Imaging microstructural dynamics and strain fields in electro-active materials in situ with dark field x-ray microscopy

Characterizing the dynamics of microstructural features under external stimuli is critical to understanding structure–property relationships in a wide range of materials. It is particularly critical for electro-active materials, such as piezoelectric, ferroelectrics, and multiferroics, where macroscopic properties depend strongly on the motion and interactions of domain walls, phase boundaries, and other defects.¹ For example, the density and mobility of domain walls in the archetypal ferroelectric lead zirconate titanate (PZT) are strongly affected by the presence of defects,² such as grain boundaries³ and dislocations.⁴ Spatially resolved models (e.g., phase field) can predict the dynamics and interactions of domain walls and heterogeneous defects; however, such models lack quantitative experimental validation.⁵ Thus, there is a strong need for experimental techniques capable of imaging the local environment around microstructural features at the relevant length scales.

The hard x-ray microscope (HXRM) on beam line ID06 at the European Synchrotron Radiation Facility (ESRF) is a newly developed instrument for multiscale in situ 3D imaging of bulk materials.^6,7 It can be operated in both transmission and reflection geometries and can change between multiple imaging and diffraction modalities to suit a given sample at the temporal, spatial, and angular resolutions required. In particular, HXRM is capable of both bright field and dark field imaging modes. In bright field, the contrast mechanisms are attenuation or scattering in the forward direction. In dark field, the contrast is produced from the scattering intensity at high angles, allowing for very high selectivity between various Bragg peaks. Uniquely, dark-field x-ray microscopy (DFXM) directly maps structural features in the sample by forming a real-space image from the scattered beam (e.g., Bragg peak) via an x-ray objective lens.^8–11 In comparison, techniques such as transmission electron microscopy (TEM) and piezoresponse force microscopy (PFM) directly measure atomic displacement and polarization of domains, but are limited to thin samples or surface measurements, respectively. While nanoprobe x-ray diffraction and polarized light microscopy overcome these limitations to some degree, DFXM offers improved temporal or spatial resolution when tracking the morphological kinetics.^7–9 Additionally, the apparatus required for DFXM is capable of collecting time-resolved reciprocal space maps (RSMs) with high angular resolution (∼10⁻⁵ Δq/q), allowing us to observe subtle perturbations to crystallographic phases. Collectively, this allows us to track and analyze morphological kinetics such as phase fronts and domain wall motion, making the HXRM well suited for studies of structure dynamics in bulk electro-active materials.

In this paper, we describe a system for imaging deeply embedded structures in bulk materials in situ, with precise temperature and electrical field control. We demonstrate how the system can be used to image the dynamics of topological features and relate these to strain maps, exemplified by studying the electric-field-induced phase transformation in single crystal BaTiO₃ using full-field HXRM in the DFXM mode. Crucially, the critical nature of the first order phase transition in BaTiO₃ at the Curie temperature requires extremely precise control of the sample temperature and electric field.¹² 

HXRM operates as a conventional Galilean microscope, in which an x-ray objective lens forms a full-field magnified image of the sample on the 2D detector (Fig. 1). In the bright field mode, the detector and objective are placed in the forward direction, and absorption contrast dominates the image. In DFXM, the objective and detector are aligned to a Bragg peak related to the feature of interest (e.g., a domain wall). Spatial maps of lattice tilts can be measured by rotating the sample away from the scattering vector, while maps of lattice strain can be measured by collectively moving the objective lens and detector through the diffracted beam.⁷ By operating at hard x-ray energies (10–30 keV), the microscope can probe embedded volumes in mm-size samples with typical spatial resolutions of 100 nm, imaging frame rates of 0.01–3 s/image and, in the dark-field mode, a strain resolution of 10⁻⁵ (Δε/ε).^6,8,11 The magnification is given by the focal power of the objective lens and the sample-to-detector distance. The aberrations of the x-ray objective lens, along with the numerical aperture and Darwin width of the crystal, ultimately limit the spatial resolution.

