are most widely used to measure crystallographic preferred orientation (e.g. Bunge (1986), Kocks et al (2000)). X-ray diffraction with a pole-figure goniometer is a routine method. For some applications synchrotron x-rays provide unique opportunities. Neutron diffraction offers some distinct advantages, particularly for large bulk samples. Electron diffraction using the transmission (TEM) or scanning electron microscope (SEM) is gaining interest, because it permits one to correlate microstructures, neighbour relations and texture. There are two distinct ways to measure orientations. One way is to average over a large volume of a polycrystalline aggregate. A pole figure collects signals from many crystals and spatial information is lost (e.g. misorientations with neighbours), but also some orientation relations (such as how x, y, and z-axes of individual crystals correlate). The second method is to measure orientations of individual crystals. In that case orientations and the orientation distribution can be determined unambiguously and, if a map of the microstructure is available, the location of a grain can be determined and relationships with neighbours can be evaluated. But compared to the bulk methods, the statistics of such measurements are limited.

2.2. X-ray pole-figure goniometer X-ray diffraction was first employed by Wever (1924) to investigate preferred orientation in metals, but only with the introduction of the pole-figure goniometer and use of electronic detectors did it become a quantitative method (Schulz 1949). Bragg’s Law for monochromatic radiation is applied. The principle is simple: in order to determine the orientation of a given lattice plane, hkl, of a single crystallite, the detector is first set to the proper Bragg angle, 2θ of the diffraction peak of interest, then the sample is rotated in a goniometer until the lattice plane hkl is in the reflection condition (i.e. the normal to the lattice plane or diffraction vector is the bisectrix between incident and diffracted beam) (see figure 1). In the case of a polycrystalline sample, the intensity recorded at a certain sample orientation is proportional to the volume fraction of crystallites with their lattice planes in reflection geometry. Determination of texture can be done on a sample of large thickness and a plane surface on which x-rays are reflected, or on a thin slab which is penetrated by x-rays. Because of defocusing effects as the flat sample surface is inclined against the beam, variations in the irradiated volume and absorption intensity corrections are necessary, particularly in reflection geometry. In reflection geometry only incomplete pole figures can be measured, usually to a pole distance of 80˚ from the sample surface normal.

2.3. Synchrotron x-rays Conventional x-ray tubes produce a broad beam of relatively low intensity (∼1 mm). A powerful new tool for texture research is synchrotron radiation. In a synchrotron a very 1372 H-R Wenk and P Van Houtte

Figure 1. Geometry of a pole-ﬁgure goniometer equipped with Soller slits and monochromator. Bragg’s Law applies to lattice planes. The sample is rotated about an axis perpendicular to the surface.

Figure 2. Synchrotron x-ray diffraction image of a sheet of rolled copper with Debye rings recorded with a CCD camera at ESRF. Intensity variations immediately display the presence of texture.

ﬁne-focused high-intensity beam of x-rays with monochromatic or continuous wavelengths can be produced. The unique advantages of high intensity, small beam size (<5 µm) and free choice of wavelength open a wide range of new possibilities (e.g. Heidelbach et al (1999), Wcislak et al (2002), Wenk and Grigull (2003)). Synchrotron diffraction images recorded by CCD detectors almost instantaneously display the presence of texture expressed in systematic intensity variations along Debye rings, as illustrated for a copper plate (ﬁgure 2). While Texture and anisotropy 1373

(a)

(b)

Figure 3. Geometry of a synchrotron x-ray diffraction experiment in transmission. (a)Foragiven reciprocal lattice vector hkl the accessible crystallite orientations lie on a cone which intersects the orientation sphere of the sample in a small circle. Diffracted x-rays lie on a cone (shaded) that intersects a planar detector along a circle. The diffraction spot of a single pole is indicated. (b) Diamond anvil cell mounted in radial geometry in the synchrotron beam to display texture in Debye cones. The sample is mounted in a boron gasket with minimal x-ray absorption. the presence of texture is immediately obvious, elaborate data processing is necessary to determine texture patterns quantitatively and interpret data in a satisfactory way. Figure 3(a) shows the geometry of a transmission diffraction experiment with incoming x-ray beam, sample and Debye cone with an opening angle 4θ, on which diffracted x-rays lie. If the sample is stationary only lattice planes hkl that are inclined by an angle (90˚−θ) to the incoming beam diffract, and the corresponding reciprocal lattice vectors lie on a cone with an opening angle of 180˚−2θ which intersects the orientation sphere of the sample in a small circle. Coverage of the pole figure from a single image is minimal but even information from such an image can be sufficient to determine the orientation distribution and then to reconstruct complete pole figures (Wenk and Grigull 2003). The use of high energy is advantageous because of good penetration and moderate absorption, as well as small 2θ angles. Synchrotron analysis is particularly valuable for compounds with weak scattering (e.g. polymers and biological materials) and for investigating local texture variations (Margulies et al 2001). Other applications are in situ observations of texture changes during deformation at high pressure with diamond anvil cells (figure 3(b), e.g. Merkel et al (2002)) and high temperature (e.g. Puig-Molina et al (2003)) and we will illustrate some examples in a later section. 1374 H-R Wenk and P Van Houtte

(a) (b)

Figure 4. (a) Schematic of the TOF neutron diffractometer HIPPO at Los Alamos National Laboratory. Multiple detector banks are arranged on rings. Each ring (at different 2θ) records reflections of differently oriented lattice planes so that the pole figure is covered simultaneously. A sketch of a (large) person is inserted for scale. (b) Pole figure coverage with thirty 2θ = 40˚, 90˚ and 150˚ detectors (Wenk et al 2003).

2.4. Neutron diffraction Neutron diffraction texture analysis is almost as old as the pole-figure goniometer. It was first applied by Brockhouse (1953) in an attempt to determine magnetic structure in steel. Neutron diffraction texture studies are done either at reactors with a constant flux of thermal neutrons, or with pulsed neutrons at spallation sources. The wavelength distribution of thermal neutrons is a broad spectrum with a peak at 1–2 Å, similar to x-rays. A disadvantage of neutrons is that the interaction of neutrons with matter is low, and long counting times are required. Weak interaction is also a great advantage because it provides high penetration and low absorption, making neutrons suitable for bulk texture investigations of large sample volumes. Because of the low absorption, environmental stages (heating, cooling, straining) can be used for in situ observation of texture changes, e.g. during phase transformations. Neutrons are also sensitive to measuring the orientation of magnetic dipoles, but though this was the original incentive for Brockhouse, to this day no satisfactory magnetic pole figures have been measured. A conventional neutron texture experiment at a reactor source uses monochromatic radiation produced with single crystal monochromators. A goniometer rotates the sample to cover the entire orientation range, analogous to an x-ray pole-figure goniometer. To improve counting efficiency position-sensitive detectors have been applied that record a 2θ spectrum with many peaks simultaneously. With the advent of pulsed neutron sources it has become customary to use polychromatic neutrons and a detector system that can identify the energy of neutrons by measuring the time-of-flight (TOF). The new TOF neutron diffractometer HIPPO at Los Alamos (figure 4(a)), dedicated to texture research, has 50 detector panels that record simultaneously diffraction spectra from crystals in different orientations (figure 4(b)).

2.5. Transmission electron microscope The TEM offers excellent opportunities to study textural details in fine-grained aggregates. Like light microscopy, the TEM not only provides information about orientation but also Texture and anisotropy 1375 about grain shape and, more importantly, about dislocation microstructures indicative of active deformation mechanisms. There are many applications of Kikuchi patterns for orientation analysis (e.g. Humphreys (1988)). The indexing procedure has been automated (Schwarzer and Sukkau 1998). A recent study of brass-type shear bands in fcc metals and their influence on texture evolution has been an excellent example to document microstructural changes with initial slip, intermediate twinning leading to a final texture (Paul et al 2004).

2.6. Scanning electron microscope Local orientations can also be measured with the SEM and this technique is becoming very popular because it does not require much background in texture theory from the user (e.g. Randle and Engler (2000)). Unlike the TEM, the SEM is not restricted to thin areas located along the edge of a hole in a small specimen (<2 mm), but enables crystal orientations to be determined on surfaces of considerable extent. Interaction of the electron beam with the uppermost surface layer of the sample produces electron back-scatter diffraction patterns (EBSPs or EBSD) that are analogous to Kikuchi patterns in the TEM. EBSPs are captured on a phosphor screen and recorded with a low intensity video camera or a CCD device. A big advance came with the automation of pattern indexing and scanning a specimen surface (Wright and Adams 1992). The sample is translated using a high precision mechanical stage or sample locations are reached by beam deﬂection in increments as small as 1 µm. At each position an EBSP is recorded. With a phosphor screen, back-scattered electrons are converted to light, this signal is transferred into a camera. The digital EBSP is then entered into a computer and indexed. Specimen coordinates, crystal orientation, parameters describing the pattern quality and a parameter evaluating the pattern match are recorded. Then the sample is translated to the next position and the procedure is repeated. A spatial resolution of less than 0.4 µm can be reached on a SEM equipped with a ﬁeld emission gun. This sounds like an ideal technique. However, it is only applicable to crystals with fairly low dislocation densities, surface preparation is critical and the automatic indexing procedure is not always reliable. Failure to index and mis-indexing of patterns are both orientation- dependent and can produce texture artefacts. Furthermore there are statistical limitations that will be discussed below.

2.7. Comparison of methods The optimal choice of texture measurements depends on many variables, such as availability of equipment, material to be analysed and data requirements. For routine metallurgical practice and many other applications in materials science and geology, an x-ray pole-figure goniometer in reflection geometry is generally adequate. It is fast, easily automated and inexpensive both in acquisition and maintenance. Transmission geometry has been successfully used for texture analysis of sheet silicates in slates and shales (Ho et al 1999). Pole figures can only be measured adequately if diffraction peaks are sufficiently separated. In geological samples and ceramics, x-ray diffraction is therefore generally limited to single phase aggregates of orthorhombic or higher crystal symmetry. Synchrotron x-rays are used for in situ experiments at high pressure and temperature, generally of very fine-grained samples. Texture changes can be recorded in real time. With neutron diffraction bulk samples rather than surfaces are measured, coarse-grained materials can be characterized, environmental cells (heating, cooling, straining) are available and angular resolution is better than for an x-ray pole-figure goniometer because no defocusing occurs. It is possible to measure complex polyphase composites with many closely spaced 1376 H-R Wenk and P Van Houtte

(a) (b)

Figure 5. (a) Deﬁnition of Euler angles α, β and γ (Roe/Matthies notation) that relate the orthogonal sample coordinate system (a, b, c) and the crystal coordinate system (x,y,z) by three rotations. (b) Corresponding rotations φ1, , φ2 used in the Bunge convention. Equal area projection. diffraction peaks. Neutron diffraction data analysis is rather complex and requires considerable expertise from the user. Also, access to facilities is limited. Electron diffraction with a TEM is most time-consuming but provides, in addition to crystal orientation, valuable information about microstructures and, at least two-dimensionally, about interaction between neighbours and about heterogeneities within grains. These are important data to interpret deformation processes. Recently EBSPs, measured with the SEM on polished surfaces, have become very popular. They allow for determination of local orientation correlations. With the possibility of automation, this technique has become comparable in expense and effort to x-ray diffraction analysis. There are limitations for samples with many lattice defects. The technique has already proved to be essential in the study of recrystallization and grain growth in metals upon annealing, as well of grain fragmentation upon plastic deformation.

3. Data analysis

3.1. Orientation distributions and texture representations In quantitative texture analysis the coordinate systems of the sample and of the crystal need to be related. This requires three quantities, such as the classical Euler angles that relate two orthogonal right-handed coordinate systems (sample: a,b,c and crystal: x,y,z) through three rotations (figure 5(a)), or an axis–angle specification that brings the two coordinate systems to coincidence through a single rotation about a specific axis. (Both these representations of orientations were originally introduced by Euler (1775).) The sample coordinate system in metals is usually defined by the forming process (e.g. a: rolling direction RD, b: transverse direction TD, c: normal direction ND). For geological samples it is often more arbitrary. The crystal coordinate system for symmetries with non-orthogonal axes follows crystallographic conventions (z: [001], y: perpendicular to [001] and [100], x: perpendicular to y and z). There are various conventions for the Euler angles. The one in figure 5(a) with α, β and γ is the Roe/Matthies convention. A first rotation (α) around c produces a system a, b, c. The second rotation (β) around b brings c to c. The final rotation (γ) around c brings b to b with a = x, b = y and z = z. The Bunge convention, used in the vast majority of papers on metals, is related to the Roe/Matthies convention: α = φ1 − 90˚, β = and γ = φ2 +90˚. Texture and anisotropy 1377

Figure 6. The three-dimensional orientation distribution of Euler angles can be viewed as a probability distribution in cylindrical space of angles α (azimuth in sample coordinates), β (pole distance) and γ (azimuth in crystal space). A pole ﬁgure (top) is a two-dimensional projection of the ODF, a peak intensity in a diffraction of a diffraction pattern (bottom) is proportional to a one-dimensional projection.

