Growth of single crystals in the (Na1/2Bi1/2)TiO3–(Sr1–xCax)TiO3 system by solid state crystal growth

Ceramics based on (Na1/2B1/2)TiO3 are promising candidates for actuator applications because of large strains generated by an electric field-induced phase transition. For example, the (1−x)(Na1/2Bi1/2)TiO3-xSrTiO3 system exhibits a morphotropic phase boundary at x = 0.2–0.3, leading to high values of inverse piezoelectric constant d*33, which can be further improved by the use of single crystals. In our previous work, single crystals of (Na1/2B1/2)TiO3-SrTiO3 and (Na1/2B1/2)TiO3-CaTiO3 were grown by the solid state crystal growth technique. Growth in the (Na1/2B1/2)TiO3-SrTiO3 system was sluggish whereas the (Na1/2B1/2)TiO3-CaTiO3 single crystals grew well. In the present work, 0.8(Na1/2Bi1/2)TiO3-0.2(Sr1−xCax)TiO3 single crystals (with x = 0.0, 0.1, 0.2, 0.3, 0.4) were produced by the solid state crystal growth technique in an attempt to improve crystal growth rate. The dependence of mean matrix grain size, single crystal growth distance, and electrical properties on the Ca concentration was investigated in detail. These investigations indicated that at x = 0.3 the matrix grain growth was suppressed and the driving force for single crystal growth was enhanced. Replacing Sr with Ca increased the shoulder temperature Ts and temperature of maximum relative permittivity Tmax, causing a decrease in inverse piezoelectric properties and a change from normal to incipient ferroelectric behavior.


Introduction 
Restrictions introduced by the European Union and several countries on the use of lead in electronic was discovered at x = 0.2-0.3 [4,5]. The inverse piezoelectric properties at compositions near the MPB were greatly improved due to the presence of a reversible electric field-induced transition between relaxor and ferroelectric phases [4,6]. Although at compositions near the MPB, the electrical properties of the materials were significantly improved, and a shifting of the depolarization temperature T d as well as the rhombohedral-tetragonal phase transition temperature T R-T to lower temperatures was also recognized. This creates a limitation in the temperature operating range of NBT-ST materials, much affecting their practical application [6].
In previous work, single crystals of 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2SrTiO 3 and 0.96(Na 1/2 Bi 1/2 )TiO 3 -0.04CaTiO 3 were grown by the solid state single crystal growth method [7,8]. Single crystal growth in the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2SrTiO 3 system was limited whereas the 0.96(Na 1/2 Bi 1/2 )TiO 3 -0.04CaTiO 3 single crystals grew well. Matrix grain growth behaviour is an important factor in solid state crystal growth, as the driving force for single crystal growth is inversely proportionate to the mean matrix grain size [9]. Numerous works have noted the effect of composition change, dopant, or sintering aid addition on grain growth behaviour in NBT-based ceramics [10][11][12], but less work has been done on the effect of composition on solid state single crystal growth behaviour [13][14][15]. Changing the composition also has an effect on the electrical properties of NBT. For example, doping (Na 1/2 Bi 1/2 )TiO 3 -BaTiO 3 with Nb causes a decrease in grain size from 2.3 to 1.8 µm, a reduction in the peak value of relative permittivity, slimmer polarization-electric field hysteresis loops, and a change of the strain-electric field curves from butterflyto sprout-shaped [16]. La addition to (Na 1/2 Bi 1/2 )TiO 3 -BaTiO 3 resulted in a gradual decrease of grain size, an increase in the temperature of maximum relative permittivity T m , a decrease in room temperature relative permittivity, and decreasing ferroelectric properties [17]. Substitution of (Na 1/2 Bi 1/2 )TiO 3 with BaTiO 3 was found to enhance the inverse piezoelectric properties of 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2SrTiO 3 [18]. By changing the composition, it may be possible to grow larger single crystals of 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2SrTiO 3 by repressing matrix grain growth as well as altering the electrical properties of the single crystal. In the present work Ca is substituted for Sr in 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2SrTiO 3 in order to study the effects on single crystal growth behaviour and electrical properties.

