A. Dynamic MA of 0.4NBT–0.6PT templates

To determine the feasibility of dynamic MA of perovskite 0.4NBT–0.6PT template particles, a sample of these platelets was dispersed in a polyester resin in a horizontal magnetic field of B = 7 T. The sample of suspended platelets was rotated at 5 rpm about a vertical axis, and the resin was allowed to cure for ∼15 min. An XRD pattern on a horizontal section through the sample is shown in Fig. 1. For reference, an XRD pattern of the as-synthesized template powder with minimal preferred orientation is also shown. XRD results show that the 0.4NBT–0.6PT platelets align very strongly with <001> perpendicular to the sample surface and perpendicular to the rotating magnetic field. Although both the <100> and <001> peaks are enhanced in the aligned template sample, the <001>/<100> peak intensity ratio is dramatically increased. Because the <001> direction is oriented perpendicular to the rotating magnetic field, the 0.4NBT–0.6PT phase is shown to have easy-plane magnetic character (χ_c < χ_a). In addition, the <110> and <111> peaks are almost nonexistent in the aligned sample, indicating that very high quality <001> orientation is obtained. The strong <001> alignment of 0.4NBT–0.6PT templates indicates that each template possesses a preferred crystallographic orientation rather than an equal mixture of the 6 ferroelectric domain variants possible for this phase. The template domain structure and its role in MA were investigated and are presented as supplemental information (published online).

FIG. 1.

XRD patterns of 0.4NBT–0.6PT platelet powder (control) and the same platelets after dispersion in polyester resin and MA in a horizontal magnetic field (B = 7 T) rotated at 5 rpm. The sample normal direction was perpendicular to the rotating field. Note that tops of the <001> and <002> peaks in the aligned sample are truncated due to saturation of the XRD detector.

The viscosities of the nonaqueous PMN–PT slips (20–40 vol% solids) are shown in Fig. 2. Nonaqueous casting slips with 35–40 vol% PMN–PT were strongly shear thinning, whereas slips with lower solid loading were nearly Newtonian in nature. At low shear rates (∼0.02 1/s), the slip viscosity increases about one order of magnitude for each 5 vol% increase in solids loading above 25 vol%.

FIG. 2.

Viscosity of nonaqueous PMN–PT slips shown as a function of the shear rate and volume percent solid loading.

PMN–PT samples were unidirectionally slip cast within the rotating magnetic field using a cylindrical sample chamber partially filled with casting plaster as shown schematically in Fig. 3. The sample holder in Fig. 3 was 1.25 cm in diameter and was charged with ∼3 mL of PMN–PT casting slip (containing 7 vol% 0.4NBT–0.6 PT templates on a powder basis). The slip was allowed to cast for at least 24 h and was removed from the sample holder. Samples produced in this way were ∼12 mm in diameter and 5–15 mm in height and could be air dried and sintered without cracking.

FIG. 3.

Cut-away schematic of the sample setup used for dynamic MA of PMN–PT slurries containing 0.4NBT–0.6PT templates. Samples were cast unidirectionally in a cylindrical sample holder, which was rotated about the vertical axis within a 2.2-T permanent magnet array.

During casting, dynamic MA samples were rotated at 3 rpm (ω = 0.05) in a horizontal magnetic field of 2.2 T. Control samples were cast using the same parameters, but without rotation or application of a magnetic field. Sample casting rates were fit to a parabolic rate law:

(6)$$h = K{t^{1/2}}\quad ,$$

where h is the height of the cast layer, t is the casting time, and K is a rate constant. The validity of this equation was confirmed by multiple slip casting experiments. Vertical position in the sintered samples was correlated with casting time using Eq. (6), after correcting for the linear z-dimension shrinkage during sintering. Both the linear shrinkage and the casting constant K were correlated with the slurry solids loading, as shown in Fig. 4. These values, along with Eq. (6), were used to relate casting time and thickness within the slip cast samples. Linear shrinkage in the casting direction decreases from ∼17% at a solid loading of 20 vol% to ∼15% at a solid loading of 42 vol%, corresponding to an increase in green density from ∼54% to ∼61%. Sintering was nearly isotropic in all cases, although low viscosity slips (<0.1 Pa s at low shear rates, corresponding to 20–30 vol%) showed slightly more shrinkage in the casting direction. The casting rate is approximately doubled as slip viscosity increases to ∼10 Pa s at 0.1 s⁻¹. At the highest values of K (15 × 10⁵ m/s^1/2 for 40 vol% solids), slip casting of a 1-cm-thick sintered sample required 106 min. In all cases, sintered samples consisted of a single perovskite phase within the resolution limits of x-ray diffraction. Previous work has found that 0.4NBT–0.6PT templates are chemically soluble in PMN–PT, resulting in textured ceramics with homogeneous composition.^{Reference Poterala, Trolier-McKinstry, Meyer and Messing9}

FIG. 4.

