New Effects in 1–3-Type Composites Based on Relaxor-Ferroelectrics Single Crystals

Abstract

Two effects of the matrix subsystem on the piezoelectric performance and hydrostatic parameters are first studied in novel 1–0–3 composites that contain relaxor-ferroelectric single-crystal rods surrounded by a ferroelectric ceramic/polymer matrix with 0–3 connectivity. First, the influence of the mutual orientation of the poling direction of the single-crystal and ceramic components on the properties of the 1–0–3 composite is discussed to demonstrate advantages concerned with the high hydrostatic piezoelectric performance and large anisotropy of squared figures of merit. In a 1–0–3 0.67Pb(Mg_1/3Nb_2/3)O₃ − 0.33PbTiO₃ single crystal/(Pb_1–x Ca _x )TiO₃ ceramic/araldite composite with x = 0.20–0.25, values of max $g_{h}^{*}$ ~ 10² mV · m/N and max($d_{h}^{*}$ $g_{h}^{*}$) ~ 10⁻¹¹ Pa⁻¹ are achieved at specific volume-fraction and rotation-angle ranges due to the new orientation effect in the presence of a highly anisotropic 0–3 matrix. Second, the influence of the aspect ratio of ceramic inclusions on the piezoelectric and hydrostatic parameters of the 1–0–3 composite based on relaxor-ferroelectric SCs is studied. In this case, the elastic anisotropy of the 0–3 matrix plays the key role in forming the large effective parameters of the composite. The studied composites can be used in piezoelectric sensor, energy-harvesting and hydrophone applications.

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1 Introduction

Composites based on relaxor-ferroelectric single crystals (SCs) with high piezoelectric activity are an important group of advanced dielectric materials wherein the effective electromechanical properties and their anisotropy can be varied and tailored [1–4] in a wide range. Among the composite systems with a high piezoelectric activity and/or sensitivity [3–5], of particular interest are those based on domain-engineered relaxor-ferroelectric SCs such as (1 − z)Pb(Mg_1/3Nb_2/3)O₃ − zPbTiO₃ (PMN–zPT) or (1 − z)Pb(Zn_1/3Nb_2/3)O₃ − zPbTiO₃ (PZN–zPT) with compositions near the morphotropic phase boundary [6–9]. These composites are of interest due to the polarization orientation effect [4, 10–12] studied for a few connectivity patterns (e.g., 1–3, 2–2 and 0–3) in the last decade. This orientation effect depends not only on the connectivity of the composite system, but also on the electromechanical properties of the SC component, and this component can be poled along a specific crystallographic direction, for instance, along [001], [011] or [111] of the perovskite unit cell.

A potential method to increase the performance of the piezo-active composites is to modify their structure by introducing a third component [13–16]. Some modifications were studied for 1–3-type composites, and their parameters were compared to those of the conventional 1–3 ferroelectric ceramic (FC) /polymer composites. The presence of an anisotropic matrix in a 1–3-type composite structure opens up new and exciting possibilities of further tailoring the effective electromechanical properties of the composite and its hydrostatic parameters [14–16].

In this chapter, we show that the use of two ferroelectric components with distinct differences in their electromechanical properties leads to important effects that promote improving a few effective parameters of the three-component 1–3-type composite. Such a composite is of particular interest due to the complex inter-relationships in the fundamental triangle of ‘composition—structure—properties’ [17]. The aim of this chapter is to discuss advantages in effective parameters that are achieved by taking into account the new effects and the active role of the matrix subsystem in the 1–3-type SC/FC /polymer composites.

2 Orientation Effect in the 1–0–3 Composite with Two Polarization Directions

2.1 Model Concepts and Effective Parameters of the Composite

It is assumed that the composite consists of long SC rods embedded in a heterogeneous matrix (Fig. 13.1). The SC rods are in the form of the rectangular parallelepiped with a square base and square arrangement in the (X ₁ OX ₂) plane. The main crystallographic axes X, Y and Z of each SC rod with the spontaneous polarization P ⁽¹⁾ _s are oriented as follows: X || OX ₁, Y || OX ₂ and Z || P ⁽¹⁾ _s || OX ₃. Spheroidal FC inclusions are surrounded by a large polymer matrix. The shape of each FC inclusion obeys the equation

