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Crystal surfaces play a pivotal role in governing various significant processes, such as adsorption, nucleation, wetting, friction, and wear. A fundamental property that influences these processes is the surface free energy, γ. We have directly calculated γ(T) for low-index faces of Lennard-Jones (LJ), germanium, and silicon crystals along their sublimation lines using the computational cleavage technique. Our calculations agree well with experimental values for Si(111) and Ge(111), highlighting the accuracy of the method and models used. For LJ crystals, we identified a premelting onset at Tpm = 0.75Tm, marked by a sharp increase in atom mobility within the second outermost surface layer. Notably, Tpm closely aligned with the endpoint of the LJ melting line at negative pressures, Tend = 0.76Tm. We hypothesize that the emergence and coexistence of a liquid film atop the LJ crystal at Tpm < T < Tm correspond to the metastable melting line under negative pressures experienced by stretched crystal surfaces. Furthermore, our study of thin LJ crystal slabs reveals that premelting-induced failure leads to recrystallization below the homogeneous freezing limit, offering a promising avenue to explore crystal nucleation and growth at extremely deep supercoolings. Finally, no evidence of premelting was detected in the model crystals of Ge and Si, which is consistent with the experimental observations. Overall, our findings offer valuable insights into crystal surface phenomena at the atomic scale. [ABSTRACT FROM AUTHOR]

We have measured diffuse scattering in a single crystal of Cd0.9Zn0.1Te using a state-of-the-art laboratory diffractometer. A large-box atomistic simulation of a model crystal is used in conjunction with Monte Carlo modeling and the Kirkwood potential. A combination of structural relaxation in the presence of the dopant and thermal motion results in good qualitative agreement between the computed diffraction patterns of the model crystal and the measured x-ray patterns. This is shown to be rather distinct from the diffuse scattering arising from purely structural relaxations or thermal motion only. The atoms are shown to displace predominantly in ⟨1,1,1⟩ and ⟨1,0,0⟩ type directions. Our approach to Monte Carlo modeling can easily be extended to more complex defect structures to incorporate, e.g., chemical ordering on the Cd/Zn sublattice, more than one species of dopant or vacancies. [ABSTRACT FROM AUTHOR]

High-fidelity constitutive models are critical for accurate meso-scale continuum modeling and prediction of shock initiation of crystalline energetic materials (EMs). While empirically calibrated or atomistic-guided anisotropic elastoplastic models of EM such as cyclotetramethylene-tetranitramine (HMX) and cyclotrimethylene-trinitramine (RDX) capture important micromechanical phenomena (such as dislocation evolution, slip-resistance, and anisotropic elasticity), the computational cost of using anisotropic single-crystal plasticity models can become prohibitive for meso-scale computations of void-collapse and hotspot formation in microstructures. Thermo-mechanically representative, isotropic, pressure, temperature, and rate-dependent material constitutive models are practical alternatives for meso-scale simulations of the shock response of microstructures. To this end, this work constructs physically consistent isotropic plasticity from anisotropic single-crystal plasticity models for HMX and RDX. State-of-the-art crystal plasticity models for HMX and RDX are used to compute the stress states in single crystals oriented in three different directions relative to shocks generated by impact at velocities ranging from 100 to 1000 m/s. Post-shock von Mises stress fields for the three orientations are then used to calibrate the strain-rate hardening coefficient and the reference strain rate for a rate-dependent Johnson–Cook (JC) yield surface model. We compare the pressures and the post-shock von Mises stresses between the JC and the anisotropic models to show that the isotropic computations closely approximate the averaged deformation response of the three different crystal orientations. We then model the interaction of a shock generated by a 500 m/s impact with a 0.5 μm void and show that the pressures and the deviatoric stresses obtained using the isotropic model closely match those computed from anisotropic models for both HMX and RDX. The resulting isotropic J2 plastic flow model for HMX and RDX can be employed to perform meso-scale simulations for energy localization due to shear bands and void collapse in the two materials. [ABSTRACT FROM AUTHOR]

The thermal hysteresis exhibited in plastic crystal compounds greatly reduces their cyclic efficiency, limiting their potential for replacing current environmentally harmful refrigerants. A mechanistic understanding of the origins of this hysteresis has yet to be established. Here, we systematically investigate the transformation kinetics of the model plastic crystal, neopentyl glycol (NPG), through microscopic and calorimetric techniques. We reveal an asymmetry between the forward (heating) and reverse (cooling) transitions. We also demonstrate that the forward transformation is rate-limited by the rate of growth of rotationally disordered domains. In contrast, the reverse transformation is rate-limited by the nucleation of the ordered crystal domain, demonstrated by the sharp exothermic peaks in calorimetry and rapid self-nucleation phenomena observed optically. This nucleation limitation is largely responsible for the large thermal hysteresis in NPG, which we observe to be as large as 16.7 °C for an approximately 10 mg sample cooled at 0.5 °C min−1. These findings demonstrate the underlying origin of the thermal hysteresis and introduce a direction to mitigate hysteresis in plastic crystal transformations. [ABSTRACT FROM AUTHOR]

The magnetocrystalline anisotropy energy of atomically ordered L10 FeNi (the meteoritic mineral tetrataenite) is studied within a first-principles electronic structure framework. Two compositions are examined: equiatomic Fe 0.5 Ni 0.5 and an Fe-rich composition, Fe 0.56 Ni 0.44 . It is confirmed that, for the single crystals modeled in this work, the leading-order anisotropy coefficient K 1 dominates the higher-order coefficients K 2 and K 3. To enable comparison with experiment, the effects of both imperfect atomic long-range order and finite temperature are included. While our computational results initially appear to undershoot the measured experimental values for this system, careful scrutiny of the original analysis due to Néel et al. [J. Appl. Phys. 35, 873 (1964)] suggests that our computed value of K 1 is, in fact, consistent with experimental values, and that the noted discrepancy has its origins in the nanoscale polycrystalline, multivariant nature of experimental samples, that yields much larger values of K 2 and K 3 than expected a priori. These results provide fresh insight into the existing discrepancies in the literature regarding the value of tetrataenite's uniaxial magnetocrystalline anisotropy in both natural and synthetic samples. [ABSTRACT FROM AUTHOR]

