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Electromechanical Anisotropy at the Ferroelectric to Relaxor Transition of (Bi_{0.5}Na_{0.5})_{0.94}Ba_{0.06}TiO_{3} Ceramics from the Thermal Evolution of Resonance Curves

^{1}

^{2}

^{3}

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## Abstract

**:**

_{0.5}Na

_{0.5})

_{0.94}Ba

_{0.06}TiO

_{3}dense ceramics were obtained from autocombustion sol-gel synthesized nanopowders and sintered at 1050 °C for 1–2 h for the study of the electromechanical anisotropy. Measurement of the complex impedance spectrum was carried out on thin ceramic disks, thickness-poled, as a function of the temperature from 16 °C up to the vanishing of the electromechanical resonances at the ferroelectric to relaxor transition near 100 °C. The spectrum comprises the fundamental radial extensional mode and three overtones of this, together with the fundamental thickness extensional mode, coupled with other complex modes. Thermal evolution of the spectrum shows anisotropic behavior. Piezoelectric, elastic, and dielectric material coefficients, including all losses, were determined from iterative analysis of the complex impedance curves at the planar, thickness, and shear virtually monomodal resonances of disks and shear plates, thickness-poled. d

_{33}was measured quasi-statically at 100 Hz. This set of data was used as the initial condition for the optimization of the numerical calculation by finite elements of the full spectrum of the disk, from 100 kHz to 1.9 MHz, to determine the thermal evolution of the material coefficients. An appropriate measurement strategy to study electromechanical anisotropy of piezoelectric ceramics has been developed.

## 1. Introduction

_{0.5}Na

_{0.5})TiO

_{3}-xBaTiO

_{3}(BNBT100x) with x = 0.06 have been widely studied in recent years due to their complex crystal structure, exhibiting a field induced phase transition at the nano-scale between short-range ordered and ferroelectric, long-range ordered, polar states [1,2]. The thermal depolarization accompanied by the inhibition of the piezoelectric activity [1] has a multiple origin [3,4]. On the one hand, there is a thermal randomization of the polarization at the ferroelectric domains. On the other hand, there is a thermally stimulated and non-abrupt transition, without macroscopic symmetry breaking, from the long-range ordered ferroelectric structure induced by the field to an ergodic relaxor at ~100 °C, well below the transition to the non-polar paraelectric phase.

**S**and stress

**T**, with two electric magnitudes, the electric field

**E**and the electric displacement

**D**. The parameters in the constitutive equations give information about macroscopic behavior, however they are related with the microscopic mechanisms. For this reason, we aim to determine all parameters in the constitutive equations as a function of the temperature in order to obtain information about the depolarization process. We use measurements of the complex impedance as a function of the frequency of resonators of a given geometry for this purpose. The analysis of these resonance spectra will be made here using alternative methods to that of current standards for measurements of piezoelectric ceramics [5], at present under revision.

## 2. Materials and Methods

^{−1}, for 30 min. Shear plates were re-electroded for the electrical measurements.

_{33}, a quasi-static measurement was carried out on a thin disk at 100 Hz using a Berlincourt piezometer (Channel products Inc., Chesterland, OH, USA).

_{31}, e

_{33}, e

_{15}), and two dielectric (${\epsilon}_{11}^{S}$, ${\epsilon}_{33}^{S}$). Using the initial seed obtained from the one-dimensional analysis, a sensitivity analysis is performed over the real part of the model. Here we obtain different information for the analysis that follows next. First, most of the sensitivity parameters are optimized first as presented by Perez et al. [5]. Second, the parameters are linked with the different resonance modes in order to found a relationship between the changes in the resonance frequency and the parameter value. Finally, we identify the less sensitive parameters to evaluate the results, for this less sensitive set the results must be compared with other samples using different geometry.

## 3. Results

#### 3.1. Impedance Measurements as a Function of the Temperature

_{max}for this mode was not sufficient to clearly show the tendency of its thermal evolution. It is here analyzed by the evolution of the overtones (R

_{2}, R

_{3}, R

_{4}).

_{2}mode. However, such an inflection point is not found for C

_{1}in the measured interval.

_{2}modes; and (iii) the C

_{1}mode, revealing the electromechanical anisotropy of the thermal depolarization process.

#### 3.2. Iterative Analysis of Impedance Curves

#### 3.3. Sensitivity Analysis

_{33}. Second, those parameters that provoke some changes in the spectrum, medium sensitivity parameters: ${c}_{12}^{E}$, e

_{15}, and ${c}_{44}^{E}$. Finally, those parameters that practically do not change the spectrum, low sensitivity parameters: e

_{31}and ${\epsilon}_{11}^{S}$.

