Piezomechanics in PZT Stack Actuators for Cryogenic Fuel Injectors

1. Introduction

Lead zirconate titanate (PZT) ceramics in special electronic devices such as structural health monitoring systems of liquid rocket engines and microvalves for space applications are subjected to cryogenic temperatures. PZT ceramics are also used in active fuel injectors under severe environments (Senousy et al. 2009a; 2009b). In the application of the PZT stack actuators to hydrogen fuel injectors, the actuators are operated under electric fields at cryogenic temperatures. Hence, it is important to understand the cryogenic electromechanical response of the PZT actuators under electric fields.

In this chapter, we address the present state of piezomechanics in PZT stack actuators for fuel injectors at cryogenic temperatures. First, we discuss the cryogenic response of PZT stack actuators under direct current (DC) electric fields (Shindo et al. 2011). A thermodynamic model is used to predict a monoclinic phase around a morphotropic phase boundary (MPB). A shift in the boundary between the tetragonal and rhombohedral/monoclinic phases with decreasing temperature is determined, and the temperature dependent piezoelectric coefficients are evaluated. Temperature dependent coercive electric field is also predicted based on the domain wall energy. A finite element analysis (FEA) is then performed, considering the shift in the MPB and polarization switching, to calculate the electromechanical fields of the PZT stack actuators from room to cryogenic temperatures. In addition, experimental results on the DC electric field induced strain, which verify the model, are presented. Next, we discuss the dynamic response of PZT stack actuators under alternating current (AC) electric fields at cryogenic temperatures (Shindo et al. 2012). Dynamic electromechanical fields of the PZT stack actuators from room to cryogenic temperatures are simulated by the FEA with MPB shift and domain wall motion effects. Dynamic strain measurements of the PZT stack actuators under AC electric fields are also presented, and a comparison is made between calculations and measurements to validate the predictions. Moreover, a parametric study using FEA is performed to investigate the factors affecting the cryogenic response of PZT stack actuators and to provide a basis for selecting desirable design details.

2. Analysis

2.1. Basic equations

Consider the orthogonal coordinate system with axes x, y, and z. The Newton's second law (the equations of motion) and Gauss' law for piezoelectric materials are given by

$$\begin{array}{l}{\sigma }_{xx,x}+{\sigma }_{yx,y}+{\sigma }_{zx,z}=\rho u_{x,tt}\\ {\sigma }_{xy,x}+{\sigma }_{yy,y}+{\sigma }_{zy,z}=\rho u_{y,tt}\\ {\sigma }_{xz,x}+{\sigma }_{yz,y}+{\sigma }_{zz,z}=\rho u_{z,tt}\end{array}$$E1

$$D_{x,x}+D_{y,y}+D_{z,z}=0$$E2

where (σ_xx, σ_yy, σ_zz, σ_xy = σ_yx, σ_yz = σ_zy, σ_zx = σ_xz) and (D_x, D_y, D_z) are the components of stress tensor and electric displacement vector, (u_x, u_y, u_z) are the components of displacement vectors, ρ is the mass density, and a comma denotes partial differentiation with respect to the coordinates or the time t. Constitutive relations for PZT ceramics poled in the z-direction can be written as

$$\begin{array}{l}\left\{\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{zz}\\ {\epsilon }_{yz}\\ {\epsilon }_{zx}\\ {\epsilon }_{xy}\end{array}\right\}=\left[\begin{array}{cccccc}{s}_{11}& s_{12}& s_{13}& 0& 0& 0\\ s_{12}& s_{11}& s_{13}& 0& 0& 0\\ s_{13}& s_{13}& s_{33}& 0& 0& 0\\ 0& 0& 0& s_{44}/2& 0& 0\\ 0& 0& 0& 0& s_{44}/2& 0\\ 0& 0& 0& 0& 0& s_{66}/2\end{array}\right]\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\\ {\sigma }_{xy}\end{array}\right\}\\ +\left[\begin{array}{ccc}0& 0& {\overline{d}}_{31}\\ 0& 0& {\overline{d}}_{31}\\ 0& 0& {\overline{d}}_{33}\\ 0& {\overline{d}}_{15}/2& 0\\ {\overline{d}}_{15}/2& 0& 0\\ 0& 0& 0\end{array}\right]\left\{\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right\}+\left\{\begin{array}{c}{\epsilon }_{xx}^r\\ {\epsilon }_{yy}^r\\ {\epsilon }_{zz}^r\\ {\epsilon }_{yz}^r\\ {\epsilon }_{zx}^r\\ {\epsilon }_{xy}^r\end{array}\right\}\end{array}$$E3

