Domain Patterns Research Articles

Deep learning and correlative microscopy for quantification of grain orientation in sintered FeNdB-type permanent magnets by domain pattern analysis

Based on a data-driven approach, a computer-assisted workflow for the quantitative analysis of optical Kerr microscopy images of sintered FeNdB-type permanent magnets was developed. By analyzing the domain patterns visible in the Kerr image with data-driven approaches such as traditional machine learning and advanced deep learning, we can quantify grain orientation and size with a better trade-off between accuracy and higher throughput than electron backscatter diffraction (EBSD). The key distinction between traditional machine learning and advanced deep learning lies in feature extraction. Traditional methods require manual, user-dependent feature extraction from input data, while advanced deep learning achieves this automatically. The predictions from the trained models were compared to the measurements from EBSD for performance evaluation. The proposed data-driven model is trained on the dataset created from the correlative microscopy technique, which requires the images of grains extracted from the Kerr microscopy and corresponding EBSD grain orientation data (Euler angles). The fine-tuned deep learning model shows better generalization ability than the traditional machine learning models trained on the manually extracted features and resulted in a mean absolute error of less than 5° for grain orientation of the anisotropic magnet samples when evaluated against the measured EBSD values. The developed approach has reduced the measurement effort for grain orientation by 5 times and have sufficient accuracy when compared to the EBSD.Further, the application of the proposed approach to determine the quality of the alignment or texture in anisotropic sintered magnets and its relationship with the magnetic remanence based on reliable statistical grain orientation data has been discussed. This approach could emerge as a tool for rapidly analyzing large-scale samples to discover and quantify heterogeneities in grain size and grain orientation.

Theoretical and numerical study of the Landau-Khalatnikov model describing a formation of 2D domain patterns in ferroelectrics

Among the numerous applications of the Ginzburg-Landau theory to the analysis of significant reaction-diffusion systems, modeling of the behavior of promising polar dielectric materials should be especially highlighted. The paper is devoted to the theoretical and numerical analysis of the Landau-Khalatnikov model describing the dynamics of 2D domain pattern formation in ferroelectrics. The unique solvability of the initial-boundary value problem for the system of 2D cubic-quintic Landau-Khalatnikov equations is proved. The proof is based on the derivation of new a priori estimates for the solution of the system of nonlinear parabolic equations. A series of computational experiments are conducted to examine both spontaneous and polar-induced domain pattern formation in biaxial ferroelectrics. Finite-element simulations allow us to visualize different types of ferroelectric domain structures depending on the varying boundary conditions.

Spontaneous pattern of orthogonal ferroelectric domains in epitaxial KNN films

Lead-free piezoelectric (K, Na)NbO3 (KNN) is considered one of the promising candidates for the replacement of Pb(ZrxTi1−x)O3. Several studies underlined the issue of K and Na volatility with increasing deposition temperatures, leading to high leakage currents in thin films, which still represents a major drawback for applications. This paper shows how epitaxial growth with concomitant preferred orientation of KNN films on niobium-doped strontium titanate (Nb:STO) depends on growth temperature and substrate strain. A preferred out-of-plane polar (001) orientation of KNN is obtained at high temperatures (>600 °C), while (100) orientation is dominant for lower ones. The (001) orientation is forced out-of-plane due to the sizeable in-plane stress derived from a negative lattice mismatch of pseudo-cubic KNN with respect to the underlying cubic (001) Nb:STO substrate. Moreover, we show that K-Na deficiency and high leakage of epitaxial KNN films deposited at high temperatures are accompanied by the appearance of a pattern of orthogonal spontaneous ferroelectric domains aligned to the [100] and [010] directions of Nb:STO. This pattern, visible in secondary electron microscopy, piezoforce response microscopy, and conductive atomic force microscopy images, is uncorrelated to the surface morphology. Supported by reciprocal space mapping by x-ray diffraction, this phenomenon is interpreted as the result of strain relaxation via ferroelectric domain formation related to K-Na deficient films displaying a sizable and increasing compressive strain when grown on Nb:SrTiO3. Our findings suggest that strain engineering strategies in thin films could be used to stabilize specific configurations of piezo- and ferroelectric domains.