Variation in the linear shrinkage during sintering and in the slip casting constant K as a function of solid loading for nonaqueous PMN–PT slips.

B. Models for texture in dynamic MA/TGG samples

The MD texture model [Eq. (1)] has been shown to apply to MA ceramics (with f = 1) in the transient alignment regime and has been used in some cases to characterize these materials.^{Reference Terada, Suzuki, Suzuki, Kitazawa, Sakka, Kaneko and Metoki25} However, the equilibrium orientation distribution function (ODF) predicted for MA is substantially different and is rewritten from Eq. (5) to give^{Reference Fujiwara, Fukui and Tanimoto19,24−26}:

(7)$$P({\rm{\theta ,\alpha }}) = 2{{\exp \left( {{\rm{\alpha }}{{\cos }^2}{\rm{\theta }}} \right)\sqrt { - {\rm{\alpha }}} } \over {{{\rm{\pi }}^{1/2}}{\rm{erf}}\left( {\sqrt { - {\rm{\alpha }}} } \right)}},\,{\rm{\alpha }} = {{V{B^2}\Delta {\rm{\chi }}} \over {2{{\rm{\mu }}_0}{k_{\rm{B}}}T}}\,{\rm{for \;static \;MA (}}\Delta {\rm{\chi > 0)}}\quad ,$$

(8)$$P({\rm{\theta ,\alpha }}) = 2{{\exp \left( { - {{\rm{\alpha }} \over 2}{{\cos }^2}{\rm{\theta }}} \right)\sqrt {{{\rm{\alpha }} \over 2}} } \over {{{\rm{\pi }}^{1/2}}{\rm{erf}}\left( {\sqrt {{{\rm{\alpha }} \over 2}} } \right)}},\,{\rm{\alpha }} = {{V{B^2}\Delta \chi } \over {2{{\rm{\mu }}_0}{k_B}T}}\,{\rm{ for \;dynamic \;MA (}}\Delta \chi {\rm{ \,\lt\, 0)}}\quad {\rm{,}}$$

where V is the particle volume, B is the applied field, and Δχ is the magnetic anisotropy χ_c − χ_a. The dimensionless field parameter α is related to the quality of particle alignment and is therefore analogous to the MD parameter r. As in the MD model, the ODF for dynamic MA may be modified to include a texture fraction f that reflects the degree of template growth during TGG, assuming only the templates are aligned by the magnetic field. This model must also be modified in the PMN–PT system to account for multiplicity of the <001>_C peak in the pseudocubic perovskite structure:

(9)$$P({\rm{\theta ,\alpha ,}}f) = \left( {1 - f} \right) + \left( {{f \over 3}} \right)2{{\exp \left( { - {{\rm{\alpha }} \over 2}{{\cos }^2}{\rm{\theta }}} \right)\sqrt {{{\rm{\alpha }} \over 2}} } \over {{{\rm{\pi }}^{1/2}}{\rm{erf}}\left( {\sqrt {{{\rm{\alpha }} \over 2}} } \right)}}\quad .$$

The similarly modified MD model is given as:

(10)$$P(f,r,{\rm{\theta }}) = (1 - f) + \left( {{f \over 3}} \right){\left( {{r^2}\cos {{\rm{\theta }}^2} + {1 \over r}\sin {{\rm{\theta }}^2}} \right)^{ - 3/2}}\quad .$$

It should be noted that Eqs. (8) and (9) assume that rotation of the magnetic field is fast with respect to randomization of particle orientation due to k _BT. In well aligned samples, the FWHM of Eq. (9) depends almost entirely on α and is approximated by:

(11)$$\sin {\left( {{{{\mathop{\rm FWHM}\nolimits} } \over 2}} \right)^2} \approx {{2\ln 2} \over { - {\rm{\alpha }}}}\,{\rm{or}}\,{\mathop{\rm FWHM}\nolimits} \approx {{2\sqrt {2\ln 2} } \over {\sqrt { - {\rm{\alpha }}} }}\quad .$$

