Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials380

(Damjanovic and Demartin, 1996; Damjanovic, 1997; Eitel et al., 2006).

This can be used for the computation of hysteresis loss energy and an effective

loss tangent (Eitel et al. 2006; Zhou et al., 2006). When the hysteresis is

small, the ferroelectric constitutive relations can be expressed using complex

variables in time harmonic form using the concepts of dielectric, elastic, and

piezoelectric loss tangents (Holland, 1967). The dielectric, mechanical, and

piezoelectric loss tangents are field and frequency-dependent along with

complex correlation, which depends on the mechanism behind them (Gerthsen

et al., 1980; Hardtl, 1982; Herbiet et al., 1989; Robels et al., 1989; Uchino

and Hirose, 2001; Eitel et al. 2006). The complex variable approach introduces

a phase lag between the applied loads and the response variables as an

approximation to the observed hysteresis. It is well known and discussed in

detail in the references cited above.

There are both dielectric and mechanical hysteretic losses in ferroelectric

materials. They are due to irreversible work done by the electric field and

stress respectively, which are directly related to the remnant electric

displacement and remnant strain changes. In ferroelectric materials, changes

of strain and electric displacement include reversible (superscript rev) and

irreversible (remnant, superscript r) components:

d = d + d

rev r

εε ε

ij ij

13.17

d = d + d

rev r

DD D

13.18

A change of internal energy can then be expressed as

d = d + d + d

rev rev

uEDTS

σε

13.19

13.8

Critical energy-temperature phase diagram for <110> poled

PMN–32%PT (Liu

et al.

, 2006). Symbols are from measurements of

electric field and stress-induced phase transition at three

temperatures (20, 40, and 60°C).

(°C)

100800604020

~ 90 –

~ 55 – 0.75

∆

(kJ/m

)

WPNL2204

Energy analysis of field-induced phase transitions 381

with

ρσεTS E D q

d = d + d + d

13.20

Here T is the temperature and, dS is change of entropy per unit mass. This

expression relates the dissipative increments of the strain and electric

displacement to the generation of thermal energy.

σε

is the irreversible

work increment done by stress and

is the irreversible work increment

done by the electric field. Models that describe the evolution of remnant

strain and remnant electric displacement may be developed and used to

compute the dissipative terms (loss) and heat generation. The second law

states that the dissipation will be non-negative:

σε

EDd+ d 0

≥

13.5 Discussion

The phase transformation behavior under combined stress, electric field, and

thermal loading is a coupled multi-field phenomenon. The work–energy

analysis shows that at a given temperature the phase transformation begins

at a critical level of internal energy density and ends at a second critical

level. A second criterion, minimization of the Gibbs free energy, determines

whether the transformation will take place. The critical energy levels were

found to be independent of the combinations of stress and electric field that

induced them.

This leads back to the analysis of the electric displacement – electric field

data, discussed initially. Collection of D-E data is straightforward and can be

readily performed at a series of temperatures. The data can be used to identify

the relative energy levels of the phases. To generate plots like Fig. 13.7 from

a limited number of measurements, a set of the elastic, piezoelectric, and

dielectric coefficients of each phase is needed. This provides sufficient

information to characterize the multi-field-driven transformation. These results

are expected to reduce the amount of experimental work required to fully

characterize multi-field-driven phase transformations.

The ‘engineered domain state’ is critical to the utilization of this approach.

In the <001> and <011> crystal cuts the domain walls are stable. This means

there is little or no ferroelectric or ferroelastic polarization reorientation. In

other crystal cuts there is domain wall motion under applied stress and

electric fields. This gives rise to hysteresis associated with domain wall

motion and energy dissipation under applied fields. In this case it would be

difficult to distinguish between energy barriers to transformation and the

highly rate dependent and dissipative process of domain wall motion.

Domain structures and domain wall motion introduce energy terms to the

total energy. Incompatible spontaneous strain and spontaneous polarization

at grain and domain boundaries give rise to local stress and local electric

field at the microstructural level. Although the volume averages of these

WPNL2204

Handbook of dielectric, piezoelectric and ferroelectric materials382

local fields are zero, the stored energy associated with these quantities is

non-zero. These must be included in the expression for the total energy.

The microstructural stress and electric fields contribute to the driving

forces for domain wall motion (evolution of the remnant strain and remnant

polarization). The microstructural stress would drive the nucleation and motion

of domain walls such that the new domain configuration would reduce the

microstructural energy. The application of an external field can thus move

domain walls in one direction and on removal of the field the microstructural

stress can move them back. The dissipation associated with this process

gives rise to minor hysteresis loops. The same arguments apply to local

electric field, domain walls with charge compatibility will form to reduce the

electrical energy.

Domain wall dynamics and phase field theory have been used to model

domain wall structures and their evolution (Bhattacharya and Ravichandran,

2003; Huber, 2005). Domain wall dynamics analyzes domain wall motion

under external loads that alter the domain wall energy balance. The phase

field modeling uses the framework of the time-dependent Ginzburg–Landau

equations. Changes of the remnant electric displacement and remnant strain

reflect domain reorientations and domain wall movements. Microscopic or

phenomenological models may be developed to describe the evolutions of

these irreversible variables (Kamlah, 2001; Landis, 2004).

13.6 Concluding remarks and future trends

Thermodynamics analysis leads to the utilization of changes of internal energy

and the Gibbs free energy as criteria that determine whether the phase

transformation will take place under combined external field loading. The

critical energy levels were found to be independent of the combinations of

applied stress and electric field. The energy criteria based on experimental

measurements provide important parameters for theoretical modelling.

Development of constitutive models for ferroelectric materials and their

implementation allow the solution of problems encountered in applications

of these materials.

Models at different length scales have their merits. From atomistic modeling

to domain wall dynamics and phase-field approaches, from microscopic

models to macroscopic (structural) computation, material parameters are

needed as inputs. Energy criteria and parameters based on measurements

seem to be the bridges connecting models at different length scales and

achieving reliable multi-scale models.

Field-induced phase transformations in ferroelectric ABO

perovskite type

oxides are critical to a number of applications, yet characterization and

modeling of this behavior has been rather limited. These transformations

occur in thin films due to large in-plane stress, in shock-driven ferroelectric

WPNL2204

Energy analysis of field-induced phase transitions 383

generators, and in large field single crystal actuators. The design of devices

that take advantage of these properties requires a finite element code with

accurate constitutive models governing this behavior. Multi-axial

phenomenological models developed within a thermodynamics framework

using internal state variables are less computationally intensive than

micromechanics-based constitutive models and are thus the most efficient

when implemented in finite element codes. Micromechanics models explicitly

link the physical behavior at smaller length scales to the observed macroscale

behavior. They are more computationally intensive, but provide the capability

of accurately simulating multi-axial loads that cannot be readily achieved

experimentally. To date, little work has been done on the development of

rate-dependent constitutive models governing ferroelectric phase

transformations. A finite element code implemented with these constitutive

models will enable exploration of the effects of geometry on the behavior of

phase transforming materials and devices.

13.7 Acknowledgement

The authors gratefully acknowledge the support of the United States Office

of Naval Research for their support of this work under grant N00014-03-1-

0987.

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related phenomena

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