Figure 5(b) illustrates the rotations for the Bunge convention. The first rotation (φ1) is around c and brings a to a . The second rotation (φ2) around a brings c to c . And the final rotation () around c brings a into a = x. In the case of a polycrystal, an orientation becomes an orientation probability distribution of the three angular quantities and is described by an orientation distribution function (ODF). A pole figure is a two-dimensional projection of this three-dimensional distribution and represents the probability of finding a pole to a lattice plane (hkl) in a certain sample direction (figure 6). Pole figures are normalized to express this probability in multiples of a random distribution (m.r.d.). Depending on the application stereographic or equal area projection of the spherical pole density distribution is used. Inverse pole figures are also projections of the ODF, but in this case the probability of finding a sample direction relative to crystal directions is plotted. This is particularly useful for axisymmetric textures (fibre textures) where only one sample direction (the symmetry axis) is of interest. Conventionally ODFs are represented as two-dimensional sections of a rectangular Euler space. Unfortunately it is very difficult to visualize orientations in this representation and the space is highly distorted. To overcome these deficiencies, cylindrical representations have been suggested to represent the crystal orientation distribution relative to sample coordinates (COD) or the sample orientation distribution relative to crystal coordinates (SOD) (Wenk and Kocks 1987, Kocks et al 2000). Matthies et al (1990) have introduced special σ -sections with minimal distortion. In this review we will mainly use pole figures and inverse pole figures. 1378 H-R Wenk and P Van Houtte

3.2. From pole ﬁgures to ODF

A pole figure is a two-dimensional distribution of a crystal direction (e.g. pole to a lattice plane hkl) relative to sample coordinates. (A direction is specified by two angles.) As mentioned above, pole figures can be considered projections of the three-dimensional ODF. There are various methods to retrieve the ODF from measured pole figures. One set of methods works in direct space and uses basically algorithms of tomography. The Williams–Imhof–Matthies– Vinel (WIMV) method, introduced by Matthies and Vinel (1982), is most widely used. Variants are the vector method (Ruer and Baro 1977), arbitrarily defined cells (ADC) (Pawlik et al 1991), and the maximum entropy method (Liang et al 1988, Schaeben 1988). Other methods work in Fourier space, most notably the harmonic method introduced by Bunge (1965) and Roe (1965). Pole figures are expanded with spherical harmonics. Harmonic coefficients from the pole figure expansion can then be used to determine harmonic coefficients of the ODF expansion. The expansion is carried to a finite order, usually between 22 and 32, providing an angular resolution of about 15–10˚. For sharp textures there remain distinct truncation effects with subsidiary oscillations, including unrealistic negative ODF values. All methods yield similar results, at least for ideal test data. However, there is some ambiguity in continuous pole figures as is most transparent for the harmonic method. Pole figures are, by their very nature, centro-symmetric, which means that odd coefficients in the harmonic expansion vanish. The ODF is not centro-symmetric and therefore requires even and odd coefficients for a full representation, but the odd coefficients cannot be obtained from pole figures (Matthies 1979). Omissions of odd coefficients can introduce errors in orientation densities in the ODF and add spurious maxima and minima, called ghosts (Kallend 2000). This uncertainty is between 10% and 30% for most textures. The uncertainty can be much reduced by exploiting the fact that the ODF cannot be negative (see, e.g. Van Houtte (1991)). For sharp textures, there is usually no uncertainty left if an angular resolution of about 10˚ is used. At higher resolution, or for weak textures with a high isotropic background (phon), additional assumptions are needed to remove the uncertainty (Mathies and Vinel 1982, Schaeben 1988). There are many software packages that calculate ODFs from pole figures and perform other operations to quantify textures in polycrystals (some examples are BEARTEX, Wenk et al (1998), LaboTex, Pawlik et al (1991), MulTex, Helming (1994), POPLA, Kallend et al (1991), TexTools, Resmat Corp.). Details can be obtained from the Internet.

3.3. Use of diffraction spectra

Traditionally texture analysis has relied on pole figure measurements. This is efficient if only a few pole figures are required for the ODF analysis and if diffraction peaks are reasonably strong (relative to background) and well separated. The method becomes increasingly unsatisfactory for complex diffraction patterns of polyphase materials and low symmetry compounds with many closely spaced and partially or completely overlapped peaks. The amount of texture information is roughly contained in the product of the number of pole figures (hkl) times the number of sample orientations used during the measurement of one pole figure. In conventional ODF analysis one relies on a few pole figures and many sample orientations. Another approach is to use many pole figures and few sample orientations. This is an obvious advantage for TOF neutron diffraction where many diffraction peaks are measured in a continuous spectrum. Rietveld (1969) proposed a method to use a continuous powder patterns to obtain crystallographic information (e.g. Young (1993)) and this method can be expanded to include texture analysis. In a powder with a random orientation of crystallites, the relative intensities are the Texture and anisotropy 1379

d [Å]

Figure 7. Spectra of experimentally deformed limestone recorded with TOF neutrons and the 40˚ detector banks of HIPPO. Relative intensity variations indicate texture. same for all sample orientations and are due to the crystal structure. In a textured material, there are systematic intensity deviations from those observed in a powder as illustrated in figure 7 for TOF neutron diffraction data of limestone. Intensities are linked to the crystal structure by means of the structure factor, they are also linked to the texture through the ODF (figure 6). As with the pole figure method described above, texture effects can be implemented in the Rietveld method either with Fourier or with direct methods. The finite number of harmonic ODF coefficients can be refined in a similar way as crystallographic parameters with a non- linear least squares procedure. With discrete methods ODF values are directly related to peak intensity values in the spectra. The Rietveld method is implemented in software packages GSAS (Von Dreele 1997) and MAUD (Lutterotti et al 1997) that can be downloaded from the Internet.

3.4. Statistical considerations of single orientation measurements With single orientation measurements that rely on surface coverage, the number of grains that can be measured is limited. This becomes apparent if we consider that a texture function (ODF) with 5˚ resolution has 181 584 cells in the case of triclinic crystal symmetry. Even if we had that many grains and a random texture, some ODF cells would have 0 grains, most would have 1 grain and there would be cells with 2, 3, 4 or more grains, i.e. the ODF would range between 0 and 4 m.r.d. For fewer grains the situation is worse. Either larger cells (and worse angular resolution) have to be used, or data have to be smoothed in a statistically correct way. For single orientations (rather than an averaged diffraction intensity) the statistical fluctuations are expressed in exaggerated pole densities and a large texture index F2. F2 has been introduced by Bunge (1982) as a bulk measure of texture strength and is equal to the volume-averaged integral of squared orientation densities over the ODF (equivalent to the ODF-weighted mean of the ODF itself). It is mainly influenced by sharp texture peaks (1 m.r.d.). Matthies and Wagner (1996) have explored the relationship between number of measured grains N and F2, and established an asymptotic 1/N dependence of F2(N) that can be used to determine the smoothing function for a given sample. It turns out that for any quantitative representation it 1380 H-R Wenk and P Van Houtte is necessary to measure a large number of grain orientations (not just data points scanned by the EBSP system). Thus, EBSP data may provide good qualitative information on texture patterns, but neutron diffraction on bulk samples is often needed to obtain quantitative information on texture strength. In most published EBSP texture analyses, arbitrary smoothing is applied (e.g. by fitting orientations with harmonic or Gauss functions) and results are, therefore, at best semi-quantitative.

3.5. From textures to elastic anisotropy If we know the orientation distribution and single crystal physical properties, then we can calculate approximate polycrystal physical properties. Of most interest have been elastic properties, described as a fourth rank tensor. Single crystal elastic constants for many materials under a variety of conditions have been compiled (e.g. Simmons and Wang (1971)). Polycrystal elastic properties are obtained by a summation over all contributing single crystals, taking into account their orientation, the relationship with neighbours and microstructure, such as shape of crystallites, presence of pores and cracks. If there is preferred orientation (non-random ODF), a macroscopic anisotropy will result. In the case of polycrystal elastic properties, the summation needs to be done to maintain continuity across grain boundaries when a stress is applied, and to minimize local stress concentrations. Recently this has been approached with finite element simulations (e.g. Dawson et al (2001)). In practice the local stress and strain distribution is usually neglected and the summation is done by simple averages. There are two extreme cases: the Voigt average assumes constant strain throughout the material and applies strictly to a microstructure composed of laths with a stress applied parallel to the laths. Values of the elastic constants are maximum. The Reuss average assumes that stress is constant and strictly applies to the case of a microstructure in which an extensional stress is applied perpendicular to the laths. In this case aggregate elastic constants are at minimum values. There are other averages that are intermediate between constant strain and constant stress. One example is the Hill average (1952), an arithmetic mean of Voigt and Reuss averages, or the geometric mean (Matthies and Humbert 1993), or a self-consistent average proposed by Kroner¨ (1961). The relationship between texture, single crystal properties and polycrystal properties is established in various ways. One application is to measure bulk elastic anisotropy and estimate from it texture. While such a determination is neither accurate nor complete, it can nevertheless be practical, e.g. to estimate crystal alignment and corresponding flow regimes in the deep earth from seismic anisotropy measurements, or take advantage of destruction-free velocity measurements of large engineering components to ascertain specifications and conceivable damage. A field that is receiving a lot of attention is to determine single crystal elastic properties from diffraction measurements of elasticity (changes in lattice spacings) on deformed textured polycrystals (e.g. Singh et al (1998), Gnaupel-Herold¨ et al (1998), Matthies et al (2001)). Another important application is the estimation of residual stresses in textured materials on the basis of diffraction data (Van Houtte and De Buyser 1993).

4. Polycrystal plasticity simulations

4.1. General comments There are several reasons why it is desirable to simulate the development of texture and anisotropy during deformation. For engineering applications an important consideration is Texture and anisotropy 1381

␾ 2

Figure 8. Hot rolled aluminium (AA5182) and texture simulations with different methods. Shown is the orientation density f(g)along the α-ﬁbre as function of φ2 (Van Houtte and Delannay 1999).

the cost of experiments compared to simulations to obtain information on properties after specific treatments. More importantly, if simulations with a physical theory can satisfactorily explain experimental results, it is likely that the underlying processes are understood and can be generalized. Another scientific reason to carry out simulations to predict deformation textures of metals is the fact, that such simulations usually provide a wealth of additional details, such as estimations of stored energy, dislocation density, and local heterogeneities of stress and plastic strain, and all this as a function of grain orientation and/or grain neighbourhood. These data are very hard to measure experimentally, but they are desperately needed to understand recrystallization upon annealing, a phenomenon that is very important in metal processing. In earth science, it is often not possible to reproduce the complex strain paths that occur in nature. Most deformation experiments on rocks are done in compression geometry, and more recently, also in torsion. Both paths are very special cases that are rarely satisfied in actual geological conditions. A second reason is that conditions in nature often cannot be reproduced in experiments, most significantly strain rates and grain size, e.g. at high pressure. If microscopic mechanisms are known that are active under a given set of conditions and if a good constitutive theory exists, then polycrystal behaviour for any strain path and conditions can be simulated. Polycrystal plasticity simulations are still far from being perfect as was recently demonstrated by Van Houtte and Delannay (1999) for such a simple system as rolled aluminium, where different theories predict different texture patterns and each different from what is actually observed (figure 8). Different mechanisms can produce or modify texture. Most important is dislocation glide during deformation, which we will discuss in the next section. Also significant is recrystallization, either dynamic or static, with nucleation of new domains and grain boundary mobility. During phase transformations features of texture patterns are often inherited. (We will illustrate examples for metals and rocks in a later section.) Textures may form during solidification from a melt, vapour deposition and electro-deposition. If fluids are present, aspects of dissolution and growth in a stress field can have a profound influence on resulting orientation patterns (e.g. Bons and den Brok (2000)). In other systems textures evolve as non- equiaxed rigid particles deform in a viscous matrix (Jeffery 1923, March 1932, Willis 1977).

4.2. Deformation Deformation of a polycrystal is a very complicated heterogeneous process. When an external stress is applied to the polycrystal, it is transmitted to individual grains. Dislocations move 1382 H-R Wenk and P Van Houtte on slip systems, dislocations interact and cause ‘hardening’, grains change their shape and orientation, thereby interacting with neighbours and creating local stresses that need to be accommodated. To realistically model these processes is a formidable task and only recently have three-dimensional finite element formulations been developed to capture at least some aspects (Mika and Dawson 1999). The difficulty is that in real materials local stress equilibrium and local strain continuity are maintained, and this requires local heterogeneity at the microscopic level. Most of the polycrystal plasticity simulations have used highly simplistic approximations, e.g. that each grain is homogeneous, and yet arrived, at least for moderate strains, at useful results. There are two extreme assumptions. Taylor (1938) suggested that in modelling plastic deformations, straining could be partitioned equally among all crystals. This hypothesis has been used extensively for fcc and bcc metals (e.g. Van Houtte (1982), Kocks et al (2000)). For this approach even to be viable, the individual crystals must each be able to accommodate an arbitrary deformation, requiring five independent slip systems. While the Taylor assumption is reasonable for materials comprising crystals with many slip systems of comparable strength, using it in other situations can lead to prediction of excessively high stresses, incorrect texture components, or both. In the Taylor model, high stresses are required to activate slip systems, even in unfavourably oriented grains, and the model is therefore known as an upper bound model. In contrast to the Taylor hypothesis, all crystals in a polycrystal can be required to exhibit identical stress, given that their behaviour is rate-dependent at the slip system level (a variant of the original Sachs (1928) assumption). The equal stress hypothesis is most effective for polycrystals comprising crystals with fewer than five slip systems. It has also been used in modelling the mechanical response of crystals that possess adequate numbers of independent slip systems, but where slip systems display widely disparate strengths. The principal drawback is that deformation often is concentrated too highly in a small number of crystals, leading to inaccurate texture predictions. With the Sachs approach only the most favourable slip systems are activated and, therefore, stresses are low. This approach is known as a lower bound model. Several other approaches have been developed for modelling the heterogeneous deformation of highly anisotropic polycrystals as is the case for many rocks. For example, Molinari et al (1987) developed the viscoplastic self-consistent (VPSC) formulation for large strain deformation in which each grain is regarded as an inclusion embedded in a viscoplastic homogeneous equivalent medium whose properties coincide with the average properties of the polycrystal. VPSC, mainly in the formulation of Lebensohn and Tome´ (1993) has been successfully applied to the prediction of plastic anisotropy and texture development of various metals (e.g. Tome´ and Canova (2000)) and geologic materials (e.g. Wenk (1999)). Self- consistent methods have to struggle with the non-linearity of the relation between stress and plastic strain or strain rate, as they implicitly use linearizations of this material model for the strain field surrounding the ‘inclusion’. The applicability of a self-consistent model to engineering studies can be refined by tuning modelling parameters to better correlate assumed microscopic interactions with macroscopic behaviour or with the results of crystal plasticity finite element simulations as explained below (Gilormini and Michel 1999, Molinari and Toth´ 1994, Molinari et al 2004). Another modelling approach is to employ finite element methodologies to compute deformations of an aggregate of crystals. In this case local heterogeneity can be taken into account, particularly if grains are discretized with many elements (Kalidindi et al 1992, Bate 1999, Mika and Dawson 1999). The boundary value problem resulting from the application of homogeneous macroscopic boundary conditions is solved to obtain the deformation of individual crystals. In this case the rotation of a grain and its deformation depends both on Texture and anisotropy 1383 the orientation and the orientations of neighbours. This method is called the crystal plasticity finite element method (CPFEM). CPFEM methods are in principle the best models for the simulation of plastic deformation of polycrystalline materials and for the prediction of deformation textures, but they still require huge calculation times, even on very powerful computers. This limits their extensive use in engineering applications. For example, in a finite element simulation of the deep drawing of a car body panel, several tens of thousands of elements need to be used, each representing an entire polycrystal. It is practically impossible to use CPFEM for each of them. The challenge then is to develop polycrystal deformation models which are 103 or 104 times faster than CPFEM models, but which reach a comparable quantitative accuracy. Much progress towards this goal has been achieved by the so-called ‘multi-grain’ models. In such models, the Taylor model is solved not for a single grain, but for an aggregate of several grains. In the LAMEL model (Van Houtte et al 2002), two grains are used whereas in the GIA model (Crumbach et al 2001), eight grains are used. This is repeated for many sets taken from the microstructure. These models capture the effect of local interactions between particular neighbouring grains on the deformation pattern, which is not possible with conventional self- consistent models, because those always ‘smear out’ the neighbours of a particular grain by averaging. Surprisingly, the quantitative accuracy of the ODFs predicted by these models for aluminium and steel was found to be comparable to the results of CPFEM models, and far better than of any other model (Van Houtte et al 2002, Li and Van Houtte 2002). It is noteworthy that Raabe and Roters (2004) have tried to solve the calculation time problem in a different way: they do not put a true CFEM model in every integration point of a FE model of a metal forming process (‘FE2 model’), but use the CPFEM code directly for the forming problem itself. The number of finite elements used for the forming process is not excessively high. They enter a simplified texture in every integration point, consisting, e.g. of a single grain or a few grains. The orientations of all these grains together do reflect the average texture of the material. All polycrystal plasticity models are comprised of two basic parts: a set of crystal equations describing properties and orientations, and a set of equations that link individual crystals together into a polycrystal. The latter set provides the means to combine the single crystal quantities to define the polycrystal response on the basis of physically motivated assumptions regarding grain interactions. As a crystal deforms by slip, it undergoes a lattice rotation. These lattice rotations are the cause of development of preferred orientations in aggregates with many component crystals. All polycrystal plasticity models deal with a highly non-linear system. A solution is obtained by working with a finite number of discrete grains (given as orientations) and deformation is applied in increments. The deformation is defined by a displacement gradient tensor that may be constant or may change with deformation. Rotations of all grains are calculated after each strain increment and the orientations, as well as their shape and slip system activities, are then updated. A comparison of the texture patterns that evolve with those that are experimentally observed is a good indicator for the applicability of a model. In addition to texture patterns microstructural and mechanical features are also predicted.

