Theoretical and computational aspects οf phase transitions in elastic ferroelectric materials

Abstract

The last fifty years a great interest from both academics and industries is shown towards materials that are capable of undergoing phase transformations. Phase transitions are an important characteristic of crystalline materials based on the fact that they can alter their crystal symmetry at various ranges of temperature. Ferroelectric materials have become important materials in a wide variety of electronics due to their pronounced dielectric and piezoelectric properties. The material macroscopic properties are related to the microscopic domain structure of the materials. To understand and predict the relation between the macroscopic properties and the domain structure, continuum models are employed in this work. More specifically, this thesis is organized as follows: In the first chapter, an introduction to the physics of ferroelectrics is presented. Explanations are given on the nature of ferroelectricity from the crystallographic point of view, but since we are interested in the modelization of these materials we present a brief introduction of the famous Landau model, which serves as a conceptual bridge between microscopic models and observed macroscopic phenomena. Moreover, we provide useful information on the behavior of these materials at very low scales. In the second chapter, we focus on the so called phase field models that use polarization as the first order parameter of the evolutionary process inside bulk ferroelectrics. The proposed phase field model is based on a theory that accumulates gradients of the Maxwelian electric field and also introduces spontaneous quadrupoles. The aim of this chapter is to reveal the role of both spontaneous and linear quadrupole polarization in the domain structure of ferroelectrics. It is proved that spontaneous quadrupoles are responsible for the thickness of domain walls and linear quadrupoles have a serious impact on low dimensional ferroelectrics, such as thin films. The third chapter introduces the level set method and we use this method to describe the kinetics of a phase boundary. Level set methods have been introduced by Osher and Sethian and soon have become a very powerful tool for tracking a moving interface within a body. The method is based on an implicit representation of the interface by considering a smooth scalar function, which changes sign across the interface. Thus the zero level set of the implicit function coincides with the interface. The introduction of a level set function results in regularization of the sharp interface model in solids. In level set methods the interfaces transform into thin transition layers where all discontinuous quantities take inhomogeneous but continuous expressions. In this way, we can make a connection to configurational mechanics proving that the forces moving the interfaces are inhomogeneity forces. Computational results provide the microstructure of ferroelectrics and they are in agreement with the results of the phase field model presented in the second chapter. In the fourth chapter, we use the previous models to study special topics in ferroelectrics. To be more precise we study the influence of point charge and dipolar defects in the motion of domain walls so as to extract useful information of damaged ferroelectrics. Moreover, we present a combined level set-phase field model to study the behavior of bi-crystals, which can be easily expanded to polycrystals