In this paper, we announce the public release of a massively parallel, graphics processing unit (GPU)-accelerated software, which is the first to combine both coarse-grained particle simulations and field-theoretic simulations in one simulation package. MATILDA.FT (Mesoscale, Accelerated, Theoretically Informed, Langevin, Dissipative particle dynamics, and Field Theory) was designed from the ground-up to run on CUDA-enabled GPUs with Thrust library acceleration, enabling it to harness the possibility of massive parallelism to efficiently simulate systems on a mesoscopic scale. It has been used to model a variety of systems, from polymer solutions and nanoparticle-polymer interfaces to coarse-grained peptide models and liquid crystals. MATILDA.FT is written in CUDA/C++ and is object oriented, making its source-code easy to understand and extend. Here, we present an overview of the currently available features, and the logic of parallel algorithms and methods. We provide the necessary theoretical background and present examples of systems simulated using MATILDA.FT as the simulation engine. The source code, along with the documentation, additional tools, and examples, can be found on the GitHub MATILDA.FT repository. [ABSTRACT FROM AUTHOR]

Classical scoring functions may exhibit low accuracy in determining ligand binding affinity for proteins. The availability of both protein–ligand structures and affinity data make it possible to develop machine‐learning models focused on specific protein systems with superior predictive performance. Here, we report a new methodology named SAnDReS that combines AutoDock Vina 1.2 with 54 regression methods available in Scikit‐Learn to calculate binding affinity based on protein–ligand structures. This approach allows exploration of the scoring function space. SAnDReS generates machine‐learning models based on crystal, docked, and AlphaFold‐generated structures. As a proof of concept, we examine the performance of SAnDReS‐generated models in three case studies. For all three cases, our models outperformed classical scoring functions. Also, SAnDReS‐generated models showed predictive performance close to or better than other machine‐learning models such as KDEEP, CSM‐lig, and ΔVinaRF20. SAnDReS 2.0 is available to download at https://github.com/azevedolab/sandres. [ABSTRACT FROM AUTHOR]

Despite their high prediction accuracy, deep learning-based soft sensor (DLSS) models face challenges related to adversarial robustness against malicious adversarial attacks, which hinder their widespread deployment and safe application. Although adversarial training is the primary method for enhancing adversarial robustness, existing adversarial-training-based defense methods often struggle with accurately estimating transfer gradients and avoiding adversarial robust overfitting. To address these issues, we propose a novel adversarial training approach, namely domain-adaptive adversarial training (DAAT). DAAT comprises two stages: historical gradient-based adversarial attack (HGAA) and domain-adaptive training. In the first stage, HGAA incorporates historical gradient information into the iterative process of generating adversarial samples. It considers gradient similarity between iterative steps to stabilize the updating direction, resulting in improved transfer gradient estimation and stronger adversarial samples. In the second stage, a soft sensor domain-adaptive training model is developed to learn common features from adversarial and original samples through domain-adaptive training, thereby avoiding excessive leaning toward either side and enhancing the adversarial robustness of DLSS without robust overfitting. To demonstrate the effectiveness of DAAT, a DLSS model for crystal quality variables in silicon single-crystal growth manufacturing processes is used as a case study. Through DAAT, the DLSS achieves a balance between defense against adversarial samples and prediction accuracy on normal samples to some extent, offering an effective approach for enhancing the adversarial robustness of DLSS. [ABSTRACT FROM AUTHOR]

Rutinosidase is a diglycosidase that catalyzes the cleavage of rutinose (α‐l‐Rhap‐(1→6)‐β‐d‐Glcp) from rutin or other rutinosides. It is also able to cleave β‐glucopyranosides, e. g., isoquercitrin. This enzyme has a strong transglycosylation activity and a remarkable substrate specificity. We have shown that rutinosidase from Aspergillus niger (AnRut) is able to cleave β‐glucopyranosides acylated at C‐6 of glucose (6′‐O‐acylisoquercitrin) with acetyl, benzoyl, phenylacetyl, phenylpropanoyl, cinnamoyl, vanillyl, galloyl, 4‐hydroxybenzoyl and 3‐(4‐hydroxy‐3‐methoxyphenyl)propanoyl. The release of the respective 6‐acylglucopyranoses was confirmed by HPLC/MS and NMR methods. Selected compounds, i. e., 6′‐O‐acetyl, 6′‐O‐benzoyl, and 6′‐O‐cinnamyl derivatives of isoquercitrin, were also tested as transglycosylation substrates. Only 6′‐acetylisoquercitrin and 6′‐O‐benzoylisoquercitrin underwent transglycosylations by AnRut to produce n‐butyl 6‐acetyl‐β‐d‐glucopyranoside and n‐butyl 6‐benzoyl‐β‐d‐glucopyranoside. Isoquercitrin 6′‐O‐cinnamate yielded on hydrolytic product. Molecular modeling based on the crystal structure of AnRut showed that large aromatic moieties at C‐6′ of isoquercitrin block the side tunnel of AnRut leading into its active site and thus hinder the entry of the acceptor substrate for transglycosylation. This study demonstrates the great substrate flexibility of rutinosidase at the glycone site. [ABSTRACT FROM AUTHOR]