Materials and method
Powders of composition 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 with x = 0.0, 0.1, 0.2, 0.3, 0.4 (labelled x = 0.0 to x = 0.4) were synthesized from Na 2 CO 3 (ACROS Organics, 99.5%) , Bi 2 O 3 (Alfa Aesar, 99.9%) , TiO 2 (Alfa Aesar, 99.8%), SrCO 3 (Aldrich, ≥ 99.9%), and CaCO 3 (Aldrich, ≥ 99.9%) starting materials by solid state reaction. To prepare each powder, the starting materials were dried in an oven at 250 ℃ for 5 h to remove adsorbed water. The stoichiometrically weighed amounts of these starting materials were mixed and ground in high-purity (99.9%) ethanol in a ZrO 2 -lined jar in a planetary ball mill (Planetary Micro Mill Pulverisette 7, Fritsch GmbH, Idar-Oberstein, Germany) for 3 h (with alternating 5 min milling and 1 min rest intervals) at 500 rpm using ZrO 2 media. After milling, most of the ethanol was evaporated using a hot plate and magnetic stirrer. Then, the slurry was dried in an oven at 80 ℃ for 12 h to totally remove any remaining ethanol. The dried slurry was ground in an agate mortar and pestle and sieved through a 180 m sieve to remove agglomerates. Each powder was put into high-purity alumina double crucibles with lids and calcined at 800 ℃ for 5 h in air, with heating and cooling rates of 5 ℃/min. The calcined powders were analyzed by X-ray diffraction (XRD, X'Pert PRO, PANalytical, Almelo, the Netherlands) using Cu K radiation, with a scan range of 20-90 2 and a scan speed of 3 ()/min. The calcined powders were planetary ball milled, ground, and sieved again to break up any agglomerates and reduce the particle size. Particle size analysis of the powders was carried out using SEM micrographs. Transmission electron microscopy (TEM) analysis was carried out on powders with x = 0.0 and x = 0.4 in a transmission electron microscope (FEI TECNAI F 20, FEI Comp., Hillsboro, OR, USA) operated at an accelerating voltage of 200 kV. Powders were dispersed in ethanol and suspended on a porous grid.
To prepare samples of each composition by solid state crystal growth, SrTiO 3 single crystal seeds (MTI Corp., CA, USA) with (001) orientation and dimensions of 5 mm  5 mm  0.5 mm were buried in 0.5 g of prepared powder of the appropriate composition in a 10 mm steel die and pressed by hand into pellets of 10 mm diameter and 4 mm thickness. Then the pellets were pressed in a cold isostatic press (CIP) at 1500 kg/cm (147 MPa). To reduce evaporation of Bi and Na, CIPed pellets were buried in packing powder of www.springer.com/journal/40145 identical composition in double alumina crucibles with lids. Samples were then sintered at 1200 ℃ for 1, 3, 10, and 20 h with heating and cooling rates of 5 ℃/min. The sintered samples were vertically sectioned using a low speed diamond wheel saw, polished to a 1 µm finish with diamond paste, and thermally etched at 1150 ℃. The etched samples were Pt-coated and observed by scanning electron microscopy (SEM, Hitachi S-4700, Tokyo, Japan) with attached energy dispersive X-ray spectrometer (EDS, EMAX energy EX-200, Horiba, Kyoto, Japan). The single crystal growth distance, mean matrix grain size, and matrix grain size distribution are analyzed from the SEM micrographs using imageJ v1.50a image analysis software (National Institute of Mental Health, Bethesda, MD, USA). To analyze single crystal thickness, the single crystal in each micrograph is divided into slices and the area of each slice is measured. By dividing the area of each slice by its width, the average growth distance of the crystal in that slice could be measured. For each sample, the mean and standard deviation of 50 slices are measured. To analyze the mean matrix grain size and grain size distribution, the equivalent 2D radii of at least 200 grains are analyzed for each sample.