Observation of Micro-Scale Domain Structure Evolution under Electric Bias in Relaxor-Based PIN-PMN-PT Single Crystals

Relaxor ferroelectrics play a vital role as functional components in electromechanical devices. The observation of micro-scale domain structure evolution under electric bias in relaxor ferroelectrics has posed challenges due to their complex domain morphology characterized by small-sized domains. The present study aims to investigate the dielectric diffusion–relaxation characteristics, domain structure, and domain switching evolution under electric bias in high-performance single crystals of Pb(In1/2Nb1/2)O3-Pb(Mg1/3Nb2/3)O3-33PbTiO3. The findings reveal the presence of strip-like domain patterns that interlock irregular small-sized nanodomains in PIN-PMN-33PT single crystals. Furthermore, the sample undergoes three distinct stages under electric bias, including the nucleation of new domains, the gradual forward expansion of domains, and the lateral expansion of domains. These observations provide valuable insights for understanding and exploring domain engineering techniques in relaxor ferroelectrics.

Patient-reported outcomes before treatment for localized prostate cancer: are there differences among countries? Data from the True North Global Registry

IntroductionSimilar Patient-Reported Outcomes (PROs) at diagnosis for localized prostate cancer among countries may indicate that different treatments are recommended to the same profile of patients, regardless the context characteristics (health systems, medical schools, culture, preferences…). The aim of this study was to assess such comparison.MethodsWe analyzed the EPIC-26 results before the primary treatment of men diagnosed of localized prostate cancer from January 2017 onwards (revised data available up to September 2019), from a multicenter prospective international cohort including seven regions: Australia/New Zealand, Canada, Central Europe (Austria / Czech Republic / Germany), United Kingdom, Italy, Spain, and the United States. The EPIC-26 domain scores and pattern of three selected items were compared across regions (with Central Europe as reference). All comparisons were made stratifying by treatment: radical prostatectomy, external radiotherapy, brachytherapy, and active surveillance.ResultsThe sample included a total of 13,483 men with clinically localized or locally advanced prostate cancer. PROs showed different domain patterns before treatment across countries. The sexual domain was the most impaired, and the one with the highest dispersion within countries and with the greatest medians’ differences across countries. The urinary incontinence domain, together with the bowel and hormonal domains, presented the highest scores (better outcomes) for all treatment groups, and homogeneity across regions.ConclusionsPatients with localized or locally advanced prostate cancer undergoing radical prostatectomy, EBRT, brachytherapy, or active surveillance presented mainly negligible or small differences in the EPIC-26 domains before treatment across countries.The results on urinary incontinence or bowel domains, in which almost all patients presented the best possible score, may downplay the baseline data role for evaluating treatments’ effects. However, the heterogeneity within countries and the magnitude of the differences found across countries in other domains, especially sexual, support the need of implementing the PRO measurement from diagnosis.

Coexistence of five domains at single propagating interface in single-crystal Ni[sbnd]Mn[sbnd]Ga shape memory alloy

Coexistence of both austenite and martensite during phase transformation is a common feature of all Shape Memory Alloys (SMAs). The martensite has different variants featuring characteristic deformations rotationally linked to each other due to the symmetries of the austenite parent phase, and the martensite variants can form twins with different mean characteristic deformations. Multiple-domain microstructures (consisting of austenite, martensite twins and individual martensite variants) evolve collectively within an SMA sample during the phase transformation, contributing thus to the material's macroscopic response (e.g., its global deformation). Particularly, the multiple domains can exist at the diffuse austenite-martensite interface nucleating and propagating in a single crystal in certain conditions. This implies an energy barrier for this interfacial structure, influencing the interface kinetics and the driving force (energy dissipation) of the phase transformation. In this paper, we report an experimentally observed interface consisting of five domains (austenite, one martensite variant and three twins) in a Ni–Mn–Ga single-crystal initially consisting of one martensite variant and subjected to a uniaxial thermal gradient. The compatibility analysis (performed from the characteristic strains of the three martensite variants having approximately a tetragonal symmetry) reveals that the five-domain interface is not a perfectly compatible pattern like the basic habit plane (consisting of only one twin compatible with austenite). However, its level of non-compatibility is similar to that of the quite common X-interface (four-domain coexistence) which is observed in many SMAs. Further, the significant effects of the thermal loading path and the material initial state (the initial martensite variant) on the domain pattern formation are demonstrated and analyzed. The experimental observation and the theoretical analysis of the domain patterns can provide hints to better understand diffuse interface kinetics and phase transformation hysteresis.