To characterize the morphological kinetics of a material, a system is required to precisely perturb the sample without compromising the imaging capabilities of the HXRM. Figure 2 shows the Stable Temperature and Applied Field (STEF) holder and gives an overview of the overall holder design and the placement of its individual components. The design is optimized to maintain a stable temperature from 22 °C (ambient) to 200 °C, simultaneously with a controlled electric field on commercially grown single crystal samples of 5 × 5 × (0.15–0.5) mm³. The microscope’s photon energy range (10–30 keV) places limits on the useful sample thickness due to attenuation. Selecting an optimum photon energy for a given experiment involves many considerations in terms of the microscope parameters and sample conditions. These considerations are beyond the scope of this article and can be found in the literature.^6–10 

Parts 3 and 5 [as shown in Fig. 2(c)] are the two most critical components because they are responsible for the fixed placement of the sample (part 4) and the control of the temperature. Part 5 is copper, with a recess on the back side that holds a coiled resistance Kanthal-wire heating element and pt-100 four-point thermal measurement probe. Parts 2, 3, and 6 are fabricated out of MACOR (2, 3) and stumatit (6) ceramic due to their good thermal-insulation and electrical-insulation properties. Parts 1 and 7 are PEEK and aluminum, respectively. The relatively large amount of material surrounding the sample provides thermal mass to improve stability, while part 5 also serves as the electrical ground for the system.

Collectively, the different parts create a large thermal mass relative to the sample. The ceramic parts insulate the sample from environmental fluctuations, and the copper enables good thermal conduction from the heating element.¹³ However, the temperature feedback loop, e.g., the measured temperature and subsequent regulation of the heating element, is the most critical component for controlling and maintaining the temperature. This is achieved by installing a small pt-100 element (platinum wire) in the recess of part 5, between the heating element and the sample. Using a four-point resistance measurement of the pt-100 element in conjunction with a Lakeshore 332/331 temperature controller and Delta Elektronika SM 120-13 power supply, we have been able to regulate the measured temperature with a precision of 0.01 K at a ramp rate of 0.5 K/min up to 250 °C (Fig. 3). The open-air nature of the holder inevitably results in convection and, thus, temperature fluctuations across the sample.¹⁴ However, our tests indicate that this does not adversely affect the precision or stability of the system. Nonetheless, due to the design with the four-point measurement close to the heating element, the STEF-holder shows a ≈9 °C higher temperature than at the sample, when measured against a calibrated thermocouple. This offset remained stable above 80 °C and decreased as the temperature was lowered toward ambient.

Many previous systems for applying electric fields in situ during x-ray measurements use a dielectric fluid (e.g., silicon oil) to prevent arcing around the sample.¹² However, this method was not used here in order to reduce complexity and avoid small-angle scattering and attenuation from the liquid, which will aberrate the image and reduce contrast.¹⁴ Applying electric fields in air reduces the maximum applicable field; however, we note that the electrical breakdown field of air is 3 kV/mm, beyond the coercive fields of many common ferroelectric materials (e.g., E_c for BaTiO₃ is 300 V/mm).¹⁶ The electric field was applied via two hand painted silver electrodes (∼4 × 4 mm² in size) on the opposing 5 × 5 mm² sides of the sample. A programmable DC power supply (Delta Electronica SM 120-13) was then used to create the electric field across the electrodes according to the user-defined voltage. For the samples used, the system had a nominal electric field precision of δE = 0.3 mV/mm. Prior to use, an ohmmeter was used to ensure there were no short circuits and that electrical resistance was minimized wherever possible (e.g., across electrodes). We also note the importance of a closed circuit across the sample to prevent the accumulation of electrical charge at the sample surfaces during x-ray measurements, which may potentially affect the local depolarization field.¹⁷ 

Figure 4(a) illustrates that the STEF apparatus allows maintaining elevated temperatures while simultaneously ramping a field across the sample. It should be noted that we did not experience any electrical breakdown at the given fields. Furthermore, the temperature could be ramped to a set temperature with an undershoot of less than 0.01 K, as shown by the variance at the given set temperature (dotted line) in Fig. 4(b).

