Section 3.3.10.7.3. Fitting problems of ferroelastic twins

The real problem of space-constrained twin textures, however, is provided by non-merohedral (ferroelastic) transformation and deformation twins (including the cubic deformation twins of the spinel law). This is schematically illustrated in Fig. 3.3.10.12 for the very common case of orthorhombicmonoclinic transformation twins (.

Figure 3.3.10.12 | top | pdf | Illustration of space-filling problems of domains for a (ferroelastic) orthorhombic monoclinic phase transition with an angle (exaggerated) of spontaneous shear. (a) Orthorhombic parent crystal with symmetry . (b) Domain pairs , and of the monoclinic daughter phase () with independent twin reflection planes (100) and (001). (c) The combination of domain pairs and leads to a gap with angle , whereas the combination of the three domain pairs , and generates a wedge-shaped overlap (hatched) of domains 3 and 4 with angle . (d) Twin lamellae systems of domain pairs (left) and (or ) (right) with low-energy contact planes (100) and (001). Depending on the value of , adaptation problems with more or less strong lattice distortions arise in the boundary region A–A between the two lamellae systems. (e) Stress relaxation and reduction of strain energy in the region A–A by the tapering of domains 2 (`needle domains') on approaching the (nearly perpendicular) boundary of domains . The tips of the needle lamellae may impinge on the boundary or may be somewhat withdrawn from it, as indicated in the figure. The angle between the two lamellae systems is .

Figs. 3.3.10.12(a) and (b) show the `splitting' of two mirror planes (100) and (001) of parent symmetry mmm, as a result of a phase transition , into the two independent and symmetrically non-equivalent twin reflection planes (100) and (001), each one representing a different (monoclinic) twin law. The two orientation states of each domain pair differ by the splitting angle . Note that in transformation twins the angle (spontaneous shear strain) is small, at most one or two degrees, due to the pseudosymmetry of the daughter phase with respect to the parent symmetry . It can be large, however, for deformation twins, e.g. calcite. The resulting fitting problems in ferroelastic textures are illustrated in Fig. 3.3.10.12(c). Owing to the splitting angle , twin domains would form gaps or overlaps, compared to a texture with , where all domains fit precisely. In reality, the misfit due to leads to local stresses and associated elastic strains around the meeting points of three domains related by two twin laws [triple junctions, cf. Palmer et al. (1988), Figs. 3–6)].

For the orthorhombicmonoclinic transition considered here, the two different twin laws often lead to two and (for small ) nearly perpendicular sets of polysynthetic twin lamellae. This is illustrated in Fig. 3.3.10.12(d). The boundaries in one set are formed by (100) planes, those in the other set by (001) planes, both of low energy. The misfit problems are located exclusively in the region AA where the two systems of lamellae meet. Here wedge-like domains (the so-called `needle domains', see below) are formed, as shown in Fig. 3.3.10.12(e), i.e. the twin lamellae of one system taper on approaching the perpendicular twin system (right-angled twins), forming rounded or sharp needle tips. The tips of the needle lamellae may be in contact with the perpendicular lamella or may be somewhat withdrawn from it. These effects are the consequence of strain-energy minimization in the transition region of domain systems, as compared to the large-area contacts between parallel twin lamellae.

The formation of two lamellae systems with wedge-like domains was demonstrated very early on by the polarization-optical study of the orthorhombicmonoclinic (2222) transformation of Rochelle salt at 297 K by Chernysheva (1950, 1955; quoted after Klassen-Neklyudova, 1964, pp. 27–30 and 76–77, Figs. 35, 38 and 100; see also Zheludev, 1971, pp. 180–185). The term `needle domains' was coined by Salje et al. (1985) in their study of the monoclinictriclinic ) transition of Na-feldspar. Another detailed description of needle domains is provided by Palmer et al. (1988) for the cubictetragonal () transformation of leucite at 878 K. The typical domain structure resulting from this transition is shown in Fig. 3.3.10.13.

Figure 3.3.10.13 | top | pdf | Thin section of tetragonal leucite, K(AlSi2O6), between crossed polarizers. The two nearly perpendicular systems of (101) twin lamellae result from the cubic-to-tetragonal phase transition at about 878 K. Width of twin lamellae 20–40 µm. Courtesy of M. Raith, Bonn.

A similar example is provided by the extensively investigated tetragonalorthorhombic transformation twinning of the high-T_c superconductor YBa₂Cu₃O_7−δ (YBaCu) below about 973 K (Roth et al., 1987; Schmid et al., 1988; Keester et al., 1988, especially Fig. 6). Here two symmetrically equivalent systems of lamellae with twin laws and meet at nearly right angles (). Interesting TEM observations of tapering, impinging and intersecting twin lamellae are presented by Müller et al. (1989). An extensive review on twinning of YBaCu, with emphasis on X-ray diffraction studies (including diffuse scattering), was published by Shektman (1993).

A particularly remarkable case occurs for hexagonalorthorhombic ferroelastic transformation twins. Well known examples are the pseudo-hexagonal K₂SO₄-type crystals (cf. Example 3.3.6.7). Three (cyclic) sets of orthorhombic twin lamellae with interfaces parallel to or are generated by the transformation. More detailed observations on hexagonal–orthorhombic twins are available for the IIIII (heating) and III (cooling) transformations of KLiSO₄ at about 712 and 938 K (Jennissen, 1990; Scherf et al., 1997). The development of the three systems of twin lamellae of the orthorhombic phase II is shown by two polarization micrographs in Fig. 3.3.10.14. A further example, the cubicrhombohedral phase transition of the perovskite LaAlO₃, was studied by Bueble et al. (1998).

Figure 3.3.10.14 | top | pdf | Twin textures generated by the two different hexagonal-to-orthorhombic phase transitions of KLiSO4. The figures show parts of plates (viewed along [001]) between crossed polarizers. (a) Phase boundary IIIII with circular 712 K transition isotherm during heating. Transition from the inner (cooler) room-temperature phase III (hexagonal, dark) to the (warmer) high-temperature phase II (orthorhombic, birefringent). Owing to the loss of the threefold axis, lamellar cyclic twin domains of three orientation states appear. (b) Sketch of the orientations states 1, 2, 3 and the optical extinction directions of the twin lamellae. Note the tendency of the lamellae to orient their interfaces normal to the circular phase boundary. Arrows indicate the direction of motion of the transition isotherm during heating. (c) Phase boundary III with 938 K transition isotherm during cooling. The dark upper region is still in the hexagonal phase I, the lower region has already transformed into the orthorhombic phase II (below 938 K). Note the much finer and more irregular domain structure compared with the IIIII transition in (a). Courtesy of Ch. Scherf, PhD thesis, RWTH Aachen, 1999; cf. Scherf et al. (1997).

Another surprising feature is the penetration of two or more differently oriented nano-sized twin lamellae, which is often encountered in electron micrographs (cf. Müller et al., 1989, Fig. 2b). In several cases, the penetration region is interpreted as a metastable area of the higher-symmetrical para-elastic parent phase.

In addition to the fitting problems discussed above, the resulting final twin texture is determined by several further effects, such as:

Systematic treatments of ferroelastic twin textures were first published by Boulesteix (1984, especially Section 3.3 and references cited therein) and by Shuvalov et al. (1985). This topic is extensively treated in Section 3.4.4 of the present volume. A detailed theoretical explanation and computational simulation of these twin textures, with numerous examples, was recently presented by Salje & Ishibashi (1996) and Salje et al. (1998). Textbook versions of these problems are available by Zheludev (1971) and Putnis (1992).