#### 3.4. Optimized Parameters

_{33}is observed, as piezoelectric activity vanishes at the ferroelectric to relaxor phase transition [1].

_{13}seems to be clear from 80 °C, in agreement with the evolution of both the resonance frequencies and impedance modulus of the radial modes (Figure 2).

_{1}) and thickness (TH) modes. It also shows that the deformation at C

_{1}and C

_{2}modes involves a great deal of shear movements. With this information, the sensitivity analysis of the Figure 4 and the supplementary material summarized in Table 3, we conclude that the thermal evolution of ${c}_{44}^{E}$ and e

_{15}, medium sensitivity parameters, and ${\epsilon}_{11}^{S}$, low sensitivity parameter, should be mostly related with that of the C

_{1}and C

_{2}modes. However, those two modes do not change in the same way as the temperature increases (Figure 2). The thermal depolarization weakly affects the real and imaginary part of ${\epsilon}_{11}^{S}$. The strong fluctuations of ${c}_{44}^{E}$ do not mask the continuous decrease as the temperature increases of its real and imaginary parts. However, the fluctuations of e

_{15}make it difficult to make conclusions about the temperature at which the piezoelectric activity linked to it vanishes, though the rate of its decrease near 100 °C is the lowest of the three piezoelectric parameters.

## 4. Concluding Remarks

_{15}is the lowest of the three piezoelectric parameters, but its fluctuations make it difficult to conclude about the temperature at which the piezoelectricity vanishes. The high sensitivity parameters associated to the thickness mode resonance change critically below 100 °C.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Experimental response at 35 °C (continuous blue line) and the numerical simulation using the optimized parameters (black dots). (

**A**) Modulus of the electric impedance; (

**B**) phase of the electric impedance; (

**C**) electric conductance G; (

**D**) electric resistance R. Fundamental radial resonance and overtones are indicated by a Rn (n = 1, 2, 3, 4), thickness resonance by a TH and the coupled modes by Cm (m = 1, 2).

**Figure 2.**Thermal evolution of resonance modes: (

**A**) Resonance frequency; (

**B**) Modulus of the complex impedance. Results are expressed as a percentage of the value at room temperature.

**Figure 3.**Experimental response at RT (symbols) plotted as R (blue) and G (black) peaks, together with the reconstructed spectra after parameter determination by iterative analysis (continuous lines), for: (

**A**) planar and (

**B**) thickness resonance modes of a thickness-poled, thin disk of BNBT6 (t = 1.91 mm and D = 10.60 mm); and for a shear resonance of a thickness-poled, thin plate of BNBT6 (9.53 × 9.43 mm): (

**C**) with t = 0.89, with strong coupling of modes and not fitted for parameters determination; and (

**D**) with t = 0.83 mm, virtually uncoupled. The regression factor of the reconstructed to the experimental spectra (R

^{2}) is shown for each analyzed resonance.

**Figure 4.**Sensitivity analysis for the real part of the elastic constants. Each curve shows the evolution of a resonance using the conductivity G. Each parameter is changed over a range ±50% from the initial value.

**Figure 5.**Result of the minimization of the numerical calculation (black dots) and experimental spectrum (continuous blue line) at 100 °C. (

**A**) Modulus of the complex electric impedance; (

**B**) Phase of the impedance.

**Figure 6.**Thermal evolution of the real part of the elastic, dielectric, and piezoelectric parameters. Parameters related with the TH mode (high sensitivity) are represented in red. Those related with the radial modes are represented in black and those related with the complex resonances C

_{1}and C

_{2}in blue.

**Figure 7.**Thermal evolution of the imaginary part of the elastic, dielectric, and piezoelectric parameters. Parameters related with the thickness mode (high sensitivity) are represented in red. Those related with the radial modes are represented in black, and those related with the complex resonances C

_{1}and C

_{2}in blue.

Mode | R_{2} | R_{3} | R_{4} | TH | C_{1} | C_{2} |
---|---|---|---|---|---|---|

Temperature (°C) | 88.5 | 90.0 | 90.2 | 95.0 | 96.3 | 97.6 |

**Table 2.**Parameters from the iterative analysis of the impedance of thin disks and plates, thickness-poled. Elastic constants are in GPa, piezoelectric constants in C/m

^{2}.