The aforementioned capabilities of the STEF apparatus are essential for the case study of phase transformations in BaTiO₃ discussed next. In comparison to a similar in situ system implemented on ceramic BaTiO₃,¹² this system offers the ability to directly image the local perturbation and dynamics of the material system. The scope of this article is to demonstrate these claims and introduce the prospective analyses our system offers.

To demonstrate and explore the capabilities of the STEF apparatus, we used it to investigate electric-field-induced phase transformations in BaTiO₃, a canonical ferroelectric. Electric field-induced phase transformations may yield large dielectric and piezoelectric response.¹⁸ Notably, when heated just above the Curie temperature (T_C), BaTiO₃ exhibits an electric-field-induced phase transition from cubic (Pm-3m) to tetragonal (P4mm) lattice symmetry. This makes it a model system for high strain lead-free piezoelectrics¹⁹ and electrocalorics.²⁰ Several previous studies in BaTiO₃ and related systems have been carried out using in situ x-ray diffraction to extract the average structure and transformations pathways.²¹ Our system provides the opportunity to image the dynamics of the local domain structure, phase distribution, and strain during these transformations.

Our experiment utilized the {200} Bragg reflection, as it shows clear splitting upon transition from the cubic and tetragonal lattice symmetries.^7,12 DFXM was used together with classical x-ray reciprocal space mapping. This provides both an overview of the average crystallographic changes occurring during the transformations (both thermally and electric-field-induced) and the local information of the strain and phase distribution within the crystal.

The BaTiO₃ single crystal was obtained commercially (Crystal GmbH, Germany) with initial dimensions of 5 × 5 × 0.5 mm³. The sample was then attached to metal stub using a low-temperature adhesive wax. The mounted sample was surrounded by four pieces of 0.14 mm-thick glass to help ensure that the sample surfaces remained flat and parallel during grinding. The sample thickness was reduced by grinding using silicon carbide paper of grit No. 800, No. 1200, No. 2400, and No. 4000, with each step inspected by optical microscopy before progressing to the next. When the sample thickness measured 0.15 ± 0.01 mm, the 5 × 5 mm² surfaces were polished with suspended silica particles of size 0.04 µm (OPS, Struers A/S, Denmark). To relax the residual stresses induced during the grinding and polishing, the sample was thermally annealed in a box furnace. Specifically, the sample was heated at a rate of 1 K/min from 26 °C to 400 °C and held at 400 °C for 2 h before being cooled to room temperature at a rate of 1 K/min.

The DFXM experiment used an x-ray energy of 17 keV (energy bandwidth ΔE/E of 10⁻⁴). A Be-based 1D compound refractive lens created a line-beam on the sample ∼400 µm wide and 1 µm high, thus illuminating only a cross section of the BaTiO₃ sample. The objective comprised a 2D Be CRL with a focal length of ∼250 mm, resulting in a total geometrical magnification of 19×. Magnified 2D images of the illuminated cross section were collected with a FReLoN CCD camera, resulting in a final effective pixel size of 70 nm. The reciprocal space maps were measured with both the condenser and objective lens removed and using a large field-of-view detector comprising a CMOS camera (Basler Ace), a scintillator, and a wide-angle lens, yielding an effective pixel size of 55 µm.^6,9

The experiments were conducted as follows: sub-T_C measurements were taken at 120 °C to confirm that the structure stayed tetragonal even at elevated temperatures. Initially, the temperature was elevated significantly above T_C to 150 °C to ensure complete transformation from tetragonal to cubic. Reciprocal space maps (RSMs) were also collected at these two temperatures. The sample then went through four heating and cooling cycles crossing T_C with each cycle to ensure that the thermal hysteresis normalized and that T_C occurs at a repeatable temperature. The sample was then cooled to a recorded 139.6 °C, ∼1 K above the measured T_C (which have slight fluctuations after normalization), and kept stable for several minutes to ensure that latent heat would not affect the measurements. At this point, a strain map was measured by rotating the sample ±0.05° and 2θ range of ±0.06°. Subsequently, a stepwise ramping of the electric field was applied across the sample. With the electric field still applied, another strain map was measured. The field ramped to 0.8 kV/mm in steps of j × 0.038 kV/mm, where j = (1, 2, 3, 4, 5, 6).

