Potential of energy harvesting in barium titanate based laminates from room temperature to cryogenic/high temperatures: measurements and linking phase field and finite element simulations

Energy harvesting, or energy scavenging, is a process that captures unused ambient energy. The ambient energy comes in the forms of sound, vibration, wind, solar radiation, heat, light, etc. Piezoelectric ceramics have been widely recognized for their potential utility in energy harvesting applications[1–3]. However, several drawbacks such as their brittleness have limited the adoption of the piezoelectric ceramics. The use of inverse magnetostriction is another possible approach to energy harvesting and requires materials with high magnetostriction. Davino et al [4] discussed the magnetostrictive and magnetic properties of Terfenol-D rods, composed of terbium, dysprosium, and iron, under various stresses and magnetic fields. They showed that the maximum magnetoelastic constants are achieved at relatively low compressive stresses and magnetic fields. Mori et al [5] examined the energy harvesting characteristics of cantilevers with resonant tuning using a Terfenol-D plate, and found that in the range of more than 130 Hz, the movable proof mass stops at a stable position and the magnetostrictive cantilever shows self-tuning capability. Recently, Narita [6] developed magnetostrictive wire/polymer composites for the first time by embedding Fe–Co wires in an epoxy matrix, and studied their inverse magnetostrictive characteristics for various residual tensile stresses. The results showed that the output voltage density of the composite due to a compressive load dramatically increases with increasing stress-rate. Narita and Katabira [7] also developed strong textured Fe–Co fiber/polymer composites, and measured the output voltage. Then, they predicted the output voltage, and discussed domain wall dynamics in relation to the macroscopic inverse magnetostrictive response (known as the Villari effect).

Applications of energy harvesting devices exist, for instance, in motors and machines that generate heat during operations where temperature gradients form. Dalola et al [8] characterized three commercial thermoelectric modules that are designed for cooling/heating applications, and measured the open-circuit voltage and output power performance for each module at different temperature gradients and load conditions. Zhu et al [9] simulated the temperature distribution, heat flux, and voltage of a thin film solar thermoelectric generators using the finite element method, and studied the effect of air convection on the performance. Chen et al [10] reported experimental studies of a thermoelectric energy harvesting system designed for integration in nuclear power plants. On the other hand, Nguyen et al [11] designed a fully instrumented pyroelectric energy converter using co-polymer poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)] and measured their temperature oscillation, charge, voltage, and the overall heat input and output experimentally. They then computed the electrical power generated and the energy efficiency. Chang and Huang [12] estimated the pyroelectric coefficient, electrothermal coupling factor, power density and efficiency of various piezoelectric laminated composites under temperature difference across the pyroelectric element. They assumed constant external electric field (short circuit condition). Yang et al [13] demonstrated the first pyroelectric nanogenerator based on ZnO nanowire arrays for harvesting thermoelectric energy, and obtained the pyroelectric current and voltage coefficients. Karim et al [14] studied a lead free alternative for lead zirconate titanate (PZT) thermal energy harvesting and storage systems using LiNbO₃ as the pyroelectric material. They found the charge density to be 2.63 mV hour⁻¹ cm⁻³ and the peak energy density to be 437.72 nW cm⁻³.

Cooling and heating can lead to power generation, and provide a means of harvesting energy. Cryogenic fuel pumps and fuel tanks will occasionally alternate along thermal cycles from cryogenic temperatures to room temperature [15]. On the other hand, a significant amount of heat is generated from a variety of applications including consumer electronics, electric vehicles and automobiles.

The objectives of this study are to report the potential of BaTiO₃ for thermal energy harvesting and to evaluate the output voltage and power as temperature is rapidly shifted from room temperature to a cryogenic temperature (77 K) or high temperature (333 K). The temperature-dependent piezoelectric coefficient and permittivity of BaTiO₃ polycrystalline ceramics were calculated using phase field simulation and conduct the finite element analysis, using these properties, to examine the experimental results.