This approximation is valid for values of f > 0.5 and α > 10 that are encountered in this work. The saturation FWHM can also be related to the order parameter $m_{\rm{3}}^{{\rm{SAT}}}$ defined by Yamaguchi et al.^{Reference Yamaguchi, Ozawa and Yamamoto24} This order parameter, defined for static MA, is given as:

(12)$$m_{\rm{3}}^{{\rm{SAT}}} = {3 \over {4{\rm{\alpha }}}}\left[ {{{\exp \left( {\rm{\alpha }} \right)} \over {\int\limits_0^1 {\exp \left( {{\rm{\alpha }}{x^2}} \right){\rm{d}}x} }} - 1} \right] - {1 \over 2}\quad .$$

An equivalent order parameter may be calculated for dynamic MA, such that equal values of $m_{\rm{3}}^{{\rm{SAT}}}$ correspond to equivalent grain orientation distributions:

(13)$$m_{\rm{3}}^{{\rm{SAT}}} = {{ - 3} \over {2{\rm{\alpha }}}}\left[ {{{\exp \left( { - {\rm{\alpha }}/2} \right)} \over {\int\limits_0^1 {\exp \left( { - {\rm{\alpha }}{x^2}/2} \right){\rm{d}}x} }} - 1} \right] - {1 \over 2}\quad .$$

As the parameters α, FWHM, and $m_{\rm{3}}^{{\rm{SAT}}}$ depend directly on the degree of particle alignment, any of these parameters are suitable for characterization of the dynamic MA process within sintered and annealed TGG samples. The parameter f in Eq. (9) has the same meaning as the MD texture fraction f in Eq. (10), in that it represents the volume fraction of material that is textured according to the selected orientation model.

C. Characterization of grain alignment in MA/TGG samples

Alignment of the 0.4NBT–0.6PT template particles by the rotating magnetic field was determined by analysis of crystallographic texture in sintered and annealed PMN–PT samples. XRD rocking curves measured on the {002} peak show that these samples have a strong <001>_C texture perpendicular to the rotating magnetic field. This texture was confirmed to be approximately uniaxial (i.e., fiber texture) by measurement of rocking curves at various sample orientations. The presence of this strong texture confirms that 0.4NBT–0.6PT templates align magnetically within the PMN–28PT matrix powder and that χ_c < χ_a for this material. To calculate alignment parameters for each sample, the {002} rocking curves were fit to Eqs. (9) and (10). An example of fitting to these models is shown in Fig. 5. As seen in Fig. 5, measured {002} rocking curves for <001>_C textured PMN–PT were found to fit the dynamic MA model [Eq. (9)] with good accuracy and fit less well to the MD model [Eq. (10)]. Similar results were seen for all solid loadings for casting times t between 5 and 30 min. This result suggests that ODFs obtained from dynamic MA represent equilibrium between MA and the randomizing effect of Brownian motion (thermal energy).

FIG. 5.

An example of a corrected {002}_C rocking curve for <001>_C textured PMN–PT made by the MA/TGG method. The sample section represents a casting time of t = 12 h from a slip with 30 vol% solid loading. Solid lines show fits to the dynamic MA model [Eq. (9)] and the MD model [Eq. (10)].

Texture evolution during MA/TGG was analyzed by tracking the FWHM of corrected {002} rocking curves as a function of slip casting time and solid loading (Fig. 6). For Newtonian slips (viscosity ≤ 0.1 Pa s at 0.1 s⁻¹ or solid loading ≤ 32 vol%), the orientation distribution FWHM appears to reach a steady value after ∼15–30 min of slip casting, in agreement with earlier evidence that dynamic MA reaches thermal equilibrium. The overall quality of alignment increases with increasing viscosity up to 0.1 Pa s, as the higher viscosity suppresses loss of alignment caused by thermal (Brownian) motion. Stable alignment was not obtained for the shear thinning slips (viscosity of 1–10 Pa s at 0.1 s⁻¹ or solid loading ≥ 35 vol%), although a sample cast at 35 vol% solids initially shows a very strong alignment (FWHM < 7°) but degrades with time. This degradation is correlated with a sharp increase in slurry viscosity and slip casting rate (Figs. 2 and 4), and all of these factors may reflect instability of particle dispersion in slips with higher solids loading.

FIG. 6.

