Relaxations near surfaces and interfaces for first-, second- and third-neighbour interactions: theory and applications to polytypism

Abstract

AbslrncL Structural relaxations near surfaces and inlerfaces are analysed in a simple, generic model with firs-, second-and lhird-layer interaclions. The relaxations have u p n e n l i a l envelopes with lhree lypes of stmuctural distonion: (i) ferrodislartive, (ii) anlifemcdislorlivc, and (iii) mcdulaled (incommensurable). meir stability conditions in the field of conlml paramelen and their relationship with stmcl~ral phase transilions is derived. A lricriliral p i n t is found far three-layer interaclions and a lransition to an incommensurate phase. All phase transitions to the ferrcdislortive phase are first-order in our model.The theory is applied to the analysis of lalliee relations near internal interfaces in ionic polytypic materials (e.g. Pbll). Diffuse x-ray scatrering and the shifl of diffraction angles are lhe typical fingerprinls for such relaxations. The relevant structure factors are calculated. mere is tentative agreemenl between lhe calculated and observed diffraction profiles.

Atomic-scale evolution of modulated phases at the ferroelectric–antiferroelectric morphotropic phase boundary controlled by flexoelectric interaction

“…The phase diagrams of the individual subsystems are well known (for example, for 2-4 potential a single solution with zero-order parameter above the transition temperature, and two symmetric non-zero solutions below). The coupled system with a biquadratic coupling 48 is remarkably more complex, but has been investigated in detail by Balashova and Tagantsev. 22 We further explore the potential origins of the observed domain wall instability, leading to the formation of spatially modulated periodic structures as the ground state of the system.…”

“…f is the flexoelectric coupling coefficient 49 , c is the component of the stiffness tensor and q is the electrostriction coefficient. The term ξ/2(P 2 A 2 ) is the quadratic coupling 48 between order-parameter fields (typical values of coupling constant ξ for Pb(Zr,Ti)O 3 (PZT) can be found in refs 50,51), while the term ζ∂P/∂xA 2 is the inhomogeneous flexoelectric coupling. Both terms are allowed by the general form symmetry (even in m3m); note that terms such as P 2 ∂A/∂x are usually not allowed owing to the additional symmetry elements, associated with A (some translations in the case of anti-ferrodistortive transitions, etc.…”

Ferroelastic phase transitions: structure and microstructure

“…Are surface relaxations in different crystals expected to have any features in common? Starting from a general model of interacting layers of atoms parallel to a free surface, Houchmandzadeh et al (1992) determined a universal exponential pro®le for the surface relaxation. The only assumption that goes into the proof is that the interactions between the planes of atoms can be expanded as a Taylor series.…”

(An)elastic softening from static grain boundaries and possible effects on seismic wave propagation

“…For thin interfaces the surface scaling is a two-dimensional subspace with m = 2 but this cannot hold for either small grains or materials close to structural instabilities where the surface relaxations extend deeply into the grain (Houchmanzadeh et al 1992;Conti and Salje 2001;Novak and Salje 1998a, b). A similar argument was put forward by Frost and Ashby (1982) where scaling in the case of rheological amplitudes and timescales, the enhanced interfacial diffusion, can lead to mixtures of grain size scaling with d -2 (1 + a/d) where the parameter a depends on the interfacial thickness and a ratio of shear constants in the bulk and in the grain boundary.…”