The ability to convert electrical energy into mechanical energy and vice versa makes piezoelectric ceramics widely used in many transduction applications. Among the most important applications are actuation devices that utilize the inverse piezoelectric effect, e.g. for common rail fuel injec-tion [1], precision positioning systems, piezohydraulic pumps [2], ultrasonic motors [3], active flow control [4], and others [5]. The most notable features of piezoactuators, as compared to other actuation techniques, are the capability of producing large forces, high accuracy, fast response time, easy control,
absence of magnetic fields or moving parts, and low power consumption. These properties, for example, allow piezoelec-tric actuators to reduce CO2 emissions and diesel consump-tion when used in common rail diesel fuel injection systems [6] or decrease the size of the small scale motors down to the millimetre length scale and below [3, 7].
The important actuator characteristics are the maximum displacement that can be achieved at a given electric field without an external mechanical load (free displacement; δf) and the maximal force that the actuator produces in a fully clamped state (blocking force; Fb). These two parameters are commonly used to define the actuator’s operational range
Journal of Physics D: Applied Physics
Enhancing the operational range of piezoelectric actuators by uniaxial compressive preloading
Jurij Koruza, Daniel J Franzbach, Florian Schader, Virginia Rojas and Kyle G Webber
Department of Materials Science, Technische Universität Darmstadt, 64287, Darmstadt, Germany
Received 20 November 2014, revised 20 March 2015Accepted for publication 27 March 2015Published 21 April 2015
AbstractThe influence of the uniaxial preload on the off-resonance actuation performance of piezoelectric ceramics was investigated for compressive preload values up to −80 MPa. The study was performed on soft-type lead zirconate titanate (PZT), being the most widely used piezoelectric material. The samples were analysed using the proportional loading method, which enables the simultaneous application of electrical and mechanical loads, thereby mimicking the real operation conditions over the full stress–strain range. An increase of the blocking stress and the longitudinal piezoelectric stress coefficient was observed for all the applied preload values. The optimum properties, a blocking stress of −56 MPa and a free strain of 0.23%, were obtained at a preload value of −40 MPa and electric field of 2 kV mm − 1. This represents an increase of 16% and 20%, respectively, as compared to the values obtained at the smallest preload. In addition, the maximum work output was increased by about 28%. Finally, the results obtained at the lowest preload of −4 MPa using the proportional loading method were compared to the operational ranges determined by other methods. The comparison revealed large discrepancies between the methods, originating from the different order of the application of electrical and mechanical fields and the inherent nonlinearity of ferroelectric materials. This discrepancy results in decreased actuator performance due to impedance mismatching, demonstrating the need for accurate determination of the actuator’s operational range.
[8–10] and, therefore, their accurate determination is of great importance for optimal device design and efficient perfor-mance. Actuators perform maximum work when impedance matched to the external linear spring load [2]. This matching, however, strongly depends on the shape of the operational range and, therefore, any deviations from linearity result in variations in the work output. Among the commonly used approaches for the determination of the operational range are the elastic method, the direct blocking force method, and the indirect blocking force method [11, 12]. While all these methods estimate the operational range from the two meas-ured or extrapolated extreme values (δf and Fb), the main difference is the order of the application of electrical and mechanical fields. For the case of linear actuator materials, the operational range determined by all the above mentioned methods will be the same. However, this was found not to be the case in ferroelectric actuator materials that display path dependence of the measured blocking force and subse-quently discrepancies in the determined characteristic values [11]. The observed differences were found to originate from the hysteretic electrical and mechanical constitutive behaviour of the ferroelectric material due to the dynamics of the ferro-electric and ferroelastic domains [13]. This problem seems to be overcome by the recently proposed measurement technique utilizing the proportional loading method [11, 14]. During measurement the electrical and mechanical loads are applied simultaneously and a linear displacement–force relationship is ensured by an external computer-controlled virtual spring. In addition to the direct determination of both extreme values (δf and Fb), this technique enables accurate determination of the actuator behaviour for any given loading regime between these extremes, thereby mimicking the realistic loading conditions.
