Determination of the Mechanical Properties of PIN–PMN–PT Bulk Single Crystals by Nanoindentation

: The present study aimed to experimentally evaluate the mechanical properties of Pb(In 1 / 2 Nb 1 / 2 )O 3 –Pb(Mg 1 / 3 Nb 2 / 3 )O 3 –PbTiO 3 (PIN–PMN–PT) bulk single crystals with di ﬀ erent crystallographic directions using the nanoindentation technique. The load – indentation depth curves, elastic and plastic deformations, hardnesses, and Young’s moduli of [100]- and [110]-oriented 0.28PIN–0.43PMN–0.29PT bulk single crystals were investigated. Our results show that with an increase in the maximum indentation depth h max , the plastic residual percentage increased for both the [100]- and the [110]-oriented single crystals. At each h max , the plastic residual percentage of the [100]-oriented PIN–PMN–PT single crystals was less than that of the [110]-oriented PIN–PMN–PT single crystals. At h max from 500 nm to 2000 nm, the plastic deformation was larger than the elastic deformation, and the plastic residual percentage was larger than 50% for both the [100]-and the [110]-oriented single crystals. This means that the plastic deformation dominated in the indentation process of PIN–PMN–PT single crystals. The indentation size e ﬀ ect on the hardness of the PIN–PMN–PT single crystals was apparent in the nanoindentation process. Both the hardness and the Young’s modulus of the [100]-PIN–PMN–PT single crystals were greater than those of the [110]-PIN–PMN–PT single crystals, which indicates that the PIN–PMN–PT single crystals had anisotropic mechanical characteristics.


Introduction
In the last 15 years, because of their excellent piezoelectric properties, relaxor-based ferroelectric single crystals such as binary Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 (PMN-PT) and ternary Pb(In 1/2 Nb 1/2 )O 3 -Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 (PIN-PMN-PT) have attracted extensive attention and become important materials in the fabrication of new-generation high-performance electromechanical devices including ultrasound transducers, actuators, sensors, and other electromechanical devices [1][2][3][4][5][6][7][8][9]. It is worth noting that due to the high toxicity of lead, the traditional lead-containing ferro-piezoelectric materials are very harmful to the environment and human health. Lead-free materials are therefore the future developmental direction of ferro-piezoelectric materials [10][11][12][13]. However, current lead-free piezoelectric materials have some defects which limit their applications, and so lead-containing materials continue to dominate in a number of applications. With its concentration of PT near the morphotropic phase boundary region, the ternary PIN-PMN-PT single crystal not only has similar piezoelectric performance to that of the binary PMN-PT single crystal but also possesses higher phase transition temperature (>120 • C) and coercive field (~6 kV/cm) than the binary PMN-PT single crystal. These properties are very crucial to many electromechanical devices, especially when applied to high-drive and high-temperature environments [14][15][16].
In practical applications, relaxor-based ferroelectric single crystals are expected to endure mechanical loading [17][18][19][20]. Moreover, mechanical loading may also degrade the performance of ferroelectric single-crystal devices, eventually leading to device failure during fabrication or service [21][22][23][24]. Understanding the mechanical properties of relaxor-based ferroelectric PIN-PMN-PT single crystals is therefore key to improving the performance and optimizing the design of these devices.
Nanoindentation is a convenient and reliable examination method to investigate mechanical properties with small volumes and has been widely used to examine various materials [25][26][27][28][29]. Nanoindentation is a continuous measurement of the load and displacement.
Then, nano-mechanical properties such as hardness, Young's modulus, indentation strain-rate sensitivity, and creep can be measured from the load-displacement curve. The mechanical behavior of the relaxor-based ferroelectric single crystals has been studied by some researchers. Gailard et al. presented a study of the deformation mechanisms of BaTiO 3 giant single crystals with known orientations using nanoindentation and an atomic force microscope [30]. Fang et al. investigated the interactions between the crack propagation and domain switching in poled BaTiO 3 single crystals under a mechanical load [31]. Babu et al. performed the Vickers microhardness test on a <001>-cut Na 1/2 Bi 1/2 TiO 3 single crystal and calculated its fracture toughness and brittleness index from the R-curve [32]. Zhou et al. investigated the electric field-dependent mechanical properties of the PMN-PT single crystal by electromechanical-coupled nanoindentation [33]. Zeng et al. investigated the deformation and mechanical properties of a (001)-oriented single crystal of PZN-0.07PT using a combination of nanoindentation, micro-indentation, and three-point bending tests [34]. These studies examined in part the mechanical behavior of the relaxor-based ferroelectric single crystals, but there has been little investigation into the mechanical properties of ternary PIN-PMN-PT single crystals using the nanoindentation technique.
In the present work, the anisotropic nano-mechanical properties of relaxor-based ferroelectric PIN-PMN-PT bulk single crystals with different crystallographic directions were experimentally investigated using nanoindentation testing. The load-indentation depth curves, elastic and plastic deformations, hardnesses. and Young's moduli of the [100]-and [110]-oriented PIN-PMN-PT bulk single crystals were evaluated.

