In Partial Fulfillment of the Requirements For the Degree of

35 Figure 3-2: Self-consistent electric displacement (top) and strain (bottom) model estimates of the response of a ferroelectric polycrystal to cyclic electric field loading [8]. An "elastic" curve is a prediction of a self-consistent model that assumes only elastic deformation.

Introduction

• Ferroelectric Materials: Definition and Application
• Piezoelectricity: Governing Equations
• Ferroelectrics: Constitutive Modeling
• Goal and Outline of Thesis

The location of the centers of charges relative to each other in the unit cell is decisive for the electromechanical properties of the material. If a small electric field is applied in the direction of the spontaneous polarization, the polarization will increase (dielectric effect) along with strain (piezoelectric effect).

Figure 1-1: The perovskite crystal structure common to many ferroelectric ceramics. For BTO the white ions at the corners are Ba +2 , the black ions on the faces are O -2 , and the central ion is Ti +4

The Self-Consistent Model

• The Eshelby Tensor
• Inhomogeneous Inclusion
• Multiple Inclusions
• Polycrystals and the Self-Consistent Model

One way to model this is to assume a background stress already present in the matrix denoted by . These models can predict the elastic and plastic deformation of the grains and, therefore, the overall stresses in the polycrystal. The term P is the plastic strain tensor in the grain, γi is the motion of the ith slip system in the grain, and i is the Schmid tensor associated with the ith slip system.

Now the stress and strain rates at the inclusion can be related to the stress and strain rate at infinity (HEM stress and strain rates) by the concentration tensor, AC, which can then be related to L*, the 'limit' of Hill's. tensor. At a certain stage of deformation, the stress, and thus the potentially active slip systems in the polycrystal components, is known.

Figure 2-1: An ellipsoidal inclusion, I, is cut out of the matrix, M.

Self-Consistent Modeling of Ferroelectrics

The Huber Model

The strain ( ) and electrical displacement (D) of the grains are taken as the volume average of L and DL over the variants, plus residual strain and polarization due to switching. The terms in the matrix (SE, d, : compliance, piezoelectricity . coefficient and electrical permittivity, respectively) are the electromechanical properties of the grain which are the volume averages of the corresponding domain properties (SE(I), d(I), ( I) ) as shown by equations (3.3-3.5). The first term on the right-hand side in equation (3.2) is the linear response of the grain while the second term on the right-hand side is the residual.

The development of the remaining part is the result of domain switching. cI is the volume fraction of each domain present in the grain. If the load were only in the form of stress, classical theories such as von Mises could be used to combine the various components of the stress tensor.

Figure 3-1: The progressive nature of ferroelectric transformation within a crystal due to domain wall motion

The New Self-Consistent Model

• Grain Selection and Average Lattice (hkl) Strain
• The Problem of Reverse Switching

If these were the only grains in the material the [002] lattice stress would be the average of the stresses within. The percentage of domains that cannot be switched can be specified as a variable parameter in the program. In the new model, any texture can be defined at the beginning of the analysis.

Therefore, only one grain is considered here, and it does not matter what happens in the rest of the material. In the first case (Figure 3-6), regardless of the sign of the applied voltage σ11, only the transformation can take place in direction “−α”.

Figure 3-3: Schematic of the experimental setup at ENGIN X. The diffracted neutrons are collected by two 2 θ = ±90º detectors with scattering vectors corresponding to the

Neutron Diffraction Experiments

Introduction

However, it will be shown below that the Rietveld method must be used with great caution when high-voltage anisotropy is involved, as is the case with ferroelectric materials. An improved Rietveld method is presented here and used to capture the anisotropic behavior of BaTiO3.

Experimental Setup and Sample Properties

Samples are oriented at a Bragg angle of θ = 45º with the incident beam (Figure 3-3) allowing the 2θ = ±90º detectors to measure the longitudinal and transverse sample response. In this thesis, all results and references relate to the longitudinal direction, unless otherwise stated. Therefore, the diffraction data consisted of multi-peak patterns from sample directions either parallel or perpendicular to the loading axis.

The x-axis is either in terms of time-of-flight. a) or d-spacing (b), while the y-axis is the intensity of the deflected neutrons in terms of counts per microsecond. Once the diffraction data has been reduced and normalized to correct for applicable instrument artifacts, the next challenge is the calculation of grating distortion.

Figure 4-2: Diffraction spectra from a time-of-flight neutron diffraction experiment at ENGIN X using an unloaded BaTiO 3 polycrystal

Calculating Lattice Strains: the Single-Peak Method

Several approaches can be used to achieve this goal. These are presented in the next section, along with their pros and cons. The peak position is used in calculating lattice voltages, and the peak intensity is used in texture analysis. The peak profile function used in this thesis is profile type number 3 in the RAWPLOT module of GSAS, which is an exponential pseudovoigt convolution (Von Dreele, 1990, unpublished) [16].

The results of this analysis are presented and discussed in Chapter 5. a) Raw data and the fitted peak with refined peak position and intensity.