$$\left\{\begin{array}{c}{D}_x\\ D_y\\ D_z\end{array}\right\}=\left[\begin{array}{cccccc}0& 0& 0& 0& {\overline{d}}_{15}& 0\\ 0& 0& 0& {\overline{d}}_{15}& 0& 0\\ {\overline{d}}_{31}& {\overline{d}}_{31}& {\overline{d}}_{33}& 0& 0& 0\end{array}\right]\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\\ {\sigma }_{xy}\end{array}\right\}+\left[\begin{array}{ccc}{\in }_{11}^{\text{T}}& 0& 0\\ 0& {\in }_{11}^{\text{T}}& 0\\ 0& 0& {\in }_{33}^{\text{T}}\end{array}\right]\left\{\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right\}+\left\{\begin{array}{c}{P}_x^r\\ P_y^r\\ P_z^r\end{array}\right\}$$E4

where (ε_xx, ε_yy, ε_zz, ε_xy = ε_yx, ε_yz = ε_zy, ε_zx = ε_xz) and (E_x, E_y, E_z) are the components of strain tensor and electric field intensity vector, ($${\epsilon }_{xx}^r,{\epsilon }_{yy}^r,{\epsilon }_{zz}^r,{\epsilon }_{xy}^r,{\epsilon }_{yz}^r,{\epsilon }_{zx}^r$$) and ($$P_x^r,P_y^r,P_z^r$$) are the remanent strain and polarization components, (s₁₁, s₁₂, s₁₃, s₃₃, s₄₄, s₆₆ = 2(s₁₁- s₁₂)) are the elastic compliances, ($${\overline{d}}_{31}$$,$${\overline{d}}_{33}$$,$${\overline{d}}_{15}$$) are the temperature dependent piezoelectric coefficients, and ($${\in }_{11}^{\text{T}}$$,$${\in }_{33}^{\text{T}}$$) are the dielectric constants. The quantities $${\epsilon }_{ij}^r$$ and $$P_i^r$$ are taken to be due entirely to polarization switching. The strain components are

$$\begin{array}{l}{\epsilon }_{xx}=u_{x,x},\begin{array}{cc}& \end{array}{\epsilon }_{yy}=u_{y,y},\begin{array}{cc}& \end{array}{\epsilon }_{zz}=u_{z,z},\\ {\epsilon }_{xy}=\frac{1}{2}\left(u_{x,y}+u_{y,x}\right),\begin{array}{cc}& \end{array}{\epsilon }_{yz}=\frac{1}{2}\left(u_{y,z}+u_{z,y}\right),\begin{array}{cc}& \end{array}{\epsilon }_{zx}=\frac{1}{2}\left(u_{z,x}+u_{x,z}\right)\end{array}$$E5

The electric field components are related to the electric potential ϕ by

$$E_x=-{\varphi }_{,x},\begin{array}{cc}& \end{array}{E}_y=-{\varphi }_{,y},\begin{array}{cc}& \end{array}{E}_z=-{\varphi }_{,z}$$E6

2.2. Temperature dependent piezoelectric coefficient

Temperature dependent piezoelectric coefficient is outlined here. Figure 1 shows the phase diagram of PZT established in Jaffe et al. (1971) and Noheda et al. (2000). As the temperature T is lowered, PZT undergoes a paraelectric-to-ferroelectric phase transition, and the cubic unit cell is distorted depending on the mole fraction X of PbTiO₃. In the Zr-rich region, the paraelectric phase changes to orthorhombic phase. An intermediate monoclinic phase exists between the Zr-rich rhombohedral perovskite and Ti-rich tetragonal perovskite phases. Compositions between Zr/Ti ratios 90/10 and 65/35 reveals a ferroelectric-to-ferroelectric transition between rhombohedral space groups. This transition involves the oxygen octahedral tilt.

The MPB between the tetragonal and rhombohedral/monoclinic phases is the origin of the unusually high piezoelectric response of PZT, and this MPB is numerically predicted. For simplicity here, we ignore the octahedral tilt transition which differentiates the high temperature (HT) and low temperature (LT) rhombohedral phases, and the orthorhombic phase.

