*2.3.1. Firing temperature dependence of dielectric and piezoelectric properties*

Figures 13(a)-(d) show the relationships between firing temperature (FT) and (a) relative dielectric constant (εr), (b) planar coupling factor (kp) for the radial mode on the disks, (c) frequency constant (fcp), and (d) piezoelectric strain d33 constant in the cases of BT02 and BT05 ceramics before and after DC poling. Although εr decreases with increasing FT, kp, fcp, and d33 with increasing FT; furthermore, there is an optimum FT of 1, 340 ℃ in BT02 for obtaining the highest kp and d33. The differences in the dielectric and piezoelectric properties vs. FT between BT02 and BT05 were due to the ceramic bulk density, as mentioned later (see the following Figure 17), and the powder particle activity during firing because the specific surface areas of BT02 and BT05 powder particles measured by the Brunauer, Emmett, and Teller (BET) method were 9.4 and 2.3 m2 /g, respectively.

### *2.3.2. Effects of firing temperature and DC poling on acoustic wave velocities and elastic constants*

Figures 14(a)-(d) show the relationships between FT and (a) longitudinal wave velocity (VL), (b) transverse wave velocity (VS), (c) Young's modulus (Y), and (d) Poisson's ratio (σ) in the cases of BT02 and BT05 ceramics. Although VL, VS, and Y before and after DC poling increase with increasing FT, σ after poling is almost independent of FT; furthermore, there is an optimum FT of 1, 340 ℃ in BT02 from the plots of FT vs. VL and Y. The increase in Y with increasing FT indicates the increase in the mechanical hardness of the ceramic disks. By comparing FT vs. VL in Figure 14(a) with FT vs. 2fcp (fcp is shown in Figure 13(c), the dependences of VL and 2fcp on FT were almost the same, because both of them correspond to longitudinal wave velocities, as shown in Figure 15. In addition, we confirmed that VL precisely corresponded to 2fct , which is twice the frequency constant (fct ) of the coupling factor (kt ) for the thickness mode on the disks measured using the typical impedance vs. frequency response (Figure 15) [29]. Therefore, it is considered that our measurement method using the ultrasonic precision thickness gauge is suitable for evaluating acoustic wave velocities, especially in piezoelectric ceramics.

piezo-d33 meter (Academia Sinica ZJ-3D). Furthermore, the acoustic wave velocities of the BT ceramics before and after poling were measured using an ultrasonic precision thickness gauge (Olympus 35DL), which has PZT transducers with a frequency of 30 MHz for longitudinal wave generation and a frequency of 20 MHz for transverse wave generation. The acoustic wave velocities were evaluated on the basis of the propagation time between the second-pulse and the third-pulse echoes in the thickness direction parallel to the DC poling field for the ceramic disks with 14 mm diameter and 0.9-1.2 mm thickness [18-20]. The sample thickness was measured using a precision micrometer (Mitutoyo MDE-25PJ). The number (*n*) of disk samples measured was *n* = 5-8, and the data in the figures indicate the average of individual measured values. Furthermore, Young's modulus (Y), Poisson's ratio (σ), modulus of rigidity (G), and bulk modulus (K) in the thickness direction of ceramic disks were calculated on the basis of the longitudinal (VL) and transverse (VS) wave velocities using the equations (1)-(4) in Section 1.2. We investigated the relationships between firing temperature and DC poling effect vs. VL, VS, Y, σ, G, and K; furthermore, we clarified the relationships between ρ, the microstructure,

Figures 13(a)-(d) show the relationships between firing temperature (FT) and (a) relative dielectric constant (εr), (b) planar coupling factor (kp) for the radial mode on the disks, (c) frequency constant (fcp), and (d) piezoelectric strain d33 constant in the cases of BT02 and BT05 ceramics before and after DC poling. Although εr decreases with increasing FT, kp, fcp, and d33 with increasing FT; furthermore, there is an optimum FT of 1, 340 ℃ in BT02 for obtaining the highest kp and d33. The differences in the dielectric and piezoelectric properties vs. FT between BT02 and BT05 were due to the ceramic bulk density, as mentioned later (see the following Figure 17), and the powder particle activity during firing because the specific surface areas of BT02 and BT05 powder particles measured by the Brunauer, Emmett, and Teller (BET)