Experimental Method
The ferroelectric 0.28PIN-0.43PMN-0.29PT single crystals used in this study were grown using a modified Bridgeman method and were provided by the Shanghai Institute of Ceramics, Chinese Academy of Sciences. High-purity (99.99%) oxide powders of PbO, In 2 O 3 , Nb 2 O 5 , MgO, and TiO 2 were mixed stoichiometrically and ground to reduce the particle size. The resulting powder was sintered for 8 hours at 1000 • C. The sintered material was put into a platinum crucible, and the crucible was sealed and placed inside a furnace. The sintered material was heated up to 1400 • C, and the melt was equilibrated for 10 hours. Lastly, the crucible was lowered at 0.5 mm/hour, and the crystal was grown. The PIN-PMN-PT single crystal was cut into a rectangular specimen with dimensions of 6 × 4 × 0.5 mm 3 . The crystallographic direction of the main plane ( For the purpose of consistency and accuracy of the nanoindentation testing results, the main planes (6 × 4 mm) of the PIN-PMN-PT single-crystal specimens were ground and mechanically and electrolytically polished to a near-mirror finish. Figure 1 shows the X-ray diffraction (XRD) patterns from the main planes of [100]-and [110]-oriented PIN-PMN-PT single crystals by using X-ray diffractometer SmartLab 9KW (Rigaku, Akishima, Tokyo, Japan). It reveals that the PIN-PMN-PT single crystals with two different crystallographic directions in this study were exactly in the [100] and [110] directions and additionally had good phase purity.
Crystals 2020, 10, x FOR PEER REVIEW 3 of 11 electrolytically polished to a near-mirror finish. Figure 1 shows the X-ray diffraction (  The mechanical properties of the PIN-PMN-PT single crystals were characterized using a Nano indenter G200 (Agilent Technologies, Santa Clara, CA, USA) with a standard diamond Berkovich indenter (Agilent Technologies, Santa Clara, CA, USA) at room temperature. For the Berkovich indenter, the centerline-to-face angle, centerline-to-edge angle, and radius at the tip were 65.3°, 77.05°, and 50 nm, respectively. The Nano indenter G200 had a maximum load of 500 mN and a maximum indentation depth greater than 500 μm. With a load resolution of 50 nN and displacement resolution less than 0.01 nm, it could accurately test the mechanical behavior of the material at both the micro-and the nano-scale. The nanoindentation measurements were conducted under continuous stiffness measurement (CSM) procedures [35][36][37]. The indentation loads acted on the main planes (6 × 4 mm) of the [100]-and [110]-oriented PIN-PMN-PT single crystal samples at a constant strain rate of 0.05 s −1 with four maximum indentation depths of 500 nm, 1000nm, 1500nm, and 2000 nm. When the indentation depth reached the maximum value, the corresponding peak load was held for 10 s and then unloaded at the same strain rate. More than five indents at each maximum indentation depth were performed on the samples to confirm the reliability and repeatability of the nanoindentation tests. For the sake of eliminating interactions among the indentation points, the spacing between each two neighboring indentation points was set to 100 µ m.
During indenting, loads and displacements were recorded to generate load-indentation depth curves. The Oliber-Pharr method was applied to determine the hardness (H) as well as the Young's modulus (E) [38,39]. The H and E were calculated using the following relationships: The mechanical properties of the PIN-PMN-PT single crystals were characterized using a Nano indenter G200 (Agilent Technologies, Santa Clara, CA, USA) with a standard diamond Berkovich indenter (Agilent Technologies, Santa Clara, CA, USA) at room temperature. For the Berkovich indenter, the centerline-to-face angle, centerline-to-edge angle, and radius at the tip were 65.3 • , 77.05 • , and 50 nm, respectively. The Nano indenter G200 had a maximum load of 500 mN and a maximum indentation depth greater than 500 µm. With a load resolution of 50 nN and displacement resolution less than 0.01 nm, it could accurately test the mechanical behavior of the material at both the microand the nano-scale. The nanoindentation measurements were conducted under continuous stiffness measurement (CSM) procedures [35][36][37]. The indentation loads acted on the main planes (6 × 4 mm) of the [100]-and [110]-oriented PIN-PMN-PT single crystal samples at a constant strain rate of 0.05 s −1 with four maximum indentation depths of 500 nm, 1000nm, 1500nm, and 2000 nm. When the indentation depth reached the maximum value, the corresponding peak load was held for 10 s and then unloaded at the same strain rate. More than five indents at each maximum indentation depth were performed on the samples to confirm the reliability and repeatability of the nanoindentation tests. For the sake of eliminating interactions among the indentation points, the spacing between each two neighboring indentation points was set to 100 µm.
During indenting, loads and displacements were recorded to generate load-indentation depth curves. The Oliber-Pharr method was applied to determine the hardness (H) as well as the Young's modulus (E) [38,39]. The H and E were calculated using the following relationships: Crystals 2020, 10, 28 where P max is the maximum applied load, and A is the projected contact area between the indenter tip and the specimen at the maximum load; h c is the contact indentation depth determined from load-indentation depth curves; E r is the reduced modulus; E i and ν i are Young's modulus and Poisson's ratio for the diamond indenter, with E i = 1141 GPa, and ν i = 0.07; S is the initial unloading stiffness determined by the slope of the unloading curve dP/dh at the maximum load, and β is the correction factor that depends on the geometry of the indenter (for the Berkovich tip, β is 1.034). Figure 2 shows the representative load-indentation depth curves at different maximum indentation depths for the [100]-and [110]-oriented PIN-PMN-PT bulk single crystals. It is observed that the load-indentation depth curves for each maximum indentation depth closely overlap when the maximum indentation depth h max is below 1500 nm. The difference between the load-indentation depth curves at the h max of 2000 nm is very small. Thus, the load-indentation depth curves exhibit a low noise level and small variation in the data. These features reflect the good phase purity and crystal uniformity of the PIN-PMN-PT bulk single crystal, which is consistent with the XRD results above.