Air surface crystallization with quenched-in Fe crystallites and associated magnetomechanical properties in Fe–Si–B amorphous ribbons

The effect of surface crystallization on the magnetomechanical properties is studied for amorphous ribbons with a nominal composition of Fe80Si9B11 that are prepared with the single roller melt-spinning method. Our results indicate that the magnetomechanical power efficiency of surface crystallization ribbons is twice as higher as that of its amorphous counterpart. The surface configuration for such ribbons has been investigated through a prism of microstructures and chemical compounds. The surface crystallization with quenched-in α-Fe(Si) crystallites is confirmed by means of x-ray diffraction measurements and the surface polishing process, inducing irregular in-plane domain patterns, which are observed by magneto-optical Kerr effect microscopy. The vibrational absorption peak for polymerized B–O bonds is observed by means of a Fourier transform infrared spectrometer, being ascribed to the presence of pre-precipitated boron atoms from the crystalline film, which is confirmed by x-ray photoelectron spectroscopy. The mechanism that enhances the magnetomechanical power conversion efficiency in FeSiB ribbons with quenched-in crystallites is quite different from the ones with annealing induced “surface-interior” configurations in the previous investigations.

Development of Fabrication Techniques for Magneto-Optical Diffractive Deep Neural Networks

Magneto-optical diffractive deep neural network (MO-D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN) is a type of diffractive deep neural network that utilizes the magneto-optical effect of magnetic materials. MO-D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN is composed of magnetic material, which is non-volatile and allows neurons to be rewritten. In our work, we fabricated and evaluated MO-D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN with 2 layers containing 100 × 100 magnetic domains with a domain size of 1 μm. Nd <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.5</sub> Bi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2.5</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> GaO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> thin films were prepared on the both sides of a Gd <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> Ga <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> substrate and magnetic domain patterns were recorded using the magneto-optical recording technique. The recording accuracies of first and second layer were 79% and 70%, respectively. We believed that magneto-optical recording technology can achieve a recording accuracy of over 90% by optimizing the fabrication and recording conditions. The optical setup was built and handwritten digit classification was carried out using fabricated MO-D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN, but the number of classifications was lower than in the case of simulation. We considered that recording errors were responsible for the low number of classifications and assessed the effect of recording errors on accuracy by simulation. When the recording error was lower than 10%, the loss in classification accuracy was found to be only ~3%. Therefore, we are convinced that MO-D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> NN has high physical implementation potential.

The polarization quaternion and its applications: a joint representationof the Q/U Stokes parameters and E/B mode polarizations

In this work, we point out that the Q/U Stokes parameters and E/B modepolarizations are the four components of a unique quaternion, which describes atthe same time the directions and the parity states of spherical linearpolarizations. We then point out that, with this polarization quaternion, themathematical form of all Q/U and E/B transforms are greatly simplified, to anextent that requires only one quaternion multiplication for each transform. Apreliminary application of the polarization quaternion is shown as an example todetect peculiar pixel domain patterns within the E- and B-families, which arethe former and latter halves of the polarization quaternion.

Creation of a Periodic Domain Structure in MgOLN by Femtosecond Laser Irradiation

The systematic imaging of the damaged tracks and domain patterns created in the MgOLN plates by one-step fs-laser irradiation at different depths was carried out. It is shown that the domains in the bulk have a spindle-like shape and start to grow in the Z− direction from the track ends. The domain shape changes from a spindle-like one with charged walls to a hexagonal prism with neutral walls after the domain reaches the polar surface. The length of the domains located in the bulk increases linearly with the pulse energy. The hexagonal domain shape at the surface is typical for the crystals of the lithium niobate family. The obtained effects have been considered in terms of the kinetic approach. After irradiation, the domains appear in the vicinity of the track ends with maximum electric field strength and grow under the action of a spatially nonuniform pyroelectric field. The key role of the pyroelectric field is confirmed by the creation of new domains at the surface without correlation with the position of the focusing point located at the vicinity of the surface. The 3D domain pattern was produced, which represented four layers of the regular matrices consisting of elongated domains about 100 μm in length.

