Waste heat energy harvesting using the Olsen cycle on 0.945Pb(Zn_1/3Nb_2/3)O₃– 0.055PbTiO₃ single crystals

The properties of PZN-PT have been studied extensively as it is a popular ferroelectric material used in sensors and actuators [14–23]. Poled single crystal PZN-4.5PT undergoes a phase transformation sequence from rhombohedral (R) to tetragonal (T) to cubic (C) during heating from 25 to 160 °C under zero electric field [19]. A small fraction of rhombohedral domains exist in the tetragonal phase [19]. The phase diagram for PZN-xPT indicates that, for PZN-5.5PT, the rhombohedral to tetragonal transition occurs between 122 and 130 °C and the tetragonal to cubic transition occurs around T_Curie = 165 °C [20]. Renault et al [23] showed that during field cooling from 177 to 27 °C [001]-oriented PZN-4.5PT has an additional orthorhombic (O) phase and undergoes a C–T–O–R phase transition sequence. The authors also demonstrated that the transition temperatures were dependent on the applied electric field which varied from 0 to 0.3 MV m⁻¹.

Ren et al [14] showed that electric-field-induced phase transitions and piezoelectric properties of PZN-xPT are strongly dependent on temperature and composition for x between 4.5 and 8%. In addition, Shen and Cao [17] reported the temperature dependence of the piezoelectric, pyroelectric and dielectric properties of PZN-xPT poled by two different methods for x equal to 4.5 and 8%. The relative permittivity and saturation polarization of PZN-4.5PT were found to be extremely sensitive to (i) the applied electric field used during poling [22] and (ii) to the poling method [17]. Although the properties of PZN-4.5PT have been reported in the literature [14–23], to the best of our knowledge, dielectric properties of PZN-5.5PT, investigated here, have not been reported.

Khodayari et al [24] studied the energy harvesting capabilities of [110]-oriented PZN-4.5PT using the Olsen cycle. The authors achieved 216.5 J/l/cycle (1 J/l/cycle = 1 mJ/cm³/cycle) by successively dipping a 1 mm thick single crystal PZN-4.5PT sample in baths at 100 and 160 °C while the electric fields varied between E_L = 0 MV m⁻¹ and E_H = 2 MV m⁻¹. Zhu et al [25] also used 1.1 mm thick [110]-oriented PZN-4.5PT single crystals to examine the energy generated with electric-field-induced phase transitions through rhombohedral, orthorhombic and tetragonal phases during the Olsen cycle. The authors obtained 101.8 J/l/cycle operating between temperatures 100 and 130 °C and electric fields E_L = 0 MV m⁻¹ and E_H = 2 MV m⁻¹ [25]. For these operating conditions, they determined that, during the isothermal processes 1–2 and 3–4 in the Olsen cycle, the samples experienced rhombohedral to orthorhombic (R–O) and orthorhombic to tetragonal (O–T) frequency-dependent phase transitions, respectively. Zhu et al [25] also found that the energy density increased with reducing the duration of isothermal process 1–2 and increasing the duration of isothermal process 3–4 which varied from 0.1 to 100 s. Finally, if the isothermal processes were of equal duration, the energy density increased with increasing duration [25]. In both of these studies [24, 25], it is unclear if these experimental results were averaged over multiple cycles and/or were repeated for different samples. In contrast, the present study performs the Olsen cycle on several PZN-5.5PT single crystal samples and assesses the sample variability in terms of generated energy.

Recently, Kandilian et al [13] derived a model estimating the energy harvesting capabilities of single crystal relaxor ferroelectric materials undergoing the Olsen cycle. The model expresses the energy density as [13]

$$\begin{eqnarray} \displaystyle \fl {N}_{\mathrm{D}}=({E}_{\mathrm{H}}-{E}_{\mathrm{L}})\left\{\vphantom {\frac {d_{33} x_3}{s_{33}}} \frac{{\varepsilon }_{0}}{2}[{\varepsilon }_{\mathrm{r}}({T}_{\mathrm{cold}})-{\varepsilon }_{\mathrm{r}}({T}_{\mathrm{hot}})]({E}_{\mathrm{H}}+{E}_{\mathrm{L}})\right.&&\displaystyle \\ \displaystyle \left.+~\left[{P}_{\mathrm{s}}({T}_{\mathrm{cold}})-{P}_{\mathrm{s}}({T}_{\mathrm{hot}})+\frac{{d}_{33}{x}_{3}}{{s}_{33}}\right]\right\}&&\displaystyle \end{eqnarray} \tag{ 2 }$$