The unimorph sample was fabricated using a BaTiO₃ ceramic layer (NEC/Tokin Co. Ltd, Japan) and a Cu substrate (see figure 1). Electrodes were coated on both sides of the BaTiO₃ ceramic layer of length 50 mm, width 25 mm, and thickness 1 mm, and the BaTiO₃ ceramic layer was bonded to the upper surface of Cu substrate of length 55 mm, width 25 mm, and thickness 0.5 mm by conductive bonding. A resistive load was then connected to the BaTiO₃/Cu laminate.

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First, the BaTiO₃/Cu laminate without the resistive load was connected to a digital multimeter (ADCMT 7461A) and was inserted into liquid nitrogen (77 K) or a constant temperature oven (333 K). The laminate was not clamped, thereby serving as the so-called stress-free sample. The capacitance of BaTiO₃ is 16.1 nF (at room temperature) and the operating frequency of digital multimeter is 2 Hz. We matched the external load impedance with internal source impedance at room temperature. Note that the matching value is not a complex conjugate but a real. The output voltage values (i.e., open-circuit output voltages) were recorded until the sample reached a settled-down state where voltage readings decreased consistently at which point the sample was taken out. The sample was then maintained again at room temperature until the output voltage settled down. Next, a resistive load was connected to the BaTiO₃/Cu laminate, and the output voltage was measured. The output power from the laminate can then be calculated using the output voltage and load resistance R. Figure 2 shows schematic of the experimental setup.

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In order to discuss further physical mechanisms of the thermal energy harvesting, we carry out finite element analysis. Let us now consider the Cartesian coordinates x_i (O-x₁, x₂, x₃). The field equations are given by [16]

$$\begin{eqnarray}&&{\sigma }_{ji,j}=0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\varepsilon }_{ijk}{E}_{k,j}=0,{D}_{i,i}=0\end{eqnarray} \tag{ 2 }$$

where σ_ij, E_i and D_i are the components of stress tensor, electric field intensity vector and electric displacement vector, respectively; ${\varepsilon }_{ijk}$ is the permutation symbol; a comma followed by an index denotes partial differentiation with respect to the space coordinate x_i; and summation convention over repeated indices is used. Constitutive relations can be written as [17]

$$\begin{eqnarray}&&{\varepsilon }_{ij}={s}_{ijkl}^{{\rm{ET}}}{\sigma }_{kl}+{d}_{kij}^{{\rm{T}}}{E}_{k}+{\alpha }_{ij}^{{\rm{E}}}{\rm{\Delta }}T\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{D}_{i}={d}_{ikl}^{{\rm{T}}}{\sigma }_{kl}+{\epsilon}_{ik}^{\sigma {\rm{T}}}{E}_{k}+{p}_{i}^{{\rm{\sigma }}}{\rm{\Delta }}T\end{eqnarray} \tag{ 4 }$$

where ε_ij is the component of strain tensor; T is temperature; ${s}_{ijkl}^{{\rm{ET}}},$ ${d}_{kij}^{{\rm{T}}}$ and ${\epsilon}_{ik}^{\sigma {\rm{T}}}$ are the elastic compliance at constant electric field and temperature, direct piezoelectric coefficient at constant temperature, and permittivity at constant stress and temperature, respectively; ${\alpha }_{ij}^{{\rm{E}}}$ is the coefficient of thermal expansion at a constant electric field; and ${p}_{i}^{{\rm{\sigma }}}$ is the pyroelectric constant at constant stress. Here, the piezoelectric coefficient and permittivity are assumed to be temperature dependent. The constitutive relations (3) and (4) can be rewritten as

$$\begin{eqnarray}&&{\varepsilon }_{ij}={s}_{ijkl}^{{\rm{ET}}}{\sigma }_{kl}+{d}_{kij}(T){E}_{k}+{\alpha }_{ij}^{{\rm{E}}}{\rm{\Delta }}T\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{D}_{i}={d}_{ikl}(T){\sigma }_{kl}+{\epsilon}_{ik}^{\sigma }(T){E}_{k}+{p}_{i}^{{\rm{\sigma }}}{\rm{\Delta }}T\end{eqnarray} \tag{ 6 }$$

One-dimensional theory of the thermal energy harvesting is summarized in appexdix A to facilitate understanding.