Hierarchical self‐assembly represents a powerful strategy for the fabrication of functional materials across various length scales. However, achieving precise formation of functional hierarchical assemblies remains a significant challenge and requires a profound understanding of molecular assembly interactions. In this study, we present a molecular‐level understanding of the hierarchical assembly of sequence‐defined peptoids into multidimensional functional materials, including twisted nanotube bundles serving as a highly efficient artificial light harvesting system. By employing synchrotron‐based powder X‐ray diffraction and analyzing single crystal structures of model compounds, we elucidated the molecular packing and mechanisms underlying the assembly of peptoids into multidimensional nanostructures. Our findings demonstrate that incorporating aromatic functional groups, such as tetraphenyl ethylene (TPE), at the termini of assembling peptoid sequences promotes the formation of twisted bundles of nanotubes and nanosheets, thus enabling the creation of a highly efficient artificial light harvesting system. This research exemplifies the potential of leveraging sequence‐defined synthetic polymers to translate microscopic molecular structures into macroscopic assemblies. It holds promise for the development of functional materials with precisely controlled hierarchical structures and designed functions. [ABSTRACT FROM AUTHOR]

Hierarchical self‐assembly represents a powerful strategy for the fabrication of functional materials across various length scales. However, achieving precise formation of functional hierarchical assemblies remains a significant challenge and requires a profound understanding of molecular assembly interactions. In this study, we present a molecular‐level understanding of the hierarchical assembly of sequence‐defined peptoids into multidimensional functional materials, including twisted nanotube bundles serving as a highly efficient artificial light harvesting system. By employing synchrotron‐based powder X‐ray diffraction and analyzing single crystal structures of model compounds, we elucidated the molecular packing and mechanisms underlying the assembly of peptoids into multidimensional nanostructures. Our findings demonstrate that incorporating aromatic functional groups, such as tetraphenyl ethylene (TPE), at the termini of assembling peptoid sequences promotes the formation of twisted bundles of nanotubes and nanosheets, thus enabling the creation of a highly efficient artificial light harvesting system. This research exemplifies the potential of leveraging sequence‐defined synthetic polymers to translate microscopic molecular structures into macroscopic assemblies. It holds promise for the development of functional materials with precisely controlled hierarchical structures and designed functions. [ABSTRACT FROM AUTHOR]

Background: In polycrystal mechanics, determination of stress is associated with diffraction methods that measure (the inherently-related) elastic strain. Microscopic digital image correlation (DIC), while commanding much higher intragranular resolution, measures total strain, and its local accuracy is typically insufficient to evaluate elastic strain magnitudes. Objective: In situ DIC measurements over a partial unload of the polycrystal, where strains are virtually elastic, are explored for grain-averaged elastic strains and then, through a posed formalism, the stresses at the point of unload. Grain averaging is functionally employed to improve the DIC accuracy. The large objective is to emulate in situ complementary diffraction. Methods: Nickel with high elastic anisotropy is chosen. The utilized highly-automated instrument offers maximal resolution for DIC with optical microscopy over a gross grain field. Orientations are predetermined for the same grain layer via electron backscatter diffraction. High-accuracy grain masks are produced to isolate the strain fields of individual grains. Results: Very promising results are shown over a number of grains with sensible apparent compliance and stress values as well as linear unload behavior. Grains with sane results are largely predicted by a posed objectivity test that relies on DIC repeated with multiple reference loads. Conclusion: Though it will require extremely careful implementations of microscopic DIC with high intragranular resolution, the premise of measuring intergranular stress fields via partial unloads seems to be viable and worthy of further exploration and verification. This capability that is superposed over strain measurement offers a more stringent validation of high-fidelity crystal plasticity models. [ABSTRACT FROM AUTHOR]

The Debye temperature is a crucial parameter in understanding various properties of solids, including their melting temperature. This study focuses on 4H‐SiC, a material renowned for its wide bandgap and high thermal conductivity, making it ideal for high‐power electronic devices. Calculating various physical parameters for 4H‐SiC, including the Debye temperature, is crucial for semiconductor fabrication. However, it is observed that existing Debye models are unsuitable for computing the Debye temperature of 4H‐SiC. Therefore, phonon calculations alongside the Debye model to establish a new model for determining the Debye temperature of 4H‐SiC are used. This research has identified an optimal temperature range, referred to as the 'T150' model, between 150 and 160 K, which yields a Debye temperature consistent with experimental values. The newly developed "T150" model, demonstrated herein, holds the potential for determining the Debye temperatures of doped 4H‐SiC, other polytypes of 4H‐SiC, and other semiconductor materials, broadening its applicability in material science. [ABSTRACT FROM AUTHOR]

Investigating fatigue failure in titanium alloys is crucial for material design and engineering. Fatigue behavior in dual‐phase titanium alloys is strongly correlated with microstructural features and microdefects. This work formulates an improved modeling method to investigate fatigue behavior of bimodal Ti–6Al–4V, emphasizing the effects of lamellar orientation and microdefects. Using an improved Voronoi tessellation method, we establish representative volume element (RVE) models with various grain size distributions. Crystal plasticity finite element modeling (CPFEM) is used to analyze fatigue deformation in bimodal Ti–6Al–4V, considering microdefects and lamellar orientation. Fatigue indicator parameters are then incorporated into CPFEM to predict fatigue life and verified with experimental data. Numerical results highlight the significant influence of lamellar orientation and microdefects on fatigue behavior, with predicted life within the 3‐error band. This method efficiently overcomes challenges in quantitatively characterizing microstructural lamellae that experiments are short of, paving the way for designing fatigue‐resistant alloy materials with similar microstructures. Highlights: The proposed method effectively addresses the effects of lamellar orientation and microdefects.The improved VT modeling can reflect grain size distribution in bimodal Ti–6Al–4V.CPFEM‐based modeling method achieves acceptable fatigue life prediction accuracy. [ABSTRACT FROM AUTHOR]