where ε₀ is the vacuum permittivity (=8.854 × 10⁻¹² F m⁻¹), ε_r(T_cold) and ε_r(T_hot) are the low frequency (∼0.1 Hz) relative permittivities of the pyroelectric material at the cold and hot operating temperatures T_cold and T_hot, respectively. The saturation polarizations of the pyroelectric material at T_cold and T_hot are respectively denoted by P_s(T_cold) and P_s(T_hot) and expressed in C m⁻² [26]. In addition, d₃₃ is the piezoelectric coefficient of the single crystal (in C N⁻¹), s₃₃ is the elastic compliance of the single crystal (in m² N⁻¹) and x₃ = α₃(T_hot − T_cold), where α₃ is the thermal expansion coefficient (in K⁻¹). Note that this model was based on the assumption that the dielectric contribution to the primary pyroelectric coefficient was negligible compared with the dipole contribution (see equation (8) in [13]). This model was validated against experimental data collected on PMN-32PT single crystals and using properties reported in the literature [27–30] for T_cold = 80 °C and T_hot varying from 130 to 170 °C, while E_L was 0.2 MV m⁻¹ and E_H ranged from 0.4 to 0.9 MV m⁻¹ [13]. The model given by equation (2) is expressed as a function of material properties typically reported in the literature. Thus, it can enable rapid identification of promising materials for waste heat harvesting without physically performing the Olsen cycle.

The present paper aims (i) to assess the performance of PZN-5.5PT single crystals in converting waste heat into electricity, (ii) to measure their dielectric properties and (iii) to further validate the above model.

In the present study, five single crystal PZN-5.5PT samples were purchased from Microfine Materials Technologies Pte Ltd, Singapore. The samples were poled in the [001] direction. Their surface area and thickness were 1 × 1 cm² and 200 µm, respectively. Each 1 × 1 cm² face of the samples was coated with a ∼10 nm NiCr bond layer and a ∼1 μm thick Au_0.68Pd_0.32 electrode. These layers were deposited by an RF sputter-deposition technique. Electrical wires were attached to the electrodes using conductive silver epoxy.

Isothermal bipolar displacement versus electric field hysteresis curves were collected at various temperatures by applying a triangular voltage with frequency of 0.1 Hz across the single crystal samples. The samples were placed in a silicone oil bath at the desired temperatures of 100, 125, 150, 175 or 190 °C. The amplitude of the voltage corresponded to an electric field varying from  − 1 to 1 MV m⁻¹. All measurements were repeated five times on each of the five different samples to assess repeatability and experimental uncertainty.

Moreover, the saturation polarization P_s(T) and the dielectric constant ε_r(T) of each sample at temperature T were evaluated by linearly fitting the section of the bipolar D–E loops corresponding to a relatively large electric field decreasing from 1 to 0.5 MV m⁻¹ according to [31]

$$\begin{eqnarray} &&D(E,T)={\varepsilon }_{0}{\varepsilon }_{\mathrm{r}}(T)E+{P}_{\mathrm{s}}(T). \end{eqnarray} \tag{ 3 }$$

The saturation polarization P_s(T) is equal to the electric displacement in the linear fit of D versus E extrapolated at zero electric field [28] and the slope of this linear fit corresponds to the product ε₀ε_r(T) as illustrated in figure 1.

Moreover, isothermal unipolar D–E loops were collected on sample 5 for the same above temperatures. They were compared with bipolar D–E loops along with the associated values of ε_r(T) and P_s(T).

