Design of PZT–Pt functionally graded piezoelectric material for low-frequency actuation applications

Abstract

In this article, simulation for PbZr_0.516Ti_0.484 (lead zirconate titanate)–Pt functionally graded piezoelectric material bimorph actuator is carried out. It consists of seven layers of lead zirconate titanate–Pt with a symmetric composition profile from the center (0% Pt, 10% Pt, 20% Pt, 30% Pt, 20% Pt, 10% Pt, and 0% Pt%). The structural parameters which include length, width, and thicknesses of piezoelectric layers and substrate layer (30% Pt) are optimized based on their first resonance frequency. The bimorph actuator based on optimized parameters shows the first resonance frequency at 960 Hz as compared to resonance frequency (7761 Hz) of original design of functionally graded piezoelectric materials. Large actuation is observed in functionally graded piezoelectric materials of proposed design. The proposed functionally graded piezoelectric material actuator can be effectively used at low frequency (960 Hz) with high tip displacement (35 µm) under 500 V. This work is helpful in fabricating functionally graded piezoelectric materials for specific applications.

Keywords

Functionally graded piezoelectric material actuator piezoelectric

Introduction

Piezoelectric actuators have been used in a wide range of commercial devices. These devices have displacement from 10 pm to 100 µm and are more commonly used in the form of diaphragms and cantilevers (Amirouche et al., 2009; Cagatay et al., 2003; Guo et al., 2008; Laser and Santiago, 2004; Maeda et al., 2004). Piezoelectric actuators can have unimorph (single piezoelectric layer) and bimorph (double layers of piezoelectric material) structures (Anton and Sodano, 2007). These actuators are mainly used in inkjet printers, control valve, micropumps, and ultrasonic motors (Amirouche et al., 2009; Cagatay et al., 2003; Laser and Santiago, 2004). Particularly, cantilever-type actuators have large displacement and have versatile applications such as accelerometer, atomic force microscopy, and biomedical applications (Guo et al., 2008; Khoshnoud and De Silva, 2012; Maeda et al., 2004). In all these applications, mechanical resonance frequency is a key parameter as these devices work efficiently near resonance frequency of the structure/device. An unimorph/bimorph cantilever consists of a metallic layer stick to the piezoelectric materials. Adhesive epoxy resin is usually used to bond the piezoelectric materials to metallic structure (Anton and Sodano, 2007; Reilly and Wright, 2009). These devices have low reliability because of multiple failure possibilities associated with adhesive in the form of creep, crack, and peel off. This is a major drawback of commercially available actuators.

In order to overcome the above-mentioned drawback, functionally graded piezoelectric materials (FGPMs) have been proposed for actuators (Maleki et al., 2012). Functionally graded materials are the materials in which the compositions gradually change (along with the thickness), resulting in a corresponding change in the properties of the material (Zhu et al., 2001). Such FGPMs can be tailored to meet numerous properties in a single material. It is a novel interfacial technique to solve the problems associated with the sharp interface between two dissimilar materials (Zhu et al., 2001). A number of researchers have developed FGPMs with spatially varying properties for a specific application (Hauke et al., 2000; Lim and He, 2001; Liu et al., 2007; Maleki et al., 2012). Lead-based piezoelectric materials are extensively investigated in the form of FGPM (Maleki et al., 2012; Zhang et al., 2008). This is due to the fact that these materials possess a large piezoelectric coefficient and electromechanical coupling.