Fig. 13.1

Schematic of the 1–0–3 realxor-ferroelectric SC/FC /polymer composite . m and 1 − m are volume fractions of the SC and surrounding 0–3 matrix , respectively. Rotation of co-ordinate axes (X ₁′X ₂′X ₃′) → (X ₁ X ₂ X ₃) is shown in inset 1, the 0–3 matrix is shown in inset 2. In the 0–3 matrix , m _i and 1 − m _i are volume fractions of the FC and polymer , respectively (reprinted from Topolov et al. [14], with permission from Elsevier)

$$(x_{1}^{\prime } /a_{ 1} )^{ 2} + \, (x_{2}^{\prime } /a_{ 2} )^{ 2} + \, (x_{3}^{\prime } /a_{ 3} )^{ 2} = 1$$

(13.1)

relative to the axes of the rectangular co-ordinate system (X ₁′X ₂′X ₃′) rotated by an angle α with respect to (X ₁ X ₂ X ₃) (inset 1 in Fig. 13.1). In (13.1), a ₁, a ₂ = a ₁ and a ₃ are semi-axes of the inclusion. To characterise its shape, we introduce the aspect ratio ρ _i  = a ₁/a ₃. Centres of the inclusions occupy sites of a simple tetragonal lattice with unit-cell vectors parallel to the OX _k ′ axes. We assume that 0 < ρ _i  < 1, and the presence of prolate inclusions facilitates poling of the FC /polymer matrix due to a weaker depolarization effect. The remanent polarization vector of the FC inclusion is P ⁽²⁾ _r ↑↑OX ₃′, and OX ₃′ is the poling axis of the matrix (inset 2 in Fig. 13.1). The FC /polymer matrix represents a composite with 0–3 connectivity in terms of work [17, 18], and the three-component composite (Fig. 13.1) is described by 1–0–3 connectivity. Assuming that the linear sizes of each inclusion in the 0–3 matrix are much smaller than the length of the side of the square being intersected the rod in the (X ₁ OX ₂) plane, we evaluate the effective properties of the 1–0–3 relaxor-ferroelectric SC/FC /polymer composite in two stages [14].

The first stage is related to the effective properties of the 0–3 composite matrix . Taking into account the electromechanical interaction between the piezo-active (poled FC ) inclusions, the effective properties of the 0–3 composite matrix are determined by either the effective field method (EFM) [4, 17] or finite element method (FEM) [17]. In the EFM, an electromechanical interaction in the FC /polymer matrix (see inset 2 in Fig. 13.1) is described using a local effective field. This effective field is determined by taking into account a system of interacting inclusions and boundary conditions based on inclusions with a spheroidal shape [4, 14, 15].

In terms of FEM modeling, COMSOL [19] is applied to obtain the volume-fraction dependence of the effective electromechanical properties of the 0–3 composite matrix . A representative unit cell, containing a spheroidal inclusion with a radius adjusted to yield the appropriate volume fraction m _i , is discretized using tetrahedral elements [4, 15]. The number of elements, depending on the aspect ratio ρ _i of the spheroidal inclusion, varies from 1,000,000 to 1,700,000. The unknown displacement and electric-potential fields are interpolated using linear Lagrangian shape functions. The corresponding number of degrees of freedom varies from 700,000 to 1,200,000. The following conditions are assumed at the inclusion—matrix interface: (i) perfect bonding (i.e., continuity of the displacement field) and (ii) continuity of the electric potential. Moreover, either Dirichlet or periodic boundary conditions are considered at the boundary of the parallelepiped representative unit cell ‘inclusion—matrix’ [14].

At the second stage, the effective electromechanical properties of the 1–0–3 composite are evaluated using the matrix method [4, 17]. The electromechanical properties of the SC rod and 0–3 composite matrix are averaged in the OX ₁ and OX ₂ directions, in which the periodic structure of the composite (Fig. 13.1) is observed, and take into account electromechanical interactions in a system of ‘piezo-active rods—piezo-active matrix’. The effective electromechanical properties of the 1–0–3 composite in the co-ordinate system (X ₁ X ₂ X ₃) are given by  

$$\left\| {C^*} \right\| = \left\| {C^*(m,\,m_{i} ,\,\rho_{i} ,\alpha )} \right\| = \left( {\begin{array}{*{20}c} {\left\| {s^{*E} } \right\|} & {\left\| {d^{*} } \right\|^{t} } \\ {\left\| {d^{*} } \right\|} & {\left\| {\varepsilon^{*\sigma } } \right\|} \\ \end{array} } \right).$$

(13.2)

In (13.1), || s* ^E ||, || d* || and || $\varepsilon_{{}}^{*\sigma }$ || are matrices of elastic compliances at electric field E = const, piezoelectric coefficients and dielectric permittivities at mechanical stress σ = const, respectively, and superscript ‘t’ denotes transposition.