The Olsen cycle was performed on the PZN-5.5PT samples for (i) different values of low and high electric fields E_L and E_H, (ii) various hot operating temperature T_hot and (iii) various cycle frequency f. The experimental set-up consisted of a thermal and an electrical subsystem. The experimental apparatus and procedure were identical to those used in our previous studies [13, 32] and need not be repeated. The cold operating temperature T_cold was fixed at 100 °C. The hot operating temperature was varied from 125 to 190 °C. The electric fields E_L and E_H ranged from 0 to 0.2 MV m⁻¹ and from 0.5 to 1.5 MV m⁻¹, respectively. The overall cycle frequency was defined as f = (τ₁₂+τ₂₃+τ₃₄+τ₄₁)⁻¹, where τ_ij corresponds to the duration of process i–j. It varied from 0.021 to 0.15 Hz by changing the duration of the isoelectric field heating and cooling processes 2–3 and 4–1, denoted by τ₂₃ and τ₄₁ (figure 1). However, the time rate of change of the electric field during the isothermal processes 1–2 and 3–4 remained the same at 0.4 MV m⁻¹ s⁻¹. Note that this rate was identical to that used to collect the D–E loops between  − 1 and 1 MV m⁻¹ at 0.1 Hz. In other words, τ₁₂ and τ₃₄ were equal and constant for a given electric field span. For example, τ₁₂ = τ₃₄ = 2.5 s for E_H − E_L = 1 MV m⁻¹. The times τ₂₃ and τ₄₁ varied between 0.5 and 20 s corresponding to the cycle frequency varying from 0.15 to 0.021 Hz.

The energy density generated per cycle N_D (expressed in J/l/cycle) is represented by the area enclosed by the cycle in the D–E diagram (figure 1). It was calculated by numerical integration of equation (1) using the trapezoidal rule. In addition, the power density P_D is the amount of energy generated by the pyroelectric material per unit volume per unit time and is expressed in W l⁻¹. It is defined as P_D = N_Df, where f is the Olsen cycle frequency.

Figure 2 shows typical isothermal bipolar D–E loops at T_cold = 100 °C as well as loops for T_hot equal to (a) 125 °C, (b) 150 °C, (c) 175 °C and (d) 190 °C obtained with sample 5. The isothermal D–E loops followed a counter-clockwise path. Figure 2 also shows that Olsen cycles corresponding to the above temperatures with E_L = 0.0 or 0.2 MV m⁻¹ and E_H = 1.0 MV m⁻¹. Results indicated that all [001] PZN-5.5PT samples were ferroelectric at 100, 125 and 150 °C. They were paraelectric at 190 °C as their saturation polarization vanished (figure 2(d)). In addition, samples 1 and 2 were paraelectric at 175 °C while samples 3, 4 and 5 were ferroelectric at 175 °C.

Zoom In Zoom Out Reset image size

Moreover, table 1 summarizes the values of P_s(T) and ε_r(T) retrieved from the isothermal D–E loops for each sample at different temperatures. The greatest inter-sample variability in D–E loops and in the resulting P_s(T) and ε_r(T) values was observed at 125 and 175 °C. The relative errors in P_s(T) and ε_r(T) among samples were less than 20% for temperatures 100, 150 and 190 °C. They were less than 25% and 30% for temperatures 125 and 175 °C, respectively. The variability at 125 °C can be attributed to the presence of mixed rhombohedral and tetragonal phases. In fact, the volume ratio of these phases may vary from one sample to another due to small chemical inhomogeneities [33]. The sample variability observed at 175 °C can be attributed to the fact that samples 3, 4 and 5 were ferroelectric with P_s ≈ 0.13 C m⁻² while samples 1 and 2 were paraelectric at this temperature.

Finally, figure 3 compares unipolar and bipolar D–E loops performed on sample 5 at 100 °C. The unipolar D–E loop, corresponding to a frequency of 0.2 Hz, had the same time rate of change in the electric field as the bipolar D–E loop measured at 0.1 Hz. Note also that the unipolar D–E loops measured at 0.1 and 0.2 Hz were nearly identical. A notable difference between the unipolar and bipolar D–E loops at 100 °C was the absence of a field-induced phase transition in the unipolar loops. Such a phase transition was responsible for the nonlinear behavior of the bipolar D–E loops at low electric field. In contrast, the unipolar D–E loops followed a nearly linear path between 0.0 and 1.0 MV m⁻¹. Figure 3 establishes that the unipolar D–E loops followed the upper curve of the bipolar D–E loop corresponding to a decreasing electric field. Thus, analysis of the unipolar D–E loops or the upper curve of bipolar D–E loops resulted in nearly identical values of saturation polarization P_s(T) and dielectric permittivity ε₀ε_r(T). The same conclusions were reached at other temperatures (not shown). Finally, the D–E loops were closed for all temperatures indicating that there was no leakage current.