Context: In order to further improve the manufacturing technology of resonator facet of GaAs (gallium arsenide)-based laser, the scratching process of GaAs was simulated by molecular dynamics. Models of GaAs crystals with different orientations, including GaAs [100], GaAs [110], and GaAs [111], were generated, followed by scratch simulations on these models. The surface characteristics of scratches, damage width, subsurface damage, stack height, and the distribution and activity characteristics of dislocations were analyzed based on the simulation results. The results show that there are obvious anisotropy in the deformation of different crystal orientation during the scratching process of GaAs. Surface features, damage width, subsurface damage, and dislocation dynamics during scraping in GaAs crystals strongly depend on crystal orientation. It was also observed that GaAs exhibits distinct characteristics of dislocation activity during the scratching process, depending on its crystal orientation. In addition, GaAs [110] crystal direction has the smallest maximum damage width and subsurface damage depth. The maximum of maximum damage width is in GaAs [100] crystal direction, and the maximum subsurface damage depth is in GaAs [111] crystal direction. In addition, the stacking height is maximum when GaAs [100] is scraped and minimum when GaAs [110] is scraped. Methods: The engraving quality of GaAs materials was investigated utilizing the LAMMPS software through molecular dynamics simulations, while observations were facilitated using the OVITO software. The MD simulation was conducted employing the NPT ensemble, with the temperature fixed at 300 K. A time step of 2 fs was utilized, and the total duration of the MD simulation spanned 600 ps. [ABSTRACT FROM AUTHOR]

The phase‐field model for the description of the solidification processes with the glass‐crystal competition is suggested. The model combines the first‐order phase transition model in the phase‐field formalism and gauge‐field theory of glass transition. We present a self‐consistent system of stochastic motion equations for unconserved order parameters describing the crystal‐like short‐range ordering and vitrification. It is shown that the model qualitatively describes the glass‐crystal competition during quenching with finite cooling speed. The nucleation of the crystalline phase at slow cooling speeds and low undercoolings proceeds by a fluctuation mechanism. The model demonstrates the tendency to amorphization with the increase of its cooling rate. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the new model of the three dimensional inertial nematic electrolyte, which models the N-species charged particles diffuse and advect in the nematic liquid crystalline matrix. This new model was proposed by [Calderer MC, Golovaty D, Lavrentovich O, et al. Modeling of nematic electrolytes and nonlinear electroosmosis. SIAM J Appl Math. 2016;76:2260–2285. doi: 10.1137/16M1056225]. The inertial effect is such that the system involves a hyperbolic feature. We first derive the local existence in large initial data when the flow is isotropy, i.e. $ \lambda =1 $ λ = 1. For the anisotropy case ( $ \lambda \gt 0 $ λ > 0 , $ \lambda \neq ~1 $ λ ≠ 1), we obtain small data local solutions. Then, by seeking some further dissipative mechanisms from the elliptic structures, the local solution can be extended globally near the constant states with electro-neutrality condition. [ABSTRACT FROM AUTHOR]

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov–Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones. [ABSTRACT FROM AUTHOR]

The stress-induced martensitic transformations and superelasticity behavior in the NiTi alloy with a single crystal model and a twist grain boundary bicrystal model at different temperatures are studied using molecular dynamics simulations. An atomic tracing method is proposed to identify specific numbers of B19′ martensite variants. Under uniaxial compressive loading, the stress-induced martensitic transformation takes place accompanied by the formation of <011>M type II twins, and the deformation process can be divided into three distinct stages based on microstructure evolution and average atomic total energy. It is found that the twist grain boundary induces an increase in the martensite start temperature, which is consistent with the experimental results. There is no residual B19′ martensite at the end of the unloading process, and the irrecoverable strain mainly results from plastic deformation at the grain boundary through the analysis of atomic local shear strains and has hardly changed with increasing deformation temperature. Remarkably, the grain boundary brings about the acceleration of martensite nucleation and an earlier occurrence of stress plateau. Further simulation results manifest that the presence of the twist grain boundary leads to weakened temperature dependence of martensitic transformation stress and a reduction in the hysteresis loop area. [ABSTRACT FROM AUTHOR]

Until quite recently, in almost all papers on crystal nucleation in glass-forming substances, it was assumed that nucleation proceeds in a completely relaxed supercooled liquid and, hence, at constant values of the critical parameters determining the nucleation rate for any given set of temperature, pressure, and composition. Here, we analyze the validity of this hypothesis for a model system by studying nucleation in a lithium silicate glass treated for very long times (up to 250 days) in deeply supercooled states, reaching 60 K below the laboratory glass transition temperature, Tg. At all temperatures in the considered range, T < Tg, we observed an enormous difference between the experimental number of nucleated crystals, N(t), and its theoretically expected value computed by assuming the metastable state of the relaxing glass has been reached. Analyzing the origin of this discrepancy, we confirmed that the key parameters determining the nucleation rates change with time as a result of the glass relaxation process. Finally, we demonstrate that, for temperatures below 683 K, this particular glass almost fully crystallizes prior to reaching the ultimate steady-state nucleation regime (e.g., at 663 K, it would take 176 years for the glass to reach 99% crystallization, while 2600 years would be needed for complete relaxation). This comprehensive study proves that structural relaxation strongly affects crystal nucleation in deeply supercooled states at temperatures well below Tg; hence, this phenomenon has to be accounted for in any crystal nucleation model. [ABSTRACT FROM AUTHOR]