4.3. Recrystallization The relationship between slip and crystal rotations is straightforward. Other processes such as climb, grain boundary sliding, diffusion in general may also affect orientation distributions. Of particular importance is recrystallization. In deformation studies, recrystallization is the development of a new grain structure with low dislocation density, either during 1384 H-R Wenk and P Van Houtte deformation (dynamic recrystallization, Guillope´ and Poirier (1979)) or after deformation (static recrystallization or annealing, Humphreys and Hatherly (1995)). It is agreed that also here energy considerations are responsible for the development of recrystallized domains. Some theories suggest that thermodynamic equilibrium in a non-hydrostatic stress field controls recrystallization (discussed in various forms by Kamb (1961), Paterson (1973), Green (1980) and Shimizu (1992)), but at least in metals it is clear that strain energies produced by accumulations of dislocations far exceed thermodynamic energies imposed by the stress field (Humphreys and Hatherly 1995). Grains with higher dislocation densities have higher stored energies than grains with low dislocation densities (Haessner 1978). Grains with higher stored energies may be consumed through boundary migration by grains with less stored energy (growth). Alternatively, dislocation-free nuclei may form in grains with high dislocation density and then grow at the expense of others. The texture which finally forms is believed to be controlled either by nucleation or by grain growth. It is possible that grain growth is ‘oriented’, i.e. for some reason grains with certain crystallographic orientations grow faster than others. In that the case the grain growth mechanism is likely to control the final texture. Otherwise ‘oriented’ nucleation may control the final texture. The texture of fresh nuclei may not be random and reflect the final texture. The last word has not yet been said on this problem (Gottstein 2002). Not only orientation distributions, but also grain size distributions are important considerations (Shimizu 1999). The deformation state of a grain depends on its orientation and its history and can thus be predicted with polycrystal plasticity theory. The changes in texture and grain size that occur during annealing, and their dependence on microstructural mechanisms provides a logical link to develop detailed recrystallization models, which couple deformation models with probabilistic laws to simulate recovery and recrystallization. Among them, a model developed by Radhakrishnan et al (1998) couples the finite element method with the Monte Carlo technique so as to account for local effects in aggregates. Deformation simulations with the self-consistent model generate a population of grains with a variation in deformation and a variation in dislocation density. Whether a grain which is ‘hard’ due to a high Taylor factor and as a results features a smaller strain, will be found to have a smaller or higher than average dislocation density depends on the value of tangent or secant modulus used in the self-consistent model, and whether an anisotropic grain-by-grain hardening model has been implemented in it. Note that according to physics, the dislocation density is a non-monotonically increasing function of the product of the Taylor factor with the local strain. Results of such models have been used in an empirical model to simulate texture changes during recrystallization (Wenk et al 1997). The microstructural hardening of slip systems during deformation provides an incremental strain energy to grains after each deformation step. Grains with a high stored energy are likely to be invaded by their neighbours with a lower stored energy. In the model the stored energy of each grain is compared with the average stored energy of the polycrystal. If the stored energy of a grain is lower than the average, it grows; if it is higher, it shrinks. If nucleation in dislocation-free domains accompanies boundary migration, a highly deformed parent grain divides upon reaching a threshold strain rate and produces a dislocation free nucleus. The nucleus (which may be a subgrain or a bulge in a grain boundary) takes on the current orientation of the parent at the time of its formation, but its dislocation density is reset to zero. This has an effect on the subsequent evolution, because these domains with low dislocation density can grow much faster. Nucleation takes place if the strain increment exceeds a threshold value. If nucleation dominates over growth, grains with high dislocation density will preferentially determine the final texture. With these parameters growth and nucleation can be balanced. Texture and anisotropy 1385

The model was applied to simulate static and dynamic recrystallization textures in hexagonal metals (Solas et al 2001, Puig-Molina et al 2003) and geologic materials such as quartz (Takeshita et al 1999), calcite (Lebensohn et al 1998), olivine (Wenk and Tome´ 1999) and ice (Thorsteinsson 2002). Some examples will be given below. Admittedly, currently available recrystallization models are highly simplistic approaches to the complex and still poorly understood process of recrystallization, but they provide methods to estimate changes in bulk anisotropy (Gottstein 2002).

5. Important texture types in metals

By far most effort has been made in the characterization of metal textures and among those aluminium and steel received much attention. A majority of texture researchers investigate conditions leading to favourable textures for particular applications. Many industrial processing facilities characterize texture patterns to achieve required standards. One of the best known examples is steel sheet or aluminium alloy sheet for car body panels, where severe requirements concerning desired anisotropy have been set by the car industry for decades. In the case of thin-walled materials such as aeroplane components and beverage cans texture optimization is crucial to ensure satisfactory mechanical properties at a minimal cost as well as minimal material loss in manufacturing. While many details are known about metal textures, there is a relatively small variety of fundamental types, compared to ceramics and geological materials. Crystal structures are basically fcc, bcc and hcp, not counting some rare ordered alloys. Deformation is principally by rolling, extrusion (of wires) and rarely compression. Deformation may occur at cold or hot conditions and is often followed by annealing. In this section we will introduce the main texture types that are observed and highlight a few issues. There are several reviews on the diversity of metal texture types that should be consulted, among them Dillamore and Roberts (1965), Hatherly and Hutchinson (1979), Mecking (1985), Rollett and Wright (2000) and Wassermann and Grewen (1962).

5.1. Fcc metals The weakest slip systems in fcc metals are {111}110, consisting of 12 symmetrically equivalent variants. Activity of these systems produce a characteristic texture pattern during rolling, which is illustrated as pole figures and orientation distribution sections for copper (figures 9(a) and 10). The large number of slip systems makes it easy to achieve compatibility and the rolling texture can be well explained with the Taylor theory, especially if individual slip systems are allowed to harden, according to their activity (Kocks and Mecking 2003) and if allowance is made for some heterogeneity across small grain boundaries (Honneff and Mecking 1981). As is obvious, the texture pattern is complex, even for such a simple case where only a single family of slip systems is active. The reasons for this complexity are obvious if one considers polycrystal plasticity with characteristic rotations of individual orientations that vary in direction and amount. Crystal orientations rotate through orientation space and those regions in orientation space where rotations are smallest and collect dynamically large grain numbers generally correspond to high orientation densities; regions where rotation increments are large correspond to depleted orientation densities. A maximum in the ODF is rarely a stable orientation where all rotations converge. More often it is a transient where orientations temporarily accumulate. In order to simplify the description and facilitate quantitative comparisons, the three- dimensional orientation distribution of the rolling texture has been divided into idealized components, defined as orientations with a lattice plane normal (hkl) in the normal direction 1386 H-R Wenk and P Van Houtte

Figure 9. Pole figures of rolled fcc metals. (a) Rolled copper determined from a single synchrotron image measured at ESRF. (b) Brass measured with an x-ray pole-figure goniometer. (c) Recrystallized copper measured with a pole-figure goniometer and displaying mostly the cube texture with a minor component due to twinning. Equal area projection, logarithmic pole density scale. Ideal components are indicated in (a): cube, ♦ Goss, × S, ◦ copper, • brass.

and a lattice direction [uvw] in the rolling direction. This description dates back to the dawn of texture analysis (e.g. Polanyi and Weissenberg (1923)), long before the ODF was introduced (Bunge 1965) and was a first step in quantifying the three-dimensional aspect of textures. Table 1 summarizes the main fcc texture components. They are also shown with symbols in figures 9(a) and 10. Some words of caution are appropriate about the ideal component description. First, components are distributions centred on the ideal orientation. Polycrystal plasticity demonstrates that the real distributions are much more complex and generally asymmetrical. Second, particularly geologists have been tempted to associate ideal components with deformation mechanisms, assuming that the slip plane normal is parallel to the normal direction and the slip direction parallel to the rolling direction. In fcc metals there is no {111}110 component, the main slip system, and in most crystals several slip systems are active simultaneously! There are a few cases of low symmetry crystals and deformation conditions where such a simplistic interpretation applies but in most cases it does not (see, e.g. Wenk and Christie (1991)). Two main fcc rolling textures are distinguished, the copper type with Cu, S and brass components (figure 9(a)), and brass with brass and Goss (figure 9(b)). In low stacking fault metals such as brass mechanical twinning on {111}112¯ accompanies slip and this produces characteristic texture differences. There is a range of intermediate types that depend on the composition and temperature, both affecting the stacking fault energy (e.g. Alam et al (1967)). More refined idealized descriptions have been introduced, such as ‘texture fibres’ corresponding to scattering about a line in orientation space. Such ‘skeleton lines’ may be Texture and anisotropy 1387

Figure 10. ODF of rolled copper (same sample as shown in ﬁgure 9(a), shown as γ -sections (Roe/Matthies convention) (COD: crystal orientation distribution relative to sample coordinates). Orthorhombic sample symmetry is assumed. Ideal components are indicated in (a): cube, ♦ Goss, × S, ◦ copper, • brass. (a) Contoured rectangular sections. (b) Polar coordinates with a less distorted orientation space.

Table 1. Ideal texture components for rolled fcc metals.

Euler angles

Roe ψ, θ, φ a Component {hkl}uvw Matthies α, β, γ Bunge , ϕ1, , ϕ2 Cube, {001}100 00 0000 Copper, ◦ {112}111¯ 180 35 135 90 35 45 S, ×{123}634¯ 211 37 117 59 37 63 Brass, • {110}112¯ 35 90 135 35 45 90 Goss, ♦ {110}001 09013504590

a Crystallographic equivalent angles inside the unit zone for orthorhombic sample symmetry (as in rolling). defined on the basis of crystal-sample geometry (and have also been used way back, e.g. Glocker (1924)) or extend irregularly through the ODF along regions of high densities. The α-fibre, with 110 parallel to the rolling direction (0˚, 0–90˚, 45˚, Bunge convention), connects brass and Goss components, the β-fibre connects copper and brass components (Hirsch and Lucke¨ 1988), the γ fibre has {111} parallel to the normal direction (0–90˚, 55˚, 45˚, Bunge convention). 1388 H-R Wenk and P Van Houtte

Figure 11. Plot of volume fractions of ideal components for copper with increasing recrystallization (Rollett and Wright 2000).

Figure 12. Inverse pole ﬁgures illustrating axial deformation of fcc metals. (a) Copper deformed in compression to a strain of 0.68 (courtesy C Necker). (b) Gold wire (Wenk and Grigull 2003). Equal area projection, logarithmic contours.

The orientation density that is associated with a fibre or spherical component can then be used to follow changes, e.g. during recrystallization, where the cube component dominates over others (figure 9(c)). This fcc recrystallization texture has been explained as preferential nucleation on shear bands with high misorientations (Beaudoin et al 1996). Often a minor {111}112¯ twin component is observed. With the component method the texture changes during annealing can be quantified (figure 11). Puig-Molina et al (2003) observed with synchrotron x-rays in a furnace the in situ evolution of the cube component, which occurred within a few minutes around 300˚C. Relatively little work has been done on fcc torsion textures (simple shear). Canova et al (1984) proposed ideal orientations based on Taylor simulations. Van Houtte (1981), Stout et al (1988) and Hughes et al (2000) documented considerable variation with material, particularly stacking fault energy. At a first glance simple shear textures resemble rolling textures but rotated against the sense of shear. In axisymmetric compression fcc materials such as copper display a strong 110 maximum parallel to the compression direction (figure 12(a)). In extension, e.g. wire drawing, a 111 maximum in the extension direction develops as illustrated for gold (figure 12(b)). Axisymmetric textures are best illustrated in inverse pole figures. Texture and anisotropy 1389

Figure 13. Texture variations for electro-deposited nickel produced by deep etch lithography (the LIGA process) as function of pH and deposition rate (Dini 1993).

There has been much interest in textures formed by electro-deposition, e.g. in the production of microactuators where orientation patterns are of considerable importance. Artiﬁcially patterned structures have been produced by deep etch lithography (the LIGA process) and applied extensively to Ni and Ni–Fe alloys. Strong texture development has been observed and variations depend on deposition conditions (ﬁgure 13, see, e.g. Van Acker et al (1994), Dini (1993) and Buchheit et al (2002)).