Zoom In Zoom Out Reset image size

Figure 4 shows the isothermal bipolar D–E loops at temperatures 100 and 190 °C for sample 4. It also depicts the experimental Olsen cycle in the D–E diagram for sample 4 at various cycle frequencies between 0.021 and 0.15 Hz. The cycle followed a clockwise path and was performed between T_cold = 100 °C and T_hot = 190 °C and electric field from E_L = 0 MV m⁻¹ to E_H = 1.0 MV m⁻¹. Note also that all the experimental Olsen cycles were closed and, unlike P(VDF-TrFE) [32], no leakage current was observed.

Zoom In Zoom Out Reset image size

First, it is interesting to note that the Olsen cycles measured at frequencies of 0.021 and 0.034 Hz overlapped. Indeed, for these frequencies, the electric displacement had reached steady state, i.e. ∂D/∂t = 0, before the electric field was varied. In this case, processes 1–2 and 3–4 followed a relatively smooth path, indicating that the four different processes in the Olsen cycle were performed under quasi-equilibrium conditions.

Moreover, for cycle frequencies larger than 0.034 Hz, the isoelectric field processes 2–3 and 4–1 were not performed under quasi-equilibrium conditions. For such cycle frequencies, the electric displacement had not reached steady state before the electric field was varied to perform processes 1–2 and 3–4. In other words, the phase transition was incomplete. In addition, the Olsen cycles did not follow a smooth path between E_L and E_H during processes 1–2 and 3–4 in the D–E diagram, as illustrated in figure 4.

Figures 5(a) and (b) respectively show the energy density and power density as functions of frequency for the Olsen cycles performed on sample 4 and reported in figure 4. Each data point corresponds to the energy density or power density averaged over five cycles. The associated error bars correspond to two standard deviations or a 95% confidence interval. The energy density reached a plateau of 140 J/l/cycle at frequencies below 0.034 Hz and decreased with increasing cycle frequency. Reducing the cycle frequency below 0.034 Hz, by increasing the duration of the isoelectric field processes 2–3 and 4–1, did not result in larger energy density because each process of the Olsen cycle was performed under quasi-equilibrium.

Zoom In Zoom Out Reset image size

Conversely, the power density increased with increasing frequency and reached a maximum of 10.1 W l⁻¹ at 0.1 Hz. For frequencies larger than 0.1 Hz, P_D decreased with increasing frequency. This can be explained by considering the expression P_D = N_D(f)f. For frequencies less than 0.1 Hz, the decrease in N_D(f), previously discussed, was compensated by the increase in frequency so that P_D increased. However, beyond 0.1 Hz, N_D decreased significantly with frequency, resulting in smaller values of P_D. In practice, the operating frequency could be adjusted according to the power needed for a given load. Note that the frequency of 0.15 Hz corresponded to τ₂₃ = τ₄₁ = 0.5 s while the time required to physically transfer the sample between the hot and cold baths was ∼0.25 s per transfer.

Furthermore, the Biot number for the pyroelectric assembly (film with electrodes) is defined as Bi = hb/k_eff, where h is the heat transfer coefficient, and b and k_eff are the sample thickness and effective thermal conductivity, respectively. The heat transfer coefficient h = 300 W m⁻² K⁻¹ corresponded to convective quenching in an oil bath [34]. The effective thermal conductivity of the assembly was estimated using the series model. The thermal conductivity of PZN-5.5PT could not be found in the literature and was approximated to be that of PZT, i.e., k ≃ 1.2 W m⁻¹ K⁻¹ at room temperature [35] while that of gold was k = 310 W m⁻¹ K⁻¹ [36]. Thus, the effective thermal conductivity of the assembly of thickness b = 202 μm was k_eff ≃ 1.21 W m⁻¹ K⁻¹ resulting in Bi ≃ 0.05 or Bi ≪ 1. Therefore, the temperature was uniform across the sample and the lumped capacitance approximation was valid for all conditions considered [36].

Finally, the thermal time constant τ associated with processes 2–3 and 3–4 can be estimated as τ = ρ_effc_p,effb/h [36], where ρ_eff and c_p,eff are the effective density and specific heat of the pyroelectric assembly, respectively. The thermal time constant was estimated to be 1.68 s for a 200 µm thick PZN-5.5PT film with ρ = 8000 kg m⁻³ [37] and c_p = 312.5 J kg⁻¹ K⁻¹ [38] sandwiched between two 1 µm thick Au_0.68Pd_0.32 electrodes having ρ = 16 951 kg m⁻³ [39] and specific heat c_p = 156.8 J kg⁻¹ K⁻¹. This time constant indicates that for cycle frequencies above 0.065 Hz the sample may have not reached thermal equilibrium during processes 2–3 and 4–1 of the Olsen cycle.

