Testing the Theory of Domain Patterns in Ferroelastics Allan Jacobs, Allan JacobsSearch for more papers by this author Allan Jacobs, Allan JacobsSearch for more papers by this author Book Editor(s):G. B. Olson, G. B. OlsonSearch for more papers by this authorD. S. Lieberman, D. S. LiebermanSearch for more papers by this authorA. Saxena, A. SaxenaSearch for more papers by this author First published: 25 January 2010 https://doi.org/10.1002/9781118803592.ch57 AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onEmailFacebookTwitterLinkedInRedditWechat Summary This chapter contains sections titled: Introduction Strains and Strain Energy Discussion of Experiment Testing the Theory at the Atomic Level Acknowledgements References G. R. Barsch and J. A. Krumhansl, Phys. Rev. Let. 53, 1069 (1984). The one-dimensional theory was worked out by F. Falk, Z. Phys. B 51, 177 (1983). 10.1103/PhysRevLett.53.1069 CASWeb of Science®Google Scholar K. Aizu, J. Phys. Soc. Jpn. 27, 387 (1969). 10.1143/JPSJ.27.387 CASWeb of Science®Google Scholar G. Arlt, J. Mater. Sci. 25, 1655 (1990). 10.1007/BF00584864 Google Scholar E. K. H. Salje, Phase Transitions in Ferroelastic and Co-elastic Crystals (Cambridge University Press, Cambridge, 1993). Google Scholar K. Bhattacharya, Microstructure of Martensite (Oxford University Press, Oxford, 2003). Google Scholar A. H. King and Y. Zhu, Phil. Mag. A 67, 1037 (1993). 10.1080/01418619308213974 CASWeb of Science®Google Scholar Y. Zhu, M. Suenaga and J. Tafto, Phil. Mag. A 67, 1057 (1993). 10.1080/01418619308224757 CASWeb of Science®Google Scholar A. E. Jacobs, Phys. Rev. B 61, 6587 (2000). 10.1103/PhysRevB.61.6587 CASWeb of Science®Google Scholar S. H. Curnoe and A. E. Jacobs, Phys. Rev. B 64, 064101 (2001). 10.1103/PhysRevB.64.064101 CASWeb of Science®Google Scholar A. E. Jacobs, S. H. Curnoe and R. C. Desai, Materials Transactions 45, 1054 (2004). 10.2320/matertrans.45.1054 CASWeb of Science®Google Scholar Ref.[10] cites 13 sources on Mg-Cd alloys, lead orthovanadate, Ta4N, Sm2O3, Mg-cordierite and other materials. Google Scholar S. H. Curnoe and A. E. Jacobs, Phys. Rev. B 63, 094110 (2001). 10.1103/PhysRevB.63.094110 CASWeb of Science®Google Scholar T. Lookman, S. R. Shenoy, K. O. Rasmussen, A. Saxena and A. R. Bishop, Phys. Rev. B 67, 024114 (2003). 10.1103/PhysRevB.67.024114 CASWeb of Science®Google Scholar Ref.[10] cites 8 sources on zirconia, leucite, BaTiO3 and several metallic alloys. Google Scholar A. E. Jacobs, S. H. Curnoe and R. C. Desai, Phys. Rev. B 68, 224104 (2003). 10.1103/PhysRevB.68.224104 CASWeb of Science®Google Scholar J. F. Nye, Physical Properties of Crystals (Oxford University Press, Oxford, 1957). Google Scholar A. E. Jacobs, Phys. Rev. B 52, 6327 (1995). 10.1103/PhysRevB.52.6327 CASWeb of Science®Google Scholar A. E. Jacobs, Phys. Rev. B 31, 5984 (1985). The different definitions of energy density and the strains require modification of the expressions for κ and α. 10.1103/PhysRevB.31.5984 CASPubMedWeb of Science®Google Scholar A. E. Jacobs, Phys. Rev. B 46, 8080 (1992). The density is the same as in Eq.(5) above, but the strains are defined differently. 10.1103/PhysRevB.46.8080 Web of Science®Google Scholar International Conference on Martensitic Transformations (ICOMAT) ReferencesRelatedInformation

