Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning

Su, Yu; Landis, Chad M.

doi:10.1016/j.jmps.2006.07.006

Cited by 265 publications

(229 citation statements)

References 38 publications

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“…The FEPF model includes mechanical equilibrium, Gauss law, and the time-dependent Ginzburg-Landau equation ͑see Ref. 20 for details͒. The three governing equations are obtained by taking the variation in the Gibbs electric energy density:…”

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confidence: 99%

Peculiar effect of mechanical stress on polarization stability in micrometer-scale ferroelectric capacitors

Gruverman

Cross

Oates

2008

Applied Physics Letters

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Piezoresponse force microscopy (PFM) has been used to study the polarization stability in micrometer size Pb(Zr,Ti)O3 capacitors. It is shown that the top electrode thickness has a profound effect on the equilibrium polarization state of poled capacitors triggering spontaneous polarization backswitching in the absence of an applied electric field and leading to the formation of an abnormal domain pattern. PFM examination of poled capacitors with thick (250 nm) top electrodes reveals domain patterns with the central regions always oriented in the direction opposite to the applied field. It is suggested that the driving force behind the observed effect is a transient response to the residual shear stress created by the top electrode in the poled capacitors during field-induced polarization switching. The proposed mechanism is quantified using finite element ferroelectric phase field modeling. The observed effect provides valuable insight into the polarization retention behavior in micrometer size ferroelectric capacitors.

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confidence: 99%

Peculiar effect of mechanical stress on polarization stability in micrometer-scale ferroelectric capacitors

Gruverman

Cross

Oates

2008

Applied Physics Letters

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“…We apply methods of homogenization to pass to the continuum limit of a very large network of rotating disks, ultimately described by a continuous polarization field ϕ(x). This is the abstraction process analogous to modeling atomistic systems by mesoscale phase field theories, ferroelectric ceramics being a prominent example [5,6,7]. Starting with Equation (S15), we define ∆x γ = a e γ where a is the side length of the discrete unit cell (i.e., the characteristic spacing between disks in the lattice), and e γ is the corresponding (not normalized) distance vector.…”

Section: Discrete Network and Continuum Limitmentioning

confidence: 99%

“…Of course, damping in the system can be arbitrarily complex, so the linear approximation chosen here is only a leading-order approximation which, however, worked excellently for our experimentally investigated 1D bistable networks [12,13]. Linear gradient flow is also the most common kinetic model used in phase field description [5,6,7].…”

Section: Discrete Network and Continuum Limitmentioning

confidence: 99%

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Atomimetic Mechanical Structures with Nonlinear Topological Domain Evolution Kinetics

Frazier

Kochmann

2017

Advanced Materials

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A mechanical metamaterial, a simple, periodic mechanical structure, is reported, which reproduces the nonlinear dynamic behavior of materials undergoing phase transitions and domain switching at the structural level. Tunable multistability is exploited to produce switching and transition phenomena whose kinetics are governed by the same Allen–Cahn law commonly used to describe material‐level, structural‐transition processes. The reported purely elastic mechanical system displays several key features commonly found in atomic‐ or mesoscale physics of solids. The rotating‐mass network shows qualitatively analogous features as, e.g., ferroic ceramics or phase‐transforming solids, and the discrete governing equation is shown to approach the phase field equation commonly used to simulate the above processes. This offers untapped opportunities for reproducing material‐level, dissipative and diffusive kinetic phenomena at the structural level, which, in turn, invites experimental realization and paves the road for new active, intelligent, or phase‐transforming mechanical metamaterials bringing small‐scale processes to the macroscopically observable scale.

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“…In general, constructing W aniso can be challenging; 7,14,15 it must have the appropriate minima in a high-dimensional space as well as reflect the symmetry of the crystal. This letter shows a simple method to construct W aniso by introducing a set of auxiliary scalar fields I ͑x͒.…”

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confidence: 99%

Formulation of phase-field energies for microstructure in complex crystal structures

Yang

Dayal

2010

Applied Physics Letters

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The unusual properties of many multifunctional materials originate from a structural phase transformation and consequent martensitic microstructure. Phase-field models are typically used to predict the formation of microstructural patterns and subsequent evolution under applied loads. However, formulating a phase-field energy with the correct equilibrium crystal structures and that also respects the crystallographic symmetry is a formidable task in complex materials. This paper presents a simple method to construct such energy density functions for phase-field modeling. The method can handle complex equilibrium structures and crystallographic symmetry with ease. We demonstrate it on a shape memory alloy with 12 monoclinic variants. © 2010 American Institute of Physics. ͓doi:10.1063/1.3319503͔The unusual behavior of shape-memory alloys, ferroelectrics, and other multifunctional materials is driven by a structural phase transformation. Below the critical phasetransformation temperature, the crystal structure changes and this leads to multiple, symmetry-related variants. The different variants can form microstructural mixtures to satisfy applied boundary conditions. Changes in the applied loads can cause microstructural rearrangements rather than the lattice distortions that are characteristic of typical materials. This ability to rearrange microstructure leads to unusual and technologically important behavior as seen in shape-memory alloys, ferroelectrics, and other multifunctional materials. [1][2][3][4] To enable design with multifunctional materials, it is important to be able to predict the microstructural patterns. Phase field models are widely used for this purpose. 5,6,[11][12][13] They obtain microstructural patterns by minimizing the free energy E of a specimen ⍀. In the case of shape-memory alloys, that are the focus of this paper, E consists of W inter , the surface energy of the interfaces between different variants, and W aniso , the anisotropy energy that penalizes deviation of the strain ⑀͑x͒ from the crystallographically preferred states.W inter = A 0 ٌ͉⑀͉ 2 / 2 penalizes the gradient of the strain and hence the interface has finite thickness to allow easy computation. In general, constructing W aniso can be challenging; 7,14,15 it must have the appropriate minima in a high-dimensional space as well as reflect the symmetry of the crystal. This letter shows a simple method to construct W aniso by introducing a set of auxiliary scalar fields I ͑x͒. We also demonstrate the approach by calculating the microstructure in thin-film shape memory alloy NiTi with monoclinic symmetry that is of technological importance to small-scale actuators. 16 We start by introducing the scalar fields I ͑x͒, I =1, ... ,N where there are N variants. Each variant is associated with a tensor ⑀ 0 I that represents the stress-free strain. At every point, we define the average stress-free strain to be ⑀ 0 ͑x͒ = ͚ I=1 N I ͑x͒⑀ 0 I . We write the free energy of a shape-memory specimen in terms of ⑀͑x͒ and the set ͕ I ͑x͖͒ as foll...

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Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning

Cited by 265 publications

References 38 publications

Peculiar effect of mechanical stress on polarization stability in micrometer-scale ferroelectric capacitors

Peculiar effect of mechanical stress on polarization stability in micrometer-scale ferroelectric capacitors

Atomimetic Mechanical Structures with Nonlinear Topological Domain Evolution Kinetics

Formulation of phase-field energies for microstructure in complex crystal structures

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