The thickness of single crystals grown on (001) SrTiO 3 seed crystals was not sufficient to separate them from the surrounding matrix grains and prepare samples for electrical property measurement. So in order to study crystal structure, dielectric, ferroelectric, and piezoelectric properties, a set of single crystals were grown by solid state crystal growth on (110) SrTiO 3 seed crystals (MTI Corp., CA, USA) at 1200 ℃ for 50 h. For crystal structure study, polycrystalline samples (i.e., without seed crystals) were sintered at 1200 ℃ for 1 h, crushed, and analyzed by powder X-ray diffraction (XRD, X'Pert PRO, PANalytical, Almelo, the Netherlands) using Cu K radiation, a 2 scan range 20-90, step size 0.013, and a scan speed 3 ()/min. Single crystal samples grown on (110) SrTiO 3 seed crystals at 1200 ℃ for 50 h were separated from the surrounding matrix grains and polished with SiC paper up to grade #4000 until the (110) faces of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystal were exposed on both sides,i.e., the SrTiO 3 seed crystal as well as the matrix grains were removed, leaving behind only the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystal. XRD was then carried out as before. For the impedance spectroscopy measurements, single crystal samples were separated from the surrounding matrix grains and were polished with SiC paper up to grade #2000 until 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystal faces with (110) orientation were exposed on both sides as above. Sample width for these samples varied from 2.5 to 4.5 mm. Sample thickness varied from 288 to 410 m, except for the x = 0.1 sample whose thickness was only 83 m. Both sides of each sample were then coated with Pt-paste and the sample was annealed at 900 ℃ for 30 min with heating and cooling rates of 5 ℃/min to prepare Pt electrodes. An impedance analyser (HP4284A, Hewlett-Packard, Kobe, Japan) was used to record the conductance and resistance of each sample at different temperatures and frequencies. Samples were measured in a hot stage (TS1500, Linkam, Tadworth, UK) in the temperature range of 30-780 ℃ in dry air with heating and cooling rates of 1 ℃/min. The AC voltage is set at 0.5 V and the frequency range from 1 MHz to 1 kHz. The data was processed using Origin 9.2 software (OriginLab Corporation, Northampton, MA, USA). The same single crystal samples were used for polarization and strain hysteresis measurements. For polarization and strain hysteresis measurements, the external electric field (E) dependencies of polarization (P) and strain (S) hysteresis were measured at room temperature in a commercial apparatus, aixPES (aixACCT systems GmbH, Aachen, Germany) using a bipolar electric field profile at 1 Hz. The samples were annealed at 200 ℃ for 5 min before measurement. The single crystal samples used for polarization and strain hysteresis measurements are shown in Fig. 12.

Results
XRD patterns of the calcined powders are shown in Fig. 1. Pattern smoothing, background, and K 2 peak removal were carried out using MDI Jade 6.5 (Materials Data Inc., CA, USA). All of the patterns are single phase perovskite and can be indexed to ICDD Card # 89-3109, cubic (Na 1/2 Bi 1/2 )TiO 3 , 3 Pm m space group. XRD patterns of the polycrystalline samples sintered at 1200 ℃ for 1 h are shown in Fig. 2(a). The polycrystalline samples also appear to be single phase perovskite without any secondary phase. All peaks were indexed using pseudocubic indices. The 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystal samples grown on (110) SrTiO 3 seed crystals at 1200 ℃ for 50 h show (110) and (220) peaks of strong intensity, confirming their single crystal nature ( Fig. 2(b)). Rietveld refinement of the XRD pattern for the x = 0.0 polycrystalline sample is shown in Fig. 3. The pattern was fitted with rhombohedral R3c and cubic 3 Pm m phases. Rietveld refinement results of the XRD patterns of the polycrystalline samples are given in Table 1. As x increases, the amount of rhombohedral phase increases.    SEM micrographs of the calcined and ball-milled 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 powders are shown in Fig. 4, along with their corresponding particle size distributions. The mean particle sizes of the powders vary from 60 to 70 nm. The particle size distributions are quite broad, with some particles up to almost 400 nm in diameter. TEM micrographs of the calcined and ball-milled 0.8(Na 1/2 Bi 1/2 )TiO 3 -0. SEM micrographs of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals grown on (001)oriented SrTiO 3 seed crystals are shown in Fig. 6. Single crystal growth distance as a function of sintering time and composition is shown in Fig. 7(a). Data for the x = 0.0 composition is taken from Ref. [15]. Single crystals of all the compositions have grown epitaxially on the (001)-oriented SrTiO 3 seed crystals. The single crystal growth distance of all the samples    A secondary phase appears to exist in the SrTiO 3 seed crystals. This may be an artefact caused by thermal etching [14]. Plate-and rod-shaped secondary phases are also visible in the matrix grains. EDS analysis of these secondary phases shows them to be TiO 2 and an Na 2 O-TiO 2 compound, possibly Na 2 Ti 6 O 13 . Peaks for these secondary phases were not visible in the XRD patterns of the sintered polycrystalline ceramics in Fig. 2(a), indicating that the amount of secondary phase is below the detection limit of the X-ray diffractometer.