*2.3.2. Effects of firing temperature and DC poling on acoustic wave velocities and elastic constants*

Figures 14(a)-(d) show the relationships between FT and (a) longitudinal wave velocity (VL), (b) transverse wave velocity (VS), (c) Young's modulus (Y), and (d) Poisson's ratio (σ) in the cases of BT02 and BT05 ceramics. Although VL, VS, and Y before and after DC poling increase with increasing FT, σ after poling is almost independent of FT; furthermore, there is an optimum FT of 1, 340 ℃ in BT02 from the plots of FT vs. VL and Y. The increase in Y with increasing FT indicates the increase in the mechanical hardness of the ceramic disks. By comparing FT vs. VL in Figure 14(a) with FT vs. 2fcp (fcp is shown in Figure 13(c), the dependences of VL and 2fcp on FT were almost the same, because both of them correspond to longitudinal wave velocities, as shown in Figure 15. In addition, we confirmed that VL precisely corresponded to

on the disks measured using the typical impedance vs. frequency response (Figure 15) [29].

) of the coupling factor (kt

) for the thickness mode

*2.3.1. Firing temperature dependence of dielectric and piezoelectric properties*

/g, respectively.

and the elastic constants.

**2.3. Results and discussion**

46 Ferroelectric Materials – Synthesis and Characterization

method were 9.4 and 2.3 m2

, which is twice the frequency constant (fct

2fct

Figures 16(a)-(b) show the changes in Y (ΔY) and σ (Δσ) during poling, respectively. ΔY decreases with increasing FT, regardless of the types of BT powder particle. This phenomenon indicates that the ceramics become mechanically softer after DC poling. It is considered that such a change is due to the ferroelectric domain (↑ ) alignment induced by the poling field, such as a change in the configuration from a random orientation (↑ ↓) before poling to a directional orientation (↑ ↑) after poling [5-7, 9-15]. Δσ abruptly increases from FT of 1, 320 ℃ and shows almost the same tendency as in the case of FT vs. kp in Figure 13(b). By comparing FT vs. kp (Figure 13(b)) with FT vs. Δσ (Fig. 16(b)), a higher kp was obtained in the case of a larger Δσ. The physical meaning of the phenomenon regarding σ was deduced, that is, a larger Δσ was needed to realize a higher kp because of the easy deformation of the bulk perpendicular to the poling field (radial direction in the disks) as well as parallel to the poling field (thickness direction in the disks).

**Figure 13.** Firing temperature (FT) vs. (a) relative dielectric constant (εr), (b) planar coupling factor (kp) for radial mode on disks, (c) frequency constant (fcp), and (d) piezoelectric strain d33 constant in BT02 and BT05 ceramics before and after DC poling.

**Figure 14.** Firing temperature (FT) vs. (a) longitudinal wave velocity (VL), (b) transverse wave velocity (VS), (c) Young's modulus (Y), and (d) Poisson's ratio (σ) in BT02 and BT05 ceramics before and after DC poling.

**Figure 15.** Comparison between longitudinal wave velocity (VL), 2fc<sup>t</sup> , and 2fcp, which are twice the frequency constants of fc<sup>t</sup> for the thickness mode and fcp for the radial mode on disks in BT02 and BT05 ceramics after DC poling; 2fc<sup>t</sup> and 2fcp correspond to acoustic wave velocities.

**Figure 16.** Changes in (a) Young's modulus (ΔY) and (b) Poisson's ratio (Δσ) in BT02 and BT05 ceramics during DC poling.

### *2.3.3. Relationship between ceramic microstructure and elastic constants*

1600

0.30 0.32 0.34 0.36 0.38 0.40

σ (-)

BT02 before poling BT05 before poling BT02 after poling BT05 after poling

BT02: VL BT02: 2fct BT02: 2fcp BT05: VL BT05: 2fct BT05: 2fcp

for the thickness mode and fcp for the radial mode on disks in BT02 and BT05 ceramics after DC poling; 2fc<sup>t</sup> and

1300 1320 1340 1360

FT (℃)

2fct 2fct

**Figure 14.** Firing temperature (FT) vs. (a) longitudinal wave velocity (VL), (b) transverse wave velocity (VS), (c) Young's

1300 1320 1340 1360

FT (℃)

1300 1320 1340 1360

FT (℃)

2fcP 2fcP

, and 2fcp, which are twice the frequency constants

2000

2400

VS (m/s)

4000

Y (1010 N/m2)

of fc<sup>t</sup>

1300 1320 1340 1360

48 Ferroelectric Materials – Synthesis and Characterization

(a) (b)

(c) (d)

modulus (Y), and (d) Poisson's ratio (σ) in BT02 and BT05 ceramics before and after DC poling.