Domain patterns in several classes of ferroelastics are studied using a Landau expansion in the strains and their derivatives. Examination of the local rotation, the non-order-parameter strains and the local energy density reveals the wedge and other disclinations responsible for the complexity of the patterns in (1) tetragonal-orthorhombic materials, and (2) hexagonal-orthorhombic and related materials. At temperatures where the parent phase is unstable and so has negative stiffness, simulations of hexagonal-orthorhombic systems yield pockets where the order parameter is much reduced; if the parent phase exists experimentally under these conditions, it might give rise to extreme damping. For cubic-tetragonal materials, perturbing the parent phase at a temperature well below its stability limit gives an inhomogeneous noncompact product.

We study domain patterns in cubic-tetragonal ferroelastics by solving numerically equations of motion derived from a Landau model of the phase transition, including dissipative stresses. Our system sizes, of up to 256^3 points, are large enough to reveal many structures observed experimentally. Most patterns found at late stages in the relaxation are multiply banded; all three tetragonal variants appear, but inequivalently. Two of the variants form broad primary bands; the third intrudes into the others to form narrow secondary bands with the hosts. On colliding with walls between the primary variants, the third either terminates or forms a chevron. The multipy banded patterns, with the two domain sizes, the chevrons and the terminations, are seen in the microscopy of zirconia and other cubic-tetragonal ferroelastics. We examine also transient structures obtained much earlier in the relaxation; these show the above features and others also observed in experiment.

A recent high-field magnetization experiment found a phase transition of unknown character in the layered, frustrated antiferromagnet RbCuCl3, in a transverse field (in the layers). Motivated by these results, we have examined the magnetic structures predicted by a model of RbCuCl3, using the classical approximation. At small fields, we obtain the structure already known to be optimal, an incommensurate (IC) spiral with wave vector q in the layers. At higher fields, we find a staircase of long-period commensurate (C) phases (separated initially by the low-field IC phase), then two narrow IC phases, then a fourth IC phase (also with intermediate C phases), and finally the ferromagnetically aligned phase at the saturation field H-S. The three-sublattice C states familiar from the theory of the triangular antiferromagnet are never optimal. The C phases and the two intermediate IC phases were previously unknown in this context. The magnetization is discontinuous at a field approximate to0.4H(S), in qualitative agreement with experiment, though we find much fine structure not reported.

Relaxor ferroelectrics (relaxors) form a special class of ferroelectric materials of which the understanding remains a challenging problem.Most technologically important relaxors crystallize in the so-called ABO3 perovskite-type structure; between those PbMg1/3Nb2/3O3 (PMN) and Na1/2Bi1/2TiO3 (NBT) which are considered as model relaxor ferroelectrics.From an application point of view, relaxor-based materials have been reported to exhibit near structural phase boundaries outstanding electromechanical properties, which point to a potential revolution in electromechanical transduction for a large range of applications.As a consequence, the potential impact of thin-film relaxor ferroelectrics has stimulated a fast growing interest.Although it has been realized that strain effects at the film-substrate interface modify dramatically their physical properties, there is incomplete understanding of the responsible mechanisms.In this presentation we will mainly focus on two innovative approaches towards the understanding of phase transitions of the relaxors PMN and NBT: on the one hand temperaturedependent birefringence imaging and on the other hand first-time high-pressure investigations of relaxor ferroelectrics by Raman spectroscopy, diffraction and diffuse scattering.As a matter of fact, the observed pressure-dependent transitions are very unusual and point, among other things, to new relaxorspecific spectral signatures and to important pressure-induced changes of the local structure and order.We further show that an external pressure of several GPa, as can be met in thin films, alters fundamentally the structural and polar properties in relaxor ferroelectrics, suggesting that intrinsic instabilities towards pressure play an important role in the unwished reduction of dielectric properties in application-designed relaxor thin films.