Zhang et al. (2006) fabricated PbZr_0.516Ti_0.484 (lead zirconate titanate (PZT))-Ag-based FGPM structure. This FGPM beam was 12 × 2 × 1 mm³ dimension and consists of three layers of PZT having 0, 5, and 15 vol.% Ag with equal thickness of 0.33 mm. The authors have thoroughly investigated electrical properties of PZT–Ag FGPM and reported their piezoelectric constant, electromechanical coupling factor, and polarization–electric field (P–E) loop (Zhang et al., 2006). Another article by the same research group reports mechanical properties of PZT–Ag FGPM beam (Zhang et al., 2008). Further, Rubio et al. (2012) worked on PZT–Ni FGPM and explored its dynamic behavior experimentally. The ceramic is graded from the top to the bottom surface (along 6 mm of thickness) and consists of five layers of PZT–5A having 0, 10, 20, 30, and 40 wt% Ni (Rubio et al., 2012). Qui et al. (2003) synthesized FGPM actuator of the Pb(Ni_1/3Nb_2/3) O₃–PbZrO₃–PbTiO₃ (PNN–PZ–PT). The authors have evaluated the performance of FGPM actuators and compared with traditional bimorph actuators using a vibration test. Li et al. (2003) has prepared the PZT/ZnO/PZT FGPM bimorph composite to reduce resistivity. However, Hauke et al. (2000) has done experiment on BaTiO₃ FGPM ceramics and demonstrated significant reduction in internal stresses. Furthermore, [Li_0.06(K_0.05Na_0.05)_0.94]NbO_3 (LKNN-Ni) and porous PZT-based FGPMs were also fabricated and reported for their performance (Li et al., 2003; Zhang et al., 2011). It is clear from all the above-mentioned studies that the number of FGPMs are fabricated and reported for their various physical properties. However, as mentioned above, the piezoelectric actuators work efficiently near resonance frequency which depends upon length, width, and thickness of the structure. Specially, in the case of FGPM, thickness is the combination of individual thickness of substrate layer and different compositions of piezoelectric layers. This study aims to optimize structural parameters in order to tune the resonance frequency of the devices. We have made an attempt to optimize structural parameters of FGPM cantilever in order to decrease the first resonance frequency. It is to be noted that low-frequency actuations are desirable for a number of devices.

In this work, simulation for PbZr_0.516Ti_0.484 (PZT)–Pt FGPM bimorph actuator is carried out. It consists of seven layers of PZT–Pt with a symmetric composition profile from the center. The structural parameters such as length, width, and thicknesses of piezoelectric layers and substrate layer are optimized based on their first resonance frequency. The tip displacements for all the proposed design (PD₁) are determined under various load and voltage conditions. Furthermore, comparison (in terms of performance) is performed between FGPM and conventional (uniform concentration) PZT bimorph structures.

Materials and modeling

The PZT–Pt based FGPM bimorph beam is fabricated by Takagi et al. (2003) using the powder metallurgy process. These powders (PZT and Pt) are mixed in a mortar according to their stoichiometric ratio of different FGPM layers. Furthermore, powder was compacted using die pressing (100 MPa) to make a green pellet, and it was sintered at 1200°C for 1 h in an alumina crucible containing an excess PbO atmosphere. Figure 1 shows schematic representation of the FGPM actuator that consists of seven layers of PZT–Pt compositions which was fabricated by Takagi et al. (2003). The middle layer of cantilever is made of PZT–30% Pt and have three graded layers (20%, 10%, and 0% Pt) on both sides of the middle layer (PZT–30% Pt). In this design, the central layer is a substrate layer (conducting material), and other layers (PZT–0% Pt, PZT–10% Pt, and PZT–20% Pt) are piezoelectric layers. Structural parameters of original design (OD₁) such as length (l), thickness of piezoelectric layers (t₁), thickness of substrate layer (t₂), and width (w) are listed in Table 1 as reported by Takagi et al. (2003). In this table PZT-0% parameters are differes as reported by Takagi et al. (2003) to validate the model. The elastic compliances (s^E), piezoelectric constants (d₃₃ and d₃₁), and dielectric constant (ε^E) are also mentioned in Table 1. Other relevant material properties (i.e. Young’s modulus (Y) and density (ρ)) of PZT–Pt composite layers are determined by using the rule of mixture methods and can be expressed as (Kou and Tan, 2007)

Y_{pzt - pt} = Y_{pzt} α_{pzt} + Y_{pt} α_{pt}

(1)

ρ_{pzt - pt} = ρ_{pzt} α_{pzt} + ρ_{pt} α_{pt}

(2)

where α denotes the volume fraction of layer. The Poisson’s ratio (υ) is estimated using following equation

G_{pzt - pt} = \frac{Y_{pzt - pt}}{2 (1 + v_{pzt - pt})}

(3)

where G is the shear modulus of the PZT–Pt layer. It is also determined by equation (1) by replacing Young’s modulus as shear modulus.