${\mathit{c}}_{11}^{\mathit{E}}$ | ${\mathit{c}}_{12}^{\mathit{E}}$ | ${\mathit{c}}_{13}^{\mathit{E}}$ | ${\mathit{c}}_{33}^{\mathit{E}}$ | ${\mathit{c}}_{44}^{\mathit{E}}$ | e_{31} | e_{15} | e_{33} | ${\mathit{\epsilon}}_{11}^{\mathit{S}}$ | ${\mathit{\epsilon}}_{33}^{\mathit{S}}$ |
---|---|---|---|---|---|---|---|---|---|

149.9 + j0.42 | 77.1 + j0.42 | 74.4 + j0.49 | 138.1 + j0.45 | 41.4 + j0.07 | −1.47 + j1.23 | 11.17 + j0.33 | 6.56 − j0.23 | 609 − j29 | 421 − j13 |

**Table 3.**Summary of the sensitivity analysis from Figure 4 and similar in the supplementary material.

c_{11} | c_{12} | c_{13} | c_{33} | c_{44} | e_{31} | e_{15} | e_{33} | ε_{1} | ε_{33} | |
---|---|---|---|---|---|---|---|---|---|---|

R_{1} | High | Med. | High | High | Null | Low | Null | Null | Null | Null |

R_{2} | High | Low | High | High | Null | Null | Null | Null | Null | Null |

R_{3} | High | Low | High | High | Low | Null | Null | Med. | Null | Null |

R_{4} | High | Null | High | High | Med. | Null | Low | Med. | Null | Null |

TH | Null | Null | Null | High | Null | Null | Null | High | Null | High |

C_{1} | High | Null | High | High | High | Null | Med. | Med. | Low | Low |

C_{2} | High | Null | High | High | High | Null | Med. | Med. | Low | Low |

T (°C) | ${\mathit{c}}_{11}^{\mathit{E}}$ | ${\mathit{c}}_{12}^{\mathit{E}}$ | ${\mathit{c}}_{13}^{\mathit{E}}$ | ${\mathit{c}}_{33}^{\mathit{E}}$ | ${\mathit{c}}_{44}^{\mathit{E}}$ | e_{31} | e_{15} | e_{33} | ${\mathit{\epsilon}}_{11}^{\mathit{S}}$ | ${\mathit{\epsilon}}_{33}^{\mathit{S}}$ |
---|---|---|---|---|---|---|---|---|---|---|

16 | 144.3 + j1.03 | 52.1 + j0.48 | 54.7 + j0.56 | 132.8 + j1.26 | 39.9 + j0.44 | −1.24 + j0.0026 | 7.72 − j0.005 | 10.09 − j0.105 | 1841 − j105 | 374 − j12 |

40 | 141.8 + j1.04 | 53.2 + j0.48 | 55.3 + j0.56 | 130.6 + j1.28 | 38.3 + j0.43 | −1.58 + j0.002 | 8.85 − j0.006 | 10.77 − j0.147 | 1853 − j101 | 431 − j15 |

60 | 139.6 + j1.01 | 53.0 + j0.47 | 55.9 + j0.55 | 128.7 + j1.29 | 37.0 + j0.44 | −1.29 + j0.0019 | 8.90 − j0.008 | 11.27 − j0.149 | 1536 − j141 | 494 − j19 |

80 | 136.7 + j1.04 | 52.4 + j0.46 | 55.8 + j0.54 | 125.1 + j1.32 | 36.3 + j0.43 | −1.83 + j0.0012j | 9.46 − j0.009 | 12.33 − j0.182 | 1632 − j155 | 590 − j26 |

100 | 134.1 + j1.05 | 49.9 + j0.47 | 53.2 + j0.51 | 125.3 + j1.37 | 33.9 + j0.42 | −1.49 + j0.0013 | 6.27 − j0.007 | 12.18 − j0.125 | 895 − j170 | 999 − j73 |

^{2}.

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**MDPI and ACS Style**

Pérez, N.; García, A.; Riera, E.; Pardo, L.
Electromechanical Anisotropy at the Ferroelectric to Relaxor Transition of (Bi_{0.5}Na_{0.5})_{0.94}Ba_{0.06}TiO_{3} Ceramics from the Thermal Evolution of Resonance Curves. *Appl. Sci.* **2018**, *8*, 121.
https://doi.org/10.3390/app8010121

**AMA Style**

Pérez N, García A, Riera E, Pardo L.
Electromechanical Anisotropy at the Ferroelectric to Relaxor Transition of (Bi_{0.5}Na_{0.5})_{0.94}Ba_{0.06}TiO_{3} Ceramics from the Thermal Evolution of Resonance Curves. *Applied Sciences*. 2018; 8(1):121.
https://doi.org/10.3390/app8010121

**Chicago/Turabian Style**

Pérez, Nicolás, Alvaro García, Enrique Riera, and Lorena Pardo.
2018. "Electromechanical Anisotropy at the Ferroelectric to Relaxor Transition of (Bi_{0.5}Na_{0.5})_{0.94}Ba_{0.06}TiO_{3} Ceramics from the Thermal Evolution of Resonance Curves" *Applied Sciences* 8, no. 1: 121.
https://doi.org/10.3390/app8010121