We study numerically the time evolution of two-dimensional (2D) domain patterns in proper tetragonal-orthorhombic (T-O) ferroelastics. Our equations of motion are derived from classical elasticity theory, augmented by nonlinear and strain-gradient terms. Our results differ from those found by other dynamical methods. We study first the growth of the 2D nucleus resulting from homogeneous nucleation events. The later shape of the nucleus is largely independent of how it was nucleated. In soft systems, the nucleus forms a flowerlike pattern. In stiff systems, which seem to be more realistic, it forms an X shape with twinned arms in the 110 and (1) over bar 10 directions. Second, we study the relaxation that follows completion of the phase transition; at these times, the T phase has disappeared and both O variants are present, separated by walls preferentially in 110-type planes. We observe a variety of coarsening mechanisms, most of them counterintuitive. Our patterns are strikingly similar to those observed in transmission electron microscopy of the improper T-O ferroelastic YBa2Cu3O7.

We study the statics and the dynamics of domain patterns in proper hexagonal-orthorhombic ferroelastics; these patterns are of particular interest because they provide a rare physical realization of disclinations in crystals. Both our static and dynamical theories are based entirely on classical, nonlinear elasticity theory; we use the minimal theory consistent with stability, symmetry and ability to explain qualitatively the observed patterns. After scaling, the only parameters of the static theory are a temperature variable and a stiffness variable. For moderate to large stiffness, our static results show nested stars, unnested stars, fans and other nodes, triangular and trapezoidal regions of trapped hexagonal phase, etc observed in electron microscopy of Ta4N and Mg-Cd alloys, and also in lead orthovanadate (which is trigonal-monoclinic); we even find imperfections in some nodes, like those observed. For small stiffness, we find patterns like those observed in the mineral Mg-cordierite. Our dynamical studies of growth and relaxation show the formation of these static patterns, and also transitory structures such as 12-armed bursts, streamers and striations which are also seen experimentally. The major aspects of the growth-relaxation process are quite unlike those in systems with conventional order parameters, for it is inherently nonlocal; for example, the changes from one snapshot to the next are not predictable by inspection.

A Landau model is used to study the phase behavior of the surface layer for magnetic and cholesteric liquid-crystal systems that are at or near a Lifshitz point marking the boundary between modulated and homogeneous bulk phases. The model incorporates surface and bulk fields and includes a term in the free energy proportional to the square of the second derivative of the order parameter in addition to the usual term involving the square of the first derivative. In the limit of vanishing bulk field, three distinct types of surface ordering are possible: a wetting layer, a nonwet layer having a small deviation from bulk order, and a different nonwet layer with a large deviation from bulk order that decays nonmonotonically as the distance from the wall increases. In particular, the large deviation nonwet layer is a feature of systems at the Lifshitz point and also those systems having only homogeneous bulk phases.

A Landau expansion of the elastic energy in the strains is used to study two-dimensional structures in tetragonal-orthorhombic ferroelastics with constraints. Local energy minima are found with respect to the components of the displacement and so the strains satisfy the compatibility relation; this interdependence of the strains, combined with the constraints, can give rise to a subtle frustration. Extraordinarily, a complex energy surface with many bulk metastable states results purely from boundary conditions, without hulk inhomogeneities (such as impurities) of any sort. Some settings require twin walls in only one set of tetragonal 110-type planes; only two variants appear, and the dilatational and shear strains are localized near the surface. Tip splitting can occur when twin walls collide with fixed boundaries. Other settings require both 110 and 1 (1) over bar 0 walls and so all four variants appear. The structures resulting from collisions of the two twin families are so complex that the ground state of a large system cannot be found with confidence. Strange walls appear between variants with identical deviatoric strain. The dilatational and shear strains are large also in the bulk. Walls wobble, bow, and bend counterintuitively, and pairs sometimes pinch in. Study of the rotation is shown to be essential for understanding some aspects of the structures, particularly collisions of orthogonal twin bands. The ferroelastic-ferromagnet analogy is found to be misleading in important respects. Tip splitting, pinching-in, wall wobbling, and other phenomena are seen in electron microscopy of YBa2Cu3O7-partial derivative and other materials.