An energy function for the solid solution between the two end-members PbTiO₃ and PbZrO₃ is given by (Bell & Furman 2003)

$$\Delta G_{PZT}=X{G}_{PT}\left(p_i\right)+\left(1-X\right)G_{PZ}\left(q_i\right)+G_C\left(p_i,q_i\right)$$E7

where p_i and q_i (i = 1,2,3) denote the polarizations of the PbTiO₃ and PbZrO₃, respectively, and

$$\begin{array}{l}{G}_{PT}\left(p_i\right)=3.74\times 10^5\left(T-T_{PT}\right)\left(p_1^2+p_2^2+p_3^2\right)-7.9\times 10^7\left(p_1^4+p_2^4+p_3^4\right)\\ +7.5\times 10^8\left(p_1^2{p}_2^2+p_2^2{p}_3^2+p_3^2{p}_1^2\right)+2.61\times 10^8\left(p_1^6+p_2^6+p_3^6\right)\\ +6.3\times 10^8\left\{p_1^4\left(p_2^2+p_3^2\right)+p_2^4\left(p_3^2+p_1^2\right)+p_3^4\left(p_1^2+p_2^2\right)\right\}-3.66\times 10^9{p}_1^2{p}_2^2{p}_3^2\end{array}$$E8

$$\begin{array}{l}{G}_{PZ}\left(q_i\right)=2.82\times 10^5\left(T-T_{PZ}\right)\left(q_1^2+q_2^2+q_3^2\right)+5.12\times 10^8\left(q_1^4+q_2^4+q_3^4\right)\\ -6.5\times 10^8\left(q_1^2{q}_2^2+q_2^2{q}_3^2+q_3^2{q}_1^2\right)+5.93\times 10^8\left(q_1^6+q_2^6+q_3^6\right)\\ +2\times 10^9\left\{q_1^4\left(q_2^2+q_3^2\right)+q_2^4\left(q_3^2+q_1^2\right)+q_3^4\left(q_1^2+q_2^2\right)\right\}-9.5\times 10^9{q}_1^2{q}_2^2{q}_3^2\end{array}$$E9

$$G_C\left(p_i,q_i\right)={\gamma }_{200}\left(p_1^2{q}_1^2+p_2^2{q}_2^2+p_3^2{q}_3^2\right)+{\gamma }_{220}\left\{p_1^2\left(q_2^2+q_3^2\right)+p_2^2\left(q_3^2+q_1^2\right)+p_3^2\left(q_1^2+q_2^2\right)\right\}$$E10

In Eqs. (8) - (10), G_PT and G_PZ are identical to a Landau-Devonshire potential (free energy of a ferroelectric crystal) up to sixth order, G_C represents the coupling energy, T_PT = 766 K and T_PZ = 503 K are the Curie temperatures of PbTiO₃ and PbZrO₃, respectively, and γ₂₀₀ and γ₂₂₀ are unknown coefficients.

Figure 1.

PZT phase diagram

The thermodynamic equilibrium state can be determined via minimization of ΔG_PZT with respect to p_i and q_i. For simplicity, only positive values of p_i and q_i are considered. For each temperature T and mole fraction X, the local minima in ΔG_PZT are systematically obtained for the following phases:

Cubic (C)

$$p_1=p_2=p_3=0,\begin{array}{cc}& q_1=q_2=q_3=0\end{array}$$E11

Tetragonal (T)

$$p_1=p_2=0,p_3\ne 0,\begin{array}{cc}& q_1=q_2=0,q_3\ne 0\end{array}$$E12

Rhombohedral (R)

$$p_1=p_2=p_3\ne 0,\begin{array}{cc}& q_1=q_2=q_3\ne 0\end{array}$$E13

Monoclinic (M)

$$p_1=p_2\ne 0,p_3\ne 0,\begin{array}{cc}& q_1=q_2\ne 0,q_3\ne 0\end{array}$$E14

The energies of the minima are then compared to define the stable state. In the simulation, γ₂₀₀ = 6$$\times$$10⁸ and γ₂₂₀ = 1.2$$\times$$10⁸ are assumed.

We now show a numerical example and comparison with experiments. Fig. 2 shows the

Figure 2.

Calculated and experimental phase diagram of PZT

calculated PZT phase diagram. We also plot the experimental phase diagrams of PZT. The open square, open circle and solid circle are the results from Jeffe et al. (1971), Noheda et al. (2000) and Pandey et al. (2008). The simulation result reasonably agrees with the experimental data. It is noted that at room temperature, the MPB is located at about X = 0.44, and that there is an apparent shift of about 13 mol% in the MPB location on cooling to 0 K. In the work of Boucher et al. (2006), the piezoelectric coefficient d₃₃₃ of PZT (Mn-doped) presented a significant decrease (about 85 %) due to a shift of 12 mol% of Zr at room temperature. Thus, from the consideration of Fig. 2, we propose the temperature dependent piezoelectric coefficient $${\overline{d}}_{ijk}$$($${\overline{d}}_{311}={\overline{d}}_{31}$$,$${\overline{d}}_{333}={\overline{d}}_{33}$$,$${\overline{d}}_{131}={\overline{d}}_{15}/2$$here) for X = 0.44, i.e.,