Figure 3 shows the analytical procedure. The temperature-dependent piezoelectric coefficient and permittivity in equations (5) and (6) were evaluated numerically. Phase field method for BaTiO₃ polycrystalline ceramics developed by our group [18, 19] was employed. For simplicity, the elastic compliance and coefficient of thermal expansion were assumed to be independent of temperature in the finite element analysis. The pyroelectric constant [20] was ignored for the simplicity of calculations since the pyroelectricity of BaTiO₃ is not likely to be the main contributor to the output voltage data as mentioned in appexdix A. Elastic compliance of BaTiO₃ ceramics can be found in 21, and the coefficient of thermal expansion is 15.7 × 10⁻⁶ K⁻¹. The elastic compliance and coefficient of thermal expansion of Cu substrate are 9.09 × 10⁻¹² m² N⁻¹ and 16.5 × 10⁻⁶ K⁻¹, respectively. Details of the phase field model for BaTiO₃ polycrystalline ceramics can be seen in appendix B.

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We will discuss the results in detail. Figure 4(a) shows the open-circuit output voltage V_out due to temperature change from room temperature to 77 K as a function of time t. Although the experimental results are for a single test, the reproducibility of the general behavior was observed. It was shown that a sharp negative voltage (approximately 10 V) is observed when the temperature was quickly decreased from room temperature to 77 K. Note that roughly 250 s after the sample was removed from liquid nitrogen, approximately 10 V was observed again. Figure 4(b) shows similar results due to temperature change from room temperature to 333 K. It is interesting to note that approximately 8 V is observed when the temperature was rapidly increased from room temperature to 333 K. A negative voltage of approximately 10 V is also observed immediately after the sample was removed from the oven. The observed voltage may be generated by the pyroelectric effect of BaTiO₃, although the value was higher than expected; the generated open circuit voltage was approximately 2.6 V for the lead-free pyroelectric material [14]. The temperature dependence of permittivity and piezoelectric coefficient is considered to be one of the causes for the higher than expected voltage that was obtained. The other contributor of output voltage higher than predicted is also contemplated deformation of BaTiO₃ based laminate due to temperature changes.