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O (τ 2) in H 1 - norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Herein, we present a new method to efficiently calculate electronically excited states in large molecular assemblies, consisting of hundreds of molecules. For this purpose, we combine the long-range corrected tight-binding density functional fragment molecular orbital method (FMO-LC-DFTB) with an excitonic Hamiltonian, which is constructed in the basis of locally excited and charge-transfer configuration state functions calculated for embedded monomers and dimers and accounts explicitly for the electronic coupling between all types of excitons. We first evaluate both the accuracy and efficiency of our fragmentation approach for molecular dimers and aggregates by comparing it with the full LC-TD-DFTB method. The comparison of the calculated spectra of an anthracene cluster shows a very good agreement between our method and the LC-TD-DFTB reference. The effective computational scaling of our method has been explored for anthracene clusters and for perylene bisimide aggregates. We demonstrate the applicability of our method by the calculation of the excited state properties of pentacene crystal models consisting of up to 319 molecules. Furthermore, the participation ratio of the monomer fragments to the excited states is analyzed by the calculation of natural transition orbital participation numbers, which are verified by the hole and particle density for a chosen pentacene cluster. The use of our FMO-LC-TDDFTB method will allow for future studies of excitonic dynamics and charge transport to be performed on complex molecular systems consisting of thousands of atoms. [ABSTRACT FROM AUTHOR]

Crystals constitute a variety of important materials from everyday life to cutting-edge fields. The properties of a crystal are determined by its structure, as demonstrated by physics theory, which is essential for understanding and designing materials. In recent years, deep learning-based methods have been proposed to predict crystal material properties and achieved satisfactory performance. However, these methods have not adequately accounted for the key composition of crystals, i.e., periodicity. To address this issue, we propose a periodicity aware crystal transformer (PACT), which uses hierarchical self-attention mechanisms to enforce periodicity constraints on the crystal structure. Specifically, it applies unit-wise self-attention and crystal-wise self-attention to ensure that the surroundings of atoms or unit cells at periodic distances are identical. Extensive benchmark experiments demonstrate that our proposed model exhibits superior performance, achieving an average improvement of 7.07% over previous methods. Additionally, ablation studies show both unit-wise self-attention and crystal-wise in the hierarchical self-attention mechanisms are effective in modeling the periodicity. [ABSTRACT FROM AUTHOR]

In this paper, we study the well‐posedness of the Cauchy problem to the compressible inertial Qian–Sheng model. Inspired by Jiang–Luo–Ma–Tang's work in [J. Hyperbolic. Differ. Equ 18:221‐256, 2021], we introduce a novel Condition (H) which ensures the energy dissipation, and then, we prove that when both the viscosity coefficients β5+β6=0$$ {\beta}_5&#x0002B;{\beta}_6&#x0003D;0 $$ and Condition (H) are satisfied, the local‐in‐time solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Finally, the local solution can be extended globally under at least one of the two conditions: (μ˜2−μ2)2<8β4μ1$$ {\left({\tilde{\mu}}_2-{\mu}_2\right)}&#x0005E;2<8{\beta}_4{\mu}_1 $$ or μ˜2=μ2$$ {\tilde{\mu}}_2&#x0003D;{\mu}_2 $$. [ABSTRACT FROM AUTHOR]

Background: A compact PET/SPECT/CT system Inliview‐3000B has been developed to provide multi‐modality information on small animals for biomedical research. Its PET subsystem employed a dual‐layer‐offset detector design for depth‐of‐interaction capability and higher detection efficiency, but the irregular design caused some difficulties in calculating the normalization factors and the sensitivity map. Besides, the relatively larger (2 mm) crystal cross‐section size also posed a challenge to high‐resolution image reconstruction. Purpose: We present an efficient image reconstruction method to achieve high imaging performance for the PET subsystem of Inliview‐3000B. Methods: List mode reconstruction with efficient system modeling was used for the PET imaging. We adopt an on‐the‐fly multi‐ray tracing method with random crystal sampling to model the solid angle, crystal penetration and object attenuation effect, and modify the system response model during each iteration to improve the reconstruction performance and computational efficiency. We estimate crystal efficiency with a novel iterative approach that combines measured cylinder phantom data with simulated line‐of‐response (LOR)‐based factors for normalization correction before reconstruction. Since it is necessary to calculate normalization factors and the sensitivity map, we stack the two crystal layers together and extend the conventional data organization method here to index all useful LORs. Simulations and experiments were performed to demonstrate the feasibility and advantage of the proposed method. Results: Simulation results showed that the iterative algorithm for crystal efficiency estimation could achieve good accuracy. NEMA image quality phantom studies have demonstrated the superiority of random sampling, which is able to achieve good imaging performance with much less computation than traditional uniform sampling. In the spatial resolution evaluation based on the mini‐Derenzo phantom, 1.1 mm hot rods could be identified with the proposed reconstruction method. Reconstruction of double mice and a rat showed good spatial resolution and a high signal‐to‐noise ratio, and organs with higher uptake could be recognized well. Conclusion: The results validated the superiority of introducing randomness into reconstruction, and demonstrated its reliability for high‐performance imaging. The Inliview‐3000B PET subsystem with the proposed image reconstruction can provide rich and detailed information on small animals for preclinical research. [ABSTRACT FROM AUTHOR]