$${\overline{d}}_{ijk}^{}=\{\begin{array}{cc}(1.7\times {10}^{-4}T+0.95)d_{ijk}& \begin{array}{cc}& 192\le T\end{array}\\ (1.0\times {10}^{-5}{T}^2+2.4\times {10}^{-3}T+0.15)d_{ijk}& \begin{array}{cc}& 0<T\end{array}\le 192\end{array}\begin{array}{c}\\ \end{array}$$E15

where d_ijk (d₃₁₁=d₃₁, d₃₃₃=d₃₃, d₁₃₁=d₁₅/2 here) is the piezoelectric coefficient at 298 K. In Eqs. (3) and (4), reduced indices for the full notations of elastic compliances s_ijkl and temperature dependent piezoelectric coefficients $${\overline{d}}_{ijk}$$ are used, with the following correspondence between the one and two indices: 1 = 11, 2 =22, 3 = 33, 4 = 23, 5 = 31, 6 = 12. We can also predict the piezoelectric coefficient for other mole fractions. For example, the temperature dependent piezoelectric coefficient for X = 0.56 can be expressed as

$${\overline{d}}_{ijk}^{}=\{\begin{array}{c}(-1.7\times {10}^{-4}T+0.25)d_{ijk}\begin{array}{cc}& 192\le T\end{array}\\ (-1.0\times {10}^{-5}{T}^2-2.4\times {10}^{-3}T+1.1)d_{ijk}\begin{array}{cc}\begin{array}{cc}& 20<T\end{array}\le 192& \end{array}\\ (1.0\times {10}^{-5}{T}^2+2.4\times {10}^{-3}T+0.95)d_{ijk}\begin{array}{cc}\begin{array}{cc}& 0<T\end{array}\le 20& \end{array}\end{array}$$E16

2.3. Polarization switching

High electromechanical fields lead to the polarization switching. We assume that the direction of a spontaneous polarization P^s of each grain can change by 180^o or 90^o for ferroelectric switching induced by a sufficiently large electric field opposite to the poling direction. The 90^o ferroelastic domain switching is also induced by a sufficiently large stress field. The criterion states that a polarization switches when the electrical and mechanical work exceeds a critical value (Hwang et al. 1995)

$$\begin{array}{l}{\sigma }_{xx}\Delta {\epsilon }_{xx}+{\sigma }_{yy}\Delta {\epsilon }_{yy}+{\sigma }_{zz}\Delta {\epsilon }_{zz}+2{\sigma }_{xy}\Delta {\epsilon }_{xy}+2{\sigma }_{yz}\Delta {\epsilon }_{yz}+2{\sigma }_{zx}\Delta {\epsilon }_{zx}\\ +E_x\Delta P_x+E_y\Delta P_y+E_z\Delta P_z\ge 2P_{}^s{\overline{E}}_c^{}\end{array}$$E17

where $${\overline{E}}_c^{}$$is a temperature dependent coercive electric field, and Δε_ij and ΔP_i are the changes in the spontaneous strain and polarization during switching, respectively. The values of Δε_ij = $${\epsilon }_{ij}^r$$ and ΔP_i =$$P_i^r$$ for 180^o switching can be expressed as

$$\begin{array}{l}\Delta {\epsilon }_{xx}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yy}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zz}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{xy}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yz}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zx}=0,\\ \Delta P_x=0,\begin{array}{cc}& \end{array}\Delta P_y=0,\begin{array}{cc}& \end{array}\Delta P_z=-2P_{}^s\end{array}$$E18

For 90^o switching in the zx plane, the changes are

$$\begin{array}{l}\Delta {\epsilon }_{xx}={\gamma }^s,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yy}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zz}=-{\gamma }^s,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{xy}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yz}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zx}=0,\\ \Delta P_x=\pm P_{}^s,\begin{array}{cc}& \end{array}\Delta P_y=0,\begin{array}{cc}& \end{array}\Delta P_z=-P_{}^s\end{array}$$E19

where $${\gamma }^s$$ is a spontaneous strain. For 90^o switching in the yz plane, we have

$$\begin{array}{l}\Delta {\epsilon }_{xx}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yy}={\gamma }^s,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zz}=-{\gamma }^s,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{xy}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{yz}=0,\begin{array}{cc}& \end{array}\Delta {\epsilon }_{zx}=0,\\ \Delta P_x=0,\begin{array}{cc}& \end{array}\Delta P_y=\pm P_{}^s,\begin{array}{cc}& \end{array}\Delta P_z=-P_{}^s\end{array}$$E20