5.2. Bcc metals The most common deformation mode in bcc metals is {110}111 slip, which is a transposition of slip plane and slip direction with respect to fcc metals. This correspondence produces some analogies: compression textures for bcc are similar to extension textures for fcc. Rolling pole figures for bcc can be obtained from those of fcc by reversing extension direction (rolling direction) and compression direction (normal direction). But in detail the situation is more complicated, because bcc metals also slip on other planes than {110} in the 111 direction (Christian 1970) and this can be described by the so-called ‘pencil glide model’ (Rosenberg and Piehler 1971, Van Houtte 1984). Table 2 lists the main bcc texture components due to rolling and annealing. They have the convenient property that they can all be represented in a ϕ2 = 45˚ section of the ODF (Bunge notation), as shown in figure 14(a). Such sections are widely used in the steel literature to represent textures and figure 14(b) shows an example of cold-rolled steel. The bcc texture development has been most extensively studied for steel and there is considerable variation with deformation conditions and composition (Ray et al 1993). Addition of 3% Si profoundly enhances the texture as illustrated by orientation density variations along the α fibre axis, both in cold and hot rolled steel (figure 15). At least qualitatively deformation textures of rolled fcc and bcc metals can be well simulated with the Taylor model, since many slip systems are available to satisfy compatibility without undue stress concentrations. However, this is not the case for axial deformation of fcc metals in compression and bcc metals in tension with a lack of slip systems to deform crystals to an axial shape. In this case the local deformation is in plane strain, resulting in a ‘curling’ microstructure (Hosford 1964). 1390 H-R Wenk and P Van Houtte

Table 2. Ideal texture components for rolled and annealed bcc metals.

Euler angles

Roe ψ, θ, φ b a Component {hkl}uvw Code Matthies α, β, γ Bunge ϕ1, , ϕ2 Cube {001}100 C 0 0 0 45 0 45 Rotated cube {001}110 H 135 0 135 0 0 45 {111}110 E1 90 55 135 0 55 45 {111}110 E2 150 55 135 60 55 45 ¯ {111}112 F1 120 55 135 30 55 45 ¯ {111}112 F2 0 55 135 90 55 45 {112}110¯ I 359013503545 Goss {110}001 G 0 90 135 90 90 45

a Crystallographic equivalent angles inside the unit zone for orthorhombic sample symmetry (as in rolling). b See ﬁgure 14(a).

Figure 14. (a) Texture components in rolled bcc metals displayed in a φ1 − , φ2 = 45˚ section of the ODF. The relative orientation of a cube is illustrated (see table 2). (b) φ2 = 45˚ section of the ODF for cold-rolled steel.

5.3. Hcp metals Texture development in hexagonal metals has received a lot of attention because of the applications of zirconium alloys in the reactor industry and titanium alloys as structural materials in aerospace. In contrast with cubic metals twinning is always signiﬁcant and correspondingly textures are more complex. Because of the low crystal symmetry several families of slip systems need to be activated. The critical shear stresses depend on composition (particularly the C/A ratio), temperature and degree of deformation (including hardening characteristics, e.g. Chin and Mammel (1969), Thornburg and Piehler (1975), Tenckhoff (1988) and Philippe (1994)). Major deformation systems are listed in table 3. A typical rolling texture of titanium is shown in ﬁgure 16(a). It is characterized by a spread-out c-axis maximum around the normal direction and (1010)¯ poles concentrated in the rolling direction. Lebensohn and Tome´ (1993) have modelled texture evolution of hcp metals with the self-consistent theory and could explain experimental patterns as a result of mainly prismatic and some pyramidal Texture and anisotropy 1391

Figure 15. Texture variations in bcc iron as function of Si content. Shown are orientation density variations along the α-ﬁbre (Sudo et al 1981).

Table 3. Main deformation systems in hcp metals. The critical resolved shear stress ratios (crss) are typical for zirconium deformed at low temperature.

System {hkil}uvw crss Twinning shear Slip Prismatic {1010¯ }12¯ 10¯ 1 Basal {0001}21¯10¯ 2.5 Pyramidal c + a{1011¯ }1¯123¯ 2.5 Twinning Tensile {1012¯ }1011¯ 1.2 0.167 Compressive {21¯12¯ }21¯1¯3¯ 1.7 0.225

slip, combined with compressive and tensile twinning. Twinned orientations have a distinct pattern and c-axes concentrate near the normal direction (figure 17), whereas grains with c-axes at large angles to the normal direction (near the transverse direction) do not twin. During recrystallization subtle changes are less dramatic than in fcc metals (figure 16(b)) and those changes are best revealed in difference pole figures (figure 16(c)). They indicate that grains with c-axes slightly tilted to the normal direction and a = (1120¯ ) parallel to the rolling direction become dominant in the recrystallization texture (see also Wagner et al 2002). These are orientations that are most heavily twinned (figure 17) and presumably twinned regions with high surface energy and heterogeneity are favoured nucleation sites and those nuclei eventually grow.

5.4. Phase transformations Metallic compounds undergo phase transformations when subjected to temperature–pressure changes. If a polycrystalline material is textured then the new phase may inherit texture information from the parent phase. Orientation relations have been suggested based on analogies in crystal structures (e.g. Kurdjomov and Sachs (1930) for bcc→fcc and Burgers (1934) for hcp→bcc) (table 4). The orientation relations suggest that in corresponding structures close-packed or nearly close-packed lattice planes are parallel, and that close-packed 1392 H-R Wenk and P Van Houtte

(a)

(b)

(c)

Figure 16. Textures of rolled titanium. In situ synchrotron measurements with a vacuum furnace. (a) 200˚C, before the onset of recrystallization, (b) at 700˚C after recrystallization, (c) difference pole ﬁgure (starting material-recrystallized material). Equal area projection (Puig-Molina et al 2003). Logarithmic contours for (a) and (b), linear contours for (c).

Figure 17. Texture simulations for hcp deformation using the VPSC theory during rolling to 50%. + symbols indicate grains that have not twinned, × are grains that have twinned at least once; symbol size is proportional to the deformation of individual grains. (001), (100) and (110) pole ﬁgures are shown in equal area projection (Wenk et al 2004).

Table 4. Orientation relations during phase transformations and number of variants. →←

fcc → hcp {111}110→(0001)1120¯ 42 bcc → fcc {110}111→{111}110 24 24 hcp → bcc (0001)1120¯ →{011}1¯11¯ 612

directions are parallel. In each case there are several symmetrically equivalent variants. If all variants were applied the texture would more or less randomize. There appears to be a strong variant selection in most of these transformations, regardless of whether the transformations occur by a martensitic (shear) mechanism or by nucleation. There is much interest, because Texture and anisotropy 1393

(a)

(b)

(c)

Figure 18. In situ observation of texture changes during phase transformations, measured with the HIPPO TOF neutron diffractometer at Los Alamos. (a)–(c) ultra low carbon iron (Wenk, unpublished), (d)–(f) zirconium (Wenk et al 2004). (a) 800˚C, bcc; (b) 950˚C, fcc; (c) 400˚C, bcc after cooling; (d) 650˚C, hcp; (e) 950˚C, bcc; (f) 650˚C, hcp after cooling. Pole figures in equal area projection. Rolling (RD), normal (ND) and transverse directions (TD) are indicated. the ‘texture memory’ is relevant for novel shape memory alloy applications (such as nitinol, e.g. Vaidyanathan et al (2001)). The selection principles are poorly understood, even for the technologically important bcc→fcc transformation in steel which has been studied in most detail (Ray and Jonas 1990). It is established that microstructure, composition and stress all have an influence. The reason for the lack of knowledge is partially the difficulty of measuring textures at high and low temperatures in situ. This is changing with the availability of new neutron and synchrotron facilities that can accommodate vacuum furnaces. We will have a closer look at the bcc→fcc transformation in ultra low carbon iron, where it has recently become possible to measure the high temperature fcc texture (950˚C) in situ by neutron diffraction. At 800˚C we find a typical rolling texture (figure 18(a)). After the transformation (800˚C) we observe a Kurdjomov–Sachs relationship with the fcc (110) pole figure roughly corresponding to the bcc (111) pole figure (figure 18(b)). Upon cooling to 400˚C the texture returns almost exactly to the initial bcc texture (figure 18(c)), documenting a texture memory and variant selection. Similar relationships have been documented for the hcp→bcc transformation. At 650˚C the texture of zirconium is similar to that of titanium (figures 16(a) and 18(d)). The high-temperature (bcc) texture, measured at 950˚C (figure 18(e)), is related to the low-temperature (hcp) texture by the Burgers relation, but with a strong variant selection. Interestingly this experimental texture is different from that predicted on the basis of models (e.g. Ciurchea et al 1996, Gey and Humbert 2002). The cubic transformation texture is best explained if preferential nucleation of the bcc phase takes place in the most highly twinned hcp grains (figure 17), similar to the recrystallization case described above. After cooling the new hcp texture closely resembles the original texture (figure 18(f )), also here with a strong memory, probably imposed by stresses from neighbouring grains. 1394 H-R Wenk and P Van Houtte

6. Ceramic textures

Polycrystalline ceramic materials are either produced in bulk or as thin coatings on substrates. Many of the bulk processing methods, such as tape casting, extrusion and injection moulding involve deformation that may produce textures. Whereas in metals most textures develop by deformation, recrystallization and casting (growth from melt), more mechanisms are active in ceramics. Processes such as cold pressing (‘green body pressing’), hot pressing (densification and phase transformation), surface grinding, epitaxial and topotaxial growth can all lead to textures in ceramics. Ceramic coatings display often clear epitaxial or topotaxial relationships between substrate and film. While textures are often weaker than in metals, ceramics have strongly anisotropic properties and optimization of textures has become increasingly important. Compared to metals and to geological materials the literature on texture development in ceramics is surprisingly sparse. Some of the pioneering work was that of Pentecost and Wright (1964) who demonstrated, with pole figures, an alignment of crystallites in pressed powders of plate-shaped Al2O3 and needle-shaped BeO. Textures in ceramics were reviewed by Bunge (1991), documenting that research in this field is growing as new materials are being manufactured with critical properties, brittle, ductile and electrical. The following discussion highlights some systems, dividing ceramics into two sections: bulk ceramics and films.

6.1. Bulk ceramics ¯ 6.1.1. α-alumina (Al2O3). Trigonal aluminium oxide (point group 32/m, mineral name corundum) is the most widely used structural ceramic, such as for lamp envelopes, spark plugs and substrates for integrated circuits. Hot forging and deep drawing experiments have documented that large scale deformation is a viable means of forming alumina ceramics. In hot forging intracrystalline slip is active and contributes to texture development (Heuer et al 1980). The most important production method is green processing of powders (die pressing, slip casting, tape casting, extrusion) and since alumina particles are generally plate-shaped, textures are produced (Dimarciello et al 1972, Bocker¨ et al 1991, Raj and Cannon 1999). The powder processing is followed by sintering to produce cohesion by grain growth, which often enhances the texture. Since thermal expansion is anisotropic, small residual stresses are introduced during thermal cycling in highly textured alumina which can lead to microfractures and eventual brittle failure (Lee et al 1993).

6.1.2. Silicon nitride (Si3N4). Silicon nitride is a ceramic material with numerous new applications in automotive components and machining tools due to its high stiffness, strength and hardness, coupled with good fracture toughness and thermal shock and corrosion resistance. It occurs in two hexagonal forms, α and β. The high-temperature β-phase forms rod-shaped needles, elongated along the c-axis, which can be oriented by hot-pressing or forging (Walker et al 1995). The textured materials have very anisotropic fracture properties (Willkens et al 1988). In accordance with the grain shape (needles and platelets, respectively), the preferred orientation is ‘inverse’ to that of alumina, i.e. (0001) poles are at high angles to the direction of principal compression, in axial and plane-strain compression. The fracture toughness generally increases and becomes more anisotropic with texture development.

6.1.3. Zirconia (ZrO2). Zirconia occurs in a cubic (when doped with Y), a tetragonal (high- temperature) and a monoclinic (lower temperature) structure (mineral name baddeleyite). The tetragonal to monoclinic phase transformation can be stress-induced (martensitic) and is greatly Texture and anisotropy 1395 inﬂuenced by crystallite orientation (Muddle and Hannink 1986). While the texture of the monoclinic phase is most relevant to generating strains and plastic work, the texture of the tetragonal phase can be advantageously tailored to enhance transformability and toughness (Bowman and Chen 1993). The tetragonal parent phase transforms into the monoclinic phase by simple shear on the plane (100) in the [001] direction. In zirconia texture is produced during the phase transformation directly by the selective stress-induced transformation and does not require the parent material to be textured.

6.1.4. Ceramic matrix composites. During many investigations of composite ceramics, it became apparent that reinforcement textures are extremely important for mechanical properties. Yet a quantitative characterization of textures is generally lacking. Experimental work on alumina–SiC whisker composites has shown that little texture development is observed in the alumina matrix under conditions that have been shown to produce strong preferred orientation in pure alumina (Sandlin and Bowman 1992). By contrast, whiskers attain strong preferred orientation due to their elongated grain shape. Zirconia ceramics have been reinforced with alumina platelets to increase fracture toughness by crack deflection and cracks arresting at platelets (Li and Sorensen 1995). These mechanisms depend on textures. Ceramic composites consisting of a SiC matrix, reinforced by SiC fibres have recently been developed for thermostructural applications. (SiC exists both in a cubic form, β, and a hexagonal form, α.) The advantage of such composites lies in the low density, the high mechanical strength and rigidity and their chemical inertness. Diot and Arnault (1991) documented that the SiC matrix has 111 directions preferentially aligned parallel to the fibre axis of the composite. Such a texture has the lowest surface energy between matrix and fibres, and is preferred.