FWHM of grain ODFs in textured PMN–28PT ceramics, plotted with respect to t ^1/2 (t is the slip casting time) and solid loading (vol%) during dynamic MA (B = 2.2 T, 3 rpm).

The onset of alignment does not appear to depend strongly on slip viscosity, although a full kinetic analysis is precluded by a lack of data at t ^1/2 < 5 min^1/2. One possible explanation is that the transient alignment state is controlled by capillary forces during slip casting and not by viscous resistance toward the MA process. The ODF also becomes more stable (less variation in FWHM or maximum texture intensity between adjacent cross sections) as viscosity increases to ∼0.1 Pa s, beyond which slips become strongly non-Newtonian. This result is attributed to a relatively low rotation rate of the suspended template particles due to viscous drag. A more rapid template rotation rate results in loss of alignment in the plane orthogonal to the magnetic field and allows any unwanted vibration of the sample chamber to interfere with the alignment process. At a slurry viscosity of 0.1 Pa s at 0.1 s⁻¹ (32 vol% solids), loss of alignment is negligible, resulting in a stable alignment of ∼7.9° FWHM through most of the sample thickness. The full time-dependent ODF at this condition (measured after sintering and TGG) was reconstructed from rocking curve measurements on sequential sections and is shown in Fig. 7. The ODF of this sample reached equilibrium after t = 20 min and remained stable for 4 h. However, a slight decrease in the FWHM (an increase in alignment quality) was observed at the end of the casting process, as shown in Figs. 6 and 7. This increase in alignment affects only a thin (∼1 mm) layer at the sample surface. A similar result was seen at 0.08 Pa s (0.1 s⁻¹, 30 vol% solids), shown in Fig. 6. This effect may be related to a slight increase in viscosity caused by evaporation of xylenes from the sample surface.

FIG. 7.

Time-dependent plot of the grain ODF in MA/TGG samples slip cast at 2.2 T, 3 rpm with a 32 vol% solid loading. Texture intensity in MRD is plotted as a function of misorientation angle θ and t ^1/2 (which is proportional to sample thickness).

The grain orientation distribution FWHM in MA/TGG textured PMN–PT was inversely correlated with solid loading and slip viscosity during dynamic MA, as shown in Fig. 6. To explain this effect, we consider the interaction between the reduced rotation frequency [ω_r = ω/(α·D _R)] and the alignment order parameter m ₃.^{Reference Yamaguchi, Ozawa and Yamamoto24} The order parameter m ₃ was calculated as a function of solid loading using averaged values of FWHM for each sample series. The rotational diffusion coefficient D _R and the reduced frequency ω_r were estimated using viscosity values measured at a shear rate of 1 s⁻¹. For this calculation, templates were modeled as discs of 8-μm diameter. The order parameter $m_{\rm{3}}^{{\rm{SAT}}}$ and FWHM values are plotted as a function of estimated ω_r in Fig. 8. As seen in Fig. 8, increasing ω_r results in an increase in m ₃ and a decrease in FWHM. This trend follows closely with the predictions of Yamaguchi et al.^{Reference Yamaguchi, Ozawa and Yamamoto24} The estimated values of ω_r obtained for slips with 25 and 30 vol% solid loading are ∼2 and 7, respectively, and are low enough that m ₃ may be noticeably reduced below $m_{\rm{3}}^{{\rm{SAT}}}$. In contrast, the 35 vol% slip shows ω_r ≈ 23, and thus, the order parameter for this sample (m ₃ = 0.992) is likely close to the true $m_{\rm{3}}^{{\rm{SAT}}}$ for the field conditions used. This $m_{\rm{3}}^{{\rm{SAT}}}$ value corresponds to a dimensionless field parameter of α ≈ −360. Although m ₃ is predicted to approach $m_{\rm{3}}^{{\rm{SAT}}}$ relatively quickly (m ₃ > 0.95$m_{\rm{3}}^{{\rm{SAT}}}$ at ω_r = 3), this order parameter is relatively insensitive to ω_r compared to the FWHM value. As shown in Eq. (6) through Eq. (12) and in Fig. 8, even a slight increase in m ₃ from 0.977 to 0.992 corresponds to a substantial change in the ODF of dynamic MA ceramics.

FIG. 8.

Variation of the order parameter m ₃ and the ODF FWHM as a function of the reduced rotation frequency ω_r. ω_r was estimated using slip viscosities measured at a shear rate of 1 s⁻¹.