Mean-field theory is insufficient to explain the magnetic order of RbFeCl3 in a magnetic field applied in the c plane. Specifically, quantum and thermal fluctuations induce magnetic structures which do not otherwise appear, as found also for CsCuCl3. We derive microscopically an expression for the free energy which describes both kinds of fluctuations on an equal basis. Our result includes mean-field theory, but goes beyond it to include terms which lift the nontrivial degeneracy. In addition to the modified 120 degrees structure at low fields and the aligned state at high fields? we find an up-up-down-type collinear structure and a coplanar structure at intermediate fields. Our theory explains qualitatively experimental results at low temperatures, but detailed confirmation requires further experiments including the neutron scattering.

We derive solutions for the twin wall linking two tetragonal variants of the cubic-tetragonal ferroelastic transformation, including for the first time the dilatational and shear energies and strains. Our solutions satisfy the compatibility relations exactly and are obtained at all temperatures. They require four non-vanishing strains except at the Barsch-Krumhansl temperature TBK (where only the two deviatoric strains are needed). Between the critical temperature and TBK, material in the wall region is dilated, while below TBK it is compressed. In agreement with experiment and more general theory, the twin wall lies in a cubic 110-type plane. We obtain the wall energy numerically as a function of temperature and we derive a simple estimate which agrees well with these values.

We study the Landau model of the class of incommensurate systems with a scalar order parameter where the modulated phase is driven by a gradient-squared term with negative coefficient. For example, theoretical studies of cholesteric liquid crystals in a field (electric or magnetic) suggest that such an modulated phase should exist at high chirality. The bulk phase diagram in the presence of a bulk external field which couples linearly to the order parameter exhibits a modulated phase inside a loop in the temperature-field plane, and a homogeneous phase outside. On analyzing the same model for a semi-infinite system, we find a surprising result; the system exhibits surface states in a region where the bulk phase is homogeneous (but close to the modulated region). These states are very different from the well-known surface states induced either by a surface field or by enhanced interactions at the surface, for they exist and are energetically favored even when the sole effect of the surface is to terminate the bulk, as expressed by free boundary conditions taken at the surface. Near the surface, the surface-state order parameter is very different from the bulk value (in fact, it has the opposite sign). When the temperature or the bulk field are varied to move away from the modulated state, we find a surface phase transition at which the surface states become energetically unfavorable, though they continue to exist as metastable states. We then study how a surface field changes the surface phase diagram.

In zero magnetic field, the stacked, triangular antiferromagnet ${\mathrm{CsCuCl}}_{3}$ has a helical structure incommensurate (IC) in the chain direction (normal to the planes). A magnetic field applied transverse to the chains distorts the helix, but the IC structure persists up to at least $0.43$ times the saturation field. The IC wave number $q$ (from neutron-diffraction experiments) decreases with increasing field, but then it has an unexpected plateau. Classical theory explains the behavior at small fields, including the temperature dependence, but it fails to explain the plateau, which we ascribe to quantum fluctuations. We find that linear spin-wave (LSW) theory also fails to explain the plateau; in fact, LSW theory fails more severely than classical theory in describing the IC phase. We introduce a phenomenological treatment of quantum fluctuations. After verifying that it describes well some known results, we apply the phenomenological theory to the IC phase of ${\mathrm{CsCuCl}}_{3}$, finding that it yields a plateau at approximately the observed value of $q$ and the observed fields; in addition, it predicts a transition to the commensurate phase so far not observed. Results depend sensitively on a weak anisotropy: A deviation of less than 1% from isotropy in the intrachain ferromagnetic exchange changes the phase diagram completely at fields above about half the saturation value.