6.1.5. Bulk high-temperature superconductors. The conductivities of the high-temperature cuprate superconductors are highly anisotropic and largely confined to the (001) Cu–O plane (e.g. Tuominen et al (1990)). Techniques for developing strong preferred orientations in macroscopic samples have been successful in improving critical current densities in many of these materials (e.g. Kumakura (1991)). Among the cuprate oxides, texture development is best documented for the Y-123 compounds (YBa2Cu3O7−x ). Significant degrees of texture can be produced by high- temperature plastic deformation of polycrystalline pellets (Wenk et al 1989, Chen et al 1993). [001] axes of Y-123 align themselves with the compression direction, due to intracrystalline slip (figure 19(a)). Other techniques to produce texturing in Y-123 are melt textured growth (e.g. Selvamanickam and Salama (1990)) and alignment in a magnetic field (e.g. deRango et al (1991)). Other compounds of interest are Bi-2223 (Bi2Sr2Ca2Cu3Ox ) and Bi-2212 (Bi2Sr2CaCu2Ox ). The plate-like crystal morphology makes these materials suitable for green- body processing (Steinlage et al 1994). Plate-like crystals are preferentially oriented with [001] axes (normal to the plane of the plate) parallel to the compression direction (figure 19(b)) and preferred orientation develops largely by rigid body rotations. Unfortunately techniques using axial stress produce a relatively low degree of crystallite orientation. More successful has been a method to sheath Bi-superconductors in silver tubes and fabricate a tape or wire and then thermally treating it (Sandhage et al 1991, Wenk et al 1996). Silver is used as the sheath material due to its chemical compatibility with Bi-2212 and 2223. It also supplies ductility and protection for the tape or wire for use in potential applications such as long power transmission lines and superconducting coils. 1396 H-R Wenk and P Van Houtte

(a) (b)

Figure 19. Texture development in high-temperature superconductors (inverse pole ﬁgures, assuming tetragonal crystal symmetry). (a) Hot compression of Y-123 (Wenk et al 1989), (b) Bi-2212 multiﬁlamentary tape encased in silver (Wenk et al 1996).

6.2. Thin films and coatings In thin films the importance of texture is obvious (e.g. Szpunar (1996)). Texture influences elastic properties and thermal expansion that are essential for the mechanical stability of films. Also, in the case of superconducting films, electrical properties are intimately linked to crystal orientation. Qualitative texture information on thin films has been extracted from powder diffractometry. Only recently have pole figure measurements and orientation distribution analysis been used to quantify crystallite orientation. There are two types of film textures. The first type consists of films and coatings on polycrystalline or amorphous substrates in which the influence of the substrate is minor and the texture is formed mainly by anisotropic growth. These are generally axially symmetric fibre textures. Of a second type are epitaxial films deposited on a structurally related single crystal. In those the texture is usually extremely strong, approaching a single crystal and is controlled by the match between the two crystal structures.

6.2.1. Silicon and diamond. The broad range of applications of polycrystalline silicon films in microelectronics and in the fabrication of micromechanical structures for use in actuators and sensors demands that the material be well characterized. From the mechanical point of view, the elastic properties and internal stresses of films during thermal cycling are texture dependent and need to be optimized for satisfactory performance. Silicon films deposited from vapour on (111) single crystal substrates on an amorphous silicon dioxide layer show a large variation in texture which is mainly controlled by temperature and silane pressure (e.g. Joubert et al (1987), Wenk et al (1990)). At high silane pressure and low temperature, Si-films are amorphous (figure 20(a)). With increasing temperature and lower pressure textures change from a strong {113/112} fibre to {110} (figure 20(b)) and then to {100} (figure 20(c)). At lowest pressure and highest temperature crystallites are aligned randomly. Low-temperature films show tensile stresses (up to 700 MPa), high-temperature films display compressive stresses (up to 600 MPa). Combinations of the texture types can be used to attain overall equilibrium. Texture types in polycrystalline Si-films can be correlated with microstructures. The {110} texture in the 620–650˚C range is due to the columnar grains, which share 110 as the growth direction (Krulevitch et al 1991). 110 and 112 are fast growth directions for silicon that crystallizes out of an amorphous state. At low Texture and anisotropy 1397

Figure 20. Si films deposited on a single crystal Si wafer, covered with amorphous silicon dioxide. (a) Variation in texture as function of temperature/silane-pressure. (b) and (c) Typical textures represented in inverse pole figures. Stereographic projection, contour interval for (a) and (c) 0.25 m.r.d., for (b) 0.5 m.r.d., shaded above 1 m.r.d. (Wenk et al 1990). temperature twinned grains oriented with the 110 direction close to the film normal survive and become columnar in shape. At higher temperatures, when twinning does not accompany crystallization, 100 is the direction of fastest crystal growth. For low stress 700˚C films deposited at 300 mTorr, there is a lower driving force for twinning, resulting in columnar grains elongated along 100 and consequently a predominantly {100} texture. Recently diamond coatings have received attention. Also here the texture varies with fabrication conditions and can be very strong (e.g. Brunet et al (1996), Helming et al (1995)).

6.2.2. Nitride, carbide and oxide coatings. Coatings are often applied to metal and ceramic substrates to improve properties for various applications. They may improve heat and wear resistance, lower frictional forces, prevent chemical corrosion resistance or simply add to the decorative appearance. Carbides and nitrides of transition metal coatings have led to significant improvements of cutting tools for turning and milling operations. Coatings are often prepared by arc evaporation and deposited from a gaseous phase on to a substrate, e.g. tungsten carbide (WC). Leonhardt et al (1982) describe strong {111} fibre textures for TiC, typical of high- temperature deposition. At lower deposition temperature, a {100} fibre is also observed. Similar textures are observed in TiN and HfN coatings on WC (figures 21(a) and (b)). Sue and Troue (1987) have documented that (100) textures reduce the erosion rate compared with (111) (figure 21(c)). Textures of oxide coatings on polycrystalline substrates appear to be mainly controlled by growth kinetics rather than by the texture of the substrate material. Zirconium oxide has a thermal expansion coefficient similar to that of stainless steel and thus only a small thermal mismatch when deposited on metallic substrates. Because of its high hardness and toughness, cubic yttrium stabilized zirconia (YSZ) is one of the most widely used protective coatings in the automotive, aeronautical and cutting tool industry (Rhys-Jones 1990). A survey of ferroelectric Pb2ScTaO6(PST) films on various substrates revealed a strong 111 fibre texture when deposited on a (100) silicon crystal coated with platinum (figure 22(a)) and a 001 fibre on (110) Al2O3 (figure 22(b)) (Chateigner et al 1997). The ferroelectric and fatigue behaviours of such films depends on crystallite orientation (Kim et al 1994). 1398 H-R Wenk and P Van Houtte

Figure 21. Textures in HfN (a) and TiN (b) tool coatings, represented as inverse pole ﬁgures, stereographic projection. (c) Dependence of the alumina particle jet erosion rate on a TiN coating as a function of texture, evaluated as (111)/(200) diffraction peak intensity ratio (Sue and Troue 1987).

Figure 22. Inverse pole ﬁgures of PST ﬁlms on substrates. (a) on platinum coated (100) Si, (b)on (110) Al2O3. Equal area projection (Chateigner et al 1997).

6.2.3. Epitaxial films. Much research has been invested in producing epitaxial superconducting films on various single crystal substrates, in an effort to obtain electronic devices with favourable electrical properties. TEM studies have documented that the films consist of a polycrystal with small crystallites but with a very high degree of preferred orientation. The orientation of the crystallites is controlled by epitaxy and growth velocity. The balance between the two factors is primarily dependent on the distance from the substrate surface (and thus the foil thickness) and the temperature and partial pressures at which the crystals grow. Because superconductivity in HTS ceramics is restricted to the Cu-plane, it is important to have the corresponding lattice plane (001) parallel to the film. The goal is to find conditions at which relatively thick films have a favourable crystal alignment with the substrate. Most work so far has been done on Y-123 films on various substrates. In this system it has been useful to measure pole figures of the combined 102 and 012 reflections of Y-123 because this reflection does not coincide with the substrate. Two orientations are of primary interest: the C-orientation has c-axes of Y-123 parallel to the foil normal and the A-orientation has c-axes in the foil plane and a-axes of Y-123 in the foil normal (Heidelbach et al 1992). The two orientations are expressed by 102 peaks at pole distances of 56˚ (C-orientation) and 34˚ (A-orientation), respectively (figure 23(a)). The a-axes of Y-123 crystallites in the C-orientation are aligned either parallel or at 45˚ to those of the substrate depending on the best match in the oxygen sublattice of film and substrate in the perovskite, periclase and fluorite structures (Tietz et al 1989). In these very strong epitaxial textures, where widths of texture peaks are less than 5˚, pole figures measured with 5˚ × 5˚ angular increments give only qualitative information about orientation relationships. To obtain quantitative values, texture Texture and anisotropy 1399

Figure 23. (a) 102 pole figure of a thin film of Y-123, laser ablated on (001) LaAlO3 at 710˚C. Peaks from A orientation (with [100] normal to the substrate) and C-orientation (with [001] normal to the substrate) are indicated. (b) Change of integrated C/A peak intensity ratio as function of deposition temperature (Wenk et al 1996). peaks need to be scanned on a much finer grid and effective peak intensities can be integrated. The ratio of the volume of crystals with c-axes in the foil plane to that of crystals with a-axes parallel to the foil plane (C/A ratio) on (001) LaAlO3 shows a steady increase with deposition temperature (figure 23(b)). A high C/A ratio is obviously preferred. Deposition on single crystal substrates is adequate for small electronic devices. Many other applications require larger structures and flexibility. Here laser deposition of thick superconducting films on textured metallic substrates has been employed, particularly biaxially textured Ni-tapes (Wu et al 1995, Yang 1998).

7. Textures in minerals and rocks

Geological polycrystals are referred to as rocks. Some rocks are monomineralic such as marble, composed of calcite, and quartzite, composed of quartz. Many rocks are polymineralic with examples such as granite, composed of feldspar, quartz and mica, and peridotite, composed of olivine and pyroxene. We will introduce a few minerals where texture development has been investigated in detail and patterns of preferred orientation have added to a better understanding of deformation in the earth. Calcite is one of the best studied minerals with strong textures developing by slip, mechanical twinning and recrystallization. Also quartz has been studied extensively, but many aspects of texture development remain enigmatic. The two trigonal minerals are also of relevance to metallurgists since they deform on similar slip systems as hexagonal metals. Orthorhombic olivine is important because of its significance for convective flow in the earth’s mantle and the seismic anisotropy that is observed. Sheet silicates are a group of minerals with a distinct platy morphology and their alignment during deformation or compaction has been modelled as rigid particle rotations in a viscous medium. Textures in ice have been investigated to better understand glacier flow and deformation of the large ice shields, as well as the rheology of the large outer planets. An interesting model system is halite (NaCl) and isostructural periclase (MgO) with high plastic anisotropy. Following a brief survey of these minerals, we will illustrate two geological applications: the use of texture in unravelling strain history and the importance for understanding seismic anisotropy. Earth materials differ from metals in several respects. Whereas ODFs of cubic metals can be represented in a small orientation volume and share many common features that can 1400 H-R Wenk and P Van Houtte

Figure 24. Graph of normalized shear strain rate plotted against shear stress, illustrating the effect of the stress exponent n on the deformation behaviour. High strain rate sensitivity, approaching a viscous behaviour, is present in minerals (n = 3–9), compared to metals (n ∼ 99). be catalogued (e.g. as component volume fractions), ODFs of low symmetry materials require large volumes, are specific in each case, with few common features. The sample coordinates of geological specimens are generally not a priori defined and it is impossible to visualize an ODF in a different orientation. For these reasons ODFs, though essential for interpretations, calculations of properties and texture simulations, are often difficult to interpret and pole figures remain an important representation. Minerals have a high strain rate sensitivity (low stress exponent; 3–9 versus >99 in metals) and differently oriented crystals may deform at very different rates and long before the critical shear stress is reached (figure 24). Minerals were the incentive to introduce strain sensitivity in polycrystal plasticity. Minerals have lower symmetry and thus fewer equivalent slip systems. This causes a high plastic anisotropy. Both features suggest that deformation of mineral aggregates is heterogeneous and this has to be taken into account in models. It must be emphasized that texture is the result of the accumulated strain history, which is particularly significant for geological situations where the strain path often changes in the course of a long history. In simulations such changes can be easily included and texture evolution, for example, in a highly heterogeneous system such as a convection cell in the earth’s mantle, can be simulated.

7.1. Calcite (CaCO3) Calcite rocks (limestone and marble) have long been a model system for experimental rock deformation. The classic Yule marble studies (e.g. Turner et al (1956)) were the basis for a quantitative interpretation of texture development in minerals by slip and twinning. Largely based on single crystal studies, the main deformation systems were established: slip on the rhombs r ={1014¯ }202¯1¯ and f ={1012¯ }022¯ 1¯220¯ 1¯ and mechanical twinning on e ={0118¯ }4041¯ . (Note that there is a preferred sense to slip on the rhombohedral systems.) In axial compression experiments, regular texture variations were observed with temperature, pressure, strain rate and grain size (Wenk et al 1973, Schmid et al 1977). At low temperature Texture and anisotropy 1401

Figure 25. Calcite deformed in axial compression and textures represented in inverse pole figures of the compression direction. (a) Experimental texture observed for fine-grained limestone at low temperature; (b) Taylor simulations for fully imposed compatibility fail to predict the experimental texture; (c) relaxation of constraints for curling are in much better agreement (Wenk et al 1986). (d) Experimental high temperature texture with significant grain growth (Wenk et al 1973). (e) Results from a recrystallization simulation that show with symbol size the degree of deformation of differently oriented grains, evaluated with the self-consistent model (Lebensohn et al 1998). a texture with a maximum of compression axes near c = (0001) and a shoulder towards the negative rhomb e = (0118¯ ) develops (figure 25(a)). Taylor simulations failed to reproduce the low temperature texture (figure 25(b)), until it was recognized that local plane strain deformation was energetically favourable over axial deformation, analogous to curling in cubic metals (Hosford 1964). When Taylor conditions are relaxed the correct texture is predicted (figure 25(c)). At high temperature, where grain growth indicates recrystallization, the texture has a maximum at high-angle positive rhombs (figure 25(d)). The high temperature texture can be explained as due to grain growth of the least deformed grains (figure 25(e)). As we will see later, this is rather exceptional in minerals. In most cases dynamic recrystallization favours the most highly deformed grains. Since calcite rocks such as limestone and marble are mechanically ductile compared to other minerals, deformation experiments other than axial compression can be performed more easily. Noteworthy are plane-strain compression experiments (referred to as ‘pure shear’ in the geological literature) with three mirror planes in the pole figure (figure 26(b)) and simple shear with only one mirror plane and a 2-fold axis in the pole figure (figure 26(c)) as compared to axial compression experiments which yield a fibre texture (figure 26(a)). These experimental pole figures are good examples to illustrate that the symmetry of pole figures reflects the symmetry of 1402 H-R Wenk and P Van Houtte

Figure 26. Calcite (0001) pole ﬁgures of experimentally deformed calcite limestone illustrating how the symmetry of the pole ﬁgure relates to the symmetry of the strain path: (a) axisymmetric compression (Wenk et al 1986), (b) plane-strain compression (Wagner et al 1982), (c) simple shear (Kern and Wenk 1983). Some contour levels (in m.r.d.) are marked. Compression, extension and shear directions are indicated by arrows. the deformation history (strain path) which has been a main criterion for interpreting textures in naturally deformed rocks (Paterson and Weiss 1961). Dynamic recrystallization was produced in torsion experiments to large strains (Pieri et al 2000). Naturally deformed calcite rocks, limestones and marbles, often display strong preferred orientation with a c-axis maximum more or less perpendicular to the schistosity plane. We will return to the interpretation of natural calcite textures later in this review.