Load-Indentation Depth Curves
Crystals 2020, 10, x FOR PEER REVIEW 4 of 11 where Pmax is the maximum applied load, and A is the projected contact area between the indenter tip and the specimen at the maximum load; hc is the contact indentation depth determined from load-indentation depth curves; Er is the reduced modulus; Ei and i  are Young's modulus and Poisson's ratio for the diamond indenter, with Ei = 1141 GPa, and i  = 0.07; S is the initial unloading stiffness determined by the slope of the unloading curve dP/dh at the maximum load, and β is the correction factor that depends on the geometry of the indenter (for the Berkovich tip, β is 1.034). With the indentation depth increasing, the load increased gradually at each hmax during the loading process, and with the hmax increasing, the corresponding peak load also gradually increased. It can be seen that when the hmax increased from 500 nm to 2000 nm, the corresponding peak load increased from 34. When the indentation depth reached the hmax, the corresponding peak load was held for 10 s and then unloaded. Only the elastic indentation depth recovered in the unloading process. After unloading, the indentation depth did not fully recover, and residual indentation depth remained, which indicated irreversible plastic deformation in the PIN-PMN-PT bulk single crystal during the loading process. In our experiment, there was no discontinuity in the load-indentation depth curves With the indentation depth increasing, the load increased gradually at each h max during the loading process, and with the h max increasing, the corresponding peak load also gradually increased. It can be seen that when the h max increased from 500 nm to 2000 nm, the corresponding peak load increased from 34. When the indentation depth reached the h max , the corresponding peak load was held for 10 s and then unloaded. Only the elastic indentation depth recovered in the unloading process. After unloading, the indentation depth did not fully recover, and residual indentation depth remained, which indicated irreversible plastic deformation in the PIN-PMN-PT bulk single crystal during the loading process. In our experiment, there was no discontinuity in the load-indentation depth curves within an indentation depth of 2000 nm, which indicated that large cracks were not formed in the PIN-PMN-PT single crystal during the loading and unloading process. within an indentation depth of 2000 nm, which indicated that large cracks were not formed in the PIN-PMN-PT single crystal during the loading and unloading process.  [40]. The formation of these non-180 domain walls may cause irreversible depolarization, relieving the internal stress built up in the PIN-PMN-PT bulk single crystals [41]. This reduces the driving force for the recovery of the PIN-PMN-PT bulk single crystal. Therefore, at the maximum indentation depth of 500 nm to 2000 nm, the plastic deformation is larger than the elastic deformation, and the plastic residual  [40]. The formation of these non-180 domain walls may cause irreversible depolarization, relieving the internal stress built up in the PIN-PMN-PT bulk single crystals [41]. This reduces the driving force for the recovery of the PIN-PMN-PT bulk single crystal. Therefore, at the maximum indentation depth of 500 nm to 2000 nm, the plastic deformation is larger than the elastic deformation, and the plastic residual percentage is larger than 50%, which means that the plastic deformation dominated in the indentation process of the PIN-PMN-PT bulk single crystal. percentage is larger than 50%, which means that the plastic deformation dominated in the indentation process of the PIN-PMN-PT bulk single crystal.   It can be seen that when the indentation depth was less than 30 nm, the fluctuation of the data was somewhat large, which may be related to the resolution of the nano-indenter, the smoothness of the specimen surface, and environmental noise when searching for the initial contact position during the nanoindentation test. However, when the indentation depth was more than 30 nm, the hardness-indentation depth curves and Young's modulus-indentation depth curves featured small variation in the data at the same range of indentation depth.   It can be seen that when the indentation depth was less than 30 nm, the fluctuation of the data was somewhat large, which may be related to the resolution of the nano-indenter, the smoothness of the specimen surface, and environmental noise when searching for the initial contact position during the nanoindentation test. However, when the indentation depth was more than 30 nm, the hardness-indentation depth curves and Young's modulus-indentation depth curves featured small variation in the data at the same range of indentation depth.