The constitutive equations after polarization switching are

$$\begin{array}{l}\left\{\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{zz}\\ {\epsilon }_{yz}\\ {\epsilon }_{zx}\\ {\epsilon }_{xy}\end{array}\right\}=\left[\begin{array}{cccccc}{s}_{11}& s_{12}& s_{13}& 0& 0& 0\\ s_{12}& s_{11}& s_{13}& 0& 0& 0\\ s_{13}& s_{13}& s_{33}& 0& 0& 0\\ 0& 0& 0& s_{44}/2& 0& 0\\ 0& 0& 0& 0& s_{44}/2& 0\\ 0& 0& 0& 0& 0& s_{66}/2\end{array}\right]\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\\ {\sigma }_{xy}\end{array}\right\}\\ +\left[\begin{array}{ccc}{\overline{d}}_{111}^{*}& {\overline{d}}_{211}^{*}& {\overline{d}}_{311}^{*}\\ {\overline{d}}_{122}^{*}& {\overline{d}}_{222}^{*}& {\overline{d}}_{322}^{*}\\ {\overline{d}}_{133}^{*}& {\overline{d}}_{233}^{*}& {\overline{d}}_{333}^{*}\\ {\overline{d}}_{123}^{*}& {\overline{d}}_{223}^{*}& {\overline{d}}_{323}^{*}\\ {\overline{d}}_{131}^{*}& {\overline{d}}_{231}^{*}& {\overline{d}}_{331}^{*}\\ {\overline{d}}_{112}^{*}& {\overline{d}}_{212}^{*}& {\overline{d}}_{312}^{*}\end{array}\right]\left\{\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right\}+\left\{\begin{array}{c}{\epsilon }_{xx}^r\\ {\epsilon }_{yy}^r\\ {\epsilon }_{zz}^r\\ {\epsilon }_{yz}^r\\ {\epsilon }_{zx}^r\\ {\epsilon }_{xy}^r\end{array}\right\}\end{array}$$E21

$$\left\{\begin{array}{c}{D}_x\\ D_y\\ D_z\end{array}\right\}=\left[\begin{array}{cccccc}{\overline{d}}_{111}^{*}& {\overline{d}}_{122}^{*}& {\overline{d}}_{133}^{*}& {\overline{d}}_{123}^{*}& {\overline{d}}_{131}^{*}& {\overline{d}}_{112}^{*}\\ {\overline{d}}_{211}^{*}& {\overline{d}}_{222}^{*}& {\overline{d}}_{233}^{*}& {\overline{d}}_{223}^{*}& {\overline{d}}_{231}^{*}& {\overline{d}}_{212}^{*}\\ {\overline{d}}_{311}^{*}& {\overline{d}}_{322}^{*}& {\overline{d}}_{333}^{*}& {\overline{d}}_{323}^{*}& {\overline{d}}_{331}^{*}& {\overline{d}}_{312}^{*}\end{array}\right]\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\\ {\sigma }_{xy}\end{array}\right\}+\left[\begin{array}{ccc}{\in }_{11}^{\text{T}}& 0& 0\\ 0& {\in }_{11}^{\text{T}}& 0\\ 0& 0& {\in }_{33}^{\text{T}}\end{array}\right]\left\{\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right\}+\left\{\begin{array}{c}{P}_x^r\\ P_y^r\\ P_z^r\end{array}\right\}$$E22

where

$${\overline{d}}_{ijk}^{*}=\{{\overline{d}}_{33}^{}{n}_i{n}_k{n}_l+{\overline{d}}_{31}^{}(n_i{\delta }_{kl}-n_i{n}_k{n}_l)+\frac{1}{2}{\overline{d}}_{15}^{}({\delta }_{ik}{n}_l-2n_i{n}_k{n}_l+{\delta }_{il}{n}_k)\}$$E23

In Eq. (23), n_i is the unit vector in the poling direction and δ_ij is the Kronecker delta.

2.4. Domain wall motion

A domain wall displacement causes changes of strain and polarization (Cao et al. 1999). For simplicity, here the applied AC electric field E_z=E₀exp(iωt) is parallel to the direction of spontaneous polarization P^s in one of the domains (see Fig. 3); E₀ is the AC electric field amplitude and ω is angular frequency(=2πf where f is frequency in Hertz). The changes of the strains and polarization due to the domain wall displacement Δl (Arlt and Dederichs, 1980) can be written as

$$\begin{array}{l}\Delta {\epsilon }_{xx}^{*}=\Delta s_{11}{\sigma }_{xx}+\Delta s_{13}{\sigma }_{zz}+\Delta d_{311}{E}_z\\ \Delta {\epsilon }_{zz}^{*}=\Delta s_{13}{\sigma }_{xx}+\Delta s_{33}{\sigma }_{zz}+\Delta d_{333}{E}_z\\ \Delta P_z^{*}=\Delta d_{311}{\sigma }_{xx}+\Delta d_{333}{\sigma }_{zz}+\Delta {\in }_{33}^{\text{T}}{E}_z\end{array}$$E24