SEM micrographs of the matrix grains are shown in Fig. 8. Mean matrix grain size as a function of sintering time and composition is shown in Fig. 7(b). Data for the x = 0.0 composition is taken from Ref. [15]. The grains are equiaxed, with larger grains appearing cubic in shape. The matrix is porous, as are the single crystals. Larger grains have pores trapped inside them. Secondary phases of TiO 2 and the same Na 2 O-TiO 2 compound as in Fig. 7  Grain size distributions of the matrix grains are shown in Fig. 9. Normalized grain size distributions (r/r mean ) are shown in Fig. 10. Data for the x = 0.0 composition is taken from Ref. [15]. For the x = 0.0 sample sintered for 1 h, the grain size distribution is relatively narrow, but some abnormal grains (defined as grains of size greater than three times the mean grain size [19]) are already present. As sintering time increases, the grain size distributions become broader and the number of abnormal grains increases. As the Ca content in the samples increases to x = 0.1, the grain size distributions become narrower and the number of abnormal grains decreases. Abnormal grain growth is on the verge of taking place after 3 h. For the samples with x = 0.2, the grain size distributions broaden again but from the normalized grain size distributions it is seen that no abnormal grains are present even after 20 h. For the samples with x = 0.3, the grain size distributions are narrow and abnormal grains appear after 3 h. For the samples with x = 0.4, the grain size distributions broaden again but again no abnormal grains are present.
Relative permittivity and loss tangent vs. temperature curves of the single crystal samples are shown in Fig. 11. All of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals show a broad peak in relative permittivity without frequency dispersion at a temperature T max and a shoulder at lower temperature T s which shows frequency dispersion. Values of T s and T max are given in Table 2. T max initially increases with replacement of Sr by Ca (x = 0.1 sample). Further substitution with Ca causes T max to decrease (x = 0.2)     Figure 12 shows polarization vs. electric field (PE) curves of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals. The x = 0.1 single crystal sample of thickness 83 m was too thin to measure due to electrical breakdown. When the sample is very thin, open pores in the sample can cause a significant reduction in thickness and a large increase in the local electric field. This is exacerbated by the large pore size in the single crystals due to pore coalescence (Fig. 6) [7,20]. Values of saturation polarization P max , remnant polarization P r , and coercivity E c are shown in Fig. 13(a). Replacement of Sr with Ca causes P max to initially decrease and then increase again for x = 0.4. Remnant polarization P r also initially decreases from   difference in strain between the upper and lower parts of the curve when E = maximum positive electric field E max /2 (x = 0.2) (Fig. 13(b)) [21]. Further substitution of Ca causes an increase in *

Discussion
Initially, (Na 1/2 Bi 1/2 )TiO 3 was determined to have a pseudocubic [22] or rhombohedral structure with R3c or R3m [23,24] space group at room temperature according to X-ray diffraction studies. The R3c space group was also favored by neutron diffraction studies [24]. Transmission electron microscopy studies, on the other hand, supported a coexistence of rhombohedral (R3c) and tetragonal (P4bm or P4 2 nm) phases [25]. Recent high resolution X-ray diffraction studies proposed a monoclinic Cc space group [26], which was further supported by optical studies [27]. However, this assignment does not fully explain the (Na 1/2 Bi 1/2 )TiO 3 structure and the room temperature phase may consist of R3c and Cc phases [28].
Formation of a solid solution with SrTiO 3 causes the structure to change from rhombohedral to pseudocubic as SrTiO 3 concentration in the NBT-ST system increases, with a morphotropic phase boundary (MPB) at x = 0.2-0.3 [4,5]. When Sr was substituted with Ca the fraction of rhombohedral phase increased (Table 1). This can be explained on the basis of the smaller ionic radius of Ca compared to Sr (0.134 nm vs. 0.144 nm for coordination number = 12) [29]. This causes a decrease in the Goldschmidt tolerance factor (using an ionic radius of 0.135 nm for Bi 3+ with coordination number = 12 [30]) from 0.979 (x = 0.0) to 0.977 (x = 0.4). The MPB composition for binary (Na 1/2 Bi 1/2 )TiO 3based solid solutions appears for tolerance factors between 0.982 and 0.986 [30,31], with similar results for ternary systems [32,33]. Hence the decrease in tolerance factor as Sr is substituted with Ca moves the structure further into the rhombohedral phase field of the (Na 1/2 Bi 1/2 )TiO 3 -SrTiO 3 phase diagram.