Figure 1.

Schematic design of FGPM cantilever actuator.

Table 1.

Functionally graded piezoelectric material properties and structural parameters (Takagi et al., 2003).

Description	PZT–0% Pt	PZT–10% Pt	PZT–20% Pt	PZT–30% Pt
Young’s modulus (GPa)	63	73.9	84.8	95.7
Poisson’s ratio	0.3	0.324	0.342	0.357
Density (kg/m³)	7500	8895	10,290	11,685
s ₁₁ (10⁻¹² m²/N)	16.5	6.91	5.53	Conductive
s ₁₂ (10⁻¹² m²/N)	−4.78	−5.32	−5.53	Conductive
s ₁₃ (10⁻¹² m²/N)	−8.45	−14.79	−12.98	Conductive
s ₃₃ (10⁻¹² m²/N)	20.7	16.38	14.47	Conductive
s ₆₆ (10⁻¹² m²/N)	43.5	39.05	35.74	Conductive
d ₃₁ (10⁻¹⁰ C/m²)	−2.74	−1.43	−0.98	Conductive
d ₃₃ (10⁻¹⁰ C/m²)	5.93	3.25	2.56	Conductive
ε ₃₃/ε₀	3400	1959	2927	Conductive
Original design (OD₁)
Length of each graded layer, l (µm)	12,000	12,000	12,000	12,000
Thickness of each graded layer (µm)	300 (t₁)	300 (t₁)	300 (t₁)	200 (t₂)
Width of each graded layer, w (µm)	3000	3000	3000	3000
Proposed design (PD₁)
Length of each graded layer, l (µm)	18,000	18,000	18,000	18,000
Thickness of each graded layer (µm)	300 (t₁)	300 (t₁)	300 (t₁)	50 (t₂)
Width of each graded layer, w (µm)	2000	2000	2000	2000

PZT: lead zirconate titanate; OD₁: original design; PD₁: proposed design.

In the simulation process, original design (OD₁) is constructed in COMSOL Multiphysics as shown in Figure 1 and assigned physical properties to each layer. Mesh is generated by a normal tetrahedral element tool. A computer work station with Xeon processor with 16 GB RAM is used for this study. The structural problem is studied under the well-established plane strain assumptions (Nimbal and Halse, 2012). Piezoelectric actuation is governed by the following equations

S = s^{E} σ + dE

(4)

D = ε^{E} E + d σ

(5)

where S, σ, D, and E are mechanical strain vector, mechanical stress vector (N/m²), electric displacement vector (C/m²), and electric field vector (V/m), respectively. For the central substrate layer, mechanical behavior is considered using the stress–strain (σ–ε) relationship as

σ = Y ε

(6)

The tip displacement (harmonic excitation) is simulated by applying vertical acceleration through body load (F), and it can be given as

F = ρ aV

(7)

where ρ, a, and V are density, acceleration constant, and volume of cantilever, respectively. The relation of cantilever length and thickness with the resonance frequency (f) can be expressed as (Prakash et al., 2012)

f = \frac{γ^{2}}{2 π l^{2}} \sqrt{\frac{D_{p}}{m}}

(8)

m = t_{p} ρ_{p} + t_{s} ρ_{s}

(9)

where γ, D_p, and m are the first-order resonance constant, bending modulus, and mass per unit area, respectively. The m is calculated using the thicknesses (t_p, t_s) and densities (ρ_s, ρ_p) as mentioned in equation (9). The subscripts for p and s denote piezoelectric and substrate layers, respectively.