Based on the || C* || matrix from (13.1), we determine the following effective parameters of the 1–0–3 composite : piezoelectric coefficients $g_{fl}^{*}$ from equation || d* || = || $\varepsilon_{{}}^{*\sigma }$ ||^.|| g* ||, squared strain–voltage figures of merit

$$(Q_{33}^{*} )^{2} = d_{33}^{*} \,g_{33}^{*} ,\,(Q_{32}^{*} )^{2} = d_{32}^{*} \,g_{32}^{*} \,{\text{and}}\,(Q_{31}^{*} )^{2} = d_{31}^{*} \,g_{31}^{*} ,$$

(13.3)

hydrostatic piezoelectric coefficients

$$d_{h}^{*} = d_{33}^{*} + d_{32}^{*} + d_{31}^{*} \,{\text{and}}\,g_{h}^{*} = g_{33}^{*} + g_{32}^{*} + g_{31}^{*} ,$$

(13.4)

and squared hydrostatic figure of merit

$$(Q_{h}^{*} )^{2} = d_{h}^{*} g_{h}^{*} .$$

(13.5)

It is assumed that electrodes applied to a composite sample (Fig. 13.1) are perpendicular to the OX ₃ axis. The piezoelectric coefficients $g_{fl}^{*}$ characterise the piezoelectric sensitivity of the composite. Squared figures of merit ($Q_{fl}^{*}$)² from (13.3) are used to characterise [4, 17] the piezoelectric sensor signal-to-noise ratio. The hydrostatic piezoelectric coefficients $d_{h}^{*}$ and $g_{h}^{*}$ from (13.4) describe [4, 17] the piezoelectric activity and sensitivity under hydrostatic loading of the composite sample for SONAR and hydrophone applications. The parameter ($Q_{h}^{*}$)² from (13.5) plays the role of the hydrostatic analog of ($Q_{3l}^{*}$)² from (13.3) and is used [17] to characterize the piezoelectric sensitivity under hydrostatic loading.

2.2 Components of the Composite

Among the components of the 1–0–3 composite , we mention [001]-poled domain-engineered PMN–0.33PT and PZN–0.08PT SCs with high piezoelectric activity, poled (Pb_1−x Ca _x )TiO₃ and modified PbTiO₃ FC s with a large piezoelectric anisotropy but moderate piezoelectric activity, and piezo-passive polymer s such as araldite and polyurethane (Table 13.1). The hydrostatic piezoelectric coefficient $d_{h}^{(1)}$ evaluated in accordance with (13.4) exhibits different signs: $d_{h}^{(1)}$ > 0 for the PMN–0.33PT SC, but $d_{h}^{(1)}$ < 0 for the PZN–0.08PT SC. Hereafter, we use superscript ‘(1)’ to denote the SC component and ‘(2)’ to denote the FC component.

Table 13.1 Elastic compliances $s_{ab}^{E}$ (in 10⁻¹² Pa⁻¹), piezoelectric coefficients d _fl (in pC/N) and relative dielectric permittivities $\varepsilon_{rr}^{\sigma }$/ε ₀ of components at room temperature

As follows from experimental data [7, 24], the coercive fields $E_{c}^{(n)}$ of the PMN–xPT SC (n = 1) and (Pb_1–x Ca _x )TiO₃ FC (n = 2) obey the condition $E_{c}^{(1)}$ ≪ $E_{c}^{(2)}$. Such a condition enables initial poling of the 0–3 matrix under a strong electric field with a subsequent poling of the SC rods in the composite (Fig. 13.1) under a lower electric field. We add that (Pb_1−x Ca _x )TiO₃ and related highly anisotropic FC s were used to form some 0–3 FC /polymer composites [17, 20, 25, 26].