7.2. Quartz (SiO2) Low quartz (α) is trigonal, point group 32. At high temperature α-quartz transforms to hexagonal β-quartz (point group 622). Both low and high quartz lack a centre of symmetry and exist in a right- and a left-handed form. The absence of a centre of symmetry is the reason for properties such as piezoelectricity in polycrystalline quartz (e.g. Parkhomenko (1971)). In investigations of preferred orientation by diffraction methods, enantiomorphism cannot be identified and a higher crystal symmetry is used by adding a centre of symmetry to point group 32. This produces point group 32/¯ m, which is that of calcite and is generally used in representations. Slip systems have been established by laboratory deformation of oriented single crystals (e.g. Blacic (1975)). Basal (0001) 1120¯ slip dominates at low temperature. At higher temperature prismatic slip {1010¯ } [0001] becomes active. These well established slip systems have a hexagonal symmetry, and if they alone were active, the orientation distribution in polycrystals should also display hexagonal symmetry and this is generally not the case. One explanation is that trigonality is introduced by mechanical Dauphine´ twinning (Zinserling and Shubnikov 1933). Dauphine´ twinning is geometrically a two-fold rotation about the c-axis but can be produced by a slight distortion of the lattice, either during the α–β phase transformation or by shear. It does not affect the orientation of c and a-axes. Dauphine´ twinning is a mechanism to produce ‘trigonal textures’. An example of a trigonal quartz texture in a naturally deformed mylonite is shown in figure 27(a)). The trigonality may have been introduced during crystallization under stress (Tullis and Tullis 1972). Twins in naturally deformed quartzite have actually been observed with EBSP orientation imaging analysis (Heidelbach et al 2000). Upon heating above 573˚C at ambient pressure, quartz transforms to hexagonal symmetry and pole figures for positive r ={1011¯ } and negative rhombs z ={0111¯ } that were different for trigonal Texture and anisotropy 1403

Figure 27. Texture changes during phase transformations in quartzite. In situ measurements with TOF neutrons. (a) Initial trigonal texture of a single crystal type with broad scattering (500˚C). (b) Hexagonal texture at 650˚C where the two rhombs 1011¯ and 0111¯ become equivalent. (c) Upon cooling to 500˚C the texture returns to the original orientation variant. Equal area projection.

quartz are now equal (figure 27(b)). Interestingly, upon cooling the material returns almost exactly to the starting texture (figure 27(c)), similar to the case of zirconium described earlier (figure 18(d)–(f)). It is speculated that the memory is imposed by stresses from neighbouring grains since α-quartz is elastically strongly anisotropic and these stresses produce a selection of the energetically favourable twin orientation. Texture development and deformation in naturally and experimentally deformed quartzites have been reviewed (Wenk 1994) and we only summarize some salient aspects. Polycrystalline quartz has been the subject of numerous experimental studies, mainly in axial compression. Figure 28 shows typical texture types developing during plastic deformation of fine grained quartz. At lower temperatures and high strain rates, c-axes concentrate near the compression axis (figure 28(a)). At higher temperature and slower strain rates a distinct maximum develops near r ={1011¯ } (figure 28(b)). The distinct difference between positive rhombs (h0hl)¯ and negative rhombs (0hhl)¯ (e.g. in figure 28(b)), can be at least partially attributed to mechanical Dauphine´ twinning. Experimental investigations also addressed the behaviour of quartz during dynamic recrystallization. Mainly based on microstructures, Hirth and Tullis (1992) identified three different mechanisms with increasing temperature, decreasing strain rate and thus decreasing flow stress: (1) strain induced grain boundary migration; (2) progressive subgrain rotation and (3) progressive subgrain rotation with rapid boundary migration at the highest temperatures. Changes in textures are characteristic of these mechanisms (Gleason et al 1993). 1404 H-R Wenk and P Van Houtte

(a) (b)

Figure 28. Typical texture patterns in axially compressed flint (a fine-grained aggregate of quartz) represented as inverse pole figures of the compression direction. At low temperature and high strain rates there is a maximum near c = 0001. At high temperature the maximum is shifted towards r = 1011¯ (Green et al 1975).

Many naturally deformed quartzite specimens have been examined. Most of these investigations relied on measurements of [0001] axes with a petrographic microscope, equipped with a universal rotation stage. The quartz texture types have been classified by Sander (1950) into c-axis maximum fabrics at low temperature with the maximum perpendicular to the foliation plane (figure 29(a)) and at higher temperature with pervasive recrystallization with the maximum in the foliation plane and perpendicular to the lineation (stretching direction) (figure 29(b)), small-circle girdles (figure 29(c)) and asymmetric crossed girdles (figure 29(d)). In all the representations the schistosity plane is horizontal and the lineation direction pointing to the right). The texture types are characteristic of formation conditions, particularly temperature (metamorphic grade). At low temperature and dominant basal slip the c-maximum perpendicular to the foliation plane develops (figure 29(a)). The crossed girdle type is characteristic of simple shear deformation with basal and prismatic slip active (Takeshita et al (1999), figure 29(d)). The type with the strong c-axis maximum in the intermediate fabric direction is characteristic of mylonites, intensely deformed rocks (figure 29(b), also 27(a)). The microstructure indicates recrystallization and plasticity simulations can explain this common texture type (Wenk et al 1997). In plane strain compression c-axes rotate towards the compression direction (figure 30(a)). However the most deformed grains are in the intermediate strain direction and if nucleation is allowed in these subordinate orientations, and the nuclei grow, they dominate the recrystallization texture (figure 30(b)). The cause of the small circle fabric is still unclear (figure 29(c)). It occurs at highest metamorphic grade (granulite facies) and may be due to activation of rhombohedral slip systems.

7.3. Olivine (Mg2SiO4) The orthorhombic mineral olivine is the major constituent of the upper mantle of the earth and has therefore been of long standing importance for understanding geodynamic processes. Natural olivine textures have been compiled by Ben Isma¨ıl and Mainprice (1998). By far the most common texture type has (010) poles nearly perpendicular to the foliation plane and [100] axes subparallel to the lineation direction (ﬁgure 31(a)). Texture development is attributed to deformation in the upper mantle at depth and the mantle rocks have later been juxtaposed in the crust where they can be sampled. Slip systems in olivine have been established in single crystal experiments (e.g. Raleigh (1968), Nicolas et al (1973), Bai et al (1991)). At high temperature (010)[100] is the Texture and anisotropy 1405

Figure 29. Typical quartz textures of naturally deformed quartzites represented as (0001) pole ﬁgures; Z is the pole to the foliation, X the lineation, and Y (centre) is the intermediate fabric direction. (a) Quartz layer in limestone (Thurnpass, Tirol), (b) Typical Y-maximum texture, common in mylonites (Melibokus, Odenwald Germany); (c) small-circle girdle observed in granulites (Burgstadt,¨ Saxony); (d) crossed girdle in quartz lens in marble (Hintertux, Tirol) (Sander 1950).

Figure 30. VPSC simulations of texture changes during deformation and recrystallization of quartz in pure shear plane strain. (0001) = c pole figures, shortening direction is vertical, extension direction horizontal. (a) Texture development during deformation. (b) and (c) Texture development if deformation is accompanied by recrystallization with significant nucleation. Initially new grains nucleate (×-symbols) and increasingly dominate the texture (c). Symbol size is proportional to grain size (Wenk et al 1997). preferred slip system and simulations reproduce the high temperature texture observed in mantle peridotites (figure 31(b)). Zhang and Karato (1995) produced simple shear textures and observed at moderate strain a [100] maximum displaced from the shear direction against the sense of shear and, at higher strain with pervasive recrystallization, a more symmetrical 1406 H-R Wenk and P Van Houtte

Figure 31. (a) Naturally deformed olivine from peridotite occurring as xenoliths in basalt from South Africa and originating from the upper mantle. (b) Based on known slip systems and assuming pure shear deformation this texture can be simulated with the VPSC theory. texture with [010] perpendicular to the shear plane and [100] in the shear direction. This texture transition can be explained with polycrystal plasticity models, again, as in the case of quartz, assuming that the most highly deformed orientations are favoured (Wenk and Tome´ 1999). We will return to olivine in the context of mantle convection.

7.4. Sheet silicates Sheet silicates such as micas (muscovite, biotite, chlorite, etc) and clay minerals (illite, smectite, kaolinite, etc) occur commonly in many deformed sedimentary (mudstone, shale, e.g. Ho et al (1999)) and metamorphic rocks (slate, schists, e.g. O’Brien et al (1987)). Sheet silicates are characterized by a plate-like morphology, parallel to the basal (001) lattice plane. This morphological anisotropy largely controls the orientation changes of sheet silicates during compaction and subsequent deformation. They are important because single crystals display extreme anisotropy of physical properties such as seismic wave propagation and rocks in which sheet silicates are oriented are also highly anisotropic (e.g. Kern and Wenk (1990)). The orientation behaviour of non-equiaxed, rigid particles in a viscous medium during deformation has been modelled by Jeffery (1923). In the extension of March (1932) and Willis (1977), in which particles are considered infinitely oblate (or infinite needles), the texture has been directly correlated with the finite strain. In spite of the severe constraints of the model which rarely apply to real systems, the predicted strain/texture relationship is often in good agreement with independent strain markers (e.g. Oertel (1983)) and experiments (Tullis 1976). The interpretation of (001) sheet silicate pole figures is based on their symmetry, their intensity and their geometrical relationship with regard to the mesoscopic structural elements such as bedding, cleavage and lineation (Weber 1981). In pure compaction, pole figures are axially symmetric with a maximum parallel to the bedding pole (figure 32(a)). In deformation textures axially symmetric pole figures, centred around the cleavage pole, indicate flattening Texture and anisotropy 1407

Figure 32. Examples of (001) sheet silicate pole ﬁgures. (a)–(c) are illite/muscovite pole ﬁgures from shales and slates in the Variscan fold-and-thrust belt in Belgium (Sintubin 1994); (d) is biotite from mylonites in Southern California which were partially deformed in simple shear (O’Brien et al 1987). Equal area projection. (a) Axisymmetric compaction texture (maximum is 6.4 m.r.d.), (b) axisymmetric deformation textures (maximum is 20 m.r.d.), (c) orthorhombic deformation texture (maximum is 10 m.r.d.), (d) asymmetric deformation texture produced by simple shear (maximum is 14.5 m.r.d.).

(figure 32(b)), whereas orthorhombic textures imply a plane-strain component (figure 32(c)). In some heavily deformed metamorphic rocks (mylonites), pole figures are often asymmetric (figure 32(d)) and this has been attributed to a simple shear component in the deformation history (O’Brien et al 1987).

7.5. Ice (H2O) Deformation of hexagonal ice–I has been the subject of numerous investigations and it has long been known that preferred orientation develops during the ﬂow of glaciers and deformation of the large polar ice sheets, with c-axes oriented perpendicular to the surface (e.g. Kamb (1959), Duval (1979), Lipenkov et al (1989), Tison et al (1994), Alley et al (1995)). These textures are largely attributed to ﬂattening in a glide and climb regime. Locally components of simple shear have been documented and in polar ice sheets there are layers with pervasive recrystallization. In experimental studies it was observed that during deformation and dynamic recrystallization c-axes align parallel to the compression direction (Duval 1981). Fabric development of ice has been modelled for regimes of deformation (Castelnau et al 1996) as well as recrystallization 1408 H-R Wenk and P Van Houtte

(Thorsteinsson 2002). Texture development greatly influences the anisotropic flow behaviour and has received a lot of interest (Duval et al 1983, Lliboutry and Duval 1985, Paterson 1991). Most of these preferred orientation studies were performed close to the melting point in the polymorph ice–I. Only recently has ice been deformed at low temperature and high pressure in the stability field of other polymorphs, relevant for the dynamics of outer planets and their satellites. Bennett et al (1997) have analysed ice textures by neutron diffraction at 77 K and observed that ice–I deformed at low temperatures also has c-axes parallel to the compression direction, whereas the high pressure polymorph ice–II has a-axes parallel to the compression direction.