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Figures 5(a) and (b) show the calculated piezoelectric coefficient d₃₃ and permittivity ${\epsilon}_{33}^{\sigma }$ versus temperature T, respectively, of poled BaTiO₃ polycrystals for average grain radius d = 0.89 μm and oxygen vacancy density N_d =0 ppm, d = 1.03 μm and N_d = 0 ppm, and d = 1.03 μm and N_d = 100 ppm. As the temperature increases, the piezoelectric coefficient increases depending on the oxygen vacancy density, as does the permittivity. Larger grain sizes result in an increase in the permittivity at higher temperatures of 300 K or more. Figure 6(a) shows the calculated open-circuit output voltage ∣V_out∣ of the BaTiO₃/Cu laminate due to temperature change from room temperature to 77 K for d = 0.89 μm and N_d = 0 ppm (blue column), d = 1.03 μm and N_d = 0 ppm (orange column), and d = 1.03 μm and N_d = 100 ppm (green column). Maximum bending stress σ_max in the BaTiO₃ layer are also shown. In the finite element simulation, the values of the other piezoelectric coefficients (d_31, d₁₅) and permittivity (${\epsilon}_{11}^{\sigma }$) at room temperature were used as they appear in 21, and these properties were presumed to have the same temperature dependence as in figure 5. For comparison, the calculated data with constant piezoelectric coefficients and constant permittivities for d = 1.03 μm and N_d = 100 ppm are displayed. The output voltage at 77 K increases with a change (decrease) in ${\epsilon}_{33}^{\sigma }$ by temperature (see equation (A4)). Conversely, the output voltage decreases with a change (decrease) in d₃₃ by temperature, because the piezoelectric effect caused by the bending stress (thermal stress) becomes small. These results suggest that if BaTiO₃ composites with large temperature dependent permittivities and piezoelectric coefficients independent of temperature are developed, even larger output voltage is likely to be obtained. The simulation was also performed using the model changing the thickness of copper layer (from 0.25 mm to 1.00 mm), but the thickness of copper layer did not affect the output voltage and the maximum stress (not shown here). Figure 6(b) shows similar results due to temperature change from room temperature to 333 K. The output voltage at 333 K seems to be decreasing with a change in ${\epsilon}_{33}^{\sigma }$ by temperature. This is due to the fact that the output voltage at 333 K is negative. In the 333 K case, if BaTiO₃ composites with permittivities–rather than piezoelectric coefficients–independent of temperature are developed, then a larger output voltage is likely to be obtained. It is interesting to note that the output voltage and the maximum bending stress (thermal stress) did not correlate perfectly. That is, electricity in the BaTiO₃ ceramic layer is generated not only by the piezoelectric effect caused by thermal stress from the change in temperature but also by the temperature dependence of the material properties, namely the piezoelectric coefficient and permittivity. It seems that differences between theoretical and experimental values are significantly large. The main reason is due to the measurement range limit of the digital multimeter. The measurement range limit is 10 V because the load resistance of the digital multimeter is 10 GΩ, which is, however, still not satisfied for high open-circuit output voltage.

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Finally, the output power of the BaTiO₃/Cu laminate due to the temperature change from room temperature to 77 K is discussed. The variations of the measured output voltage V_out of the BaTiO₃ layer with changing load resistance R are shown in figure 7 for the BaTiO₃/Cu laminate at 77 K. The negative output voltage increases with load resistance, as is expected. Figure 7 also shows the output power P for the BaTiO₃/Cu laminate at 77 K. It can be seen that the maximum output power reaches about 2.8 μW when the resistance is 25 MΩ. The corresponding output power density is approximately 2240 nW cm⁻³.

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In conclusion, the unique discovery of these experiments and multiscale simulations lie in the observation that the BaTiO₃/Cu laminate is able to display piezoelectric energy harvesting behavior from undergoing temperature change. Large voltage outputs were generated experimentally and numerically when the temperature of the BaTiO₃/Cu laminate was rapidly shifted. The effect of direct pyroelectricity, the production of voltage caused by a change in spontaneous polarization due to a temperature change, is likely too small to be the single major contributor to the observed behavior. As experimental results and calculation suggest, electricity in the BaTiO₃ ceramic layer is generated not only through the piezoelectric effect caused by a thermally-induced bending stress but also by the temperature dependence of the BaTiO₃ piezoelectric coefficient and permittivity. Using the stress-free laminate, approximately 2240 nW cm⁻³ was observed. By use of a strain-free laminate, an even larger power output can be expected, and work in the area is currently being pursued. Through the incorporation of these findings in future research, this information can further the field of energy harvesting materials and the production of self-powering devices.

This work was supported by JSPS KAKENHI Grant Number 16H04227.

Here, we discuss the thermal energy harvesting for a one-dimensional piezoelectric plate poled in the x₃-direction. Dimensions of length, width and thickness are assumed along the x₁, x₂ and x₃ axis, respectively. In this case, the plate is free to expand vertically, which implies that the stress σ₃₃ is equal to zero. Also, because the plate is considered to be long and slender, we may assume that the stress σ₂₂ is zero. As the polarization is parallel to the electric field E₃ and electric displacement D₃, no shear stresses will develop, and we may conclude that no shear strains are present, which will reduce the nontrivial stress to σ₁₁. Thus, the constitutive relations (5) and (6) for the BaTiO₃ ceramics with respect to the reference axes of the plate (x₁, x₃) becomes