For practical purposes and for validating the model, it is important to assess sample variability. Figure 6 shows the power density generated experimentally as a function of high electric field E_H at cycle frequency of 0.1 Hz for four different samples. Here also, each data point corresponds to the power density averaged over five cycles and the associated error bars correspond to two standard deviations. In these cycles, T_cold and E_L were set to be 100 °C and 0.2 MV m⁻¹, respectively. The temperature T_hot varied from 125 to 190 °C while E_H ranged from 0.5 to 1.5 MV m⁻¹. Figure 6 demonstrates that, for all samples, the power density increases with increasing electric field E_H and with hot source temperature T_hot. The maximum power obtained was 11.7 W l⁻¹ for T_cold = 100 °C, T_hot = 190 °C, E_L = 0.2 MV m⁻¹ and E_H = 1.5 MV m⁻¹. Increasing the electric field E_H beyond 1.5 MV m⁻¹ during the Olsen cycle led to sample failure caused by excessive thermo-electro-mechanical stress.

Zoom In Zoom Out Reset image size

Furthermore, the maximum relative error in P_D among samples for all values of E_H was 18.3% for T_hot = 125 °C. It decreased to 18.0%, 9.8%, and 6.5% as T_hot increased from 150, 175 to 190 °C, respectively. The larger variability observed at lower temperature can be attributed to the large differences in the rhombohedral and tetragonal volume fractions among samples. This was already observed in the bipolar D–E loops and in the retrieved properties P_s(T) and ε_r(T). On the other hand, the value of E_H was found to have no significant effect on sample variability.

Figures 7 and 8 show the generated energy density N_D as a function of electric field E_H for sample 5 with low electric field E_L equal to 0.0 and 0.2 MV m⁻¹, respectively. In all cases, the temperature T_cold was 100 °C while T_hot was (a) 125, (b) 150, (c) 175 and (d) 190 °C. The cycle frequency was 0.034 Hz corresponding to Olsen cycles with quasi-equilibrium processes, as previously discussed. Results for E_H = 1.0 MV m⁻¹ correspond to the Olsen cycles shown in figure 2 for each value of T_hot.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figures 7 and 8 indicate that the energy density increased with increasing values of T_hot and E_H. They demonstrate that operating the cycle with T_hot above T_Curie = 165 °C yields significantly higher energy densities than cycles operating below T_Curie. In fact, a maximum energy density of 150 J/l/cycle was achieved for operating temperatures between T_cold = 100 °C and T_hot = 190 °C and electric fields E_L and E_H equal to 0.0 and 1.2 MV m⁻¹, respectively.

In addition, increasing E_L resulted in a decrease in the generated energy density. For example, the maximum energy density decreased from 150 to 124 J/l/cycle as E_L increased from 0.0 to 0.2 MV m⁻¹ for the same T_cold = 100 °C, T_hot = 190 °C and E_H = 1.2 MV m⁻¹. This indicates that the PZN-5.5PT sample did not become depoled when lowering the applied electric field to 0.0 MV m⁻¹, unlike observations made with PMN-32PT [13]. Since PZN-5.5PT retains its polarization at zero electric field, the isoelectric field cooling process 4–1 can be performed at E_L = 0 MV m⁻¹ which may simplify the practical implementation of the Olsen cycle.

Moreover, it is interesting to compare the maximum energy density of 150 J/l/cycle obtained with PZN-5.5PT with 100 J/l/cycle obtained with PMN-32PT for the same temperature difference (90 °C) and for E_L and E_H equal to 0.2 and 0.9 MV m⁻¹, respectively. However, note that T_cold and T_hot were 80 and 170 °C for PMN-32PT [13] instead of 100 and 190 °C for PZN-5.5PT. This suggests that these two materials should be operated in slightly different temperature ranges for optimum performance. This could be useful in a multistage pyroelectric converter as envisioned by Olsen et al [9]. Another material that could be included in the same multistage device is [110]-oriented PZN-4.5PT whose Curie temperature is around 157 °C [20] compared with 150 °C and 165 °C for PMN-32PT and PZN-5.5PT, respectively.