Billiard based optimization with deep learning driven anomaly detection in internet of things assisted sustainable smart cities

Internet of Things (IoT) technology involves a network of interconnected devices and sensors that gather and exchange information. In smart cities, IoT devices were utilized in several fields including energy, transportation, waste management, healthcare, etc., to improve the overall quality of life and sustainability of the populace. But, as the usage of IoT increases, the cybersecurity and data privacy become a concern of safety. An anomaly detection system helps to identify possible data breaches or cyber-attacks by identifying abnormal data patterns. Deep learning (DL) driven anomaly detection has emerged as an effective and powerful method for identifying abnormal behaviours or patterns in the data domain. This technique leverages the abilities of a deep neural network for automated learning of complex patterns and representations from data, which make it better for anomaly detection task where irregularities cannot be easily defined by handcrafted rules. This paper establishes a new Billiard Based Optimization with Deep Learning Driven Anomaly Detection and Classification (BBODL-ADC) technique in IoT-assisted Sustainable Smart Cities. The goal of the BBODL-ADC technique lies in the proper recognition and classification of anomalies in the IoT-assisted smart city. To obtain that, the BBODL-ADC system applies a binary pigeon optimization algorithm (BPEO) algorithm for the effectual selection of features. Besides, the BBODL-ADC technique utilizes Elman recurrent neural network (ERNN) approach for the recognition and classification of anomalies. Moreover, the BBO system can be used for better parameters chosen by the ERNN algorithm. The stimulation value of the BBODL-ADC algorithm can be executed benchmark database. The achieved outcomes demonstrate the remarkable outcome of the BBODL-ADC methodology of 95.69% and 99.21% compared to existing models under dataset-1 and dataset-2.

Combined Funnel, Concentrator, and Particle Valve Functional Element for Magnetophoretic Bead Transport Based on Engineered Magnetic Domain Patterns.

Controlled actuation of superparamagnetic beads (SPBs) within a microfluidic environment using tailored dynamic magnetic field landscapes (MFLs) is a potent approach for the realization of point-of-care diagnostics within Lab-on-a-chip (LOC) systems. Making use of an engineered magnetic domain pattern as the MFL source, a functional LOC-element with combined magnetophoretic "funnel", concentrator, and "valve" functions for micron-sized SPBs is presented. A parallel-stripe domain pattern design with periodically decreasing/increasing stripe lengths is fabricated in a topographically flat continuous exchange biased (EB) thin film system by ion bombardment induced magnetic patterning (IBMP). It is demonstrated that, upon application of external magnetic field pulses, a fully reversible concentration of SPBs at the domain pattern's focal point occurs. In addition, it is shown that this functionality may be used as an SPB "funnel", allowing only a maximum number of particles to pass through the focal point. Adjusting the pulse time length, the focal point can be clogged up for incoming SPBs, resembling an on/off switchable particle "valve". The observations are supported by quantitative theoretical force considerations.

Electric field-controlled deterministic magnetization reversal in nanomagnet Fe3Si/PMN-PT multiferroic heterostructures

Pure electric field-controlled 180° magnetization switching plays a vital role in low-power magnetoelectric memory devices. Using micromagnetic simulation, we engineered a square-shaped epitaxial Fe3Si nanomagnet on a PMN-PT piezoelectric substrate to make the magnetic easy axis slightly deviate 18° from the piezostrain axis, aiming to break the symmetry of the magnetization distribution and achieve deterministic magnetization reversal paths. Under the coaction of a magnetic field and an electric field, the simulated magnetic hysteresis loops and magnetic domain patterns reveal a fourfold to twofold magnetic anisotropy transition and magnetization reversal paths. Stimulated by pure electric field-induced piezoelectric strain, deterministic 180° magnetization reversals are accomplished by the two successive clockwise 90° switching process. The results help to comprehend electrically regulated deterministic magnetization reversal and pave an avenue for designing multistate spintronics devices.