Single crystal and matrix grain growth behaviour in the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 system is controlled by the structure of the interfaces between the grains, which can be disordered (rough) or ordered (smooth/faceted) on the atomic scale [34,35]. The structure of the grain interfaces in turn decides whether grain growth is controlled by diffusion of atoms between the shrinking and growing grains or by interface reaction at the surface of the growing grains. The faceted shape of the powder particles (Fig. 5) indicates that the interfaces will be ordered (faceted), as has been previously found in NBT-based systems [15,19,20]. The grain growth behaviour will then be controlled by interface reaction. For a system with solid/liquid interfaces, the driving force G for grain growth of a particular grain is given by [36,37]: where γ is the solid/liquid interfacial energy, r the radius of the grain, V m the molar volume, and r the radius of the critical grain (a grain with G = 0, i.e., that is neither growing nor shrinking). For a system where growth is controlled by diffusion, r is the mean grain radius r mean , but for a system where growth is controlled by interface reaction, r is smaller than r mean [37]. The seed crystal has r >> r and so the driving force for the single crystal is approximately [9]: If the system has disordered interfaces, easy atom attachment to or detachment from the surface of the grains means that the rate of atomic diffusion across the solid/liquid interface between the shrinking and growing grains limits grain growth [35,36]. The growth rate of any particular grain will increase linearly with its driving force (the dotted black line in Fig. 14(a)) and any grain with G > 0 can grow. In this case, the normalized size distribution of the grains ( mean / r r ) is invariant with sintering time [35]. If the system has ordered interfaces, atoms that attach to the surface of the grain have a large number of broken bonds. They are unstable and will detach from the surface unless they can migrate to a favorable site such as a two-dimensional nucleus or a screw dislocation. The steps formed by such sites contain many kinks where atoms can attach without changing the interfacial energy of the grain [38]. The rate at which atoms can attach to the kink sites now limits grain growth, causing it to have a nonlinear relationship with driving force [37,39,40]. For a system where grain growth is controlled by the nucleation and growth of 2D nuclei, the rate of formation of stable 2D nuclei is very low below ΔG c , the critical driving force. Grain growth is almost negligible because there are few kink sites available for atoms to attach to. At the critical driving force ΔG c , the rate at which stable nuclei form increases exponentially. The grain can now grow because many kink sites are present. The grain growth rate therefore depends exponentially on ΔG (the dashed red line marked 2DNG in Fig. 14(a)) [41,42]. For ΔG > ΔG c , kinetic roughening takes place [38,41,43]. Atoms can easily attach to the grain surface due to the presence of many nuclei, causing grain growth to become limited by diffusion. The grain growth rate is almost zero at ΔG < ΔG c , increases exponentially at ΔG  ΔG c and then increases linearly with further increase in ΔG (the solid red line in Fig. 14(a)). Grains with G < 0 will shrink following a linear dependence with G, as atoms can detach from each corner of the grain without increasing the grain's interfacial energy and multilayer dissolution can occur [35,36,39]. The value of ΔG c for 2D nucleationcontrolled growth is [35,39]: where ε is the step free energy (the excess energy caused by the edge of the step), k is Boltzmann's constant, h is the step height of the 2D nucleus, and n 0 is the number density of atoms in the liquid. ψ = n * νexp(Δg m /kT), where n * is the number of atoms in a position near to a critical 2D nucleus, ν is the vibration frequency of atoms in the liquid, and Δg m is the activation energy for jumping across the liquid/solid interface. ε varies with composition, dopant addition, sintering temperature, and atmosphere [44][45][46]. This type of grain growth behviour, interfacereaction limited below ΔG c and diffusion-limited above ΔG c , is known as mixed control growth. When grain growth is controlled by mixed control growth, the grain growth behaviour depends on the relative values of G c and G max , where G max is the maximum driving force for the largest grain in the system [35,36,47]. If G c  G max , a large number of grains have G > G c and can grow. As a result, the growth behaviour is similar to normal grain growth behaviour (pseudo-normal). If G c  G max , the grain growth is abnormal. Most of the matrix grains have G < G c and grow slowly or not at all. A small number of grains have G ≥ G c . These grains grow rapidly and can become much larger compared to other grains, becoming abnormal grains. If G c  G max , none of the grains can appreciably grow and grain growth is stagnant. In the solid state crystal growth method, the seed crystal behaves like an abnormal grain with driving force > G c [48][49][50]. A single crystal of the matrix composition grows on the seed crystal. Matrix grains that are large enough to have G ≥ G c can also grow to form abnormal grains. The samples sintered in the present work were single phase and nominally free of a liquid phase. However, previous work in NBT-based compositions showed an amorphous phase at multi-grain junctions [15,19], indicating that a liquid phase may be present during sintering. Similar grain growth behaviour has also been found to occur during solid-state sintering [39,[51][52][53][54][55][56][57][58][59][60][61].