The RSMs in Figs. 4(a) and 4(b) show that we have a cubic phase above T_C and a tetragonal phase below T_C. The cubic phase possesses a fourfold rotation symmetry around all three axes with equal lattice spacing. When aligning the single crystal to the (200) reflection, a single peak is observed, as seen in Fig. 4(a). The tetragonal phase only possesses a fourfold symmetry along the elongated axis, producing two unique lattice spacings. The domain formation, and hence the polarization orientation, produces the two main peaks, annotated as 1 and 2 in Fig. 4(b). Twinning along the domain walls gives rise to four new peaks (1a, 1b, 2a, and 2b), arising from the lattice distortion associated with the requirement for mechanical compatibility.^1,22 The tetragonal phase is characterized by its six different peaks, representing three crystallographic twins about 110-type lattice planes, seen in Fig. 4(b). The tails of the 1a and 2b peaks in Fig. 4(b) are due to strain associated with the domain wall. When applying the electrical field above T_C, the (200) peak associated with the cubic phase becomes extinct, and a single peak appears in the lower angle position (i.e., the same as position 1 in the zero-field tetragonal RSM). This indicates the absence of 90° domain walls, implying either the formation of either a monodomain or that only 180° domain walls exist in the diffracting volume of the sample.²³ In Fig. 4(d), it can be seen that removing the electric field from the sample did not return the sample back to the original cubic phase, as two diffraction spots appear in the RSM. This indicates that the electric-field-induced cubic-to-tetragonal phase transition is irreversible at this temperature.¹⁵ The peak indicated with “?” does not correspond to any of the reflections seen in Fig. 4(b). However, it has a higher q_y value, indicating lower d-spacing. We speculate that the polarization and, thus, the spontaneous strain might have diminished in some part of the crystal after the field was removed. The RSM method probes the average structure of the illuminated region and does not provide information about the local environment in real space. This establishes a clear need for carrying out microscopic investigations of the dynamic behavior of these phenomena.

To this end, Fig. 5(a) shows a DFXM image of the microstructure of the BaTiO₃ sample in the cubic phase prior to the application of the electric field. The indicated temperature is 139.6 °C, with the T_C at 138.5 °C. Although the RSM did not show any indication of tetragonal phase above T_C, heterogeneous structures are clearly evident in the DFXM images. Furthermore, the wavy, fringe-like features in the images are uncharacteristic of the domain structures normally seen in the tetragonal phase of BaTiO₃.²⁴ These structures could be related to local lattice strains and misorientations due to dislocations, crystal growth defects, or accumulations of atomic defects.²⁵ We note that these features were observed to remain very stable over time, even close to T_C. The image stability is taken to indicate very high thermal stability of the system, as even subtle changes (e.g., strains less than 10⁻⁴) in lattice strain/orientation would be readily detectable in the DFXM image.

The images shown in Fig. 5 are taken from a dataset of 422 images, acquired while ramping the field to 0.8 kV/mm. Slow ramping of the electrical field changed these heterogeneous wavy structures considerably, even at very low electric fields of 0.02 kV/mm. The cubic-to-tetragonal phase transition occurred when the field reached a critical value of 0.4 kV/mm, identified by a sudden drop in intensity accompanied by changing wavy patterns in Fig. 5(b). Since we were imaging the cubic peak, the intensity reduces further in Fig. 5(c) as the transition continues. In the sequence of images from Figs. 5(d)–5(i), a phase front propagates through the sample from the left-hand side, diagonally toward the upper right corner of the image. The phase front propagates through the field of view, until no more intensity from the cubic phase can be observed (not showing the final, completely dark image). During this propagation, the structure inside the illuminated area changes its internal structure, with 45° domain-like structures appearing at the cubic side of the phase front. The periodic pattern could potentially be a lattice distortion or intermediate phase or structure, as suggested in previous studies.²⁴ We note that the wavy, fringe-like features present in the cubic phase remain clearly visible throughout the phase transformation.