Figure 4-3: Single-peak fitting to reflections (200) and (002) of BaTiO 3 . ( a) Raw data and the fitted peak with peak position and intensity refined

Calculating Lattice Strains: the Rietveld Method

Typically, multiple Bragg reflections can contribute to the observed intensity at any point in the pattern. The least-squares minimization procedure leads to a set of normal equations involving the derivatives of all the calculated intensities, yci; with respect to each adjustable parameter and the inversion of the normal matrix Mjk gives the solution. Rietveld refinement, unlike single-peak fitting, is dependent on the physics of the problem, and a full intensity of the diffracted neutron intensity versus time-of-flight or d-distance is calculated based on the refinement of any or all of the following material data: convergence criteria, number of phases , lattice symmetry for each phase, lattice parameters for each phase, atoms in unit cells, atomic positions in unit cells, atomic occupancy, thermal motion parameters, background function type, background function coefficients, diffractometer constants, absorption/reflectivity correction, phase fractions, histogram scaling factors, peak profile type, and.

Preferred orientation (or texture) is one of the common problems affecting material properties and diffraction data analysis. It characterizes the strength of the preferred orientation and is often related to the amount of sample distortion.

Table 4-1: Initial values used as a starting point for Rietveld refinements in BaTiO 3

Calculating Lattice Strains: the Improved Rietveld Method

One way to relax the assumption of constant voltage is to consider the grid voltage as a function of a series of powers of Ahkl, leading to equation (4.17) below. A better representation of elastic strain anisotropy can be obtained by using higher order terms of cosφ, which will lead to more parameters fitting, as shown in equation (4.21). A more sensible approach could be the use of equation (4.19) and rearranging it to obtain (again with the Reuss assumption.

The results are also compared with those from the currently used approach denoted by equation (4.20). The new approach represented by equation (4.22) successfully estimates the elastic anisotropy of both materials.

Figure 4-5: Different grains have different plane-specific elastic moduli, hence will exhibit different lattice strains

Conclusions and Future Work

Experimental Conclusions: Texture Evolution

However, this does not mean that Rietveld analysis cannot provide any information about the texture development of the sample. Second, the switching process is slow and gradual and lasts up to the breaking stress of the ceramic. This should be compared with Figure 5-1, which shows the superiority of the single-peak method over Rietveld in texture analysis.

In these diagrams, the y-axis is applied voltage (which is the independent variable), while the x-axis is the ratio of the areas of the integrated peak intensities in the doublet. This means the maximum value of the resolved shear stress will be at α = 0 and will decrease as α increases, which exactly matches the observations.

Figure 5-1: (002)/(200) peaks of BaTiO 3 : (a) is the unloaded sample, and (b) is under –100 MPa compressive stress

Experimental Conclusions: Lattice Strains

Leaving aside the possibility of scattering at high voltages, a comparison of various grid strains obtained from single peak analysis is appropriate (Figure 5-10). A comparison of the macroscopic strain values ​​with the lattice strains shows that the effect of strain due to switching (permanent strain) is at least an order of magnitude greater than the lattice strains (elastic strain). The results of Figure 5-10 are best appreciated when compared to those of Figure 5-12 which shows the lattice strains of domains with different hkl reflections if no domain conversion had occurred and the system had acted.

For example, consider the lattice stresses at the ultimate load (–400 MPa) for different hkl reflections in both the experimental results (Figure 5-10) and the imaginary elastic material (Figure 5-12). In other words, because of the linear relationship between the grain stress (domain strain) and lattice strain (elastic domain strain), when the lattice strain in a specific hkl increases, one can conclude that the strain in the grains containing these domains also increased.

Figure 5-8: The evolution of (002) / (200) peak doublets under loading analyzed with the single-peak fitting method

Modeling Conclusions

As load increases, the number of domains in the (002) curve decreases and exactly the same number of domains are added to the (200) curve. If we consider, for example, the (200) and (002) domains, one must realize that they both reside in the same grain. Consequently, the elastic strain (lattice strain) will not increase in these domains because lattice strain is linearly related to stress in the grain.

But it should be noted that what is plotted is actually the lattice stress versus the applied stress, and not the stress in the grain. Since applied stress is the independent variable here, it will increase regardless of what happens in the material/model, leading to apparent hardening.

Table 5-1: Input data for the self-consistent program to model the BaTiO 3 experiment

Summary and Conclusions

Future work should expand on these schemes while seeking to widen the horizon of materials that can be explored within the model and to further improve the level of accuracy achieved, especially by extending it to the inelastic deformation regime. 8], a number of advances were made, including an innovation that corrects for the inverse coupling problem, a method that calculates lattice strain by selecting appropriate grains and averaging their contributions to specific reflections, a capacity to track the number of domains contributing for a chosen reflection, an ability to enter texture (initial grain orientation distribution) and a mechanism to enable locking of domain switching. The level of agreement between the model and the experimental data was satisfactory, especially considering the relative simplicity of the model.

While these tools involve substantial assumptions and should be used only when appropriate, they may nevertheless be useful in future studies of ferroelectrics as they opened up a new avenue of research in this field.

Future Work

This can only be achieved by a combination of mechanics modeling (eg SCM) and appropriate modifications of the profile functions in Rietveld. If successful, this effort will have a major impact on the field of diffraction stress/strain analysis. Mori, In-situ neutron diffraction study of the rhombohedral to orthorhombic phase transformation in lead zirconate titanate ceramics produced by uniaxial compression.

Leffers, Self-consistent modeling of the plastic deformation of FCC polycrystals and its implications for diffraction measurements of internal stresses. Von Dreele, Use of Rietveld refinement to fit a hexagonal crystal structure in the presence of elastic and plastic anisotropy.