The results in Fig. 8 were also checked by calculating the time required for MA at each viscosity. Nearly complete alignment is achieved at t _r > 5, where t _r is the reduced time given by t _r = D _R·α·t. Minimum alignment times (t _r = 5) were initially calculated using α ≈ −360 and using viscosities measured at 1 s⁻¹. The times calculated for alignment of 25 vol% and 30 vol% slips are 11 and 24 min, respectively. These times cannot be validated as they are both shorter than the first measurements shown in Fig. 6. However, the calculated time to align a 35 vol% slip (t = 49 min) is substantially longer than the observed alignment time (t < 6 min). In contrast, estimation of α using the observed alignment time and measured slurry viscosity yields a much higher value (α ≈ −3000).

The $m_{\rm{3}}^{{\rm{SAT}}}$ value of 0.992 and the corresponding value of α ≈ −360 were used to estimate a lower bound for the magnetic anisotropy of 0.4NBT–0.6 PT template particles using volume average template dimensions of ∼8 μm in diameter and ∼1 μm in thickness. The magnetic anisotropy calculated in this way is χ_c − χ_a = −1.5 × 10⁻⁸ in SI units or roughly half the value for α-Al₂O₃ (3.5 × 10⁻⁸ in SI units). This value underestimates the true magnetic anisotropy of the 0.4NBT–0.6PT phase because it does not take into consideration the polycrystalline multidomain microstructure of these particles (see supplemental information). As a result, the FWHM values achieved (∼7°) are likely dictated by grain misalignment within the template particles, or misalignment between the <001>_C direction and the template magnetic axis due to the template domain structure. This conclusion is supported by the rapid alignment kinetics in Fig. 8, which suggests a higher magnetic anisotropy than was estimated from an equilibrium alignment of 6–7° FWHM. Using a kinetic approach to estimate the magnetic anisotropy results in a higher value for χ_c − χ_a of −1.3 × 10⁻⁷ or about 4 times greater than α-Al₂O₃. This anisotropy value would be predicted to yield a FWHM of ∼2.5° at thermal equilibrium if the template particles were monocrystalline in nature. In any case, the success of dynamic MA at low field levels (2.2 T) is not attributed to strong magnetic characteristics of the NBT–PT templates. These results are primarily attributed to the use of coarse templates, for which the MA driving force is 3–4 orders of magnitude greater than k _BT. In contrast, conventional MA approaches utilize a lower alignment force due to the smaller size of aligned particles, while the thermal energy k _BT (the force opposing alignment) remains constant.

Dielectric and piezoelectric properties were measured on well-textured samples (FWHM ∼ 8°) cast from a slip with 32 vol% solid loading. Samples 1, 2, and 3 represent the center slices of the sample depicted in Fig. 7 and correspond to approximately 33, 90, and 182 min after the start of slip casting (distances of 3.5, 5.9, and 8.4 mm from the mold surface). Although a lower FWHM (6–7°) could be achieved at a solid loading of 35 vol%, this texture quality was not stable enough with respect to casting time to produce thick homogeneous samples. The properties of three samples and the texture parameters determined from Eq. (10) are shown in Table I.

TABLE I.

Properties of <001>_C textured PMN–PT samples made by a combined MA/TGG process.

All three samples show similar texture parameters (FWHM, α, and f). The texture fraction f is almost identical to values measured in textured PMN-PT made by tape casting with the same template particles. These samples also have a lower FWHM (∼7.9°) than tape cast samples (∼12°).^{Reference Poterala, Trolier-McKinstry, Meyer and Messing9} It should be noted that the ODF resulting from MA has lower tails than the MD texture model (Fig. 6), and thus, MA-derived samples contain fewer strongly misaligned grains than tape cast samples with the same FWHM and f parameters.

The relative permittivity of MA/TGG samples (3332) was found to be similar to that of well-textured PMN–PT produced by tape casting and TGG (3490), and both materials show an average dielectric loss tan(δ) of 0.010. However, MA samples have a 20% higher piezoelectric coefficient d ₃₃ (1222 pC/N) than tape cast samples (1016 pC/N), which may be attributed to the narrower FWHM obtained via MA. This value approaches the piezoelectric coefficient of <001> PMN–28PT single crystal (1470 pC/N) and demonstrates that combined MA/TGG methods have promise for producing very high performance textured ceramics.