A Multi-Label Learning Framework for Predicting Chemical Classes and Biological Activities of Natural Products from Biosynthetic Gene Clusters.

Natural products (NP) or secondary metabolites, as a class of small chemical molecules that are naturally synthesized by chromosomally clustered biosynthesis genes (also called biosynthetic gene clusters, BGCs) encoded enzymes or enzyme complexes, mediates the bioecological interactions between host and microbiota and provides a natural reservoir for screening drug-like therapeutic pharmaceuticals. In this work, we propose a multi-label learning framework to functionally annotate natural products or secondary metabolites solely from their catalytical biosynthetic gene clusters without experimentally conducting NP structural resolutions. All chemical classes and bioactivities constitute the label space, and the sequence domains of biosynthetic gene clusters that catalyse the biosynthesis of natural products constitute the feature space. In this multi-label learning framework, a joint representation of features (BGCs domains) and labels (natural products annotations) is efficiently learnt in an integral and low-dimensional space to accurately define the inter-class boundaries and scale to the learning problem of many imbalanced labels. Computational results on experimental data show that the proposed framework achieves satisfactory multi-label learning performance, and the learnt patterns of BGCs domains are transferrable across bacteria, or even across kingdom, for instance, from bacteria to Arabidopsis thaliana. Lastly, take Arabidopsis thaliana and its rhizosphere microbiome for example, we propose a pipeline combining existing BGCs identification tools and this proposed framework to find and functionally annotate novel natural products for downstream bioecological studies in terms of plant-microbiota-soil interactions and plant environmental adaption.

Magneto-optical diffraction of visible light as a probe of nanoscale displacement of domain walls at femtosecond timescales.

Using diffraction of femtosecond laser pulses of visible light by a magnetic domain pattern in an iron garnet, we demonstrate a proof of concept of time-resolved measurements of domain pattern movements with nanometer spatial and femtosecond temporal resolution. In this method, a femtosecond laser (pump) pulse initiates magnetization dynamics in a sample that is initially in a labyrinth domain state, while an equally short linearly polarized laser pulse (probe) is diffracted by the domain pattern. The components of the diffracted light that are polarized orthogonally to the incident light generate several concentric diffraction rings. Nanometer small changes in the relative sizes of domains with opposite magnetization result in observable changes in the intensities of the rings. We demonstrate that the signal-to-noise ratio is high enough to detect a 6nm domain wall displacement with 100fs temporal resolution using visible light. We also discuss possible artifacts, such as pump-induced changes of optical properties, that can affect the measurements.

A Decade of Research Into Second Language Acquisition: A State-of-the-Science Review (2009-2019)

Abstract This study investigated the knowledge domain of second language acquisition (SLA) research in 2009-2019. It selected about a decade of studies in the SLA knowledge domain and applied an available data analysis tool CiteSpace to show a picture of SLA research. The 11-year period of studies was retrieved from the Web of Science database, totally 4, 660 bibliographic records. By employing the embedded algorithm analyzing literature in CiteSpace, the present study found the factors influencing SLA and statistical models were critical aspects in L2 research, and research topics were conducted in psychological, neuroscience, linguistic, and language teaching frameworks. Additionally, research patterns in the SLA domain have become more diversified, and empirical studies of cognitive science have gradually taken a dominant position in recent years. Spikes of increased numbers of detected papers in the period of 2009-2019 showed research focus and methodology would possibly continue to be interdisciplinary in future research.