2.3 Volume-Fraction and Orientation Dependence of Hydrostatic Parameters

As seen from Fig. 13.2, a and b, max $g_{h}^{*}$ and max[($Q_{h}^{*}$)²] of the PMN–0.33PT-based composite are achieved at a rotation angle α ≈ 90° with a volume fraction of FC m _i  = 0.12. Local max $g_{h}^{*}$ relates to the volume fraction of SC 0.01 < m < 0.12, however, the fabrication of a composite sample at m < 0.03 may be problematic in terms of the manufacturing tolerance [1, 2, 4, 17, 25]. The largest value of $g_{h}^{*}$ at 0.1 ≤ ρ _i  ≤ 0.5 relates to m _i  ≈ 0.12 and 0.01 < m < 0.03, and in this m range, a local max $g_{h}^{*}$ is observed at various values of m _i , ρ _i and α [14]. Our evaluations based on the EFM (0–3 matrix ) and matrix method (1–3-type composite) lead to absolute max $d_{h}^{*}$ = 305 pC/N at m = 0.532, m _i  = 0.12, ρ _i  = 0.1, and α = 90° [14].

Fig. 13.2

Examples of the hydrostatic piezoelectric response of the 1–0–3 PMN–0.33PT SC/(Pb_0.75Ca_0.25)TiO₃ FC /araldite composite at ρ _i  = 0.1: a near local max $g_{h}^{*}$(0.05, m _i , 0.1, α), b near absolute max{[$Q_{h}^{*}$(m, 0.12, 0.1, α)]²}. Electromechanical properties of the 0–3 matrix at the first stage of averaging were determined by means of the EFM (reprinted from Topolov et al. [14], with permission from Elsevier)

Using the matrix method, we find for comparison, that the 1–3 PMN–0.33PT SC/araldite composite is characterized by absolute max $g_{h}^{*}$ = 158 mV · m/N, absolute max[($Q_{h}^{*}$)²] = 8.27 · 10⁻¹² Pa⁻¹ and absolute max $d_{h}^{*}$ = 274 pC/N at m = 0.016, 0.103 and 0.509, respectively [14]. At m = 0.05 for the 1–3 PMN–0.33PT SC/araldite composite, we obtain $g_{h}^{*}$ = 115 mV^.m/N, and this value is considerably smaller than $g_{h}^{*}$ near the maximum point in Fig. 13.2a.

The large values of ($Q_{h}^{*}$)² (Fig. 13.2b) and $d_{h}^{*}$ in the 1–0–3 composite are achieved due to the presence of a 0–3 matrix based on FC with piezoelectric coefficients $d_{3f}^{(2)}$ that obey the condition [20] $d_{33}^{(2)}$/| $d_{31}^{(2)}$ | ≫ 1. At α = 90°, the remanent polarization vector P ⁽²⁾ _r of each FC inclusion (see inset 2 in Fig. 13.1) is parallel to OX ₂, and this P ⁽²⁾ _r orientation leads to a decrease in | $d_{32}^{*}$ | with minor changes in $d_{31}^{*}$ and $d_{33}^{*}$ (or $g_{31}^{*}$ and $g_{33}^{*}$, respectively) as a result of the weak lateral piezoelectric effect in the 0–3 matrix . As a consequence of the reduced | $d_{32}^{*}$ |, we observe an increase in both $d_{h}^{*}$ and $g_{h}^{*}$.

The elastic anisotropy of the 0–3 matrix with highly prolate inclusions is an additional factor in increasing the hydrostatic parameters from (13.4) and (13.5). Ratios of the elastic compliances in the 0–3 (Pb_0.75Ca_0.25)TiO₃ FC /araldite composite matrix are represented as follows:

$$\begin{aligned} & s_{11,0 - 3}^{E} /s_{12,0 - 3}^{E} = - 2. 2 1,s_{11,0 - 3}^{E} /s_{13,0 - 3}^{E} = - 6. 5 2 { }\,{\text{and}}\,s_{11,0 - 3}^{E} /s_{33,0 - 3}^{E} = 1. 9 5 {\text{ at}}\,\rho_{i} = \, 0. 1 {\text{ and}}\,m_{i} = \, 0. 10, \\ & {\text{and}} \\ & s_{11,0 - 3}^{E} /s_{12,0 - 3}^{E} = - 2. 5 2,\,s_{11,0 - 3}^{E} /s_{13,0 - 3}^{E} = - 3. 5 1 { }\,{\text{and}}\,s_{11,0 - 3}^{E} /s_{33,0 - 3}^{E} = 1. 2 5 {\text{ at}}\,\rho_{i} = 0. 3 {\text{ and}}\,m_{i} = 0. 10. \\ \end{aligned}$$

This means that a significant decrease of | $s_{11,0 - 3}^{E}$/$s_{13,0 - 3}^{E}$ | and | $s_{11,0 - 3}^{E}$/$s_{33,0 - 3}^{E}$ | is observed with a weakening of the piezoelectric activity of the 0–3 composite.