7.6. Halite (NaCl) and periclase (MgO) The deformation behaviour of halite (‘rock salt’, NaCl) and related minerals has been extensively investigated, mainly because salt mines are being considered as suitable nuclear waste repositories and their long term stability needed to be evaluated. Most of these studies emphasize mechanical properties but texture plays obviously an important role. The review of Kern and Richter (1985) still gives a fairly up-to-date overview to which interested readers are referred. In compression experiments {110} fibre textures are generally produced (Kern and Braun 1973) and in extrusion experiments at low temperature a {100} fibre dominates (Skrotzki and Welch 1983). Salt textures in natural settings described by Goeman and Schumann (1977), Kampf¨ et al (1986) and Scheffzuk¨ (1996) document a {100} maximum perpendicular to the foliation plane. Ductility of single crystals and deformation mechanisms were investigated by Carter and Heard (1970). At low temperature {110}110¯ slip is dominant. At higher temperature {100}011 and {111}110¯ become equally active. Salt deformation is of interest for polycrystal plasticity because, though this mineral is cubic and has many slip systems, it is plastically highly anisotropic. This is because the favoured slip system {110}110¯ has only two independent variants and harder slip systems are required to deform an aggregate homogeneously. In lower bound models all deformation is concentrated on soft systems, whereas upper bound models equally activate hard systems to achieve compatibility. Thus different plasticity models give entirely different texture results (Wenk et al 1989). FEM has recently been applied to halite (Lebensohn et al 2003) and predictions from different models have been compared with new experiments performed in axial extension. Differences between the models are best visible in the grain shape distribution as illustrated for d23 strain rate components in figure 33. For homogeneous deformation (Taylor) all grains have the same strain rate (circle in centre). For VPSC there is a very large spread, some grains deforming over 10 times more than others. For FEM the spread is considerably smaller and similar to that which is actually observed. The halite crystal structure is common to many ionic compounds. Of particular interest to geophysics is periclase (MgO) because it is a likely component of the earth’s lower mantle and significant for its rheology. With radial diamond anvil experiments, texture development in axial deformation can be studied in situ at high pressure. Figure 32(a) is an inverse pole figure for MgO deformed at 47 GPa. The axial stress component is estimated to be 8.5 GPa. The texture displays a strong (001) fibre component (figure 34(a)) which can only be explained if exclusively {110}110¯ slip is active as simulated with the self-consistent model (figure 34(b)). If the Taylor model is used, compatibility requires that harder slip systems become activated and the texture changes profoundly (figure 34(c)). It is to be expected that at lower mantle conditions, in analogy to halite at higher temperature, several slip systems become active. Indeed, texture patterns produced in torsion experiments on magnesiowuestite (Mg, Fe)O by Texture and anisotropy 1409

Figure 33. Polycrystal plasticity simulations for halite. Shown are the off-diagonal strain rate components d23 predicted at the initial step of each simulation. For Taylor assumption all grains deform at the same rate (dot in centre), for VPSC there is a large spread and for FEM (HEP for hybrid element polycrystal) there is a moderate spread (Lebensohn et al 2003).

Figure 34. Inverse pole ﬁgures for MgO, deformed in axial compression. (a) In situ diamond anvil experiment at 20 GPa. To the right are simulations with {110}[110]¯ slip highly favoured with (b) the self-consistent theory and (c) the Taylor theory (Merkel et al 2002). Logarithmic contours.

Stretton et al (2001) and shear experiments by Yamazaki and Karato (2002) suggest activity of {111}, {110} and {100} slip systems, all with the [110] slip direction.

7.7. Polymineralic rocks Most texture research in geology was done on monomineralic rocks such as quartzites and marbles. By contrast most of the naturally occurring deformed rocks are polymineralic and this adds great complexities to characterization and interpretation. For example, granite, consisting of feldspars, quartz and mica, shows initially an almost random orientation distribution 1410 H-R Wenk and P Van Houtte

Figure 35. Deformation of granite in the Santa Rosa mylonite zone in Southern California. Pole ﬁgures of (a) (001) biotite (a mica mineral) and (b) (11−20) quartz in granite which was progressively deformed to mylonite and phyllonite. Determination by neutron diffraction. Equal area projection. Contours for biotite are 0.5, 1, 1.5, 2, 2.5, 3, 4,...14 m.r.d., for quartz 0.5, 0.75, 1, 1.25,...m.r.d., dot pattern below 1 m.r.d. (Wenk and Pannetier 1990).

(figure 35(a)). With increasing deformation at metamorphic conditions mica and quartz attain a strong texture in mylonite (figure 35(b)). During progressive deformation to phyllonite, which includes grain size reduction, the mica texture further increases, whereas the quartz texture attenuates (figure 35(c)). The texture change in quartz was attributed to a change in deformation mechanisms: dislocation glide accompanied by recrystallization during the first stage, and superplastic flow during the second stage. In polymineralic eclogite from high pressure rocks of Dabie Shan, China, omphacite (pyroxene) develops a strong texture whereas the garnet component has a random orientation distribution (Wenk et al 2001). A challenge for texture research has been to determine preferred orientation of triclinic plagioclase, a common constituent in many rocks (Dornbusch et al 1994, Siegesmund et al 1994, Xie et al 2003). All of these studies relied on neutron diffraction and some on the Rietveld method to deconvolute the very complex diffraction patterns. There are few experiments on texture development in polymineralic rocks. Tullis and Wenk (1994) found that addition of mica to quartz reduces the strength of preferred orientation of quartz, presumably due to preferential sliding on sheet silicates and a similar behaviour was observed in limestone–halite aggregates (Jordan 1987). The understanding of plasticity in polyphase materials is still very rudimentary. Different phases may have very different mechanical properties and corresponding aggregates will deform very heterogeneously. Microstructural investigations of granitic rocks (Tullis 2002) distinguish two regimes, one with interconnected strong or weak phases and another one with interconnected weak layers. It appears that with increasing deformation granites switch from Texture and anisotropy 1411

Figure 36. Texture of calcium hydroxide on cement paste-aggregate interface. The plot of 0001 pole density against angle to surface interface illustrates the effect of adding 5% silica fume to the paste (none) and of ageing (increase in texture strength) (Detwiler et al 1988). a frame structure to a layer structure in which strain is localized in bands (e.g. Herwegh and Handy (1996)). Similar processes have been described in engineering materials (Raj and Ghosh 1981). Texture development in polymineralic metamorphic rocks cannot be explained purely with polycrystal plasticity. Models must include processes of dissolution and growth (Spiers and Takeshita 1995), interaction with aqueous solutions (Karato et al 1986), chemical reactions and phase transformations. Apart from orientation state and dislocation microstructure, the chemical composition, grain size and shape are important parameters (Wang 1994, Heilbronner and Bruhn 1998, Shimizu 1999). Because of these complexities there are still many unresolved questions about texture development in such common rocks as gneiss and amphibolite. Are mica and hornblende crystals aligned due to rigid body rotations of particles with anisotropic grain shape, is the alignment the result of crystal plasticity or is it due to growth, perhaps under stress?

7.8. Cement minerals Concrete consists of particles of gravel and sand (‘aggregate’) that are connected by a variety of hydrated minerals occurring in the cement paste. Of particular importance for the strength of concrete is a thin layer of calcium hydroxide (portlandite) directly adjacent to aggregate particles. Detwiler et al (1988) have documented that this layer shows a strong growth texture with c-axes aligned perpendicular to the aggregate surface. The strength of the texture increases with ageing (figure 36) which appears to be preferable. Other cement minerals, such as ettringite may also show local preferred orientation that influences not only the strength but also the permeability by solutions and thus affects the durability. Rather unexpectedly it was observed that the durability of concrete correlates amazingly with texture characteristics of aggregate rocks (Monteiro et al 2001). Deformation of granite, such as that illustrated in figure 35 greatly enhances the alkali silica reaction that causes concrete to prematurely deteriorate. Texture is a bulk measure of the deformation state and it appears that highly deformed quartz is very unstable. 1412 H-R Wenk and P Van Houtte

Figure 37. Structure of Earth. Depth, pressure and temperature are indicated as well as main phases in the different units.

7.9. Earth structure

The Earth is composed of compositionally distinct units with characteristic properties. They are summarized in figure 37, which also gives information on temperature and pressure. In the hydrosphere ice is significant and texture forms during flow of glaciers and the large polar ice caps. The crust shows the greatest diversity of minerals (over 3000) that are accessible to direct observation in igneous, metamorphic and sedimentary rocks. The minerals of most interest are calcite, quartz and mica. They have been discussed earlier. Structural geologists use textures to unravel the deformation history during mountain building and we will illustrate this with an example below. The deeper earth (mantle) is dominantly composed of Mg, Si, and O: olivine (Mg2SiO4) in the upper mantle, spinel-like structures ringwoodite (Mg2SiO4) and wadsleyite (Mg2SiO4) in the transition zone, perovskite (MgSiO3) and periclase (MgO) in the lower mantle. At higher pressure phases tend to have simpler crystal structures than minerals in the crust, as established by high pressure experiments and theory (e.g. Fiquet (2001)). Textures have become of special significance with the discovery of seismic anisotropy. Seismic anisotropy was first recorded in the shallow upper mantle beneath Hawaii (Hess 1964), observing that surface waves travel about 10% faster in the E–W direction than N–S. Since then maps of azimuthal anisotropy have been constructed for the uppermost mantle beneath oceanic lithosphere (figure 38). Fast wave propagation directions overall correspond to flow directions as implied from plate motions. The anisotropy pattern has been refined and it became apparent that anisotropy in the upper mantle varies greatly with depth (e.g. Silver (1996), Montagner and Guillot (2002)). Texture and anisotropy 1413

Figure 38. Distribution of velocities for Rayleigh waves. The lines correspond to the maximum velocities at a depth of about 100 km. Spreading ridges are indicated (Montagner and Tanimoto 1990).

Anisotropy was documented in layers of the transition zone (400–700 km) where olivine breaks down to high pressure phases. Phase transformations in this zone are associated with large volume changes and this may be the cause of deep-focus earthquakes (Green and Houston 1995). Just above the 660 km discontinuity anisotropy is pronounced and may be related to intense deformation associated with those phase transitions (Trampert and Heijst 2002). But contrary to the regular pattern of anisotropy in the uppermost mantle, anisotropic regions in the transition zone are confined, vary in extent and depth and are to some degree associated with subduction (Fouch and Fisher 1996). There is little evidence for anisotropy between 700 and 2700 km, but this may be partially due to limited crossing ray coverage. However, the deep mantle, in the vicinity of the boundary with the core (D layer), reveals itself as a fascinating and heterogeneous area of the earth. Geodynamic modelling suggests very strongly deformed subducting slabs (McNamara et al 2001) and seismologists have observed anisotropy that could be due to texturing (e.g. Fouch et al (2001), Vinnik et al (1998)). The outer core is liquid and thus isotropic, but it is firmly established, both from body waves and free oscillation observations, that the solid inner core is again anisotropic. Compressional P-waves travel 3–4% faster along the (vertical) axis of the earth, than in the equatorial plane (e.g. Song (1997)). In the next sections we will first show an example of how texture can be used by structural geologists to infer the tectonic deformation history in the crust. We will then explore how texture analysis and polycrystal plasticity contribute to a better understanding of deformation and anisotropy in the deep earth.

7.10. Textures as indicators of strain history

For structural geologists the aim is often to unravel the detailed strain history and textures are signiﬁcant because they are generally sensitive to the path. For example, has a tectonic zone 1414 H-R Wenk and P Van Houtte

Figure 39. Cartoon comparing crustal deformation by coaxial thinning (top) and non-coaxial shearing (bottom). The ﬁnite strain (shape of ellipse) may be the same, but in simple shear the ellipse is inclined by an angle θ to the shear plane.

Figure 40. (0001) pole figures of calcite obtained with the Taylor model for 100% equivalent strain and using resolved shear stress ratios corresponding to low temperature deformation. Pure shear on left, mixed modes in the centre, and simple shear on right. Note the increasing asymmetry of the maximum (inclined by ω to the shear plane normal). Sense of shear is indicated (from Wenk et al 1987). been subject to non-coaxial shearing in a shear zone (figure 39, bottom) or coaxial crustal thinning (figure 39, top)? Both paths can lead to an identical finite strain. Textures provide a means to quantify the amount of non-coaxial deformation. We have already above seen how the deformation geometry affects the symmetry of the texture pattern of calcite (figure 26). For coaxial deformation one expects orthorhombic pole figures, whereas a non-coaxial path is likely to produce monoclinic pole figures. Texture patterns of calcite in deformed marbles can be used to determine the partitioning of deformation into a coaxial deformation component and a non-coaxial (simple shear) component. The philosophy is to first develop a deformation model and test it by comparing resulting texture patterns with experiments. Once mechanisms are established and the model adequately predicts the experiment, one can then apply the model to any arbitrary strain path, including those that cannot be approached experimentally. For calcite, calculated (0001) pole figures for plane strain document a symmetrical (orthorhombic) pure shear pole figure and an asymmetrical (monoclinic) simple shear pole figure (figure 40) and this agrees with experiments (Kern and Wenk 1983, Pieri et al 2000). Simulations can provide intermediate Texture and anisotropy 1415

Figure 41. (a) Determinative diagram with angle of asymmetry ω against strain partitioning factor as obtained from Taylor simulations (figure 40). Results for (b) marble mylonites from core complexes of the American Cordillera (Erskine et al 1993), and (c) various limestones from Alpine spreading nappes (Ratschbacher et al 1991). states and the relative amount of simple shear can be quantified by measuring the angle of asymmetry ω between the (0001) maximum and the shear-plane normal. One can construct an empirical determinative diagram to assess the amount of simple shear from the asymmetry of the (0001) texture maximum (figure 41(a)). In practice, geologists collect oriented rock samples in the field, then measure pole figures in the laboratory relative to geological coordinates, such as schistosity plane and lineation direction which define the macroscopic shear plane and shear direction, respectively. From the asymmetry of the 0001 pole figure maximum relative to the shear plane the sense of shear can be inferred. From the angle of asymmetry ω and using the determinative diagram in figure 41(a), the strain partitioning can be estimated. Whereas many marbles in core complexes of the Western United States show almost symmetrical patterns of (0001) axes (Erskine et al (1993), figure 41(b)) and presumably formed largely by coaxial crustal extension, limestones from the spreading nappes in the Alps have generally highly asymmetric texture patterns attributed to shearing on thrust planes (Dietrich and Song (1984), Ratschbacher et al (1991), figure 41(c)).

7.11. Anisotropy in the deep earth The mantle is not exposed on the surface of the earth. However, during solid state convection of the mantle and tectonic activity, smaller and larger parts of upper mantle rocks have been locally juxtaposed within the crust and can be sampled. The rocks are mainly peridotite, composed largely of olivine and subordinate pyroxene. Locations where mantle peridotites can be sampled are in Oman (studied extensively by Boudier and Nicolas (1995)), as inclusions in volcanic rocks from Africa, and several other places. As has been described earlier by far the most common olivine texture type has (010) poles nearly perpendicular to the foliation plane and [100] axes subparallel to the lineation direction (ﬁgure 31(a)). Deformation mechanisms in olivine have been studied in the laboratory and, knowing slip systems and their activity, it is possible to predict texture development for a given strain path. In ﬁgure 31(b) we illustrated this for plane strain compression. We can now apply the same method to the larger system of upper mantle deformation. In the mantle large cells of convection are induced by instabilities and driven by temperature gradients (e.g. Bunge et al 1416 H-R Wenk and P Van Houtte

Figure 42. Simulated texture development during convective flow in the upper mantle. Streamline with (100) olivine pole figures (1000 grains) at five different locations. During upwelling (left) a strong texture develops and is modified during spreading and subduction (Dawson and Wenk 2000).