$$\begin{eqnarray}&&{\varepsilon }_{11}={s}_{11}^{{\rm{ET}}}{\sigma }_{11}+{d}_{31}^{{\rm{T}}}(T){E}_{3}+{\alpha }_{1}^{{\rm{E}}}{\rm{\Delta }}T\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{D}_{3}={d}_{31}^{{\rm{T}}}(T){\sigma }_{11}+{\epsilon}_{33}^{\sigma }(T){E}_{3}+{p}_{3}^{{\rm{\sigma }}}{\rm{\Delta }}T\end{eqnarray} \tag{ A2 }$$

We consider two special cases. The first case is σ₁₁ = 0, and the second is ε₁₁ = 0. In the experiment, the open-circuit output voltage was measured so that D₃ = 0.

We first assume that σ₁₁ = 0. Substituting D₃ = 0 into Equation (A2), we obtain the electric field

$$\begin{eqnarray}&&{E}_{3}=-\displaystyle \frac{{p}_{3}^{{\rm{\sigma }}}}{{\epsilon}_{33}^{\sigma }(T)}{\rm{\Delta }}T\end{eqnarray} \tag{ A3 }$$

The output voltage of the stress-free BaTiO₃ ceramic plate with thickness h becomes

$$\begin{eqnarray}&&{V}_{{\rm{out}}}^{{\rm{\sigma }}0}=\displaystyle \frac{{p}_{3}^{{\rm{\sigma }}}}{{\epsilon}_{33}^{\sigma }(T)}h{\rm{\Delta }}T\end{eqnarray} \tag{ A4 }$$

Equation (A4) means that the output voltage is obtained due to the pyroelectric effect.

Next, we assume that ε₁₁ = 0. Eliminating σ₁₁ from equations (A1) and (A2) and taking into account D₃ = 0, we have

$$\begin{eqnarray}&&{E}_{3}=-\left(\displaystyle \frac{{\alpha }_{1}^{{\rm{E}}}{d}_{31}(T)-{s}_{11}^{{\rm{ET}}}{p}_{3}^{{\rm{\sigma }}}}{{d}_{31}^{2}(T)-{s}_{11}^{{\rm{ET}}}{\epsilon}_{33}^{\sigma }(T)}\right){\rm{\Delta }}T\end{eqnarray} \tag{ A5 }$$

On the other hand, the thermal stress is given by

$$\begin{eqnarray}&&{\sigma }_{11}=\displaystyle \frac{{\epsilon}_{33}^{\sigma }(T){\alpha }_{1}^{{\rm{E}}}-{d}_{31}(T){p}_{3}^{{\rm{\sigma }}}}{{d}_{31}^{2}(T)-{s}_{11}^{{\rm{ET}}}{\epsilon}_{33}^{\sigma }(T)}{\rm{\Delta }}T\end{eqnarray} \tag{ A6 }$$

From equation (A5), the output voltage of the strain-free BaTiO₃ ceramic plate is

$$\begin{eqnarray}&&{V}_{{\rm{out}}}^{{\rm{\varepsilon }}0}=\left(\displaystyle \frac{{\alpha }_{1}^{{\rm{E}}}{d}_{31}(T)-{s}_{11}^{{\rm{ET}}}{p}_{3}^{{\rm{\sigma }}}}{{d}_{31}^{2}(T)-{s}_{11}^{{\rm{ET}}}{\epsilon}_{33}^{\sigma }(T)}\right)h{\rm{\Delta }}T\end{eqnarray} \tag{ A7 }$$

Equation (A7) means that the output voltage is obtained due to the pyroelectric effect and coupling between thermal stress and electric field (see figure A1). When d₃₁(T) → 0, equation (A7) reduces equation (A4).