The orientation effect considered in the 1–0–3 composite favours an increase in ($Q_{h}^{*}$)² and $d_{h}^{*}$ near its maxima by approximately 29 % and 11 %, respectively, in comparison to a ‘traditional’ two-component 1–3 PMN–0.33PT SC/araldite composite [14]. The studied 1–0–3 composite is attractive due to large values of local maxima of $d_{h}^{*}$, $g_{h}^{*}$ and ($Q_{h}^{*}$)² at m _i  = const, especially at m _i  < 0.15.

2.4 Anisotropy of Squared Figures of Merit

An important feature of the studied 1–0–3 composite consists in a large anisotropy of squared figures of merit from (13.3). For instance, the inequality

$$(Q_{33}^{*} )^{2} /(Q_{3j}^{*} )^{ 2} \ge 10 \, \left( {j = 1 {\text{ and 2}}} \right)$$

(13.6)

holds at volume fractions of SC m _Q1 ≤ m ≤ m _Q2 which depend on the rotation angle α (Fig. 13.3). The validity of condition (13.6) is achieved due to the 0–3 FC /polymer matrix with a significant elastic and piezoelectric anisotropy at m _i  = 0.50 and ρ _i  = 0.1. The presence of prolate FC inclusions has a significant influence on the electromechanical properties of the 0–3 matrix , and the anisotropy of these properties promotes the validity of condition (13.6) for the 1–0–3 composite . Values of m _Q2 < 0.1 may be a result of the high piezoelectric activity of the SC while $d_{33}^{(1)}$/$d_{33}^{(2)}$ ≈ 100. We note that for the PMN–0.33PT SC, condition ($Q_{33}^{(1)}$)²/($Q_{31}^{(1)}$)² = ($Q_{33}^{(1)}$)²/($Q_{32}^{(1)}$)² = ($d_{33}^{(1)}$/$d_{31}^{(1)}$)² ≈ 4.5 holds (see Table 13.1).

Fig. 13.3

Region of validity of condition (13.6) (hatched area) in the 1–0–3 PMN–0.33PT SC/(Pb_0.75Ca_0.25)TiO₃ FC /araldite composite at ρ _i  = 0.1 and m _i  = 0.50. Electromechanical properties of the 0–3 matrix at the first stage of averaging were determined by means of the EFM (reprinted from Topolov et al. [14], with permission from Elsevier)

2.5 Comparison of Effective Parameters

The effective parameters obtained using different methods are given for comparison in Table 13.2. To predict the effective properties of the 0–3 composite matrix at the first stage of calculations, we use either the EFM or one of two FEM models. Either Dirichlet (FEM-1) or periodic (FEM-2) boundary conditions are enforced on the boundary of the representative unit cell of the 0–3 FC /polymer matrix . At the second stage of calculations, we use the matrix method that is applicable to the 1–3 composite structure with planar interfaces [14]. The use of Dirichlet boundary conditions gives rise to a higher piezoelectric activity of the 0–3 matrix , whereas periodic boundary conditions lead to a lower piezoelectric activity than that obtained using the EFM method. The EFM results are close to average values obtained from the FEM models. Relatively small differences between the parameters obtained using the EFM, FEM-1 and FEM-2 (Table 13.2) are accounted for by the very high piezoelectric activity of the SC rod in comparison to the 0–3 matrix .