(1998)). The strain distribution in a convection cell is highly heterogeneous. Dawson and Wenk (2000) have used the finite element method that incorporates as a constitutive equation polycrystal plasticity to investigate the development of anisotropy during mantle convection. Figure 42 follows texture development of olivine in the upper mantle along a streamline in [100] pole figures. A strong texture develops rapidly during upwelling (B). The preferred orientation stabilizes during spreading (C, D) and attenuates during subduction (E). The pole figures are distinctly asymmetric due to the component of simple shear. While the finite strain along a streamline increases monotonically, the texture does not. Knowing textures patterns over the upper mantle, one can then average anisotropic elastic properties and from those evaluate seismic wave velocities. The model suggests large variations, up to 15%, comparable to those that have actually been observed by seismologists. Even though in detail texture development of olivine is complex and not simply an alignment of slip directions with flow lines, the overall seismic anisotropy of the uppermost mantle can be reasonably explained as a result of texturing during upwelling along ridges (Blackman et al 2002). Much less is known about the deeper Earth. Deformation experiments are more difficult because pressures are beyond conditions reached by ordinary mechanical devices. One approach has been to use analogue systems with phases of similar structures and bonding but different composition that are stable at lower pressure and deform at lower temperature. For example, for MgSiO3 perovskite CaTiO3 has been used (Karato et al 1995). An analogue for periclase (MgO) and magnesiowuestite (FeMgO2) is halite (NaCl). But analogues are of questionable value when it comes to deformation mechanisms, since slip depends on the detailed electronic structure and bonding characteristics. For non-quenchable phases in situ observations are required and here texture comes to play a crucial role because, contrary to microstructure, texture can be measured in situ at high pressure, for example, with diamond anvil experiments (Merkel et al 2002). If we know Texture and anisotropy 1417

Figure 43. Temperature distribution (grey shades) during simulated subduction of a slab into the lower mantle (dark: cold, white: medium, dark: hot, as indicated) (McNamara et al 2001).

Figure 44. Simulation of texture development of periclase (MgO) during slab subduction into the lower mantle along a streamline of the McNamara et al (2001) model (ﬁgure 43) at 4 different depths. {100} pole ﬁgures. It is assumed that {110} and {111) slip are equally active.

texture patterns, we can infer deformation mechanisms. During subduction of upper mantle slabs into the lower mantle, geodynamic modelling suggests heterogeneous deformation with complicated streamlines (figure 43). Using slip systems that are assumed to be active at high temperature and pressure, we can then again predict texture evolution along such a streamline in the subducting slab with increasing depth (figure 44). Simulated textures for MgO (represented as {100} pole figures) are strong at depths of 2000 km, but to produce seismic anisotropy, substantial single crystal elastic anisotropy is also required. At high pressure and temperature MgO is in fact strongly anisotropic (e.g. Karki et al (1999)). Curiously, at intermediate pressures single crystal anisotropy of MgO is minimal, which may explain the absence of significant anisotropy in the intermediate lower mantle. The main component of the inner core is an iron-rich alloy, probably with a hexagonal close packed (ε-iron) structure (e.g. experiments by Shen et al (1998), and first principles calculations by Wasserman et al (1996)). As was mentioned, seismic waves travel about 3–5% faster along the N–S axis of the inner core than in the equatorial plane. There is general agreement that the reason for seismic anisotropy is an alignment of crystals but there are 1418 H-R Wenk and P Van Houtte

Figure 45. Inverse pole figures for ε-iron (hcp) deformed in compression. (a) In situ diamond anvil texture determination at 220 GPa. (b) Texture simulation with the self-consistent theory for conditions that favour basal slip, 50% strain. Equal area projection, logarithmic contours (Wenk et al 2000). many ideas about the processes that lead to such an alignment. Wenk et al (2000) presented a model for core texturing during convection. Another possibility is solidification texturing at the boundary with the liquid outer core (Bergman 1997). Finally the earth’s magnetic field may produce stresses that deform the material and thus produce crystal rotations (Karato 1999, Buffett and Wenk 2001). Since the core is so remote, and many conditions are poorly known, convincing cases can be made for each mechanism. Here we will only illustrate the example of the magnetic field. The largest electromagnetic (Maxwell) shear stresses in the Earth’s geodynamo arise from the combined influence of the radial and azimuthal components of the magnetic field and are on the order of several pascals. Strain gradually accumulates to about 50% over 1 million years as the inner core grows by solidification. The azimuthal component of the Maxwell stress, which is about an order of magnitude larger than the radial component, imposes a strong simple shear deformation. A prerequisite for modelling this deformation is the knowledge about slip systems that are active in ε-iron. This phase is not stable at ambient conditions and deformation experiments need to be performed at high pressure. This can be done again with diamond anvil cells. A comparison of texture patterns that were observed at pressures close to those in the inner core (220 GPa) (figure 45(a)) with polycrystal plasticity simulations (figure 45(b)) can be used to infer slip systems. A strong c-axis maximum near the compression direction is only compatible with significant basal slip, consistent with ab initio predictions (Poirier and Price 1999). With such information about intracrystalline mechanisms one can then apply the Maxwell stresses to the solid core and predict orientation patterns for different locations. Indeed a strong texture develops, particularly in the outer parts of the inner core (Buffett and Wenk (2001), figure 46(a)). A next step is to average single crystal elastic properties over the simulated orientation distributions. Ab initio calculations of Steinle-Neumann et al (2001) for core temperature (6000 K) and pressure reveal strong anisotropy for the single crystal, with fastest P-wave velocities perpendicular to the c-axis and more than 15% slower than that parallel to the c-axis (figure 46(b)). These single crystal elastic constants can then be averaged with the texture predicted from Maxwell stresses (figure 46(c)) and the pattern suggest indeed a small anisotropy with faster velocities parallel to the N–S axis. The example illustrates that even for such remote places as the centre of the earth a combination of experimental techniques Texture and anisotropy 1419

Figure 46. Anisotropy development in the inner core of the earth assuming that the material is ε-iron (hcp) and was deformed by Maxwell stresses. (a) (0001) pole ﬁgure for one location in the core. (b) and (c) Anisotropy of velocities of compressional elastic waves (P). (b) Single crystal P-velocities for 330 GPa and 6000 K based on ﬁrst principles calculations (Steinle-Neumann et al 2001); (c) P-velocities for the aggregate shown in (a) obtained by averaging of the single crystal tensor. The range of grey shades is 5%, equal area projection, linear contours (Buffett and Wenk 2001). and theories that are commonly applied in materials science can be used to better understand structural features.

8. Textures in mineralized biological materials

Texture research has focused on engineering and earth materials such as metals, ceramics, polymers and rocks. However, there is increasing awareness that texture is signiﬁcant in organisms, both in the organization of protein crystals as well as in mineralized skeletons. In such skeletons texturing has a direct mechanical support function and organisms optimize texture patterns for particular environments. We will brieﬂy illustrate a few examples of mollusc shells and bones.

8.1. Nacre of mollusc shells (aragonite) Mollusc shells are composed either of calcite or aragonite, trigonal and orthorhombic polymorphs of CaCO3, respectively, and show a wide variety of textures. In many molluscs an outer shell is composed of calcite and the inner shell of aragonite. This aragonite nacre (‘mother of pearl’) displays a brick-like microstructure. Texture analysis indicates that in nacre of most molluscs, c-axes are oriented more or less perpendicular to the surface of the shell but a-axes display characteristic patterns, either resembling a single crystal as in the bivalve Pinctada (figure 47(a)) and shells of most land snails, or they display a texture pattern with {110} twinning as in Nautilus (figure 47(b)), or they spin randomly about the c-axis with a [001] fibre texture as in Haliotis (Abalone, figure 47(c)). Chateigner et al (2000) surveyed a large number of mollusc species and concluded that the diverse texture patterns are related to phylogeny and presumably the protein type. Since [100] is the stiffest crystal direction, such a texture produces an optimal strength of the shell. Mollusc shells have received considerable attention because of their bio-mimetic properties.

8.2. Bones (apatite) Texture patterns in biological materials, including carbonates, silica minerals and phosphates, composing shells, skeletons, bones and teeth are a new ﬁeld of endeavour, with very few quantitative investigations (Lowenstam and Weiner 1989). The structure of bone and teeth 1420 H-R Wenk and P Van Houtte

(a) (b) (c)

Figure 47. Growth textures of aragonite in nacre from mollusc shells. (010), and (001) pole ﬁgures, projected on the shell surface. Equal area projection, logarithmic contours 0.5, 0.7, 1, 1.4 m.r.d., etc, dot pattern below 1 m.r.d. (a) Nacre of the bivalve Pinctada maxima (Oyster) with more or less a single component texture, (b) nacre of Nautilus macromphalus with a pseudohexagonal pattern due to twinning on 110, (c) nacre of Haliotis cracherodis (Abalone) with an axial texture (Hedegaard and Wenk 1997).

Figure 48. Pole figures of hexagonal hydroxyapatite in tissue of a dinosaur tendon. The texture was measured by synchrotron radiation and analysed with the Rietveld method (Lonardelli et al 2004). The tendon axis is in the centre of the pole figure. Contour levels logarithmic; some are indicated (maximum 3.2 m.r.d.). Equal area projection. is locally very heterogeneous and crystallite size is very small. Only with the advent of microfocus x-ray beams at synchrotron sources have these materials become accessible to quantitative analysis. While in turkey and dinosaur tendon c-axes of pseudohexagonal needle- shaped hydroxyapatite crystals are aligned parallel to the tendon axis, a bovine ankle bone displays a strong alignment of c-axes parallel to the surface of the bone (Wenk and Heidelbach 1998). Figure 48 shows a c-axis pole figure of apatite in a well-preserved dinosaur tendon, with Texture and anisotropy 1421 a very strong crystallite alignment. Texturing is important in bone implants as was documented by Benmarouane et al (2004) in a neutron diffraction study. Single-cell organisms, called Foraminifera, exploit in their calcite skeleton all possible preferred orientation patterns, and also a random arrangement of calcite laths (Haynes 1981).

9. Conclusions

While we were preparing this review, we surveyed a large amount of literature. It has been impressive to see how much information the classic books of Wassermann on metals (1939, 1962) and Sander on rocks (1950) already contain. Different ideas on texture-forming processes and mechanisms had been proposed over many years (such as Sachs (1928), Kurdjomov and Sachs (1930), Burgers (1934), Taylor (1938), Calnan and Clews (1950), Hill (1952), Bishop (1954), Eshelby (1957) to name just a few). The next twenty years, between 1960 and 1980, brought enormous advances, mainly because of the new possibilities of quantifying texture data. This was only possible through introduction of high-speed computers for data processing and digital electronics for diffraction measurements. These same developments produced a big leap in structural crystallography. The years will be remembered for the introduction of the ODF (Bunge 1965, Roe 1965), the first polycrystal plasticity simulations (Hutchinson 1970, Siemes 1974, Van Houtte and Aernoudt 1975, Lister et al 1978) and TEM investigations to establish deformation mechanisms (e.g. Groves and Kelly (1963), Weertman (1968), Kocks (1970), Ashby (1972)). Between 1980 and 2000 came a period of refinements. It is also the time when the quantitative methods, developed in the previous years, became available for a large group of texture researchers and became generally accepted. Calculations of the ODF were refined (e.g. Matthies and Vinel (1982), Van Houtte (1983, 1991), Schaeben (1988), Dahms (1992)), polycrystal plasticity models allowed for heterogeneity (e.g. Molinari et al (1987), Canova et al (1992), Asaro and Needleman (1985), Mathur and Dawson (1989), Van Houtte et al (2002)). On the experimental side, a new tool was introduced, EBSPs (Dingley 1981) with the SEM and its automation (Wright and Adams 1992). With this method not only bulk texture but local textures and misorientation analysis became available for characterization. What will the future bring? Of course predicting directions of research is purely speculative. Nevertheless there are numerous new opportunities to extend texture research into new directions. With computers available in every lab that are many orders of magnitude faster that the mainframes of the 1960s, polycrystal plasticity modelling can now be much more sophisticated, and with finite element models local intragranular heterogeneity can be accounted for (e.g. Mika and Dawson (1999)). Such models will probably replace the conventional one-grain one-orientation models and can be applied to polyphase materials for which still no satisfactory polycrystal plasticity simulations exist. A weak link of simulations is the uncertainty about microstructural hardening (Kocks and Mecking 2003). Quantification on the local scale of slip systems is generally beyond experiments and here first principles simulations of dislocation dynamics and dislocation interaction will contribute greatly to a realistic assessment of slip activity (e.g. Tang et al (1998)). So far the texture community and the internal stress community have been separated, yet both effects are highly correlated and satisfactory interpretation relies on a combined approach. Particularly with new neutron diffraction facilities simultaneous texture and strain measurements are now within reach (e.g. Walther et al (2000)). With new instrumentation at neutron diffraction and synchrotron facilities in situ texture measurements at high and low temperatures and high pressure can now be performed routinely. This is significant to understand phase transformations, memory effects and variant selection. 1422 H-R Wenk and P Van Houtte

For geophysics the unprecedented possibilities to perform deformation experiments at any conditions in the earth and study in situ texture changes is revolutionizing a field and mineralogists that were previously mainly concerned with phase relations and crystal structures are becoming aware of preferred orientation. We predict that a whole new community of synchrotron users will become texture clients. Many images on inorganic and organic materials reveal texture and this needs to be taken into account by crystallographers that rely on normalized diffraction intensities. With modern Rietveld codes that are capable of processing such images and extract texture information, crystal structure analysis can be quantified. Also with synchrotrons and a fine-focused beam of 1–10 µm local texture variations can be investigated such as in biomaterials. Data acquisition is extremely fast and the method is available for so many in situ applications, including stress, that there will be breakthroughs, similar to EBSD in the 1980s. The discussion has illustrated that texture research is indeed a wide multidisciplinary field dealing with an intrinsic material property. It connects many domains of science with metallurgy, ceramics, polymer science, geology, geophysics and biology as just a few highlights. The field is old but has advanced tremendously during the last decade, thanks to new experimental techniques and to sophisticated modelling that became possible with modern computing resources. Yet it remains a field with many unsolved problems that will keep scientists occupied for years to come.

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