Screw dislocations terminating at the grain surface form constantly regenerating ledges for easy atom attachment [40,43]. The grain growth rate then increases parabolically with ΔG (the dashed blue line marked SDG in Fig. 14(a)) [43]. The growth rate of the grain varies parabolically with ΔG for ΔG < ΔG c and linearly with ΔG for ΔG > ΔG c (the solid blue line in Fig. 14(a)). At ΔG > ΔG c , kinetic roughening takes place as before and grain growth becomes diffusion controlled [40]. The difference in growth rate between grains with ΔG < ΔG c and grains with ΔG > ΔG c will be less drastic than in the case for 2D nucleation controlled growth.
The step free energy  is important as its magnitude determines the value of ΔG c and the grain growth behaviour. The magnitude of  can be estimated from the macroscopic grain shape [62][63][64]. The grain shape in the 0.95(Na 1/2 Bi 1/2 )TiO 3 -0.05BaTiO 3 system was found to be cubic with faceted {100} faces and rounded corners [19]. Our previous work on the NBT-ST system showed that the grains are cubic in shape, with flat faces and rounded edges and corners [15,20]. This indicates that NBT-based compositions have a moderate value of  [62,64]. Grain growth behaviour in NBT-based systems tends to be initially pseudo-normal but changes to abnormal after extended sintering [15,19]. Figure 14(b) shows a schematic plot of growth rate vs. driving force for the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 compositions in the present work. Grain growth is assumed to be 2D-nucleation controlled growth due to the absence of screw dislocations (Fig. 5). From the particle size distributions in Fig. 4, it can be seen that the initial mean grain size is submicron for all the compositions and the particle size distributions are broad. The small values of r mean indicates that the grains with r > r mean initially have a high driving force, as G is inversely proportional to r mean (Eq. (1) and Refs. [9,36]). For all the samples, at the beginning of sintering there are many grains with ΔG > ΔG c and so the matrix grain growth behaviour is initially pseudo-normal. For the sample with x = 0.0, as sintering proceeds the mean grain size r mean increases, causing G max and ΔG for all the grains to decrease. As ΔG for a particular grain decreases, its growth rate decreases linearly. Once ΔG drops below ΔG c , the growth rate for that grain no longer decreases linearly but exponentially. As the number of grains with ΔG > ΔG c decreases, the grain growth behaviour shifts from pseudo-normal to abnormal [19,36]. Most of the grains now have ΔG < ΔG c and are growing very slowly. A few grains have ΔG > ΔG c and can grow relatively quickly to form abnormal grains. For the samples with x = 0.0, this has begun to take place by 1 h. As sintering continues, both the normal grains and the abnormal grains continue to grow (very slowly and quickly, respectively) and the grain size distribution broadens and its peak shifts further to the right. The value of r mean will level off with sintering time, as can be seen in Fig. 7(b).
The behaviour of the samples with x = 0.1 is consistent with a decrease in the step free energy . The value of the critical driving force ΔG c shifts to the left ( Fig. 14(b)). The value of ΔG max is not affected, as this depends on the particle size distribution of the starting powder, which does not change much with composition (Fig. 4). The lower value of ΔG c means that pseudonormal grain growth can continue for longer before the driving force of most of the grains drops below ΔG c and abnormal grain growth begins. Abnormal grain growth is just about to start after 3 h of sintering (Fig. 10).
Because the value of ΔG c is reduced, all the grains will have shifted to lower values of driving force and growth rate by the time that abnormal grain growth starts. Hence after abnormal grain growth starts, grain growth is more limited than in the case of the x = 0.0 samples. The matrix grain size remains smaller than in the case of the x = 0.0 samples and the grain size distributions remain narrower. As Ca concentration increases to x = 0.2, step free energy  decreases further and ΔG c shifts to even lower values. Because the value of ΔG c is very low, grain growth behaviour is pseudo-normal even up to 20 h. The grain size distribution changes slowly and retains a unimodal appearance. It is notable that the mean grain size after 1 h is larger than the x = 0.0 and x = 0.1 samples. This may be due to the larger number of grains The seed crystal acts as an abnormal grain and a single crystal of the matrix composition grows on the seed during sintering. The driving force for growth of the single crystal is inversely proportional to the mean grain size (Eq. (2)) and will always be larger than ΔG max , the driving force for growth of the largest matrix grain.