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We compare the contributions of pyroelectricity and piezoelectricity to the output voltage. Material properties ${s}_{11}^{{\rm{ET}}}$ = 13.9 _× 10⁻¹² m² N⁻¹, ${d}_{31}$ = −144 × 10⁻¹² mV⁻¹, ${\epsilon}_{33}^{\sigma }$ = 129 × 10⁻¹⁰ C V⁻¹m⁻¹, ${\alpha }_{1}^{{\rm{E}}}$ = 15.7 × 10⁻⁶ K⁻¹, and ${p}_{3}^{\sigma }$ = 1.0 × 10⁻⁴ Cm⁻²K⁻¹ for the BaTiO₃ ceramics at room temperature were used. We obtained that the ratio ${V}_{{\rm{out}}}^{{\rm{\varepsilon }}0}/{V}_{{\rm{out}}}^{{\rm{\sigma }}0}$is approximately equal to 3. Pyroelectricity is defined as the linear interaction between electrical and thermal systems. Since BaTiO₃ is ferroelectric, the material ought to demonstrate pyroelectricity as well. However, as the electric potential across the stress-free BaTiO₃ ceramic plate due to pyroelectricity is usually not observed to be so large, pyroelectricity is not likely to be the main contributor to the output voltage data that was observed during our experiments. Therefore, piezoelectricity, in conjunction with some effect from pyroelectricity, is likely to be the mechanism causing the voltage output.

The phase field model is outlined here. The temporal evolution of the component of polarization vector P_i and thus the domain structure is governed by the following time-dependent Ginzburg–Landau equation [22]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {P}_{i}({x}_{i},t)}{\partial t}=-M\displaystyle \frac{\delta F}{\delta {P}_{i}({x}_{i},t)}\,(i=1,2,3)\end{eqnarray} \tag{ B1 }$$

where F is the total free energy of the ferroelectric polycrystal, M is the kinetic coefficient related to the domain mobility, and t is the time. The total free energy is given by

$$\begin{eqnarray}&&F=\displaystyle {\int }_{V}({f}_{{\rm{bulk}}}+{f}_{{\rm{grad}}}+{f}_{{\rm{elas}}}+{f}_{{\rm{elec}}})dV\end{eqnarray} \tag{ B2 }$$

where f_bulk is the local bulk free energy density, f_grad is the gradient energy density which is only non-zero around domain walls and grain boundaries, f_elas is the elastic strain energy density, f_elec is the electrostatic energy density which is dependent on the oxygen vacancy density N_d, and V is the volume of the polycrystal.

The oxygen vacancy density N_d is governed by the following diffusion equation [23, 24]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {N}_{{\rm{d}}}}{\partial t}={\rm{\nabla }}\cdot \left[{\beta }_{{\rm{d}}}{N}_{{\rm{d}}}{\rm{\nabla }}\left(\displaystyle \frac{\partial {W}_{{\rm{d}}}}{\partial {N}_{{\rm{d}}}}+eZ\phi \right)\right]\end{eqnarray} \tag{ B3 }$$

where β_d = 5 × 10⁵ m² s⁻¹J⁻¹ is the defect's mobility, W_d is the contribution to the free energy due to defects, e = 1.6021773 × 10⁻¹⁹ C is the coulomb charge per electron, Z = 1 is the valency of donors (oxygen vacancies), and ϕ is the electric potential. If the mole fraction of vacancies is very small, equation (B3) is simplified to

$$\begin{eqnarray}&&\displaystyle \frac{\partial {N}_{{\rm{d}}}}{\partial t}={\rm{\nabla }}\cdot ({\beta }_{{\rm{d}}}{k}_{{\rm{B}}}T{\rm{\nabla }}{N}_{{\rm{d}}}+{\beta }_{{\rm{d}}}eZ{N}_{{\rm{d}}}{\rm{\nabla }}\phi )\end{eqnarray} \tag{ B4 }$$

where k_B = 1.3807 × 10⁻²³ J K⁻¹ is the Boltzmann constant. To generate a two-dimensional grain structure, the grain growth model developed by Krill and Chen [25] is employed.