From the material point of view, the global market for pie-zoactuators, which is expected to exceed the size of 12 bil-lion US-$ in 2014, is currently dominated by lead zirconate titanate (Pb(Zr,Ti)O3; PZT) based materials [15]. It should be noted that soft-type piezoelectrics are typically preferable for off-resonance actuation applications because of their large piezoelectric coefficients [5]. This is obtained by donor doping with Bi3+ or La3+ onto Pb2+ sites or Nb5+, Ta5+, or Sb5+ onto the Zr4+ and Ti4+ sites, resulting in A-site vacancies, which facilitate domain wall motion, resulting in lower coercive stress and coercive electric field values, as compared to hard-type piezoelectrics [16]. Besides their excellent piezoelectric properties, the critical feature, important for the applications of these ceramics, is their low fracture toughness (~1 MPa m−1/2) [17]. This means that these piezoceramics have to be protected against undesirable tensile and shear forces, which can easily induce crack formation [8, 18]. Despite attempts to increase the fracture toughness, e.g. by inclusion of whiskers [19] or other second-phase particles [20, 21], the fracture toughness remains relatively low, requiring other design-based approaches to overcome this material deficiency. Among them is preloading with uniaxial compressive stress to avoid the development of tensile stresses upon the application of external electric fields during operation. Furthermore, the preloaded actuators are able to operate in a push-pull mode, to some degree. As a rule of thumb, the uniaxial preload value is often chosen to be about
20% of the compressive load limit or up to 50% of the blocking force value, but the exact values can vary depending on the intended operation conditions of the actuator [8]. The shear forces can be prevented by the use of external supports, such as flexure guides.
Besides protection against undesirable stresses, the appli-cation of preloads was found to have a positive influence on piezoactuator performance. Lynch was among the first to inves-tigate the effect of the uniaxial stress on the electromechanical properties of lanthanum-doped PZT, demonstrating consider-able hysteresis and nonlinearities due to ferroelastic and ferro-electric domain switching [22]. A more detailed explanation of the phenomena was given by Chaplya and Carman, who inves-tigated the polarization and strain response of another com-mercially available soft PZT composition [23]. Interestingly, maximum values of dielectric and piezoelectric unipolar responses were not achieved in a stress-free state, but at a uniaxial compressive preload of −50 to −60 MPa, dependant on composition and maximum electric field, amongst others. Observed enhanced actuation performance was attributed to the increased extrinsic contributions from non-180° domain wall motion. The application of a compressive preload increases the amount of domains perpendicular to the applied uniaxial stress [24] and thus more non-180° domains are available for switching during the application of the electric field, resulting in increased unipolar strain and polarization values. However, upon increasing the uniaxial preload above the peak stress value the non-180° domain switching becomes restricted, reducing the obtainable strain [25]. Furthermore, this peak was found to shift to higher stress values upon increasing the amplitude of the electric field, demonstrating the dependence on the balance between the electrical and mechanical fields. Similar observations were reported by others investigating soft-type PZT ceramics [25–27], while the peak preload value seems to be even higher in hard-type PZT ceramics [28].
As evident from previous studies, large signal electrical and mechanical loads significantly influence polarization and strain response of piezoceramic materials and, therefore, this behaviour needs to be accurately determined under actual application conditions. The aim of this study is to investigate the effect of uniaxial compressive preloads on the force gen-eration and the operational range of soft-type PZT ceramics. Besides being a representative ferroelectric material and the frequent choice for actuation applications, this material was selected since it represents a benchmark for the evaluation of performance of the newly emerging lead-free materials [29–32]. The force–displacement behaviour was studied using the proportional loading method, which enables the simulation of external loads with any selected spring constant revealing the full operational range of the material.
2. Experimental work
All measurements were performed on commercially available PIC151 ceramics (PI Ceramic, Lederhose, Germany), which is a soft-type Pb(Zr,Ti)O3 piezoelectric. The softening effect is achieved by adding Pb(Ni1/3Sb2/3)O3, whereby the Sb5+ acts
as a donor. The composition is adjusted to be in the vicinity of the morphotropic phase boundary, slightly on the tetragonal side. The samples were drilled from large blocks to obtain cylinders with a diameter of 5.8 mm and a height of approxi-mately 6 mm, ground on both circular faces and annealed at 600 °C for 1 h to depole the material and relax internal stresses, possibly induced during the preparation procedure. Finally the circular faces were sputter-coated with silver (SCD 050, Bal-Tec, Germany).
The measurements were performed using a custom designed experimental arrangement that is described in detail in [11]. The sample was mounted in a screw-type loading frame (Z030, Zwick, Ulm, Germany), used to apply uniaxial compressive preloads. The electric field was applied by a high-voltage amplifier (20/20C, TREK Inc., Medina, NY, USA). The force was measured by a 30 kN load cell with an accuracy of ±0.5% for forces >30 N, while the displacement was detected by a custom built linear variable differential transformer (LVDT) with a resolution of ±20 nm. All signals were recorded and evaluated by a LabVIEW computer program with an inte-grated proportional-integral-derivative (PID) feedback loop that controls the large stack actuator (PI Ceramic, Lederhose, Germany) positioned in the loading axis above the sample. The stack actuator acts as a virtual spring with a predefined spring constant, which can be adjusted by the driving voltage to simu-late different external linear spring loads.
The measurement procedure was as follows (figure 1): the sample was initially preloaded with a uniaxial compressive preload, poled by applying two sinusoidal unipolar cycles with the amplitude of 2 kV mm−1 and a frequency of 10 mHz, and subsequently driven by the same sinusoidal signal. The spring constant of the virtual spring was adjusted between k = 0 and k ≈ ∞ , whereby 14 spring constant values between these two extremes were used to assess the full operational range. Note that the poling step was repeated prior to each measurement in order to compensate for possible mechanical depoling during the previous steps.
The stress dependent small signal d33 coefficient was measured by sinusoidally unloading the sample with an amplitude of 0.49 ± 0.01 MPa and a frequency of 10 mHz. For that purpose, the stack actuator was used to apply the small mechanical signal, controlled by a specially designed LabVIEW-program, while the global stress on the sample was applied by the load frame. The resulting sinusoidal change in polarization was measured with a Saywer-Tower electrical circuit and all signals were acquired, filtered and evaluated by LabVIEW. From the determined stress and polarization ampli-tudes the piezoelectric coefficient d33 could be calculated. For
this measurement, the sample was annealed at 400 °C for 1 h and cooled down with a maximum rate of 1 K min−1. After annealing, the sample was poled in silicone oil at 120 °C for 5 min at 2 kV mm−1 and cooled down to room temperature with the electric field still applied. Before testing, a minimum waiting time of 24 h after poling was used. The complete measurement setup will be described in detail in a subsequent publication.
3. Results and discussion
The blocking stress of soft PZT was measured at different values of the uniaxial compressive preload in the range of −4 to −80 MPa using the proportional loading method and elec-tric fields up to 2 kV mm−1. For these measurements the sample displacement in the direction of the uniaxial stress was fully clamped, simulating the conditions of operating against an infinitely stiff external load (spring constant k ≈ ∞). The blocking stresses obtained for different electric fields are depicted in figure 1. At the lowest preload value of −4 MPa and an electric field of 2 kV mm − 1 the sample developed a blocking stress of −47 MPa, which is lower compared to the values measured by the elastic (−68 MPa) or the direct blocking stress method (−51 MPa) and higher than those reported by the indirect blocking stress method (−36 MPa). Despite the possibility of slight batch-to-batch sample variations, the main reason for this discrepancy lies in the path-dependence of the blocking stress [11], resulting from the different order of the application of the electrical and mechanical fields and the inherent nonlinearity of the ferroelectric materials. As the proportional loading method applies both loads simulta-neously, the obtained blocking stress value is believed to be closer to the values developed by the material under real appli-cation conditions. Upon further increase of the compressive preload the blocking stress increased up to −40 MPa, where a maximum blocking stress of −56 MPa was reached, and sub-sequently started to decrease. Note that the blocking stress at 2 kV mm−1 peaks at a similar uniaxial stress value as other piezoelectric and ferroelectric properties, such as the small (d33) and large signal (d*33) piezoelectric coefficients [23, 25, 27, 33]. Application of a uniaxial compressive stress results in ferroelastic switching of non-180° domains in the direc-tion perpendicular to the direction of the applied mechanical field, thus increasing the amount of domains available for fer-roelectric switching during the application of the electric field and resulting in an increase in the electromechanical proper-ties. However, at a certain preload value the external electric field cannot overcome the applied mechanical energy and the blocking stress values start to decrease. Interestingly, the blocking stress obtained at the highest measured preload value of −80 MPa was −52 MPa, which is higher than the value obtained at the smallest measured preload value of −4 MPa. This difference will be further discussed later. In addition, it should be noted that an electric field dependence of the peak blocking stress value was observed (dashed circles in figure 2), as previously reported for piezoelectric properties [25]. The electrical energy supplied at electric fields below 2 kV mm−1
Figure 1. The measurement procedure for the determination of the full operational range of actuators by the proportional loading method.
is namely only enough to switch a comparably smaller amount of non-180° domains, which were previously ferroelastically switched by the mechanical energy, thus developing smaller blocking stresses.
The coupling between elastic and dielectric parameters in piezoelectric materials can be described using the thermo-dynamic approach, whereby the choice of appropriate set of independent variables (depending on the elastic and electric boundary conditions) leads to the constitutive piezoelectric equations [34]. For the loading conditions used during the blocking stress measurement the main equation for the stress tensor σij can be formulated as follows:
σ = −c S e E ,ij ijklE
kl kij k (1)
whereby cijklE is the stiffness tensor at a fixed electric field, Skl
is the elastic strain tensor, ekij is the piezoelectric stress tensor, and Ek is the electric field vector. Note that this is valid for isothermal conditions and that the higher order terms were neglected. During the blocking stress measurement the sample is assumed to be fully clamped and the total strain Skl is zero, therefore the equation (1) is reduced to:
σ = −e E .ij kij k (2)
The piezoelectric stress coefficient ekij can be used as a figure of merit for material’s ability to develop (blocking) stress, similarly as the piezoelectric strain coefficient dijk indi-cates the ability to develop strain. In fact both coefficients can be related trough the following equation:
=e d c .ijk ilm jklmE (3)
The large signal values of the longitudinal piezoelectric stress coefficient e*33 can be determined from the electric-field-induced stress curves measured under clamped conditions, as shown in figure 3(a) for the preload of −4 MPa. Note that the number of indices was reduced due to symmetry reasons and the Voigt notation was used, whereby the first of the two indices indicates the direction of the applied electric field and the second one the direction of the induced stress. The
longitudinal large signal piezoelectric stress coefficient can be calculated for a given electric field as follows:
ΔσΔ
=eE
* .33 (4)
The values of the e*33 coefficient for all the investigated preloads are presented in figure 3(b) and compared to the values of the large-signal longitudinal piezoelectric strain coefficients reported for the same material in [25] (all at 2 kV mm − 1).
Although the blocking stress is an important figure of merit, actuators are typically not operated in a fully clamped state, where the work output is zero, but rather perform work against an external load with a defined linear or non-linear stiffness. It is, therefore, important to assess the full operational stress–strain range at different values of the compressive uniaxial preload. This was done by utilizing the proportional loading method (figure 4), whereby the spring constant of the virtual external load was varied in the range from k = 0 (horizontal line, parallel to the x-axis; free strain conditions, no force is produced) to k = ∞ (vertical line, parallel to the y-axis; fully clamped conditions, zero strain and maximum blocking stress is produced). Note that here the term ‘free strain’ implies the unipolar strain at a constant preload. The measurements were carried out for three characteristic preload values: −4 MPa (lowest preload),−40 MPa (peak blocking stress preload), and −80 MPa (highest preload). The free strain values (k = 0) obtained at an electric field of 2 kV mm − 1 were found to be 0.18%, 0.23%, and 0.13%, respectively, which is comparable to previous reports for soft PZT [23, 25, 27]. An increase of the preload stress from −4 to −40 MPa resulted in an increase of the materials operational stress–strain range (figure 4(b)), as the additionally available ferroelastically-switched domains enabled the sample to develop larger strains and stresses at all measured values of the spring constant. The blocking stress and free strain values increased by approximately 16% and 20%, respectively. This result clearly demonstrates the ben-eficial effect of the compressive uniaxial preload on the actu-ator response under real application conditions and confirms previous findings about the stress-dependent behaviour of ferroelectrics [22–27]. Upon further increase of the uniaxial preload to −80 MPa, a drastic decrease and an asymmetrical change of the operational range was observed (figure 4(c)), characterized by a large 42% decrease in the free strain and a relatively smaller decrease of the blocking stress by about 8%. The smaller strain values throughout the whole stress–strain operational range can be explained by the dominance of the external mechanical stresses over the applied electric field, restricting the non-180° domain switching. It should be noted that this preload value is higher than the coercive stress, which has been determined to be approximately −50 MPa [24, 35], and, therefore, the majority of the domains are assumed to be aligned perpendicular to the applied mechanical load prior to the application of the electric field [23].
Interestingly, despite the large external loads and partial clamping of the domains the preloaded sample is still able to produce relatively large stresses, for example −52 MPa in the fully clamped state (blocking stress) at the preload
Figure 2. Dependence of the blocking stress on the uniaxial compressive preload for soft PZT piezoceramics at different electric fields up to 2 kV mm −1. The dashed circles mark the peak values.
of −80 MPa. In order to further investigate this phenomenon we measured the stress dependence of the small signal d33 direct piezoelectric coefficient. These values are plotted in figure 5, together with the stress dependent values of the Q11 electrostrictive coefficient, taken from the work of Dittmer et al [25]. The piezoelectric coefficient reaches a maximum value at approximately −30 MPa and starts to decrease dra-matically upon further increase of the mechanical stress, as previously observed by others [23, 27, 33]. Note that the d33 value obtained at −80 MPa is more than 2 times lower than the initial value. In contrast, the electrostrictive coef-ficient exhibits a peak value at a slightly higher preload of approximately −40 to −50 MPa, but more important, its value at the preload of −80 MPa is still almost two times higher than at −4 MPa. This was previously explained by the fact that strain mostly depends on the non-180° domain wall motion, while electrostriction is influenced by the polariza-tion, which depends on non-180° and 180° domain dynamics [23, 33]. This could indicate that different mechanisms are
Figure 3. (a) Electric-field-induced stress of soft PZT in the fully clamped state (blocking stress) and the calculated large-signal longitudinal piezoelectric stress coefficient (e33
* ) at a preload of −4 MPa and (b) the e33* values calculated for all investigated preloads,
compared to the large-signal longitudinal piezoelectric strain coefficient at 2 kV mm − 1 (the latter taken from [25]).
Figure 4. Full operational stress–strain behaviour of soft PZT, as measured by the proportional loading method at preload stress values of −4,−40, and −80 MPa (Emax = 2 kV mm − 1).
0
-10
-20
-30
-40
-50
-60
-40
-50
-60
-70
-80
-90
0.00 0.05 0.10 0.15 0.20 0.25
-80
-90
-100
-110
-120
-130
Preload -4 MPa
)aP
M(ssert
S)a
PM(
ssertS
Preload -40 MPa
c)
b)Preload -80 MPa
)aP
M(ssert
S
Strain (%)
a)
Figure 5. Stress dependence of the direct piezoelectric d33 coefficient and the electrostrictive Q11 coefficient of the soft PZT ceramics. The values for the latter are taken from [25]. The dashed vertical lines represent the preload values, at which the operational ranges in figure 2 were measured.
predominantly responsible for the strain and force develop-ment in these materials, resulting in large blocking stresses even at compressive preloads as high as −80 MPa. However, additional work is needed to further investigate these mechanisms.
The differences in size and shape of the operational ranges presented in figure 4 are important for the determination of actuator’s work output per unit volume (W), performed when operating against an external load. This can be calcu-lated for the present case of a linear spring by the following relationship:
σ ε=W12
· · , (5)
where σ is the stress and ε is the strain produced by the mate-rial at a given electric field. The work output throughout the operational range was calculated for the three characteristic preload values and is presented as a function of the spring con-stant in figure 6. In all cases the sample does not perform any work when operated either in the freely displaced (k = 0) or fully clamped (k = ∞) states, as the developed stress or strain is zero, respectively. Note that the work done against the initial preload was neglected for this comparison. For the spring con-stant values between these extremes the work output could be fitted using a pseudo-Voigt function. At the preload of −4 MPa the maximum work of 10.1 kJ m − 3 was obtained at a spring constant of 26.0 GPa (116.4 MN m − 1). Upon increasing the preload value to −40 MPa the output work increased due to the increase of the operational range, reaching a maximum value of 12.9 kJ m − 3 at a lower spring constant 23.6 GPa (105.5 MN m − 1). As expected, further increase of the preload resulted in a decrease of the work output with a maximum value of 7.2 kJ m − 3. Interestingly, this value was reached at a spring constant of 38.9 GPa (174.1 MN m − 1), which is higher than in the previous two cases, indicating the change in the shape of the operational range.
As demonstrated in figure 6, both the maximum work and the corresponding spring constant are strongly preload-dependent values. From the application point of view this is an important conclusion, as actuators are often designed to operate at spring constant values where the maximum work output is achieved, also referred to as impedance matching [2]. However, these conditions are typically determined by measuring the free strain and blocking stress using one of the principal methods described in the introduction (elastic method, direct method or indirect method) and subsequently extrapolating the operational range using linear approxima-tion. The operational ranges of soft PZT at an electric field of 2 kV mm − 1 determined by different methods are presented by the shaded areas in figure 7(a). The Young’s modulus used for the elastic method was taken from Fett et al [36]. While the elastic and the direct method overestimate the opera-tional range, the indirect method tends to underestimate it. This comparison demonstrates large discrepancies between the different methods, resulting from the differences in the determined blocking stress values. These differences origi-nate from the inherent hysteretic electromechanical behaviour of ferroelectrics and the different order of application of the electrical and mechanical loads by each method. Furthermore,
Figure 6. Work as a function of the applied spring constant for selected values of the uniaxial compressive preload (Emax = 2 kV mm − 1). The experimental points obtained by the proportional loading method (symbols) were fitted using a pseudo-Voigt function, represented by the connecting lines.
9 10 11 120
2
4
6
8
10
12
14 -4 MPa-40 MPa-80 MPa
kroW
(m/Jk
3 )
log(k)
Figure 7. Operational stress–strain ranges (a) and the calculated corresponding work output (b) of soft PZT ceramics at Emax = 2 kV mm − 1 and a preload value of −4 MPa, as determined by different methods. The dashed lines represent the corresponding optimal spring constant values at which the maximum work is obtained (impedance matched conditions).
it can be observed that the different methods give different optimum spring constant values for impedance matched con-ditions (dashed lines in figure 7(a)) and different values for the work output (figure 7(b)). While a maximum work output of 10.1 kJ m − 3 was determined by the proportional loading method, the elastic and direct methods for the same operating conditions predicted a work output of 15.7 and 11.8 kJ m − 3, which is an overestimation of 55% and 17%, respectively. On the other hand the indirect method predicted a maximum work output of 8.3 kJ m − 3; an underestimation of 18%. These results once again demonstrate the need for accurate determi-nation of the true operational range.
During operation the piezoelectric materials are often exposed to elevated temperatures, which may be induced by the environment (e.g. car engine) or by the actuator itself due to self-heating. Full characterization of the operational range therefore inevitably includes the measurements of the charac-teristic values at elevated temperatures, which are presented in figure 8. Both values show a small increase up to a tem-perature of 100 °C, however, above this temperature the free strain starts to decrease while the blocking stress increases further. The same trend was previously observed during the direct blocking force measurements, where the decrease of the free strain above 100 °C was related to the deviation from the slightly curved morphotropic phase boundary, while the increased blocking stress was related to the increased coercive stress resulting in increased sample stiffness [37].
4. Summary
The operational range of a soft-type commercially available PZT piezoceramics was investigated at different uniaxial stress preloads using the proportional loading method. The optimal operating conditions were found at a compressive preload of −40 MPa, whereby a maximum blocking stress of −56.2 MPa, a free strain of 0.23%, and a maximum output work of 12.9 kJ m −3 were reached at 2 kV mm − 1. This rep-resented an increase of the operational range by 16─20%. At higher preload values the free strain was found to decrease drastically, while the obtained blocking stresses were still relatively high, indicating a difference between the strain- and
stress-developing mechanisms in these materials. Finally the measured operational range was compared to the results obtained by other methods, revealing large discrepancies, which can be explained by the different order of the appli-cation of external loads and the nonlinear behaviour of fer-roelectrics. The deviations in the determined operational ranges were shown to result in inaccurate calculation of the maximum work output conditions, thereby influencing the actuator’s efficiency.
Acknowledgments
This work was financially supported by the Deutsche Forsc-hungsgemeinschaft under SFB 595/D6 and benefited from the support of the DFG under WE 4972/1-1.
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