For the x = 0.0 samples, pseudo-normal grain growth initially takes place in the matrix. Although the single crystal is also able to grow, it does not have a large advantage in growth rate over the matrix grains and so its growth will be limited [34]. Only when the driving force of most of the grains drops below ΔG c will the single crystal have a significant growth advantage and grow rapidly. This corresponds with the start of abnormal grain growth in the matrix. From Fig. 7 and Fig. 9 it can be seen that this has happened by 1 h. As the mean matrix grain size increases with sintering time, the growth rate of the single crystal decreases, as can be seen in Fig. 7(a) for the x = 0.0 samples.
The crystal growth distance of the x = 0.1 sample is lower than that of the x = 0.0 sample, despite the fact that the mean matrix grain size is similar or smaller. This may be because the onset of abnormal grain growth is delayed to 3 h. During the pseudo-normal growth stage, the single crystal has to compete with a larger number of growing matrix grains for material from the shrinking grains, limiting its growth. By the time abnormal grain growth starts, the driving force for growth of the single crystal has shifted to a lower value than that of the x = 0.0 sample, reducing the growth rate of the single crystal even after abnormal grain growth starts. However, the small value of r mean compared to the x = 0.0 samples allows the single crystal to continue growing even up to 20 h and eventually its growth distance overtakes that of the x = 0.0 sample. For the x = 0.2 samples, the large values of r mean limit the single crystal growth rate. For the x = 0.3 samples, the small values of r mean allow the single crystal to grow more rapidly than the other compositions due to its increased driving force for growth (Eq. (2)). For the x = 0.4 samples, the large values of r mean limit single crystal growth, similar to the x = 0.2 samples.
The change in  with change in composition is difficult to predict. The effect of solid solution formation on  has been studied in several systems [34,46,65,66].  contains enthalpy h s and entropy s s terms and is given by  = h s -Ts s [64]. Substitution of Sr with Ca initially increases configurational entropy, causing an increase in s s and a decrease in  [67].
Composition changes also affect interatomic bond strengths and specific surface energies, further affecting [68] will decrease h s , also causing a decrease in . Initially, substitution of Sr with Ca will decrease .
Further addition of Ca has less effect on configurational entropy [67], causing the rate of decrease of  to slow down. The reason for the apparent increase in  for the x = 0.3 sample is not known.
All of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals show behaviour typical of (Na 1/2 Bi 1/2 )TiO 3 -type ferroelectric relaxors, i.e., a broad peak in relative permittivity without frequency dispersion at a temperature T max and a shoulder at lower temperature T s which shows frequency dispersion [69][70][71]. The x = 0.0 sample shows a typical PE hysteresis loop for a normal ferroelectric material (Fig. 12). Similar to other NBT-based materials, the x = 0.0 sample at room temperature is in a non-ergodic relaxor state and can be transformed into a stable ferroelectric state by the electric field [70][71][72]. The sample then behaves like a normal ferroelectric material. The effect of Ca substitution on the dielectric and electromechanical behaviour of the single crystals appears contradictory at first sight. As Ca content increases from x = 0.0 to x = 0.3, the PE loops become increasingly pinched and the SE curves change from butterfly-to sprout-shaped (Fig. 12). This indicates that the material is changing from a non-ergodic relaxor to an ergodic relaxor and that Ca substitution is destabilizing the ferroelectric ordering of the material [73,74]. However, T s and T max also increase with Ca substitution (Fig. 11 and Table 2), which indicates the opposite. The origin of T s is as yet unclear. It may be caused by a transition from the rhombohedral phase to tetragonal or mixed rhombohedral and tetragonal phases [24,75,76], thermal evolution of mixed rhombohedral and tetragonal polar nanoregions [70], a transition from non-ergodic to ergodic relaxor behaviour [77] or from a ferroelectric to relaxor ferroelectric state [78]. A decrease in T s is usually associated with the change of the bipolar strain hysteresis curves from butterfly to sprout-shaped and with the change from a non-ergodic to ergodic relaxor material [4,79,80]. However, in the present case T s is increasing.