To investigate the performance of FGPM, mechanical and electrical boundary conditions are also shown in Figure 1 for the frequency optimization and mechanical load–displacement study. Mechanical boundary conditions are considered for cantilever-type design, where the left side is fixed while all the other faces are free for displacement or bending (Figure 1). The electrical behavior is investigated by applying upper and lowest layer faces as the floating potential condition while the middle layer is grounded (Figure 1). On the other hand, electrical potential is applied in d₃₁ mode for piezoelectric actuation characterization. Here, side faces of piezoelectric and middle layers are acted as an electrode and grounded, respectively. To investigate piezoelectric actuation performance, only the bottom edge of left side of cantilever is fixed. These boundary conditions are considered to validate the results with the actuator proposed by Takagi et al. (2003). The OD₁ is validated by applying the voltage potential (100 V), and the corresponding tip displacement is found to be 2.17 µm. However, for the same structure, the tip displacement is 2 µm under 100 V as reported by Takagi et al. (2003), which validates our formulations.

Results and discussion

Structural parameters of FGPM cantilever are optimized for their first resonance frequency. In order to know the resonance frequency of OD₁, a frequency-domain study is conducted. The first resonance frequency is computed as 7761 Hz. It indicates that OD₁ FGPM can be efficiently worked near this first resonance frequency (7761 Hz). However, their performance decreases drastically at other frequencies. Original FGPM (OD₁) was fabricated with l = 12,000 µm, w = 3000 µm, t₁ = 300 µm, and t₂ = 200 µm. In order to tune resonance frequency, the above-mentioned parameters need to be optimized. In this context, frequency-based optimization of FGPM cantilever is carried out by varying length, width, and thickness of each layer. In the first step, cantilever length and width were varied, and the first resonance frequency was simulated as shown in Figure 2. In this process, piezoelectric and substrate layers thicknesses are kept constant as 300 and 200 µm, respectively. The relation between frequency and length is mentioned in equation (8). It indicates that as the length of cantilever increases, the resonance frequency decreases, which is also clear from Figure 2. However, there is no clear trend observed between width and frequency. Three lowest frequency configurations are selected with lengths of 10,000, 15,000, and 18,000 µm with the corresponding widths of 3000, 4000, and 2000 µm, respectively (Figure 2). These configurations are selected for further optimization of piezoelectric and substrate layer thicknesses of FGPM.

Figure 2.

Plots of the first resonance frequency versus length for various cantilever width (lines are only for guiding eyes).

In order to further optimization, the thickness of piezoelectric layer is varied from 50 to 600 µm as shown in Figure 3 (lines are only for guiding eyes in Figure 3), while the thickness of substrate layer is kept constant (t₂ = 200 µm). Data of resonance frequency are scattered; however, an increasing trend is observed in the length scale under study (Figure 3). It indicates that as the thickness of piezoelectric layer increases, the resonance frequency also increases. It is clear from Figure 3 that the thickness (300 µm) of piezoelectric layers gives lowest frequency for all the width under study. Hence, the thicknesses of piezoelectric layer (all six layers) are fixed as 300 µm for the rest of the simulation. The thickness effect of substrate layer (PZT–30% Pt) is also simulated which is shown in Figure 4. The substrate thickness is varied from 50 to 500 µm at t₁ = 300 µm. Lowest resonance frequency 960 Hz is achieved at a substrate thickness of 50 µm (Figure 4). Therefore, proposed design (PD₁) of FGPM contains parameters such as l = 18,000 µm, w = 2000 µm, t₁= 300 µm, and t₂ = 50 µm as listed in Table 1. FGPM with the optimized parameters can efficiently work at 960 Hz, which is significantly lower than resonance frequency (7761 Hz) of original design (Takagi et al., 2003). It is important to note that volumes of the both design (OD₁ and PD₁) are almost the same; however, their resonance frequencies are of different order. In this study, a number of trials were carried out for the frequency optimization, and the results are within the ±5% limit.

Figure 3.

First resonance frequency versus thickness of piezoelectric layers (lines are only for guiding eyes).

Figure 4.

Resonance frequency versus thickness of substrate layers (lines are only for guiding eyes).