where all terms with Δ are the contributions from the domain wall motion, and

$$\begin{array}{l}\Delta s_{11}=\frac{{({\gamma }^s)}^2}{2l{f}_D},\begin{array}{cc}& \end{array}\Delta s_{13}=-\frac{{({\gamma }^s)}^2}{2l{f}_D},\begin{array}{cc}& \end{array}\Delta s_{33}=\frac{{({\gamma }^s)}^2}{2l{f}_D},\\ \Delta d_{311}=-\frac{{\gamma }^s{P}^s}{2l{f}_D},\begin{array}{cc}& \end{array}\Delta d_{333}=\frac{{\gamma }^s{P}^s}{2l{f}_D},\begin{array}{cc}& \end{array}\Delta {\in }_{33}^{\text{T}}=\frac{{(P^s)}^2}{2l{f}_D}\end{array}$$E25

In Eq. (25), l is the domain width and f_D is the force constant for the domain wall motion process. The strains ε_xx, ε_zz and electric displacement D_z are given by

Figure 3.

Schematic drawing of many grains which in turn consist of domains and basic unit of a piezoelectric crystallite with a domain wall.

$$\begin{array}{l}{\epsilon }_{xx}^{}=s_{11}^{*}{\sigma }_{xx}+s_{12}{\sigma }_{yy}+s_{13}^{*}{\sigma }_{zz}+d_{31}^{*}{E}_z\\ {\epsilon }_{zz}^{}=s_{13}^{*}{\sigma }_{xx}+s_{13}^{*}{\sigma }_{yy}+s_{33}^{*}{\sigma }_{zz}+d_{31}^{*}{E}_z\\ D_z^{}=d_{31}^{*}{\sigma }_{xx}+d_{31}^{*}{\sigma }_{yy}+d_{33}^{*}{\sigma }_{zz}+{\in }_{33}^{\text{T}*}{E}_z\end{array}$$E26

where

$$\begin{array}{l}{s}_{11}^{*}=s_{11}+\Delta s_{11},\begin{array}{cc}& s_{13}^{*}=s_{13}+\Delta s_{13}\end{array},\begin{array}{cc}& s_{33}^{*}=s_{33}+\Delta s_{33}\end{array}\\ d_{31}^{*}={\overline{d}}_{31}+\Delta d_{311},\begin{array}{cc}& \end{array}{d}_{33}^{*}={\overline{d}}_{33}+\Delta d_{333},\\ {\in }_{33}^{\text{T}*}={\in }_{33}^{\text{T}}+\Delta {\in }_{33}^{\text{T}}\end{array}$$E27

Experimental studies on PZT ceramics have shown that 45-70% of dielectric and piezoelectric moduli values may originate from the extrinsic contributions (Luchaninov et al. 1989, Cao et al. 1991). The extrinsic dielectric constant Δ is approximately estimated as the two thirds of the bulk properties (Li et al. 1993). Here, the following equation (Narita et al. 2005) is utilized to describe Δ in terms of AC electric field amplitude E₀ and temperature dependent coercive electric field $${\overline{E}}_c^{}$$:

$$\Delta {\in }_{33}^{\text{T}}={\in }_{33}^{\text{T}}\frac{2E_0}{3{\overline{E}}_c^{}}$$E28

By substituting Eq. (28) into the sixth of Eq. (25), lf_D = 3(P^s)²$${\overline{E}}_c^{}$$/(4$${\in }_{33}^{\text{T}}$$E₀) is obtained. By eliminating lf_D, the changes in the elastic compliances and piezoelectric coefficients in Eq. (25) can be rewritten in terms of AC electric field amplitude etc.

2.5. Finite element model

Consider a PZT stack actuator with 300 PZT layers of width W_p = 5.2 mm and thickness h_p = 0.1 mm, thin electrodes, and elastic coating layer of thickness h_e = 0.5 mm as shown in Fig. 4. A rectangular Cartesian coordinate system O-xyz is used and the origin of the coordinate system coincides with the center of the stack actuator. Each PZT layer is sandwiched between thin electrodes. An external electrode is attached on both sides of the actuator to address the internal electrodes.

In order to discuss the electromechanical fields near the internal electrode, the problem of the stack actuator is solved using the unit cell model (two layer piezoelectric composite with |x|$$\le$$W_p/2 + h_e, |y|$$\le$$W_p/2 + h_e, 0$$\le$$z$$\le$$2h_p) shown in Fig. 5. Electrodes of length a and width W_p = 5.2 mm are attached to the PZT layer of thickness h_p = 0.1 mm, and a W_p - a tab region exists on both sides of the layer. Because of the geometric and loading symmetry, only the half needs to be analyzed. The electrode layers are not incorporated into the model.