Hardness and Young's Modulus
percentage is larger than 50%, which means that the plastic deformation dominated in the indentation process of the PIN-PMN-PT bulk single crystal.   Figures 5 and 6 show the hardness-indentation depth curves and Young's modulus -indentation depth curves, respectively, of the [100]-and [110]-oriented PIN-PMN-PT single crystals at the maximum indentation depths from 500 nm to 2000 nm. It can be seen that when the indentation depth was less than 30 nm, the fluctuation of the data was somewhat large, which may be related to the resolution of the nano-indenter, the smoothness of the specimen surface, and environmental noise when searching for the initial contact position during the nanoindentation test. However, when the indentation depth was more than 30 nm, the hardness-indentation depth curves and Young's modulus-indentation depth curves featured small variation in the data at the same range of indentation depth.           Figure 7, it can be seen that with increasing indentation depth from 30 nm to 500 nm, the hardness decreased from 16.92 GPa to 7.50 GPa for the [100]-oriented PIN-PMN-PT bulk single crystal and from 8.46 GPa to 6.47 GPa for the [110]-oriented PIN-PMN-PT bulk single crystal. This indicated that the hardness decreased dramatically with the increase of the indentation depth from 30 nm to 500 nm. When the indentation depth exceeded 500 nm, the decreasing tendencies gradually slowed. When the indentation depth was more than 1300 nm, the hardness tended to be stable. Therefore, the indentation size effect on the hardness of the PIN-PMN-PT bulk single crystal appeared during the nanoindentation process. This indentation size effect is usually attributed to the large surface strain gradient and work hardening [42,43]. From Figure 8, it can be seen that the Young's modulus was almost independent of the indentation depth in the nanoindentation process.

Hardness and Young's Modulus
The average hardness and Young's modulus were evaluated in the indentation depth range from 1400 to 1600 nm, as shown in Figures 7 and 8

Conclusions
The load-indentation depth curves, elastic and plastic deformations, hardnesses, and Young's moduli of the [100]-and [110]-oriented relaxor-based ferroelectric 0.28PIN-0.43PMN-0.29PT bulk single crystals were investigated by the nanoindentation technique and continuous stiffness measurements. The conclusions are as follows: (1) The load-indentation depth curves for the PIN-PMN-PT bulk single crystals featured a low noise level and small variation in the data, which reflects the good phase purity and uniformity of the PIN-PMN-PT bulk single crystals. There was no discontinuity in the load-indentation depth curves within an indentation depth of 2000 nm, which indicates that large cracks were not formed in the PIN-PMN-PT single crystals during the loading and unloading processes.
(2) With increasing the maximum indentation depth h max from 500 nm to 2000 nm, both the elastic and the plastic deformations increased for both the [100]-and the [110]-oriented single crystals. However, with the increasing h max , the elastic recovery percentage decreased, and the plastic residual percentage increased for both the [100]-and [110]-oriented single crystals. At each h max , the elastic recovery percentage of the [100]-oriented PIN-PMN-PT bulk single crystal was larger than that of the [110]-oriented PIN-PMN-PT bulk single crystal, and the plastic residual percentage of the [100]-oriented PIN-PMN-PT bulk single crystal was less than that of the [110]-oriented PIN-PMN-PT bulk single crystal. In the h max from 500 nm to 2000 nm, the plastic deformation was larger than the elastic deformation, and the plastic residual percentage was larger than 50% for both the [100]-and [110]-oriented single crystals, which means that the plastic deformation dominated in the indentation process of the PIN-PMN-PT bulk single crystals.
(3) The indentation size effect on the hardness of the PIN-PMN-PT bulk single crystals was apparent, which is usually attributed to the large surface strain gradient and work hardening. The Young's modulus was almost independent of the indentation depth in the nanoindentation process. Both the hardness and the Young's modulus of the [100] PIN-PMN-PT bulk single crystal were a little larger than those of the [110] PIN-PMN-PT bulk single crystal, which indicates that the PIN-PMN-PT bulk single crystal had anisotropic mechanical characteristics. These anisotropic mechanical characteristics were mainly attributed to the arrangement of atoms in different crystallographic directions.