Landau theory of structures in tetragonal-orthorhombic ferroelastics

A. E. Jacobs

doi:10.1103/PhysRevB.61.6587

Access by Xinjiang University

Landau theory of structures in tetragonal-orthorhombic ferroelastics

A. E. Jacobs^*

Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel 76100

^*Electronic address: jacobs@physics.utoronto.ca

PDF

Phys. Rev. B 61, 6587 – Published 1 March, 2000

DOI: https://doi.org/10.1103/PhysRevB.61.6587

Abstract

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 bulk 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 $11 ¯ 0$ 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 ${YBa}_{2} {Cu}_{3} O_{7 - δ}$ and other materials.

References (27)

K. Aizu, J. Phys. Soc. Jpn. 27, 387 (1969).
E. K. H. Salje, Phase Transitions in Ferroelastic and Co-elastic Crystals (Cambridge University Press, Cambridge, 1993).
E.K.H. Salje and Y. Ishibashi, J. Phys.: Condens. Matter 8, 8477 (1996). A comprehensive discussion of needle twins is given by E.K.H. Salje, A. Buckley, G. Van Tendeloo, Y. Ishibashi, and G.L. Nord, Jr., Am. Mineral. 83, 811 (1998).
F. Falk, Z. Phys. B: Condens. Matter 51, 177 (1983).
G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett. 53, 1069 (1984).
J. Sapriel, Phys. Rev. B 12, 5128 (1975).
A.E. Jacobs, Phys. Rev. B 31, 5984 (1985).
A.E. Jacobs, Phys. Rev. B 46, 8080 (1992).
S. Kartha, T. Castán, J.A. Krumhansl, and J.P. Sethna, Phys. Rev. Lett. 67, 3630 (1991).
S. Kartha, J.A. Krumhansl, J.P. Sethna, and L.K. Wickham, Phys. Rev. B 52, 803 (1995).
A.E. Jacobs, Phys. Rev. B 52, 6327 (1995).
W.C. Kerr, M.G. Killough, A. Saxena, P.J. Swart, and A.R. Bishop, Phase Transit. 69, 247 (1999).
S.R. Shenoy, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B 60, 12 537 (1999).
S. Semenovskaya, Y. Zhu, M. Suenaga, and A.G. Khachaturyan, Phys. Rev. B 47, 12 182 (1993), and references therein.
A.M. Bratkovsky, E.K.H. Salje, and V. Heine, Phase Transit. 52, 77 (1994), and references therein.
P. Klouček and M. Luskin, Continuum Mech. Thermodyn. 6, 209 (1994); Math. Comput. Modell. 20, 101 (1994); B. Li and M. Luskin, Mater. Sci. Eng. A 273, 237 (1999).
A.H. King and Y. Zhu, Philos. Mag. A 67, 1037 (1993).
Y. Zhu, M. Suenaga, and J. Tafto, Philos. Mag. A 67, 1057 (1993).
References and also found a complex energy surface, but for completely unrelated reasons; the density in Refs. contains explicit bulk inhomogeneities.
A convenient reference is L. Brillouin, Tensors in Mechanics and Elasticity (Academic, New York, 1964).
The parameters a and $T_{0}$ could, in principle, be fitted from the transition temperature $T_{c}$ and the stability limits; the T state is unstable for $A_{2} < 0,$ and the O state is unstable for $A_{2} > B_{2}^{2} {/ (4 C}_{2}) .$
S.F. Borg, Matrix-Tensor Methods in Continuum Mechanics (Van Nostrand, New York, 1963). This gives a particularly thoughtful derivation of Eqs. (2.6) and (A2), both the linearized versions and the leading nonlinear corrections.
The compatibility relation for finite strains is given, for example, by W. Jaunzemis, Continuum Mechanics (MacMillan, New York, 1967), Eq. (15.20).
G.R. Barsch and J.A. Krumhansl, Metall. Trans. A 19, 761 (1988).
J.L. Ericksen, Int. J. Solids Struct. 22, 951 (1986).
J. Lajzerowicz, Ferroelectrics 35, 219 (1981).
Centering the derivatives causes the strains to extend two grid points into a region where the displacement vanishes identically.