It is not readily possible to quantitatively characterize the distortions created by defects or interfaces with single-exposure images such as those in Fig. 5. However, such dynamically acquired image sequences can be an excellent means to characterize the dynamics of topological features, such as domain wall motion, domain growth, domain density, and phase front speed.⁷ 

Quantitative characterization of these defects requires maps of strain and misorientation, which can be achieved by acquiring several images at different 2θ angles of the objective-detector arm. At present, such scans are time-consuming and, therefore, not suitable for dynamic studies of processes occurring within seconds or less.^6,7 Nonetheless, strain maps acquired before and after the transformation provide unique insight into the underlying defect structure inside the reached state. By observing the strain map in Fig. 6(a), taken above T_C with zero applied field, wavy patterns similar to those seen in Fig. 5(a) are clearly visible. Since they are measured under the same conditions, it seems reasonable to attribute both patterns to local lattice distortions. Figure 6(b) shows a similar strain map, acquired with the same experimental parameters but now under a field of 0.8 kV/mm. The difference in strain is clear, with some of the wavy features weakly present in the map. The large change in strain is a strong indication of the phase transformation that has been observed with RSMs and in DFXM image series. The changes in strain along the wavy patterns and sample interphase at the surface are clear and can potentially be attributed to the alignment of misorientations with the electric field and compressive strains at the surface induced by the electric field.²⁶ 

The sample environment system we describe has demonstrated the ability to acquire diffraction, time-dependent, and strain-dependent datasets in a single experimental setting. Moreover, experiments can be carried out in bulk samples under highly controlled thermal and electrical boundary conditions. Ultimately, this makes it possible to detect bulk dynamics in real time. Phase fronts, domain walls, and other features can be detected and tracked with the DFXM mode, and the associated strain from the transformations can be characterized. The thermal stability of the system has proven to be of high importance, and STEF has been shown to possess the capability of temperature control of ΔT = ±0.01 K in the 20–250 °C range. At the same time, we have applied fields at least up to 0.8 kV/mm without short-circuiting. This makes it a very versatile holder system, which can be used for many thermal and electroactive materials.

The possibility to probe different elements of thermodynamic parameter space could make it a tool for improving and validating phase-field and other thermodynamic models. While Landau–Ginsburg–Devonshire (LGD) theory²⁷ predicts phase transformations similar to what we have shown here, it is strongly dependent on the accuracy of the material data it uses and the assumptions it makes regarding symmetry and the presence of local heterogeneity. Thus, providing LGD models with quantitative data under specific thermal and/or electrical boundary conditions could provide a more precise prediction of material properties and behavior using LGD-type approaches. Furthermore, such theories may provide further insight into how domain and phase fronts might be propagated in directions non-parallel to the electric field.²⁸ 

The natural evolution for the STEF system presented here would be to implement computer vision methods of tracking topological features and classifying datasets.²⁹ We foresee that this will lead to the development of either machine or deep-learning based methods that can be integrated with this or similar in situ systems. This could lead to a much more profound method of studying materials and create a new type of synergy between theoretical models and experiments.

The authors are grateful for beam times and the use of the facilities at the ESRF. The authors are particularly grateful for the help from the machine workshops at DTU Physics, NTNU Gløshaugen, and sample environment team at ESRF. They thank Julia Glaum for providing them with excellent samples for their beam time. They specially thank Julian Walker for motivational discussions. They also thank DANSCATT for support of travels. H.S. acknowledges support from ERC Starting Grant No. 804665 “3D-PXM.”