Understanding and characterizing semi-crystalline models with crystalline and amorphous segments is crucial for industrial applications. A coarse-grained molecular dynamics (CGMD) simulations study probed the crystal network formation in high-density polyethylene (HDPE) from melt, and shed light on tensile properties for microstructure analysis. Modified Paul–Yoon–Smith (PYS/R) forcefield parameters are used to compute the interatomic forces among the PE chains. The isothermal crystallization at 300 K and 1 atm predicts the multi-nucleus crystal growth; moreover, the lamellar crystal stems and amorphous region are alternatively oriented. A one-dimensional density distribution along the alternative lamellar stems further confirms the ordering of the lamellar-stack orientation. Using this plastic model preparation approach, the semi-crystalline model density ( ρ c r ) of ca. 0.913 g·cm−3 and amorphous model density ( ρ a m ) of ca. 0.856 g·cm−3 are obtained. Furthermore, the ratio of ρ c r / ρ a m ≈ 1.06 is in good agreement with computational (≈1.096) and experimental (≈1.14) data, ensuring the reliability of the simulations. The degree of crystallinity ( χ c ) of the model is ca. 52% at 300 K. Nevertheless, there is a gradual increase in crystallinity over the specified time, indicating the alignment of the lamellar stems during crystallization. The characteristic stress–strain curve mimicking tensile tests along the z-axis orientation exhibits a reversible sharp elastic regime, tensile strength at yield ca. 100 MPa, and a non-reversible tensile strength at break of 350%. The cavitation mechanism embraces the alignment of lamellar stems along the deformation axis. The study highlights an explanatory model of crystal network formation for the PE model using a PYS/R forcefield, and it produces a microstructure with ordered lamellar and amorphous segments with robust mechanical properties, which aids in predicting the microstructure–mechanical property relationships in plastics under applied forces. [ABSTRACT FROM AUTHOR]

During the plastic deformation of metallic materials, part of expended mechanical energy diffuses as heat. The fraction of plastic work converted into heat is called the Taylor–Quinney Coefficient (TQC), which is often assumed to be a constant parameter of about 0.9. The remaining portion of the plastic work is called stored energy. The stored energy is known as the main driving force for dynamic or static recovery and recrystallization. Therefore, numerical predictions and experimental measurements of the stored energy and TQC are essential to optimize thermomechanical material processing. An adequate prediction of the stored energy and the TQC using existing crystal plasticity models in line with the experimental measurements remains a challenging problem. In this work, a thermodynamic class of crystal plasticity models is used to predict the stored energy and TQC of copper and aluminum single crystals. Then, the numerical stored energy predictions are extended to polycrystalline austenitic steel 316L and compared with the experimental measurements from the literature. An ad-hoc factor is introduced in the numerical expression of stored energy in order to compensate for the difference with the experimental measurement. To this end, the contributions of statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) for the stored energy prediction are analyzed to understand the physical origin of the ad-hoc factor. The contribution of GNDs to stored energy and enhanced hardening is accounted for by means of a strain gradient plasticity model. The present systematic finite element crystal plasticity simulations also include specific interface conditions at grain boundaries. The presented computational analysis indicates that, compared to the experiment, there remains dark energy in the evaluation of energy storage as predicted by the proposed thermodynamically consistent crystal plasticity framework. [ABSTRACT FROM AUTHOR]

Cylindrical Haynes 282 tensile specimens fabricated by laser powder bed fusion (L-PBF Haynes 282) have shown anomalous tensile deformation behavior, characterized by anisotropic lateral contraction and elliptical tensile fracture surfaces. In this work, crystal plasticity finite element simulations are conducted to explore the underlying mechanisms of such anomaly. Cylindrical models of grain aggregates are created according to the experimentally observed grain structure of L-PBF Haynes 282. The initial crystallographic orientations are assigned to generate a range of textures ranging from single crystalline to completely random. The evolution of the shapes and crystallographic textures of the simulated grain aggregates during tensile deformation have been analyzed and compared with experiments. It is found that the near-single crystal texture of L-PBF Haynes 282 is responsible for the deformation anomaly. Specifically, the simulated single crystal under tension along the [110] axis exhibits strong anisotropic lateral contraction; and the deformation of the model with random texture is symmetric about the loading axis. While both models evolve a relatively strong [111] pole along the loading axis during deformation, the orientation of the single crystal model splits about the [ 1 1 ¯ 0 ] direction and the random-texture model evolves into a strong fiber texture. When small, random misorientations are introduced into the single crystal model, the deformation anisotropy and texture evolution appear to be the intermediate of the two extreme cases; and resemble what is observed from the experiment. [ABSTRACT FROM AUTHOR]

We study the counting problem of BPS D-branes wrapping holomorphic cycles of a general toric Calabi-Yau manifold. We evaluate the Jeffrey-Kirwan residues for the flavoured Witten index for the supersymmetric quiver quantum mechanics on the worldvolume of the D-branes, and find that BPS degeneracies are described by a statistical mechanical model of crystal melting. For Calabi-Yau threefolds, we reproduce the crystal melting models long known in the literature. For Calabi-Yau fourfolds, however, we find that the crystal does not contain the full information for the BPS degeneracy and we need to explicitly evaluate non-trivial weights assigned to the crystal configurations. Our discussions treat Calabi-Yau threefolds and fourfolds on equal footing, and include discussions on elliptic and rational generalizations of the BPS states counting, connections to the mathematical definition of generalized Donaldson-Thomas invariants, examples of wall crossings, and of trialities in quiver gauge theories. [ABSTRACT FROM AUTHOR]

We introduce a class of 4-dimensional crystal melting models that count the BPS bound state of branes on toric Calabi-Yau 4-folds. The crystalline structure is determined by the brane brick model associated to the Calabi-Yau 4-fold under consideration or, equivalently, its dual periodic quiver. The crystals provide a discretized version of the underlying toric geometries. We introduce various techniques to visualize crystals and their melting configurations, including 3-dimensional slicing and Hasse diagrams. We illustrate the construction with the D0-D8 system on 4. Finally, we outline how our proposal generalizes to arbitrary toric CY 4-folds and general brane configurations. [ABSTRACT FROM AUTHOR]