Probing the photo induced micro actuation properties of optimized Cu doped Co-34at%-Ni-35at%-Al-31at% ferromagnetic shape memory alloys

The strange Photo induced micro actuation (PIMA) properties associated with Cu doped Co-Ni-Al Ferromagnetic Shape Memory Alloys (FSMA) has showed enhancement of mechanical properties in our early studies. This enhancement and optimized PIMA properties have opened a scope of various laser-controlled real engineering applications. In this study, we have explored the microstructural scaling and determined the evolved force due to the laser action in optimized Cu doped Co-Ni-Al alloy system with a detailed insight. This unique PIMA property is rare in literature and actual reason are still not explored. EBSD, TEM, AFM/MFM based techniques were further adopted to understand the role of microstructure in more detail. The evolved magnetic domain patterns for actuating and non-actuating alloys have been analyzed in details. Atomic Force Microscopy was adopted to estimate the constitutive mechanical properties of the thin alloy samples by mechanical spectrometry. This estimation was directly used to predict a general trend in force distribution pattern of the Cu-Co-Ni-Al alloys strips through FEM analysis. The crux of FEM based model and its simulated results can be applied for real time PIMA oriented engineering application and device manufacturing. The detail microstructural exploration and its effect in the PIMA response along with generalized deflection-force model have been presented in the current study.

A novel compact semicircular defected ground structure ultrawideband monopole antenna integrated with Ku band for breast cancer detection

SummaryA compact and novel ultrawideband (UWB) antenna integrated with Ku band for microwave imaging applications is reported. The reported antenna consists of a novel patch structure along with a tapered feed structure on top of FR‐4 substrate having a compact dimension of 16 × 22 mm2 and novel semicircular defected ground structure (SC‐DGS) etched below the substrate. The proposed antenna covers a wide impedance bandwidth of 15.07 GHz spanning from 3.51 to 18.58 GHz with a fractional bandwidth (FBW) of 136%. The intended antenna performance is analyzed for S11, peak gain, efficiency, and radiation patterns in frequency domain, and group delay, isolation (S21), and linear phase response in time domain. The simulated and measured results of reported antenna are well matched for the usage in practical portable UWB applications. More importantly, the proposed antenna system is simulated for the detection of breast cancer tissue by comparison of S21 results, current densities, and specific absorption rate with and without tumor conditions. These findings show that the radiator and the whole system work well in detecting the breast tumor.

PyProtif: a PyMol plugin to retrieve and visualize protein motifs for structural studies.

Proteins often possess several motifs and the ones with similar motifs were found to have similar biochemical properties and thus related biological functions. Thereby, multiple databases were developed to store information on such motifs in proteins. For instance, PDBsum stores the results of Promotif's generated structural motifs and Pfam stores pre-computed patterns of functional domains. In addition to the fact that all this stored information is extremely useful, we can further augment its importance if we ought to integrate these motifs into visualization software. In this work, we have developed PyProtif, a plugin for the PyMOL molecular visualization program, which automatically retrieves protein structural and functional motifs from different databases and integrates them in PyMOL for visualization and analyses. Through an expendable menu and a user-friendly interface, the plugin grants the users the ability to study simultaneously multiple proteins and to select and manipulate each motif separately. Thus, this plugin will be of great interest for structural, evolutionary and classification studies of proteins.

Domain pattern formation in tetragonal ferroelectric ceramics

We present the results of high-resolution simulations of the ferroelectric domain evolution in polycrystalline lead zirconate titanate (PZT), using a phase-field framework that accounts for thermal fluctuations. Leveraging the parallel efficiency of a Fourier spectral scheme, we model micron-sized ceramic samples with thousands of grains at atomic-unit-cell spatial resolution. We introduce a method to automatically identify and track different types of domain walls from phase-field data, which we exploit to study their role during polarization reversal under applied electric fields. Results indicate that the density of domain walls and the domain width obey the Kittel-Mitsui-Furuichi-Roitburd square-root law with implications on the macroscopically observable piezo- and dielectric material properties. Moreover, analyzing the statistics of the domain pattern formation in simulated samples reveals correlations between the average polarization and strain within a grain and its crystallographic orientation, which is in agreement with high-energy x-ray diffraction experiments. Furthermore, we study the occurrence of the two predominant types of domain patterns – monodomain and laminate/twin domain structures – whose emergence within gains of a polycrystal is traced back to the grain orientation. Phase-field statistics are supported by a simple analytical model, which is based on minimizing the electric enthalpy and accurately predicts some of the reported correlations and allows us to further study the behavior of monodomains vs. laminate patterns in ferroelectric ceramics.