Table 13.2 Hydrostatic piezoelectric coefficient $g_{h}^{*}$ (in mV · m/N) and squared hydrostatic figure of merit ($Q_{h}^{*}$)² (in 10⁻¹² Pa⁻¹) of the 1–0–3 PMN–0.33PT SC/(Pb_0.80Ca_0.20)TiO₃ FC /araldite composite at ρ _i  = 0.1 and α = 90°

We add for a further comparison, that in a 1–3 PZT FC /epoxy composite with aligned FC rods [27], values of max $d_{h}^{*}$ ≈ 110 pC/N and max[($Q_{h}^{*}$)²] = 6.0 · 10⁻¹² Pa⁻¹ are considerably lower than the maximum values of these parameters of the studied 1–0–3 composite (see, for instance, Fig. 13.2 and Table 13.2). According to data [17], a 1–3 PMN–0.33PT SC/araldite composite (a limiting case of the 1–0–3 composite at m _i  = 0) is characterized by absolute max $d_{h}^{*}$ = 274 pC/N (at m = 0.509), absolute max $g_{h}^{*}$ = 130 mV · m/N (at m = 0.016) and absolute max[($Q_{h}^{*}$)²] = 7.45 · 10⁻¹² Pa⁻¹ (at m = 0.115). The aforementioned values are also smaller than those related to the 1–0–3 composite .

3 Aspect-Ratio Effect in the 1–0–3 Composite

3.1 Two Examples of the High Performance

In Sect. 13.3, we discuss the role of the aspect ratio ρ _i of FC inclusions in forming the large effective parameters of the 1–0–3 SC/FC /polymer composite . Hereafter, we consider the composite shown in Fig. 13.1 at α = 0° and vary ρ _i in the 0–3 FC /polymer composite matrix to show its influence on the effective parameters of the 1–0–3 composite .

Our recent results suggest that large changes in the piezoelectric coefficient $g_{33}^{*}$ and squared figures of merit ($Q_{3j}^{*}$)² are observed (Fig. 13.4) at

Fig. 13.4

Aspect-ratio (ρ _i ) dependence of the piezoelectric coefficient $g_{33}^{*}$ and hydrostatic piezoelectric coefficient $g_{h}^{*}$ (a, in mV · m/N), and squared figures of merit ($Q_{3j}^{*}$)² (b, in 10⁻¹² Pa⁻¹) of the 1–0–3 PMN–0.33PT SC/modified PbTiO₃ FC /polyurethane composite at volume fractions m _i  = const (FC inclusions in the 0–3 matrix ) and m = const (SC rods in the composite). At the first stage of averaging, electromechanical properties of the 0–3 matrix were evaluated by EFM (reprinted from Topolov et al. [15], with permission from Elsevier)

$$0.0 1< \rho_{i} < 2.$$

(13.7)

In the range (13.7), the shape of the FC inclusions changes from highly prolate (ρ _i  ≪ 1) to oblate (ρ _i  > 1). Such changes in the microgeometry of the 0–3 composite matrix give rise to significant changes in its elastic properties and ratios of elastic compliances $s_{1b}^{(m),E}$/$s_{pq}^{(m),E}$ that then have a strong influence on the piezoelectric properties and their anisotropy in the 1–0–3 composite even at the small volume fraction of FC m _i  = 0.05 (Fig. 13.4).

The correlation between the elastic compliance $s_{33}^{(m),E}$ of the 0–3 matrix and the squared figure of merit ($Q_{33}^{*}$)² of the composite was recently studied in work [15]. This correlation stems from the important role of $s_{33}^{(m),E}$ in the formation of the piezoelectric response of a 1–3 composite along the poling axis OX ₃. In the case of the 1–0–3 composite (Fig. 13.1), the elastic anisotropy of its 0–3 matrix leads to a stronger link between $s_{33}^{(m),E}$ and ($Q_{33}^{*}$)², especially in the range (13.6).

Now we consider the hydrostatic piezoelectric coefficients of the 1–0–3 PZN–0.08PT SC/modified PbTiO₃/polyurethane composite (Fig. 13.5). Graphs in Fig. 13.5 show the important role of the matrix subsystem in forming the hydrostatic response of the composite wherein the highly piezo-active SC component is characterized by $d_{h}^{(1)}$ = –20 pC/N. Even at a relatively small volume fraction of the FC inclusions m _i , changes in the aspect ratio ρ _i give rise to considerable changes in the hydrostatic parameters , especially at ρ _i  > 1. Moreover, max $g_{h}^{*}$ (Fig. 13.5b) correlates with max $d_{h}^{*}$ (Fig. 13.5a) because the relation [17] $g_{h}^{*}$ = $d_{h}^{*}$/$\varepsilon_{33}^{*\sigma }$ is valid. In Fig. 13.5b we show a restricted range of volume fractions of SC m due to the significant decreasing $g_{h}^{*}$ at m > 0.2 irrespective of the FC content. Changes in sgn $d_{h}^{*}$ (Fig. 13.5a) take place at m ≈ 1, when the SC rods play the dominating role in forming the piezoelectric properties of the composite. The use of the highly oblate FC inclusions (ρ _i  ≫ 1) leads to a lower dielectric permittivity of the 0–3 FC /polymer matrix at m _i  = const and to a larger piezoelectric coefficient $g_{h}^{*}$ of the composite at m = const (cf. curves 1 and 5 in Fig. 13.5b). We note that changes in the matrix elastic properties and their anisotropy become appreciable at aspect ratios ρ _i  > 1.