This study belongs to an emerging area of research seeking ways to depolarize societies in the short run (around events such as elections) as well as in a sustainable fashion. We approach the depolarization process with a model of three homophilic groups (US Democrats, Republicans, and Independents interacting in the context of upcoming federal elections). We expand a previous polarization model, which assumed that each individual interacts with all other individuals in its group with mean-field interactions. We add a depolarization field, which is analogous to the Blume–Capel model's crystal field. There are currently numerous depolarization efforts around the world, some of which act in ways similar to this depolarization field. We find that for low values of the depolarization field, the system continues to be polarized. When the depolarization field is increased, the polarization decreases. [ABSTRACT FROM AUTHOR]

Based on the model of magnonic crystals (MCs) with honeycomb structure, we propose another model of two-dimensional MCs with honeycomb–kagome structure that is a periodic magnetic composite system composed of Fe, Co, or Py ferromagnetic cylindrical scatterers arranged in the EuO matrix as the honeycomb–kagome structure. The band structures of magnons in these systems are studied numerically by using the plane-wave expansion method. The results show that the Dirac points of magnons will be generated at the Brillouin region points if the scatterers are close-packed, that is to say, the edges of cylindrical scatterers are in contact with each other. The frequency of Dirac points can be indirectly adjusted by changing the radius ratio of close-packed cylinders. In addition, in the case of a large difference in the radius between the close-packed cylindrical scatterers, there will be a magnonic flat band in the band structure, which is a phenomenon of so-called compact localized states different from the impurity state in the crystal, and it is formed by the highly interference superposition of spin waves in the honeycomb–kagome structure. The research on the generation and modulation of magnonic Dirac points and flat bands not only expands the research content of condensed matter topological physics but also provides a promising platform for the application of artificial MCs in the fabrication of spin-wave topological devices. [ABSTRACT FROM AUTHOR]

Surface integrity influences the material fatigue performance significantly, but most micromechanical models based on crystal plasticity only consider the microstructure of the matrix. In this paper, a crystal plastic finite element modeling method was proposed, which simultaneously considered surface roughness, residual stress, and gradual microstructure. It was verified through multiaxial fatigue experiments of materials processed by different multistage machining processes. The results show that larger axial residual compressive stress and smaller grain size can lead to higher multiaxial fatigue life. The existence of residual compressive stress causes the location of material crack initiation to decrease from the surface layer to the subsurface layer. The proposed model can predict the multiaxial fatigue life of materials within a 50% error band. Highlights: A crystal plasticity model considering surface integrity was established.The multiaxial fatigue life of turning specimens is higher than that of grinding.The residual compressive stress cause crack initiation at the subsurface layer.A fatigue life prediction method was proposed within a 50% error band. [ABSTRACT FROM AUTHOR]

Large-sized crystals are fundamental materials in many fields such as semiconductors, communication, military, and radiation detection. However, preparing large-sized and high-quality single crystals for high technology demands is still very difficult. Theory is an important part of crystal growth research. In this review, we briefly summarize the crystal growth models and propose the research directions for the future to break through the preparation of large crystals. In terms of application, chemical bonding theory is currently an effective model for the growth of large-sized crystals. In the future, three aspects, such as utilizing artificial intelligence technology to obtain parameters that accurately describe system properties, establishing cross-scale simulation methods, and calculating interface positions accurately, may be the keys to promote the development of large-sized crystal preparation. [ABSTRACT FROM AUTHOR]

Predicting the crystal structure is essential to address the reliance on serendipity for facilitating the discovery and design of high‐performance high‐entropy oxides (HEOs). Here, three classic algorithms‐based machine learning models to predict the crystal structure of HEOs are successfully established and analyzed by combining five metrics, and the XGBoost classifier shows excellent accuracy and robustness with ACC and F1 scores up to 0.977 and 0.975, respectively. SHAP summary plot indicates that the anion‐to‐cation radius ratio (rA/rC) has the greatest impact on crystal structure, followed by difference in Pauling and Mulliken electronegativities (ΔχPauling and ΔχMulliken). It is noteworthy that the rA/rC, ΔχPauling, and ΔχMulliken lower than 0.35, 0.1, and 0.2, respectively, tend to lead to a fluorite crystal structure, whereas rock‐salt and spinel crystal structures are always formed. This work is expected to facilitate the discovery and design of HEOs with tailorable crystal structures and properties. [ABSTRACT FROM AUTHOR]

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity. [ABSTRACT FROM AUTHOR]

The review concerns methods for modeling the crystal structure of polynitrogen compounds and their cocrystals. The state of the art in chemical research into (co)crystals of energetic compounds is described. Various structure modeling methods (including original ones) are considered and analyzed, and criteria for cocrystal formation at different co-former ratios are proposed. The results are discussed of studies on the search for crystal packings of various polynitrogen compounds and their cocrystals, on prediction of the possibility of cocrystal formation, and targeted synthesis of new substances. The predictive power of the developed structure modeling methods is confirmed by the results of X-ray diffraction studies of the synthesized cocrystals. Prospects for application of certain cocrystals as a basis for energetic materials are outlined. [ABSTRACT FROM AUTHOR]