VL VL

FT (℃)

1300 1320 1340 1360

FT (℃)

4200

**Figure 15.** Comparison between longitudinal wave velocity (VL), 2fc<sup>t</sup>

4600

5000

Acoustic wave velocity (m/s)

2fcp correspond to acoustic wave velocities.

5400

5800

4400

4800

VL (m/s)

5200

5600

2800

3200

Figure 17 shows the FT dependence of the bulk density (ρ) of the ceramic disks. The relative bulk density (ρ/ρ0) became over 94% for both BT02 and BT05 during firing at 1, 340-1, 360 ℃, where ρ0 (6.02 g/cm<sup>3</sup> ) is the theoretical density of BT ceramics [25]. From our previous study on the relationships between kp vs. Y and σ in piezoelectric ceramics, a higher kp was realized at a lower Y and a higher σ [18-20]. However, there is no correspondence between the lower Y (Fig. 14(c)), the higher σ (Fig. 14(d)), the lower kp (Fig. 13(b)), and the lower d33 (Fig. 13(d)) during firing at 1, 300-1, 320 ℃ because of the low relative bulk density of ρ/ρ<sup>0</sup> (< 0.94). On the other hand, while Y became almost constant during firing at 1, 340-1, 360 ℃ (Fig. 14(c)), σ before poling decreased with increasing FT (Fig. 14(d)). It was considered that the decrease in σ with increasing FT was due to the increase in the density of coarse grains with average diameters of 50 μm as mentioned later. As FT vs. σ before poling (Fig. 14(d)) shows the same tendency as FT vs. ε<sup>r</sup> (Fig. 13(a)), the εr of which is directly related to ferroelectric domain structures, the σ dependence of FT was considered to be due to the domain structures and anisotropy of coarse grains being different from those of fine grains with average diameters of 1.2 μm. Since the ferroelectric domain size increases with increasing grain size of BT ceramics, the domain density decreases in the case of coarse grains [30-33]. Therefore, the ceramics with coarse grains exhibit a lower εr and a larger crystal anisotropy than the ceramics with fine grains [34-38]. Furthermore, σ decreases with increasing FT because it approaches a behavior similar to that of a BT single crystal, σ of which is 0.29 while σ of the ceramics is 0.31 [25]. In order to evaluate the physical characteristics of ceramic grains, the modulus of rigidity (G) and bulk modulus (K) were calculated by the equations (3) and (4) in Section 1.2. [23, 24]: G and K before poling increase with increasing FT during firing at 1,300-1,360 ℃, as shown in Figures 18(a)-(b), respectively, because the ceramic bulk density is improved in BT ceramics as shown in Figure 17; as a result, G and K are obtained from the equations (3) and (4), as well as Y in Figure 14(c) from equation (1). On the other hand, σ is independent of the ceramic bulk density, as shown in equation (2). Figures 19(a)-(d) show the relationships between ρ/ρ<sup>0</sup> vs. Y, σ, G, and K, respectively. Y, G, and K depend on ρ/ρ0 because they are affected by increasing the mechanical strength, especially the hardness of the ceramics, with FT. As mentioned previously regarding Figure 14(d), σ is independent of ρ/ρ<sup>0</sup> except when ρ/ρ<sup>0</sup> is above 0.94, as shown in Figure 19(b), which corresponds to ceramics with a larger domain size and a lower domain density with coarse grains. G decreased and K increased after poling owing to the domain alignment as shown in Figure 18. In addition, the change in G (ΔG) in BT ceramics during poling linearly decreased with increasing FT, and peaks of the changes in K (ΔK), which correspond to the peaks of Δσ (Figure 16(b)), were obtained at FT of 1340 ℃ in the BT02 ceramics and at FT of 1, 350-1, 360 ℃ in the BT05 ceramics, as shown in Figures 20(a)-(b). Therefore, a higher kp could be realized at FT at the peaks of ΔK and Δσ. Since ceramic grains with a higher K due to domain alignment during DC poling, indicating a high ceramic bulk density for obtaining a higher kp, are difficult to change in terms of their volume while applying external stress, the deformation of grains is practically transferred from the parallel direction to the perpendicular direction toward the direction of the applied stress; as a result, a higher σ is achieved. We believe that the increase in σ (Δσ) as a result of the increase in K (ΔK) indicates fundamental issues regarding the poling of ceramic grains to obtain a higher kp; therefore, the R&D of piezoelectric ceramics with high piezoelectricity must be focused on to realize a lower G and a higher K during DC poling from the viewpoints of elastic constants. From the abovementioned results, it was considered that the values of FT vs. G and K in BT02 and BT05 ceramics after poling (Figures 18(a)-(b)) correspond to the values of kp vs. FT (Figure 13(b)).