Fig. 13.5

Volume-fraction (m) dependence of hydrostatic piezoelectric coefficients $d_{h}^{*}$ (a in pC/N) and $g_{h}^{*}$ (b in mV · m/N) of the 1–0–3 PZN–0.08PT SC/modified PbTiO₃ FC /polyurethane composite at m _i  = const (reprinted from Topolov et al. [16], with permission from World Scientific)

3.2 Comparison of Data Obtained Using Different Methods

In Sect. 13.3.2, we consider a few effective parameters that were calculated for the 1–0–3 SC/FC /polymer composite by means of different methods. We also compare the effective parameters of the 1–0–3 composite to those known for related piezo-active composites.

First, for the PZN–0.08PT-based composite with oblate FC inclusions in the 0–3 matrix , we see good consistence between the hydrostatic piezoelectric coefficients calculated in two different ways (Table 13.3).

Table 13.3 Hydrostatic piezoelectric coefficient s calculated for the 1–0–3 PZN–0.08PT SC/modified PbTiO₃ FC /polyurethane composite at ρ _i  = 100

Second, we consider a specific case of a performance of a 1–0–3 SC/FC /polymer composite with a piezo-passive 0–3 matrix . It is assumed that the FC inclusions in the polymer matrix (Fig. 13.1) were not previously poled and, therefore, remain piezo-passive. By varying the aspect ratio ρ _i of these inclusions, we see changes in the effective parameters of the 1–0–3 composite (Table 13.4), however these parameters become larger than those in the case of the poled 0–3 matrix . This is a result of a decrease in the dielectric permittivity of the piezo-passive 0–3 matrix that leads to an appreciable decrease of the dielectric permittivity of the 1–0–3 composite $\varepsilon_{33}^{*\sigma }$ at relatively small SC volume fractions m. Results shown in Table 13.4 suggest that the elastic anisotropy in achieving high piezoelectric performance for the 1–0–3 composite remains a dominant factor irrespective of the piezoelectric activity of its 0–3 matrix . Thus, during the manufacture of samples of the 1–0–3 composite at α = 0° (Fig. 13.1), there is no need to pole the 0–3 FC /polymer matrix under a fairly strong electric field, as applied, for instance, to the PbTiO₃-type FC samples [24]. Changes in the volume fraction of FC , m _i , in the 0–3 matrix give rise to weaker changes in the effective parameters of the composite (Table 13.4) in comparison to changes caused by the volume fraction of SC, m. These weaker changes are concerned with the high piezoelectric activity of the SC rods and their parallel orientation along the poling axis: at such an arrangement, even minor changes in m at m ≪ 1 give rise to considerable changes in the piezoelectric performance and figures of merit of the 1–0–3 composite [4, 15–17].

Table 13.4 Effective parameters of the 1–0–3 PMN–0.33PT SC/modified PbTiO₃ FC /polyurethane composite in a case of a piezo-passive 0–3 matrix ^a (reprinted from Topolov et al. [15], with permission from Elsevier)

Third, our results on the performance of the 1–0–3 composites based on SCs are compared to literature data on the 1–3-type piezo-active composites. For instance, work [28] reports a piezoelectric performance of a 1–0–3 FC /FC /polyurethane composite, where the PCR-7 M FC (a PZT-type composition [17]) is the main component. The 1–0–3 PCR-7 M FC /PCR-7 M FC /polyurethane composite is characterized [28] by large piezoelectric coefficients , e.g., $g_{33}^{*}$≈ 400 mV · m/N, $g_{h}^{*}$ ≈ 200 mV · m/N and $d_{h}^{*}$ ≈ 350 pC/N. The aforementioned parameters are comparable to those given in Tables 13.3 and 13.4. According to data [2], max $g_{33}^{*}$ related to a 1–3 PMN–0.30PT SC/epoxy composite is 440 mV · m/N (at the volume fraction of SC m = 0.018) and is comparable to values of $g_{33}^{*}$ from Tables 13.3 and 13.4. However, a 1–3 PMN–zPT-based composite from work [29] is characterized by $d_{h}^{*}$ = 111 pC/N, $g_{h}^{*}$ = 37 mV · m/N and ($Q_{h}^{*}$)² = 4.12 · 10⁻¹² Pa⁻¹, and these effective parameters are smaller than those shown in Tables 13.3 and 13.4.