The change in the behaviour of the materials from a non-ergodic to ergodic relaxor material may be due to the reduction in tolerance factor caused by replacing Sr with Ca [81,82]. Reduction of the Goldschmidt tolerance factor destabilizes the non-ergodic ferroelectric behaviour [83]. The lower polarizability of Ca compared to Sr [1] may also weaken ferroelectric ordering in the material, making it easier for the electric field induced ferroelectric phase to return to an ergodic relaxor phase upon removal of the electric field, leading to a reduction in S neg . The asymmetry of the PE loop for the x = 0.2 sample and the asymmetry of the SE curves may be caused by defects such as vacancies which pin the domain walls of the electric field induced ferroelectric phase [84]. Although the chemical composition of the crystals in this study was not measured, previous studies have shown that NBT-based single crystals grown by solid state crystal growth are Na deficient [15,20,85].
The piezoelectric properties of the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals are inferior to those of other NBT-based single crystals such as (Na 0.5 Bi 0.5 )TiO 3 -BaTiO 3 * 33 (d = 200-450 pm/V) [76,86] and (Na 0.5 Bi 0.5 )TiO 3 -BaTiO 3 -(K 0.5 Na 0.5 )NbO 3 (S pos = 0.57%-0.9%) [21,[87][88][89][90]. This may partially be because the 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals have (110) orientation, which has relatively poor inverse piezoelectric properties in NBT-based single crystals [76,85,88,90]. It may also be due to the effect of Ca addition on the structure of the single crystals. From the Rietveld refinement results of the polycrystalline samples, it can be seen that replacing Sr with Ca increases the amount of rhombohedral phase in the material (Table 1). This is also consistent with the increase in T s if T s is assumed to be the temperature of transition from the rhombohedral phase to mixed rhombohedral and tetragonal phases ( Fig. 11 and Table 2). This may explain the reduction in * 33 d and S pos as Ca is substituted for Sr. Several workers have studied the electric field-induced phase transition in NBT-based materials. Hinterstein et al. [91] used in-situ X-ray and neutron diffraction techniques to study the electric field-induced phase transition in 0.92(Bi 1/2 Na 1/2 )TiO 3 -0.06BaTiO 3 -0.02(K 1/2 Na 1/2 )NbO 3 . They found a fieldinduced phase transition from a nonpolar pseudocubic (tetragonal) P4bm to a polar rhombohedral R3c phase. Chen et al. [90] studied the orientation dependence of the electric field-induced phase transition in single crystals of the same composition using in-situ Raman scattering and X-ray diffraction measurements. They found that application of the electric field in the [001] direction caused a phase transition from nonpolar tetragonal P4bm to polar tetragonal P4mm, with a corresponding large strain. Application of the electric field in the [111] direction however caused a partial transition to a rhombohedral phase with very little induced strain. Ge et al. [76] studied the electric fieldinduced phase transition of a 0.944(Bi 1/2 Na 1/2 )TiO 3 -0.056BaTiO 3 single crystal at different temperatures. At temperatures close to T s , an electric field induced phase transition from a nonpolar tetragonal P4bm phase to a polar tetragonal phase was observed, with a corresponding large electric field induced strain. From these studies it appears that the presence of a pseudocubic or a tetragonal P4bm phase is necessary to obtain large electric field induced strains in NBT-based single crystals. In the present work, replacing Sr with Ca reduces the amount of pseudocubic or tetragonal phase, thereby reducing the inverse piezoelectric properties.

Conclusions
The effect of replacing Sr with Ca on the matrix grain growth, single crystal growth, and electrical properties of 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 1-x Ca x )TiO 3 single crystals (with x = 0.0-0.4) has been studied. An increase in the Ca content causes changes in matrix grain growth behaviour from abnormal to pseudo-normal behaviour, with an increase in mean matrix grain size and a corresponding reduction in single crystal growth. The 0.8(Na 1/2 Bi 1/2 )TiO 3 -0.2(Sr 0.7 Ca 0.3 )TiO 3 composition shows anomalous behaviour, with reduced mean matrix grain size and enhanced single crystal growth. XRD of polycrystalline samples shows them to contain mixed rhombohedral and cubic phases. Increasing Ca content causes an increase in the amount of rhombohedral phase. Increasing Ca content causes an increase in the shoulder temperature T s and temperature of maximum relative permittivity T max of the single crystals, a decrease in inverse piezoelectric properties, and a change from normal to incipient ferroelectric behaviour. This is likely due to the reduction of the Goldschmidt tolerance factor and the increased amount of rhombohedral phase in the single crystals.