Figure 21(a) shows the FT dependence of microstructures in the BT02 ceramics. With the increase in kp with FT, ceramic grains grew from 1.2 μm (fine grains shown as white parts in the figures) to 50 μm (coarse grains shown as black parts in the figures) in diameter; moreover, the ratio of black parts to whole parts (black parts plus white parts) increased. The border between fine and coarse grains is shown in Figures 21(b)-(c). Furthermore, it was found that the black parts consist of several coarse grains, as shown in Figure 21(b). This phenomenon was almost the same as in the case of the microstructure in the BT05 ceramics. Figure 22 shows the relationship between kp and the area ratio of coarse grains (black parts) measured by imageanalyzing software (WinROOF [39]). In this figure, when the area ratio increases, kp increases because of the increase in the density of coarse grains. Therefore, the coarse grains in the dense ceramic bulk contribute to a higher kp and VL, VS, Y, and σ in the coarse grains correspond to those at FT of 1, 360 ℃ in Figures 14(a)-(d), which almost agree with the values previously reported [25].

grains being different from those of fine grains with average diameters of 1.2 μm. Since the ferroelectric domain size increases with increasing grain size of BT ceramics, the domain density decreases in the case of coarse grains [30-33]. Therefore, the ceramics with coarse grains exhibit a lower εr and a larger crystal anisotropy than the ceramics with fine grains [34-38]. Furthermore, σ decreases with increasing FT because it approaches a behavior similar to that of a BT single crystal, σ of which is 0.29 while σ of the ceramics is 0.31 [25]. In order to evaluate the physical characteristics of ceramic grains, the modulus of rigidity (G) and bulk modulus (K) were calculated by the equations (3) and (4) in Section 1.2. [23, 24]: G and K before poling increase with increasing FT during firing at 1,300-1,360 ℃, as shown in Figures 18(a)-(b), respectively, because the ceramic bulk density is improved in BT ceramics as shown in Figure 17; as a result, G and K are obtained from the equations (3) and (4), as well as Y in Figure 14(c) from equation (1). On the other hand, σ is independent of the ceramic bulk density, as shown in equation (2). Figures 19(a)-(d) show the relationships between ρ/ρ<sup>0</sup> vs. Y, σ, G, and K, respectively. Y, G, and K depend on ρ/ρ0 because they are affected by increasing the mechanical strength, especially the hardness of the ceramics, with FT. As mentioned previously regarding Figure 14(d), σ is independent of ρ/ρ<sup>0</sup> except when ρ/ρ<sup>0</sup> is above 0.94, as shown in Figure 19(b), which corresponds to ceramics with a larger domain size and a lower domain density with coarse grains. G decreased and K increased after poling owing to the domain alignment as shown in Figure 18. In addition, the change in G (ΔG) in BT ceramics during poling linearly decreased with increasing FT, and peaks of the changes in K (ΔK), which correspond to the peaks of Δσ (Figure 16(b)), were obtained at FT of 1340 ℃ in the BT02 ceramics and at FT of 1, 350-1, 360 ℃ in the BT05 ceramics, as shown in Figures 20(a)-(b). Therefore, a higher kp could be realized at FT at the peaks of ΔK and Δσ. Since ceramic grains with a higher K due to domain alignment during DC poling, indicating a high ceramic bulk density for obtaining a higher kp, are difficult to change in terms of their volume while applying external stress, the deformation of grains is practically transferred from the parallel direction to the perpendicular direction toward the direction of the applied stress; as a result, a higher σ is achieved. We believe that the increase in σ (Δσ) as a result of the increase in K (ΔK) indicates fundamental issues regarding the poling of ceramic grains to obtain a higher kp; therefore, the R&D of piezoelectric ceramics with high piezoelectricity must be focused on to realize a lower G and a higher K during DC poling from the viewpoints of elastic constants. From the abovementioned results, it was considered that the values of FT vs. G and K in BT02 and BT05 ceramics after poling (Figures 18(a)-(b)) correspond to the values of kp vs. FT (Figure 13(b)).