Fourth, data on the squared hydrostatic figure of merit ($Q_{h}^{*}$)² ~ 10⁻¹² Pa⁻¹ (see Fig. 13.2b and Table 13.4) are to be compared to ($Q_{h}^{*}$)² of the 1–3-type FC -based composites. We note that the ($Q_{h}^{*}$)² values achieved at small volume fractions of the FC inclusions m _i (see Table 13.4) are approximately equal to ($Q_{h}^{*}$)² of a 1–3–0 PZT FC /foamed polymer composite [30].

4 Conclusions

In the present chapter, we have analyzed the new effects in the piezo-active 1–3-type composites. The first effect is the new orientation effect in the 1–0–3 SC/FC /polymer composite (Fig. 13.1) with two contrasting ferroelectric components (a highly piezo-active SC and highly anisotropic FC ) poled on different directions. The second effect is concerned with the active role of the 0–3 matrix and the aspect ratio of its FC inclusions in forming the large hydrostatic parameters of the 1–0–3 SC/FC /polymer composite . Our main research results are concluded as follows.

• 1. Changes in the rotation angle α = (P ⁽¹⁾ _s ^P ⁽²⁾ _r ) (Fig. 13.1) give rise to changes in the elastic and piezoelectric anisotropy of the 0–3 FC /polymer matrix . The (Pb_1−x Ca _x )TiO₃ FC component plays an important role in the orientation effect due to the large anisotropy of the piezoelectric coefficients $d_{3j}^{(2)}$ in comparison to the anisotropy of $d_{3j}^{(1)}$ of the PMN–0.33PT SC, and due to the high degree of the elastic anisotropy.

• 2. The orientation effect studied in this 1–0–3 composite leads to a considerable anisotropy of squared figures of merit (5) and a relatively high piezoelectric activity, and such characteristics are of significant interest for the design of high performance composite structures for hydroacoustic, piezoelectric energy harvesting and transducer applications.

• 3. An important feature of the studied 1–0–3 composite is that elastic properties of its 0–3 matrix considerably depend on the aspect ratio ρ _i of FC inclusions therein and influence the effective electromechanical properties of the 1–0–3 composite in wide aspect-ratio and volume-fraction ranges. This influence becomes important in the presence of two contrasting piezoelectric components. There is no need to pole the 0–3 matrix in a strong electric field, and the high piezoelectric sensitivity is achieved in the case of the non-poled 0–3 matrix with aligned FC inclusions.

• 4. The electromechanical interaction between the highly piezo-active SC component and the anisotropic 0–3 matrix leads to the creation of high-performance 1–0–3 composite structures. The large values of $g_{33}^{*}$ ~ 10² mV · m/N are of significant interest for sensor and receive-type transducer applications, and the large values of $d_{h}^{*}$ ~ 10² pC/N, $g_{h}^{*}$ ~ 10² mV · m/N and ($Q_{h}^{*}$)² ~ 10⁻¹¹ Pa⁻¹ are of value for hydrophone and related hydroacoustic applications. The large values of the squared figure of merit ($Q_{33}^{*}$)² ~ 10⁻¹⁰ Pa⁻¹ and anisotropy ($Q_{33}^{*}$)²/($Q_{31}^{*}$)² ≈ 8–9 are important for piezoelectric energy-harvesting applications.

• 5. Calculations of effective parameters of the studied 1–0–3 composite were performed using the EFM, FEM and matrix method. Changes in the boundary conditions of the unit cell of the 0–3 matrix with spheroidal FC inclusions, i.e., either Dirichlet (FEM-1) or periodic (FEM-2) boundary conditions, do not give rise to considerable changes in $g_{h}^{*}$ and ($Q_{h}^{*}$)² as predicted by means of the FEM. Results obtained by different ways of averaging the properties at the first and second stages are in agreement.