Figure 4.

Schematic drawing of PZT stack actuator.

First, we consider the PZT stack actuator under DC electric fields. The electric potential on two electrode surfaces (-W_p/2$$\le$$x$$\le$$-W_p/2+a, 0$$\le$$y$$\le$$W_p/2, z = 0, 2h_p) equals the applied voltage, ϕ = V₀. The electrode surface (W_p/2-a$$\le$$x$$\le$$W_p/2, 0$$\le$$y$$\le$$W_p/2, z = h_p) is connected to the ground, so that ϕ = 0. The mechanical boundary conditions include the traction-free conditions on the coating layer surfaces at x =$$\pm$$(W_p/2 + h_e), y = W_p/2 + h_e and the symmetry conditions on the xz plane at y = 0 and xy planes at z = 0, 2h_p. In addition, the origin is constrained against the displacement in the x-direction, to avoid rigid body motion. We next consider the PZT stack actuator under AC electric fields. The electric potential on two electrode surfaces (-W_p/2$$\le$$x$$\le$$-W_p/2+a, 0$$\le$$y$$\le$$W_p/2, z = 0, 2h_p) equals the applied voltage, ϕ = V₀exp(iωt), and the electrode surface (W_p/2-a$$\le$$x$$\le$$W_p/2, 0$$\le$$y$$\le$$W_p/2, z = h_p) is connected to the ground.

Figure 5.

Unit cell of the PZT stack actuator.

Each element in ANSYS is defined by eight-node 3-D coupled field solid for the PZT layers and eight-node 3-D structural solid for the coating layer. P^s = 0.3 C/m² and γ^s = 0.004 are used. In order to calculate the electromechanical fields, we need the temperature dependent coercive electric field $${\overline{E}}_c^{}$$. First-principles free energy calculations for ferroelectric perovskites (Kumar and Waghmare 2010) have shown that the domain wall energy increases linearly about 50% as the temperature T decreases from room temperature to 260 K. Since higher domain wall energy leads to higher coercive electric field, the temperature dependent coercive electric field is assumed to be the following equation:

$${\overline{E}}_c^{}=(4.84-0.0129T)E_c$$E29

where E_c is a coercive electric field at 298 K.

3. Experiment

The stack actuator is fabricated using 300 soft PZT N-10 layers (NEC/Tokin Co. Ltd., Japan) of width W_p = 5.2 mm and thickness h_p = 0.1 mm (see Fig. 6). A rectangular Cartesian

Figure 6.

Experimental setup.

Elastic compliance					Piezoelectric coefficient			Dielectric constant		Density
(×10^-12 m^2/N)					(×10^-12 m/V)			(×10^-10 C/Vm)		(kg/m^3)
s11	s33	s44	s12	s13	d31	d33	d15	$${\in }_{11}^{\text{T}}$$	$${\in }_{33}^{\text{T}}$$	ρ
14.8	18.1	44.9	-5.1	-5.8	-287	635	930	443	481	8000

Table 1.

Material properties of N-10.

coordinate system O-xyz is used and the origin of the coordinate system coincides with the center of the actuator. The electrode length is a = W_p = 5.2 mm (full electrodes), and the actuator is coated with epoxy layer of thickness h_e = 0.5 mm. The total dimensions of the specimen are width of 6.2 mm and length of 40.5 mm. Table 1 lists the material properties of N-10. The coercive electric field of N-10 at room temperature is approximately E_c = 0.36 MV/m. Young's modulus, Poisson's ratio and mass density of epoxy layer are 3.35 GPa, 0.214 and 1100 kg/m³, respectively.

The actuator is bonded to the test rig of a SUS304 stainless steel plate using epoxy bond, and DC voltage (0 Hz) and AC voltage (400 Hz) are applied using a power supply. Two strain gages are attached at the center of the y =$$\pm$$3.1 mm planes, and the magnitude of strain is measured. To control the temperature of the actuator, an automated helium refill system (TRG-300, Taiyo Toyo Sanso Co. Ltd., Japan) is used. For the reliability of the test, two specimens are experimented, and four strain values are obtained.