50 Ferroelectric Materials – Synthesis and Characterization

Figure 21(a) shows the FT dependence of microstructures in the BT02 ceramics. With the increase in kp with FT, ceramic grains grew from 1.2 μm (fine grains shown as white parts in the figures) to 50 μm (coarse grains shown as black parts in the figures) in diameter; moreover, the ratio of black parts to whole parts (black parts plus white parts) increased. The border between fine and coarse grains is shown in Figures 21(b)-(c). Furthermore, it was found that the black parts consist of several coarse grains, as shown in Figure 21(b). This phenomenon was almost the same as in the case of the microstructure in the BT05 ceramics. Figure 22 shows the relationship between kp and the area ratio of coarse grains (black parts) measured by imageanalyzing software (WinROOF [39]). In this figure, when the area ratio increases, kp increases because of the increase in the density of coarse grains. Therefore, the coarse grains in the dense

**Figure 17.** Firing temperature (FT) dependence of bulk density (ρ) in BT02 and BT05 ceramic disks.

**Figure 18.** Firing temperature (FT) vs. (a) modulus of rigidity (G) and (b) bulk modulus (K) in BT02 and BT05 ceramics before and after DC poling.

**Figure 19.** Relative bulk density (ρ/ρ0) dependences of (a) Young's modulus (Y), (b) Poisson's ratio (σ), (c) modulus of rigidity, and (d) bulk modulus (K) in BT02 and BT05 as-fired (before poling) ceramics sintered during firing at 1, 300-, 360 ℃, where ρ and ρ0 (6.02 g/cm<sup>3</sup> ) are the measured bulk density and theoretical density [25] of BT ceramics, respec‐ tively.

### *2.3.4. Firing temperature dependence of elastic constants in barium titanate ceramics*

Figure 23 shows that the relationships between kp vs. Y, σ, G, and K in BT02 and BT05 ceramics fired at different temperatures were inserted into the relationships between k<sup>p</sup> vs. Y, σ, G, and K in lead-containing and lead-free piezoelectric ceramics (Figure 3), which were fired at the optimal temperatures for each composition to realize the maximum kp. Higher kp is obtained in the cases of lower Y and G, and furthermore, higher σ and K, which are indicated by yellow arrows in Figure 23. This figure also indicates the firing temperature dependence of the bulk density vs. Y, σ, G, and K in BT02 and BT05 ceramics in Figure 19. kp increases with the increase of Y, G, and K because of the increase in bulk density with increasing firing temperature. On the other hand, σ was independent of kp because σ is an intrinsic material constant. These phenomena are indicated by green arrows in Figure 23. It is predicted the same phenomena regarding the firing temperature and bulk density dependences on k<sup>p</sup> vs. Y, σ, G, and K in cases of lead-containing and lead-free ceramics.

Origin of Piezoelectricity in Piezoelectric Ceramics from the Viewpoints of Elastic Constants Measured by… http://dx.doi.org/10.5772/60793 53

360 ℃, where ρ and ρ0 (6.02 g/cm<sup>3</sup>

G (109 N/m2)

tively.