Finally, Khodayari et al [24] obtained 216.5 J l⁻¹ with PZN-4.5PT single crystal samples with operating conditions T_cold, T_hot, E_L and E_H equal to 100 °C, 160 °C, 0.0 MV m⁻¹ and 2.0 MV m⁻¹, respectively. The larger energy density obtained for PZN-4.5PT can be attributed to the larger electric field span (E_H − E_L) and to the lower Curie temperature. Indeed, the lower Curie temperature of PZN-4.5PT allowed for the tetragonal to cubic phase transition to occur for smaller temperature swing but with the same value of T_cold. This results in reduced thermal stress on the samples. It may also be why the PZN-4.5PT samples were able to withstand electric fields as high as 2.0 MV m⁻¹ without sample failure [24, 25] compared with up to 1.5 MV m⁻¹ in the present study.

Figures 7 and 8 also compare experimental energy density generated by sample 5 with predictions of the model given by equation (2). The saturation polarization and dielectric constant of sample 5 at T_cold and T_hot retrieved from D–E loops are given in table 1. Here, the last term of equation (2), corresponding to the contribution of thermal expansion to the energy density, was ignored. Indeed, the Olsen cycles performed on PZN-5.5PT fell on or within the bounds of the isothermal D–E loops at T_cold and T_hot (figure 2), indicating that thermal expansion did not contribute to the generated energy.

Figures 7 and 8 also report the average and maximum relative error between experimental data and model predictions denoted by δ_avg and δ_max, respectively. Relatively good agreement was observed between model predictions and experimental data, particularly for temperature T_hot less than 175 °C and E_L = 0.2 MV m⁻¹. Then, the average relative error was less than 24%. Both the average and maximum relative errors were larger for E_L = 0.0 MV m⁻¹ than for E_L = 0.2 MV m⁻¹ for any given temperature T_hot. This can be explained by considering the isothermal D–E loops shown in figures 2 and 3. For low electric fields (E < 0.2 MV m⁻¹) and for temperatures below T_Curie = 165 °C, the material is ferroelectric and the electric displacement is a nonlinear function of the electric field. This nonlinearity was also observed by Zhu et al [25] for [110]-oriented PZN-4.5PT and was attributed to electric-field-induced phase transitions. This phenomenon was not accounted for by the above model which treated ε_r(T) as a function of temperature only [13]. However, ε_r(T) could be assumed to depend only on T for electric fields larger than 0.2 MV m⁻¹ as previously discussed.

Moreover, figures 7(d) and 8(d) indicate that the average difference between experimental and model predictions reached 44 and 33% for E_L = 0.0 and 0.2 MV m⁻¹ at T_hot = 190 °C, respectively. For such a large value of T_hot, the model systematically overpredicted the experimental data. For T_hot = 175 °C, the discrepancies can be attributed to the fact that the quasi-equilibrium Olsen cycles did not follow the isothermal D–E loop as the electric field was reduced from 1 to 0 MV m⁻¹ (process 3–4) as illustrated in figure 2(c). This suggests that the sample did not undergo the same phase transition during the isothermal D–E loops and the Olsen cycle. However, for T_hot = 190 °C process 2–3 of the Olsen cycles did not span the same electric displacement as the isothermal D–E loop as illustrated in figure 2(d). An electric displacement extending beyond the bounds of the D–E loops was attributed to positive thermal expansion by Kandilian et al [13]. Conversely, the reduced electric displacement span can be attributed to negative thermal expansion. The lattice parameters of PZN-5.5PT were reported to be constant between 190 and 100 °C [18] indicating zero thermal expansion, whereas the thermal expansion coefficient of Au_0.75Pd_0.25 was reported to be 12 μm m⁻¹ K⁻¹ [39]. The mismatch in thermal expansion coefficients between the electrode and the PZN-5.5PT sample induced a tensile stress in the material, which may have resulted in a decrease in sample thickness. This phenomenon was not observed for T_hot = 125 and 150 °C as is evident in figures 2(a) and (b) where the Olsen cycles tended to follow the path of the unipolar D–E loops measured at T_hot. Then, good agreement was observed between model predictions and experimental data.

The samples used in this study broke after 100–250 cycles. The successive thermal stress in combination with the electrically induced strains contributed to the samples eventually cracking and breaking. This explains why data were not reported for all samples under all conditions (figure 6). Strategies to increase the sample durability include (i) pre-stressing the sample such as in thin layer unimorph ferroelectric driver and sensor actuators [40] and (ii) applying a conformal coating (e.g. Parylene HT) to the sample [41].