4. Results and discussion

We first consider the PZT stack actuators under DC electric fields. Figure 7 shows the predicted normal strain ε_zz versus temperature T at x = 0 mm, y = W_p/2 + h_e = 3.1 mm and z = 0 mm of the PZT stack actuators for mole fraction X = 0.44 and 0.56 with a = W_p = 5.2 mm (full electrodes) under DC electric field E₀ = V₀/h_p = 0.1 MV/m (f = 0 Hz). Also shown are the measured data (average of four values) for X = 0.44. The electric field induced stain of the PZT stack actuator for X = 0.44 decreases with decreasing temperature due to a shift in the MPB. We see that the trend is sufficiently similar between the prediction and measurement. On the other hand, the electric field induced stain of the PZT stack actuator for X = 0.56 tends to increase with decreasing temperature reaching a peak at about T = 20 K and then decrease in magnitude. Figure 8 shows the predicted normal strain ε_zz versus DC electric field E₀ at x = 0 mm, y = 3.1 mm and z = 0 mm (strain measurement location) of the PZT stack actuators for X = 0.56 with a = 5.0 mm (partial electrodes) and a = W_p = 5.2 (full electrodes) at liquid hydrogen temperature (20 K). Strain versus electric field curves show nonlinear behavior due to the polarization switching under high negative DC electric fields. Small difference is observed in the DC electric field induced strains for a = 5.0 and 5.2 mm. Figure 9 shows the polarization switching zones near the electrode tip (x = 2.4 mm) at y = 0 mm plane of the PZT layer for X = 0.56 with a = 5.0 mm (partial electrodes) at 20 K. The coercive electric field at 20 K is about 1.65 MV/m. As the negative DC electric field increases, the area of the switched region grows.

We next consider the PZT stack actuators under AC electric fields. Figure 10 shows the normal strain ε_zz versus electric field amplitude E₀ at x = 0 mm, y = 3.1 mm and z = 0 mm of the PZT stack actuators for X = 0.44 and 0.56 at frequency f = 400 Hz and T = 20 K. The dashed line represents the strain computed by the FEA without domain wall motion effect, and the solid line represents the strain after the domain wall motion effect has been applied. The open circle denotes the experimental data. As AC electric field increases, the strain increases gradually away from the linear curve. This is due to the domain wall motion under the influence of AC electric fields. It can be seen that agreement between analysis with domain wall motion effect and experiment is fair. Figure 11 shows the distribution of the normal component of stress σ_zz as a function of x at y = 0 mm and z = 0.025 mm of the PZT stack actuators for X = 0.44 and 0.56 with a = 5.0 mm (partial electrodes) and a = W_p = 5.2 mm (full electrodes) under E₀ = 1.65 MV/m at f = 400 Hz and 20 K. The coercive electric field at 20 K is about 1.65 MV/m. In the case of the PZT stack actuator with full electrodes, small difference is observed for X = 0.44 and 0.56. For the PZT stack actuators with partial electrodes, a high normal stress occurs near the electrode tip (x = 2.4 mm) as is expected. Figure 12 shows the electric field distribution E_z as a function of x at y = 0 mm and z = 0.025 mm for the PZT stack actuators with a = 5.0 mm (partial electrodes) and a = 5.2 mm (full electrodes) under E₀ = 1.65 MV/m at f = 400 Hz and 20 K. For the PZT stack actuators with partial electrodes, a high electric field is observed near the electrode tip.

Figure 7.

Strain vs temperature of PZT stack actuators for X = 0.44 and 0.56 under DC electric field E₀ = 0.1 MV/m.

Figure 8.

Strain vs DC electric field of PZT stack actuators for X = 0.56 at 20 K.

Figure 9.

Polarization switching zones at y = 0 mm plane of PZT layer for X = 0.56 at 20 K, induced by DC electric field (a) E₀ = -1.5 MV/m, (b) E₀ = -1.64 MV/m and (c) E₀ = -1.72 MV/m.

Figure 10.

Strain vs AC electric field of PZT stack actuators for X = 0.44 and 0.56 at 20 K.

Figure 11.

Variation of normal stress σ_zz as a function of x at y = 0 mm and z = 0.025 mm for PZT stack actuators under AC electric field E₀ = 1.65 MV/m at T = 20 K.

Figure 12.

Variation of electric field E_z as a function of x at y = 0 mm and z = 0.025 mm for PZT stack actuators under E₀ = 1.65 MV/m at T = 20 K.

5. Conclusions

Numerical and experimental examination on the electromechanical response of PZT stack actuators at cryogenic temperatures is reported. It is found that the electric field induced strain decreases or increases with decreasing temperature depending on the mole fraction. That is, in the case of high performance PZTs for X = 0.44, the electric field induced strain is very high at room temperature, whereas in the case of PZTs for X = 0.56, the electric field induced strain at cryogenic temperatures will seem to be higher than at room temperature. It is also shown that the stress and electric field in the PZT layers are very high near the electrode tip for the PZT stack actuators with partial electrodes, although the electric field induced strains at the center of the surface for the partially and fully electroded PZT stack actuators have the same level. This study may be useful in designing high performance hydrogen fuel injectors.