Y (1010 N/m2)

0.86 0.90 0.94 0.98

52 Ferroelectric Materials – Synthesis and Characterization

(a) (b)

(c) (d)

0.30 0.32 0.34 0.36 0.38 0.40

6

) are the measured bulk density and theoretical density [25] of BT ceramics, respec‐

8

10

K (1010 N/m2)

BT02 before poling BT05 before poling

**Figure 19.** Relative bulk density (ρ/ρ0) dependences of (a) Young's modulus (Y), (b) Poisson's ratio (σ), (c) modulus of rigidity, and (d) bulk modulus (K) in BT02 and BT05 as-fired (before poling) ceramics sintered during firing at 1, 300-,

Figure 23 shows that the relationships between kp vs. Y, σ, G, and K in BT02 and BT05 ceramics fired at different temperatures were inserted into the relationships between k<sup>p</sup> vs. Y, σ, G, and K in lead-containing and lead-free piezoelectric ceramics (Figure 3), which were fired at the optimal temperatures for each composition to realize the maximum kp. Higher kp is obtained in the cases of lower Y and G, and furthermore, higher σ and K, which are indicated by yellow arrows in Figure 23. This figure also indicates the firing temperature dependence of the bulk density vs. Y, σ, G, and K in BT02 and BT05 ceramics in Figure 19. kp increases with the increase of Y, G, and K because of the increase in bulk density with increasing firing temperature. On the other hand, σ was independent of kp because σ is an intrinsic material constant. These phenomena are indicated by green arrows in Figure 23. It is predicted the same phenomena regarding the firing temperature and bulk density dependences on k<sup>p</sup> vs. Y, σ, G, and K in

*2.3.4. Firing temperature dependence of elastic constants in barium titanate ceramics*

12

14

σ (-)

0.86 0.90 0.94 0.98

ρ/ρ<sup>0</sup>

0.86 0.90 0.94 0.98

ρ/ρ<sup>0</sup>

ρ/ρ<sup>0</sup>

0.86 0.90 0.94 0.98

ρ/ρ<sup>0</sup>

cases of lead-containing and lead-free ceramics.

**Figure 20.** Changes in modulus of rigidity (ΔG) and bulk modulus (ΔK) in BT02 and BT05 ceramics during DC poling.

**Figure 21.** (a) Firing temperature (FT) dependence of microstructures in BT02 ceramics, (b) border between fine grain [see (c), average grain size of 1.2 μm] and coarse grain [see (b), average grain size of 50 μm], and relationship between black parts and coarse grains.

**Figure 22.** Planar coupling factor (kp) vs. area ratio of coarse grains of black parts measured by an image-analyzing software program (WinRoof [39]). Figure 23 Replace with the figure below because oflack of letter (σ) in the figure of σvs. kp.

23 3 bulk modulus vs firing temperature bulk modulus vs. firing temperature **Figure 23.** Planar coupling factor (kp) vs. Y, σ, G, and K in BT02 and BT05 ceramics fired at different temperatures. These data were inserted into the relationships between kp vs. Y, σ, G, and K in lead-containing and lead-free piezo‐ electric ceramics (Figure 3), which were fired at the optimal temperatures for each composition to realize maximum kp.

### **2.4. Conclusions in this part**

Figure 12 Replace with the figure as below because of lack

0.0

0.1

0.2

0.3

σ (-)

0.4

0.5

25 13 Devices -Practice and Applications- Devices ‒Practice and Applications‒

4 2 the material constants the elastic constants

18 9 firing at 1,\_300-1,\_360℃ firing at 1,300-1,360℃

4 5 material constant in "the caption of Figure 2" elastic constants

22 Fig.

12 Fig.

12

23

25 4 (ed.) Ferroelectrics‒Applications- (ed.) Ferroelectrics‒Applications‒ The effects of firing temperature and DC poling on the longitudinal and transverse wave velocities in barium titanate ceramics were investigated using an ultrasonic precision thickness

0.2 0.3 0.4 0.5 0.6 0.7

VS/V<sup>L</sup>

of letter (σ) in the figure of VS/VL vs.σ.

Polyethylene Synthetic rubber

Pb

 

 

2 1

σ

<sup>1</sup> <sup>1</sup>

Au

Nylon Ag

soft PZT hard PZT

 

1

− −=

2 S L V V

 

Cu Brass

AlW

Lead-free

Polystyrene Duralumin Ni Ice Sn Constantan Ti Mg

Pt Polymethyl Stainless Steel

PbTiO<sup>3</sup>

Fe Zn

Glass

Quartz

Be

3

gauge with high-frequency pulse generation. The results could explain the relationships between acoustic wave velocities, Young's modulus, Poisson's ratio, the modulus of rigidity, and the bulk modulus vs. firing temperature, and the changes in elastic constants during DC poling.
