The domain size dependence of piezoelectric properties of ferroelectrics is investigated using a continuum Ginzburg-Landau model that incorporates the long-range elastic and electrostatic interactions.

We devise a two dimensional model that mimics the recently observed power law distributions for the amplitudes and durations of the acoustic emission signals observed during martensitic transformation [ Vives et al, Phys. Rev. Lett. 72, 1694]. We include a threshold mechanism arising from the athermal nature of transformation, long-range interaction between the transformed domains, inertial effects, and dissipation arising due to the motion of the interface. The model exhibits thermal hysteresis of the transformation, and more importantly, it shows that the energy is released in the form of avalanches with power law distributions for their amplitudes and durations. Computer simulations also reveal morphological features similar to those observed in real systems.Many spatially extended driven systems naturally evolve to a marginally stable state characterized by avalanches with power law distributions for their amplitudes and durations reflecting lack of intrinsic length scales and time scales in the system. Such a state is termed as self-organized critical (SOC) state by Bak et al [1]. Several physical systems exhibit SOC features; for example, earthquakes [2], acoustic emission from volcanic rocks [3], and stress drops during jerky flow [4], to name a few. Recently, Vives et al [5], measured the acoustic emission (AE) signals during martensitic transformation of Cu-Zn-Al single crystals under thermal cycling. They reported power law statistics for the amplitudes and durations of the AE signals both during cooling and heating runs. To the best of our knowledge, there is no strain (or displacement) based model of martensitic transformation which explains these results. More over, any prospective model has to take into account the non-equilibrium nature of the hysteresis. Even though extensive theoretical studies exist on martensitic transformations [6][7][8], the influence of dissipation and defects on hysteresis has received very little attention. Here, we propose a simple phenomenological model which captures the power law distribution of AE signals along with the thermal hysteresis of the transformation. Below, we will briefly collect SOC type dynamical features of the martensitic transformation relevant for modelling the system. Martensitic transformation is a first-order, solid-solid, diffusionless, structural phase transition. On cooling, the unit cell gets distorted [6][7][8] leading to the nucleation of thin plate-like product domains with a twinned structure. This induces internal strains, which in turn induce longrange fields, that block the transformation leaving the system in a two phase metastable state. ( Note that this implies the existence of built-in threshold mechanism.) Thus, the amount of the transformed phase is entirely determined by the excess free energy and, an additional undercooling is required for further growth. This implies that thermal fluctuations have little role in the transformation kinetics. Thus, the transformation is athermal and hence the nucleation is athermal, usually occu...

By including the contributions of dipolar defects in the time-dependent Ginzburg-Landau theory, we have simulated the domain switching process in ferroelectrics. The model incorporates elastic effects in the form of an anisotropic long-range interaction that is obtained by integrating out the strain fields, subject to the elastic compatibility constraint. The defects are simulated by considering an inhomogeneous electric field due to randomly placed coarse-grained dipoles. It is shown that these defects act as nuclei for the formation of 90°t winned structures, resulting in a lower coercive field compared to the defect-free case. Due to these defects, the simulated polarization switching occurs by two successive 90°rotations, rather than a single 180°flipping as in the defect-free case.

Freestanding BaTiO3 nanodots exhibit domain structures characterized by distinct quadrants of ferroelastic 90° domains in transmission electron microscopy (TEM) observations. These differ significantly from flux-closure domain patterns in the same systems imaged by piezoresponse force microscopy. Based upon a series of phase field simulations of BaTiO3 nanodots, we suggest that the TEM patterns result from a radial electric field arising from electron beam charging of the nanodot. For sufficiently large charging, this converts flux-closure domain patterns to quadrant patterns with radial net polarizations. Not only does this explain the puzzling patterns that have been observed in TEM studies of ferroelectric nanodots, but also suggests how to manipulate ferroelectric domain patterns via electron beams.

Flexoelectric effect is the response of electric polarization to the mechanical strain gradient. At the nano-scale, where large strain gradients are expected, the flexoelectric effect becomes appreciable and may substitute piezoelectric effect in centrosymmetric materials. These features make flexoelectricity of growing interest during the last decade. At the same time, the available theoretical and experimental results are rather contradictory. In particular, experimentally measured flexoelectric coefficients in some ferroelectric materials largely exceed theoretically predicted values. Here, we determine the upper limits for the magnitude of the static bulk contribution to the flexoelectric effect in ferroelectrics, the contribution which was customarily considered as the dominating one. The magnitude of the upper bounds obtained suggests that the anomalously high flexoelectric coupling documented for perovskite ceramics can hardly be attributed to a manifestation of the static bulk effect. V C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4865208] Flexoelectric effect is the response of electric polarization to the mechanical strain gradient. It can be viewed as higher-order effect with respect to piezoelectricity, which is the response of polarization to strain itself. However at the nano-scale, where large strain gradients are expected, the flexoelectric effect becomes appreciable. Besides, in contrast to piezoelectric effect, flexoelectricity is allowed by symmetry in any material. Due to these features flexoelectricity has attracted growing interest during the last decade. On the other hand, the available theoretical and experimental results are rather contradictory, attesting to a limited understanding in the field. In particular, often experimentally measured flexoelectric coefficients largely exceed theoretically predicted values. It is important to distinguish different contributions to the effect: bulk and surface contributions; static and dynamic contributions. The relative magnitude of these contributions is discussed in a recent review article.

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Materials which can undergo extremely fast displacive transformations as well as very slow diffusive transformations are studied using a Ginzburg-Landau framework. This simple model captures the essential physics behind microstructure formation and time-temperature-transformation diagrams in alloys such as steels. It also predicts the formation of mixed microstructures by an interplay between diffusive and displacive mechanisms. The intrinsic volume changes associated with the transformations stabilize mixed microstructures such as martensite-retained austenite (responsible for the existence of a martensite finish temperature) and martensite-pearlite.

Recently, it has been demonstrated that martensitic transformation in nanocrystalline shape memory alloys can be suppressed for small grain sizes. Motivated by these results, we study the grain size dependence of martensitic transformations and stress-strain response of nanocrystalline shape memory alloys within the framework of the Ginzburg-Landau (GL) theory. A GL model for a square to rectangle transformation in polycrystals is extended to account for grain boundary effects. We propose that an inhibition of the transformation in grain boundary regions can occur, if the grain boundary energy of the martensite is higher than that of the austenite phase. We show that this inhibition of transformation in grain boundary regions has a strong influence on domain patterns inside grains. Although the transformation is inhibited only at the grain boundaries, it leads to a suppression of the transformation even inside the grains as grain size is decreased. In fact, below a critical grain size, the transformation can be completely suppressed. We explain these results in terms of the extra strain gradient cost associated with grain boundaries, when the transformation is inhibited at grain boundaries. On the other hand, no significant size effects are observed when transformation is not inhibited at grain boundaries. We also study the grain size dependence of the stress strain curve. It is found that when the transformation is inhibited at grain boundaries, a significant reduction in the hysteresis associated with stress-strain curves during the loading-unloading cycles is observed. The hysteresis for this situation reduces even further as the grain size is reduced, which is consistent with recent experiments. The simulations also demonstrate that the mechanical behavior is influenced by inter-granular interactions and the local microstructural neighbourhood of a grain has a stronger influence than the orientation of the grain itself.