## 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

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

(3) |

(4) |

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, (*s*_{11}, *s*_{12}, *s*_{13}, *s*_{33}, *s*_{44}, *s*_{66} = 2(*s*_{11}- *s*_{12})) are the elastic compliances, (

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

### 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 PbTiO_{3}. 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 PbTiO_{3} and PbZrO_{3} is given by (Bell & Furman 2003)

where *p*_{i} and *q*_{i} (*i* = 1,2,3) denote the polarizations of the PbTiO_{3} and PbZrO_{3}, respectively, and

(8) |

(9) |

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 PbTiO_{3} and PbZrO_{3}, respectively, and *γ*_{200} and *γ*_{220} are unknown coefficients.

The thermodynamic equilibrium state can be determined via minimization of Δ*G*_{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 Δ*G*_{PZT} are systematically obtained for the following phases:

Cubic (C)

Tetragonal (T)

Rhombohedral (R)

Monoclinic (M)

The energies of the minima are then compared to define the stable state. In the simulation, *γ*_{200} = 6^{8} and *γ*_{220} = 1.2^{8} are assumed.

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

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 *d*_{333} 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 *X* = 0.44, i.e.,

where *d*_{ijk} (*d*_{311}=*d*_{31}, *d*_{333}=*d*_{33}, *d*_{131}=*d*_{15}/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 *X* = 0.56 can be expressed as

### 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)

where *ε*_{ij} and Δ*P*_{i} are the changes in the spontaneous strain and polarization during switching, respectively. The values of Δ*ε*_{ij} = *P*_{i} =^{o} switching can be expressed as

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

where ^{o} switching in the *yz* plane, we have

The constitutive equations after polarization switching are

(21) |

(22) |

where

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}=*E*_{0}exp(i*ωt*) is parallel to the direction of spontaneous polarization *P*^{s} in one of the domains (see Fig. 3); *E*_{0} is the AC electric field amplitude and *ω* is angular frequency(=2π*f* where *f* is frequency in Hertz). The changes of the strains and polarization due to the domain wall displacement Δ*l* (Arlt and Dederichs, 1980) can be written as

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

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

(26) |

where

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 *E*_{0} and temperature dependent coercive electric field

By substituting Eq. (28) into the sixth of Eq. (25), *lf*_{D} = 3(*P*^{s})^{2}*E*_{0}) 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*|*W*_{p}/2 + *h*_{e}, |*y*|*W*_{p}/2 + *h*_{e}, 0*z**h*_{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.

First, we consider the PZT stack actuator under DC electric fields. The electric potential on two electrode surfaces (-*W*_{p}/2*W*_{p}/2+*a*, 0*y**W*_{p}/2, *z* = 0, 2*h*_{p}) equals the applied voltage, *ϕ* = *V*_{0}. The electrode surface (*W*_{p}/2-*a**x**W*_{p}/2, 0*y**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* =*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, 2*h*_{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*W*_{p}/2+*a*, 0*y**W*_{p}/2, *z* = 0, 2*h*_{p}) equals the applied voltage, *ϕ* = *V*_{0}exp(*iωt*), and the electrode surface (*W*_{p}/2-*a**x**W*_{p}/2, 0*y**W*_{p}/2, *z* = *h*_{p}) is connected to the ground.

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/m^{2} and *γ*^{s} = 0.004 are used. In order to calculate the electromechanical fields, we need the temperature dependent coercive electric field *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:

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

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/m^{3}, 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* =

## 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 *E*_{0} = *V*_{0}/*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 *E*_{0} 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 *E*_{0} 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 *E*_{0} = 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 *E*_{0} = 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.

## 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.