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Smart Mater. Struct. 21 (2012) 025021 F Y Lee et al Table 3. Critical electric field Ecr(T), spontaneous polarization Ps(T), remnant polarization Pr(T) and relative permittivity εr(T) of 8/65/35 PLZT (sample 4) retrieved for two piecewise regions of isothermal bipolar D–E loops in the temperature range between 45 and 160 ◦ C. T (◦C) 45 65 100 110 120 130 160 Ecr(T) (MV m−1) 0.4 0.6 1.2 1.5 1.6 1.8 — EH = 2.5 MV m−1 to Ecr(T) εr (T ) 2624 3700 3977 3714 3700 3849 6439 Ps(T) (C m−2) 0.265 0.220 0.155 0.151 0.139 0.118 0.0145 Ecr(T) to EL = 0.2 MV m−1 εr (T ) 10850 26050 15040 12547 11243 9651 6439 Pr(T) (C m−2) 0.232 0.0856 0.0221 0.0188 0.0181 0.0148 0.0145 , Figure 7. Electric displacement versus electric field diagram containing four experimental Olsen cycles (sample 7). The electric field was cycled between 0.2 and 7.5 MV m−1. The cold source temperature Tcold and hot source temperature Thot were 25 and 160 ◦C, respectively. The average energy density over four cycles was 887.5 J l−1 at 0.0178 Hz, corresponding to the largest energy harvested by 8/65/35 PLZT in this study. 240 and 320 ◦C in the absence of an electric field for the 65/35 Pb/Ti ratio and x = 8, 7, 6 and 2%, respectively [32]. 5. Model predictions As previously discussed, the electric displacement D(E,T) of 8/65/35 PLZT varies nonlinearly when the electric field decreases from EH to EL. Therefore, the dielectric properties required in equation (4) were retrieved for two piecewise regions of the isothermal bipolar D–E loops corresponding to isothermal field reduction (i) from EH to Ecr (T ) and (ii) from Ecr (T ) to EL . The critical electric field Ecr (T ) was estimated from the inflection point in the isothermal D–E loop. It was found to increase with increasing temperature. Table 3 summarizes the critical field Ecr (T ), saturation and remnant polarizations Ps (T ) and Pr (T ), and relative permittivity εr (T ) of sample 4 for temperatures between 45 and 160 ◦ C. The energy density predicted by the piecewise model can be expressed as the sum of two components: ND =ND(EL,Ecr,Tcold,Thot)+ND(Ecr,EH,Tcold,Thot) (5) where the function ND(EL/H, Ecr, Tcold, Thot) is given by equation (4). The contribution from the region of decreasing electric field from EH to Ecr (T ) can be predicted by equation (4) using the saturation polarization Ps(T). The contribution from the region of decreasing electric field from Ecr(T) to EL = 0.2 MV m−1 can be predicted by equation (4) using the remnant polarization Pr(T) instead of Ps(T). Figures 5 and 6 compare systematically the energy den- sity obtained experimentally with predictions of equation (5) for four hot source temperatures Thot = 100, 120, 130 and 160◦C and EH ranging from 0.4 to 2.5 MV m−1. The cold source temperature Tcold was set as 65 ◦C (figure 5) and 45 ◦C (figure 6). Note that the thermal expansion term d33x3/s33 [31] corresponding to the secondary pyroelectric coefficient was ignored in the model predictions for PLZT. Indeed, Kandilian et al [19] observed that, for PMN-32PT, the Olsen cycle extended beyond the electric displacement bounded by the isothermal D–E loops. This was attributed to the contribution of thermal expansion to the energy density [19]. However, in the case of 8/65/35 PLZT, the Olsen cycles fell within the bounds of the isothermal D–E loops at both Tcold and Thot. Therefore, the thermal expansion did not contribute to the energy density harvested. Similar observations were made for PZN-5.5PT [20]. Figures 5 and 6 also report the range and the average value of the relative error between experimental data and model predictions, denoted by δ and δavg, respectively. For example, the average relative error reached 39% and 28% for Tcold = 65◦C and Tcold = 45◦C at Thot = 100◦C, respectively. For such a low value of Thot = 100 ◦C, a small absolute difference in energy density resulted in a large relative error. However, the model predicted the experimental data reasonably well, for Tcold = 45 ◦C and Thot ≥ 100 ◦C and for Tcold = 65◦C and Thot ≥ 130◦C. In these cases, the average relative error between model predictions and experimental data was less than 30%. Figure 8 shows the isothermal bipolar D–E loops collected on sample 4 at Tcold = 45 ◦C and Thot equal to (a) 100 ◦C, (b) 120 ◦C, (c) 130 ◦C and (d) 160 ◦C overlaid 9PDF Image | Pyroelectric waste heat harvesting using relaxor ferroelectric
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