In order to further study the performance of original design (OD₁ and proposed design (PD₁) of FGPM, we have compared their resonance frequencies with piezoelectric bimorph of PZT–30% Pt layer as substrate laminated with PZT layers of equal thickness. For this analysis, piezoelectric layers (PZT–0% Pt, PZT–10% Pt, and PZT–20% Pt) of OD₁ and PD₁ are considered as made of pure PZT only. These new designs are denoted by OD₂ (corresponding to OD₁) and PD₂ (corresponding to PD₁). However, the thickness of substrate (conducting) layer is kept the same as considered in OD₁ and PD₁. Six resonance frequencies of these four designs are shown in Figure 5. The first resonance frequency of OD₁(FGPM), OD₂, PD₁(FGPM), and PD₂ are 7761, 6670, 960, and 2800 Hz, respectively. It is clear that the resonance frequencies of OD₁ and OD₂ are almost of the same order. It interprets that FGPM (OD₁) and OD₂ have no significant difference in the first resonance frequency. However, the resonance frequency of PD₁ is three times less than that of PD₂. Therefore, FGPM cantilever (PD₁) can be performed better for low-frequency application as compared to simple PZT cantilever (PD₂).

Figure 5.

The effect of FGPM and design on the resonance frequency (lines are only for guiding eyes).

For further investigation, the displacement (harmonic excitation) performance of these cantilever designs based on OD₁, OD₂, PD₁, and PD₂ are studied. For this purpose, we have applied boundary load of 0.12mg, 0.3mg and 0.6mg on the free end of cantilever, and their tip displacements are shown in Figure 6. These displacements are compared near the first resonance frequencies of respective designs (OD₁, OD₂, PD₁, and PD₂). The maximum tip displacement 9.6 µm occurs in the case of PD₁ (Figure 6(a)) while the minimum displacement of 25 × 10⁻³µm obtained in the case of OD₂ (Figure 6(d)). PD₁ shows almost 19 times and 40 times large displacement than that of the PD₂ and OD₁, respectively, at 0.6mg load. Therefore, proposed design (PD₁) has highest tip displacement with low operating frequency.

Figure 6.

Comparison of tip displacement under 0.12, 0.3 and 0.6 mg load for (a) FGPM OD₁, (b) FGPM PD₁, (c) conventional bimorph OD₂, and (d) conventional bimorph PD₂.

The electrically induced displacement of the all configurations is estimated under various voltages (100–500 V). Figure 7 shows the tip displacement of cantilever under 100–500 V voltage potential for OD₁, OD₂, PD₁, and PD₂. The electric potential is applied in the d₃₁ mode. The displacement is proportional to an applied voltage; therefore, as the voltage increases, displacement also increases. It is observed that PD₁ has maximum tip displacement while OD₂ has minimum tip displacement (Figure 7). PD₁ and OD₁ have large displacement as compared to PD₂ and OD₂. It shows that FGPM (PD₁ and OD₁) can perform better than the traditional bimorph actuators (PD₂ and OD₂), which are made of PZT layer without any graded composition. In summary, FGPM actuators can be effectively used with better performance as compared to conventional bimorph. Moreover, their structure parameters can be optimized for specific applications.

Figure 7.

Tip deflection of bimorph actuator under various electric potentials.

Conclusion

FGPM bimorph of PZT–Pt composites is optimized for their structural parameters. The FGPM’s parameters length (l), width (w), thickness of piezoelectric layers (t₁), and thickness of substrate layer (t₂) are optimized based on their first resonance frequency. The proposed design (PD₁) shows seven times less resonance frequency as compared to original design (OD₁), which is favorable in low-frequency applications. It also shows 40 times large tip displacement than that of OD₁ under similar mechanical load. The proposed design of the FGPM actuator can be effectively used at low frequency (960 Hz) with high tip displacement (35 µm) under 500 V. This study is one of the first attempts to optimize structural parameters of FGPMs, which includes thickness of each graded layers. Such studies are helpful in fabricating FGPM for specific applications.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

One of the authors (Rahul Vaish) received funding from the Indian National Science Academy (INSA), New Delhi, India, through a grant by the Department of Science and Technology (DST), New Delhi, under INSPIRE faculty award—2011 (ENG-01) and INSA Young Scientists Medal—2013.

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