Material properties of N-10.
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.
2.1. Basic equations
Consider the orthogonal coordinate system with axes
The electric field components are related to the electric potential
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
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)
In Eqs. (8) - (10),
The thermodynamic equilibrium state can be determined via minimization of Δ
The energies of the minima are then compared to define the stable state. In the simulation,
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
2.3. Polarization switching
High electromechanical fields lead to the polarization switching. We assume that the direction of a spontaneous polarization
where is a temperature dependent coercive electric field, and Δ
For 90o switching in the
where is a spontaneous strain. For 90o switching in the
The constitutive equations after polarization switching are
In Eq. (23),
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
where all terms with Δ are the contributions from the domain wall motion, and
In Eq. (25),
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
By substituting Eq. (28) into the sixth of Eq. (25),
2.5. Finite element model
Consider a PZT stack actuator with 300 PZT layers of width
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 |
First, we consider the PZT stack actuator under DC electric fields. The electric potential on two electrode surfaces (-
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.
The stack actuator is fabricated using 300 soft PZT N-10 layers (NEC/Tokin Co. Ltd., Japan) of width
|Elastic compliance||Piezoelectric coefficient||Dielectric constant||Density|
|(×10-12 m2/N)||(×10-12 m/V)||(×10-10 ||(kg/m3)|
coordinate system O-
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
4. Results and discussion
We first consider the PZT stack actuators under DC electric fields. Figure 7 shows the predicted normal strain
We next consider the PZT stack actuators under AC electric fields. Figure 10 shows the normal strain
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.