[c1] K. Aizu, J. Phys. Soc. Jpn. 27, 387 (1969).

[c2] E. K. H. Salje, Phase Transitions in Ferroelastic and Co-elastic Crystals (Cambridge University Press, Cambridge, 1993).

[c3] E.K.H. Salje and Y. Ishibashi, J. Phys.: Condens. Matter 8, 8477 (1996). A comprehensive discussion of needle twins is given by E.K.H. Salje, A. Buckley, G. Van Tendeloo, Y. Ishibashi, and G.L. Nord, Jr., Am. Mineral. 83, 811 (1998).

[c4] F. Falk, Z. Phys. B: Condens. Matter 51, 177 (1983).

[c5] G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett. 53, 1069 (1984).

[c6] J. Sapriel, Phys. Rev. B 12, 5128 (1975).

[c7] A.E. Jacobs, Phys. Rev. B 31, 5984 (1985).

[c8] A.E. Jacobs, Phys. Rev. B 46, 8080 (1992).

[c9] S. Kartha, T. Castán, J.A. Krumhansl, and J.P. Sethna, Phys. Rev. Lett. 67, 3630 (1991).

[c10] S. Kartha, J.A. Krumhansl, J.P. Sethna, and L.K. Wickham, Phys. Rev. B 52, 803 (1995).

[c11] A.E. Jacobs, Phys. Rev. B 52, 6327 (1995).

[c12] W.C. Kerr, M.G. Killough, A. Saxena, P.J. Swart, and A.R. Bishop, Phase Transit. 69, 247 (1999).

[c13] S.R. Shenoy, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B 60, 12 537 (1999).

[c14] S. Semenovskaya, Y. Zhu, M. Suenaga, and A.G. Khachaturyan, Phys. Rev. B 47, 12 182 (1993), and references therein.

[c15] A.M. Bratkovsky, E.K.H. Salje, and V. Heine, Phase Transit. 52, 77 (1994), and references therein.

[c16] P. Klouček and M. Luskin, Continuum Mech. Thermodyn. 6, 209 (1994); Math. Comput. Modell. 20, 101 (1994); B. Li and M. Luskin, Mater. Sci. Eng. A 273, 237 (1999).

[c17] A.H. King and Y. Zhu, Philos. Mag. A 67, 1037 (1993).

[c18] Y. Zhu, M. Suenaga, and J. Tafto, Philos. Mag. A 67, 1057 (1993).

[c19] References and also found a complex energy surface, but for completely unrelated reasons; the density in Refs. contains explicit bulk inhomogeneities.

[c20] A convenient reference is L. Brillouin, Tensors in Mechanics and Elasticity (Academic, New York, 1964).

[c21] The parameters a and $T_{0}$ could, in principle, be fitted from the transition temperature $T_{c}$ and the stability limits; the T state is unstable for $A_{2} < 0,$ and the O state is unstable for $A_{2} > B_{2}^{2} {/ (4 C}_{2}) .$

[c22] S.F. Borg, Matrix-Tensor Methods in Continuum Mechanics (Van Nostrand, New York, 1963). This gives a particularly thoughtful derivation of Eqs. (2.6) and (A2), both the linearized versions and the leading nonlinear corrections.

[c23] The compatibility relation for finite strains is given, for example, by W. Jaunzemis, Continuum Mechanics (MacMillan, New York, 1967), Eq. (15.20).

[c24] G.R. Barsch and J.A. Krumhansl, Metall. Trans. A 19, 761 (1988).

[c25] J.L. Ericksen, Int. J. Solids Struct. 22, 951 (1986).

[c26] J. Lajzerowicz, Ferroelectrics 35, 219 (1981).

[c27] Centering the derivatives causes the strains to extend two grid points into a region where the displacement vanishes identically.

Physical Review B

Export citation

Export citation

Landau theory of structures in tetragonal-orthorhombic ferroelastics

Abstract

References (27)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation