Piezoelectric materials

Mateusz Griner, July 2016

Piezoelectric materials (piezein (greek): press, squeeze) are used for engineering applications as actuators and sensors generating and sensing mechanical deformation in materials.
The phenomenon of the piezoelectric effect was discovered by Pierre and Jacques Curie in 1880 ^[1]. Two separate effects are differed:

a) Direct piezoelectric effect (sensor): Applied pressure leads to a mechanical deformation of the piezoelectric material and induces a separation of charges within the material,
which can be measured as a difference in electric potential between opposite sides of the solid object.

b) Inverse piezoelectric effect (actuator): Applying a voltage to the surface of the piezoelectric material the solid object changes its shape and dimension (mechanical deformation).

Piezoelectric Materials:

Typical materials used in applications are natural monocrystalline materials like Rochelle salt (used for microphones, pick-up heads of record players, ultrasonic sensors), turmalin crystals and quartz crystals (used for oscillators, filters, sensors). ^[2]

Compared to these, polycrystalline ferroelectric ceramics like barium titanate and lead zirkonate titanate (PZT) (mostly used for sensors and actuators) show a higher electric-mechanical coupling. Moreover polycrystalline ceramics are produced synthetically, what allows a more flexible design of their shape. Contrary to PVDF (described below), piezoelectric ceramics are very hard and brittle, therefore not very resistant to high tension stresses, insensitive to chemical influences and moisture. ^[2]
Adding alloys to the PZT ceramic, the lattice structure can be optimized for better piezoelectric properties. Depending on the lattice structure, soft and hard PZT ceramics are differed. Without going into details, their relevant piezoelectric properties are compared in the following part. Soft PZT ceramics are characterized by easier polarization, higher piezoelectric coefficients, higher relative permittivity, higher dielectric losses, higher insulating resistance, higher coupling factors, lower mechanical quality factors and lower resistance against depolarization compared to hard PZT ceramics. Typical application for soft PZT ceramics are sound generators and receivers (acoustics), sensors (metrology), ultrasonic diagnostics (medicine) and deformation elements (actuation). Hard PZT ceramics are used for high power ultrasound applications such as ultrasound cleaning and sonar. ^[3]
Before polycrystalline piezoelectric materials can be used, they need to be polarized, what is described within the next chapter ^[2].

There are also some partially crystalline polymers like polyvinylidene fluoride (PVDF), which are piezoelectric after having been polarized. Compared to the piezoelectric ceramics, PVDF is more elastic, has a lower specific weight and thinner piezoelectric components can be built. In comparison to piezoelectric ceramics it’s less chemical resistant and sensors using PVDF are less sensitive than sensors using
PZT ^[4]. PVDF is used for example in transducers of ultrasonic devices up to 24GHz, hydrophones or infrared sensors (Infrared Thermography). ^[2]

Since ferroelectrets show a similar behavior to piezoelectric materials, they are introduced within this section, but not discussed within the following chapters. Cellular thermoplastic polymers, such as poly-propylene (PP), poly-ethylene-therephtalate or cyclo-olefines, can be tailored to obtain a piezoelectric-like behavior. Other than the piezoelectric effect, this effect is based on ellipsoidal voids (porosity) within the material, where positive and negative charges are located on opposite surfaces within the voids. These voids can be generated by stretching or foaming the material, while the electrical charge is obtained by placing the material in an electrical field. Cellular PP is the most investigated ferroelectret, since it is very flexible, lightweight and cheap. Its electromechanical coefficient d₃₃ (see chapter electro-mechanical coupling in sensors and actors) is 20-40 times larger than the one of PVDF. Ferroelectrets are under investigation for applications like speakers, microphones, ultrasonic transducers,
actuators etc. ^[5]

The Polarization Process:

Polycrystalline materials such as PZT consist of many crystallites with a cubic lattice structure above Curie-Temperature (250-500°C depending on the material). Since the cubic lattice structure is symmetric, the positive and negative centers of charge (resulting from positive and negative Ions) share the same location and equalize each other. Below Curie-Temperature the material changes its lattice structure into a tetragonal structure, which is not symmetric anymore, but has a lower energy level. Due to its asymmetry the centers of charge do not share the same location and a dipole within the elementary cell occurs, resulting in a dipole moment. Since there are a lot of crystallites which are orientated randomly within the material, the dipole moments equalize and the external effect of negative and positive charge is cancelled. This way the piezoelectric can’t be measured nor used for sensing or actuating applications. That’s why polycrystalline materials require an external polarization in advance. ^[2]

Figure 1 shows the process of polarization. Using high currents to induce an electric field (>2kV/mm) and temperatures close to Curie-temperature, but still below, the material polarizes. This means that the dipole moments align their orientation. After removing the electric field, a residual polarization remains which can be used for the inverse piezoelectric effect. ^[2]

The polarization characteristic for each material is described within polarization curves, also called butterfly loops, as shown in Figure 2. It shows the behavior during the polarization process and how it expands in different electrical fields. Typically, piezoelectric actuators are used after their polarization within the red hysteresis. Increasing or reducing the electrical field it expands nonlinear up to a certain length. Once polarized, the expansion doesn’t return to zero after removing the electrical field. This point is called remanent elongation εr. Applying a strong enough negative electrical field (field in other direction than the electrical field used for the first polarization) the material depolarizes or rather polarizes in the other way, what may lead to destruction of the actuator’s function. That's why actuation of the material against its polarization has to be avoided. For safety reasons temperatures at least 80°C below Curie-Temperature or an offset voltage has to be used, so that a depolarization is not possible ^[7]. ^[6]

The butterfly curve is only valid below Curie-Temperature, since the piezoelectric effect doesn’t exist above this temperature.

Electro-Mechanical Coupling in Sensors and Actuators:

As already described above, piezoelectric materials react to applied mechanical stress (force devided by surface [N/mm]) inducing a voltage (and other way round) due to the electro-mechanical coupling. The strength of the electro mechanical-coupling depends on the direction of the applied stress or applied electrical field. Without using complicated equations, the main effects are presented briefly in this section. In Figure 3 the direction of polarization is the same for all examples, while the direction of the electrical field and therefore the deformation of the piezo element differs. The numbers 4, 5, 6 of the axis indicate shear on axis 1, 2, 3.

Typical values for the electro-mechanical coupling of PZT show that the d15 effect (Fig. 3 c) is the strongest, but it can't be used for most applications, since it is a shearing effect. The amount of the electro-mechanical coupling d₃₃ (longitudinal effect) (Fig.3 a) is two times bigger than the d₃₁ effect (transversal)(Fig.3 b), but in the other direction. Both effects are coupled and appear simultaneously. While the piezo element expands in direction 3 it contracts in direction 2, and other way. This happens, because the volume of the structure remains constant. Therefore the material has to contract orthongonally to the axis of expansion. Both effects are used for sensors and actuators, because of the defined change of shape. This way longitudinal, transversal and shear waves can be generated.

Typical values for PZT: ^[8]
d₃₃= 380…590 x 10^-12 m/V (electrical field in direction 3 leads to expansion in 3)       (Fig.3 a)
d₃₁= -60...-270 x 10^-12 m/V (electrical field in direction 3 leads to contraction in 2/1)  (Fig.3 b)
d₁₅= 265…765 x 10^-12 m/V (electrical field in direction 1 leads to shearing in 5)         (Fig.3 c)

In general the tendency of the behavior is similar, but the absolute values differ for different materials.

Direct Piezoelectric Effect and Setup:

Exerting pressure deforms the structure of the piezoelectric ceramic, causing separation of positive and negative gravity centers generating dipoles. Inside, the facing gravity centers may cancel each other, but an external polarization appears at the material's surface, what generates an electric field. If both surfaces of the piezoelectric ceramic are connected to an electrical circuit, the electric charge induced by the mechanical load on the material can be measured. ^[1]

Figure 4 shows the principle of a) the charge amplifier circuit which is used for applications at very low frequencies, but still bigger than 0 Hz. Static measurement is not possible, because the electrical loading would creep away from the capacitor, used in the circuit. The magnitude of the signal is proportional to the magnitude of movement. This is beneficial for measurements of elongations of materials, what may happen due temperature expansion or mechanical loads. The current amplifier circuit b) generates a signal which is proportional to the velocity of the load. This is interesting for high frequency applications as well the measurement of accelerations, since the magnitude of the signal increases with higher frequencies.

Figure 5 shows the schematic connection of the piezoelectric ceramic to the cirquitry within a sonotrode. Its main parts are an oscillating element, which is the piezoelectric plate, a damping mass for noise reduction and a matching layer for adjustment of the acoustic impedance of the test specimen ^[9].

The measured electrical charge is recalculated by physical equations into a mechanical load. This way the load in the piezoelectric material and its expansion is derived.

Physical Equations and Coefficients:

Some basic equations for the direct piezoelectric effect, regarding only linear behavior are explained in this section:

• Applying a mechanical load, a stress is produced within the material. This has two effects: elastic stress proportional to mechanical load is generated by Hooke’s law: ε = ʛ / E', with the expansion ε [-], mechanical stress ʛ [N/mm²], the Young Modulus E' [N/mm²].^[1][10]
• The piezoelectric polarization D = ε₀ x ε_r x E occurs, with ε₀ the vacuum permittivity (8,854 x 10-12 As/Vm), the relative permittivity ε_r [As/Vm] and the electrical field E [V/m]. Considering both effects, the following equation derives for the dielectric displacement [C/m²]:
  D = d₃₁ x ʛ + ε₃₃ x U / h_p = d₃₁ x ʛ + ε₃₃ x E' where d₃₁ is the piezoelectric strain constant [m/V] of the material in 31 direction, ʛ the mechanical stress [N/mm²], ε₃₃ die electrical permittivity [As/Vm] of the piezoelectric ceramic in 33 direction, U the applied Voltage [V] and h_p the thickness [mm] of the piezoelectric element.^[1][10]
• The piezoelectric voltage constant g_ij = d_ij / ε_r [Vm/N] characterizes the electrical field E = g_ij x ʛ_ij [V/m], which is induced by an applied mechanical stress on the piezoelectric material. ^[6]

For the inverse piezoelectric effect:

• The expansion of the piezoelectric plate is ε = (1/E’) x ʛ + d x E, with the expansion ε [-], the Young Modulus E’ [N/mm²], the mechanical strain ʛ [N/mm²], the piezoelectric strain constant d [m/V] and the electric field E [V/m]. The dimensionless free elongation of a piezoelectric ceramic is ʌ = d₃₁ x U / h_p. This equation is valid for an actuator which is mounted on one side while the other side is free. ^[1][10]

The following section explains some important piezoelectric factors:

• The electromechanical coupling factor k describes how well the material transfers mechanical to electric energy and the other way. It is as a relation between stored mechanical or electric energy to the input electrical or mechanical energy. The equation for the coupling factor is k² = d / (ε₀ x ε x s) with the elastic compliance s = 1 / E' [mm²/N]. ^[6]
• The frequency constant N =f_s x l [Hz x m] describes the relation between the resonant frequency and the shape of the piezoelectric component. l is a length or diameter characterizing the size of the structure. If l increases, the resonant frequency of the structure decreases. ^[1]

Application of Piezoelectric Materials

They are used as actuators in fuel injection systems, alignment of mirrors of super large telescopes, for ultrasound based devices like sonar, air ultrasound, ultrasonic welding and weldseam inspection, ultrasonic cutting and mixing, precision positioning systems with inchworm actuators, active damping of structure vibrations as well as for medical screenings. Furthermore they are used as very sensible sensors to measure expansion, pressure, vibration of mechanical structures or accelerations of parts. Civil engineers are using them in ultrasonic inspection technologies to inspect concrete structures, looking for concrete reinforcements and cracks (acoustic emission analysis), thickness measurement as well as for inspection of composite materials, inspection of steel and inspection of wood.

Types of Sensors

Different types of sensors made of piezoelectric materials exist. Some examples are piezo blocks, embedded ceramics, piezoelectrical foils, microphones and embedded piezo fibres. Figure 6 shows some examples of sensors. ^[2]

Types of Actuators:

To give an idea of piezoelectric actuators, some examples are listed below. Piezoelectric actuators can act only on very short distances, but with forces, which are high enough to extend steel. Some examples are the bimorph actuator, the telescopic actuator, the inchworm actuator, the stack actuator and the rhombic actuator. ^[10]

Resonant and Non-Resonant Sensors

Piezoelectric sensors may be used at their resonant frequencies or at other frequencies than their resonant frequency. Resonant sensors are sensible within a narrow domain of frequencies. Their advantage is a high sensitivity for certain frequencies showing high magnitudes of the signal. This is beneficial for measurements in high damping materials or thickness measuring of structures. In comparison to the resonant sensors, broadband sensors may be used for a wide range of frequencies showing a linear behavior of their transfer function, but with lower magnitudes. A combination of their advantages is gained using sensors with crystals showing different resonant frequencies. But such multi-resonant sensors don’t show a linear behavior anymore, that’s why they and the resonant sensors are not suitable to record the full range of frequencies transferred by a structure, which could be used for a Fourier-Transformation to characterize the material. Therefore it is necessary to know the behavior of the sensors used and to use the correct sensors for each type of measurement. Figure 7 compares the magnitudes at different frequencies for different types of sensors. ^[11][12]

To clarify the effect, here is a more specific example of application: Ultrasonic waves are recorded via piezoelectric sensors which transfer applied mechanical loads into analogue signals. Using the inverse piezoelectric effect, ultrasonic sensors can also emit ultrasonic waves. Since their emitted signals are narrowband and nonlinear (certain frequencies) they are called resonant sensors. By using multiple layers of piezoelectric material (with different thicknesses) or additional damping masses, their frequency range may be changed and extended. Additionally a good coupling without air gaps between the sensor and the sample has to be ensured. ^[11]

Moreover sensors react angle-depended on the incoming waves. This has to be considered in applications, where the magnitude has to be measured accurately ^[13]. But for most applications a relative measurement is enough to detect discontinuities.

Conclusion:

Positive aspects of piezoelectric materials are the high range of frequencies (0 Hz <f up to f > 10 kHz), within they may be used. As shown above, the test setup is relatively uncomplicated and due to the small mass and volume of sensors they don’t require heavy equipment. Sensors can also be mounted to the structure for health monitoring applications. Some negative aspects are that piezoelectric ceramics are brittle and in high electric fields their polarization may be destroyed. Actuators need high voltages for actuation.

Literature

• DIN EN 50324: Piezoelectric properties of ceramic materials and components
• Piezokeramische Materialien und Bauelemente, CAT125D Piezokeramische Werkstoffe 11/03/14.3,0 ; Physik Instrumente (PI) GmbH & Co. KG (2011)

References

1. Arnau, Antonio: Piezoelectric Transducers and Applications, Springer publ., Heidelberg (2008), Second Edition.
2. Flemming, Manfred, W.J. Elpass M.: Aktive Funktionsbauweisen – Eine Einführung in die Struktronik, Springer (1998).
3. Janocha Hartmut: Adaptronics and Smart Structures, Springer-Verlag Berlin Heidelberg (2007), Second Edition.
4. Balagaes, Daniel; Fritzen, Claus-Peter; Güemes, Alfreado: Structural Health Monitoring with Piezoelectric Sensors, ISTE Ltd. (2006), p. 288-377.
5. Mohebbi, Abolfazil; Mighri, Frej; Ajji, Abdellah; Rodrigue, Denis: Cellular Polymer Ferroelectret: A Review on Their Development and Their Piezoelectric Properties, Wiley Periodicals, Inc. (2016), 16 pages.
6. Heimann, Robert B.: Electroceramic Materials, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2010), p. 253-318.
7. Baier, Horst: Adaptive Strukturen/Funktionsstrukturen, Lehrstuhl für Leichtbau, TUM, 2015, p. 40-49.
8. Frühauf, Joachim: Werkstoffe der Mikrotechnik, Carl Hanser Verlag GmbH & Co. KG (2005).
9. Kohout Benedikt: Optimierung von Anpassschichten für Ultraschallwandler, KIT (2009), 109 pages .
10. Donald, J. Leo: Engineering Analysis of Smart Material Systems, Piezoelectric Materials, John Wiley & Sons, Inc. (2007), p. 122-204.
11. Grosse, Christian U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen, Centrum Baustoffe und Materialprüfung, TUM, 2015, p. 57-59.
12. Krautkrämer, J. & Krautkrämer, H.: Werkstoffprüfung mit Ultraschall, Springer Verlag, Heidelberg (1986), 5th Edition.
13. Langen, A: Ein Verfahren zur Konstruktion anwendungsoptimierter Ultraschallsensoren auf der Basis von Schallkanälen. Dissertation, Fakultät der Konstruktions- und Fertigungstechnik der Universität Stuttgart (1993), 140 pages.