Ericksen and Leslie proposed a hydrodynamic model for liquid crystals in the format of conservation laws in the 1960s. Their original model includes inertial and compressibility effects, which makes the model a coupled parabolic–hyperbolic system. In this paper we build up the connection between the compressible and incompressible parabolic–hyperbolic liquid crystal model in the framework of classical solutions. We first derive the scaled Ericksen–Leslie system with dimensionless numbers, including Mach, Reynolds, and Ericksen numbers. In particular, we introduce the so-called inertial constant χ which characterizes the inertial effect of the liquid crystal molecular. Next, we establish the energy estimates uniform in the Mach number ϵ for both the compressible system and its time-derivative system with small data. Then, we pass to the limit ϵ → 0 in the compressible system, so that we establish the global classical solution of the incompressible system by the compactness arguments. Moreover, we also obtain the convergence rates associated with L 2 -norm in the case of well-prepared initial data. This is the first result on the incompressible limit of the compressible parabolic–hyperbolic liquid crystal model and confirms the relations of different parabolic–hyperbolic liquid crystal model rigorously. [ABSTRACT FROM AUTHOR]

Context: Under the ReaxFF/lg force field, the multiscale shock technique (MSST) was employed to investigate the decomposition behavior of perfect, dislocated, and twinned octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) at a velocity of 11 km/s. This study aimed to analyze the changes in system temperature, bond formation and breaking, variations in the number of small molecules, and the number of clusters. The results indicated that the sensitivity of dislocated HMX was the lowest, while the sensitivity of twinned and perfect HMX was comparable. Comparing the formation and breaking of bonds in HMX during the shock process, it was found that the change in the number of bonds in dislocated HMX was similar to that in perfect HMX, whereas twinning accelerated the breaking of bonds. By analyzing the changes in small molecular fragments (CO, CO2, H, H2, H2O, N2, N2H, NH2, NO, NO2, and O) during the shock process of HMX, it was found that dislocation had a relatively minor effect on the small molecular fragments, while twinning promoted the generation of CO, H, NO, and O and accelerated the decomposition of NO. A comparison of the number, weight, and atomic ratio of clusters under perfect, dislocated, and twinned conditions revealed that under the influence of shock, the number of clusters initially increased sharply and then decreased slowly. Meanwhile, compared to the perfect and dislocated explosives, the number of clusters under the twinned structure was significantly fewer, indicating that the twinned structure could reduce cluster formation. The proportion of oxygen to carbon in the twinned HMX was lower than that in the perfect and dislocated explosives, possibly due to the higher content of small molecular fragment O in twinned HMX. Methods: Different structures of HMX crystals were constructed, including twinned defect structure (with a supercell containing 6458 atoms), dislocation defect structure (with a supercell containing 2352 atoms), and perfect structure (also with a supercell containing 2352 atoms). The modeling of defect crystal structures was carried out using the Atomsk software. For the twinned defect structure, we first constructed a mirror symmetric structure of the original configuration and then merged these two structures together. For the dislocation defect structure, we shifted a segment of the originally ordered perfect crystal structure by a certain distance using Atomsk. Before conducting the simulations, we performed geometric optimization of the models using the conjugate gradient (CG) algorithm and carried out 10 ps of NVT and NPT simulations to equilibrate the energy, temperature, and other parameters within the system. Finally, a 50-ps MSST impact simulation was performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). [ABSTRACT FROM AUTHOR]

Context: Molecular dynamics simulations are performed to clarify the deformation mechanism of β-HMX crystal in the P 2 1 / n space group setting under uniaxial compression. Nanoscale shear bands whose internal structure is regular enough to form twin with parent structure were found under high strain rate loading in the [010] direction. These deformation twins are formed by the change of lattice orientation due to atomic translation under shear stress, with 2 3 ¯ 1 ¯ (or 23 1 ¯ ) twin planes and ⟨ 11 1 ¯ ⟩ (or ⟨ 1 1 ¯ 1 ¯ ⟩) twin directions. Molecular rotation can significantly reduce the activation barrier of twin systems; meanwhile, the change of lattice direction is accomplished by a serial of fractional translation steps. Our results implicate that these factors, such as decreasing the activation energy barrier of twin systems via molecular rotation and new twin systems introducing shear bands, should be considered via applying the crystal plasticity model to investigate the hot spot formation in energetic explosive crystals. Methods: All simulations were carried out with the MD code package LAMMPS. The non-reactive and flexible molecular force field proposed by Smith and Bharadwaj was adopted to simulate the uniaxial compression of the monoclinic β-HMX molecular crystal in the P 2 1 / n space group setting on (010) plane along [010] direction. In addition, the Shake algorithm was used to constrain all C-H bonds to the equilibrium length. Two methods, i.e., the Von Mises strain and the relative displacements of molecules, were applied to analyze the structure of twin type shear bands of β-HMX during compression. Visualization analysis for atomistic simulations was performed by using OVITO. [ABSTRACT FROM AUTHOR]

We study a discrete version of a biaxial nematic liquid crystal model with external fields via an approach based on the solution of differential identities for the partition function. In the thermodynamic limit, we derive the free energy of the model and the associated closed set of equations of state involving four order parameters, proving the integrability and exact solvability of the model. The equations of state are specified via a suitable representation of the orientational order parameters, which imply twoorder parameter reductions in the absence of external fields. A detailed exact analysis of the equations of state reveal a rich phase diagram where isotropic versus uniaxial versus biaxial phase transitions are explicitly described, including the existence of triple and tricritical points. Results on the discrete models are qualitatively consistent with their continuum analogue. This observation suggests that, in more general settings, discrete models may be used to capture and describe phenomena that also occur in the continuum for which exact equations of state in closed form are not available. [ABSTRACT FROM AUTHOR]

The article discusses two events: the 3rd International Symposium on "Modeling of Crystal Growth Processes and Devices" and the 2nd Indo-Japan Joint Workshop on Photovoltaics. The symposium focused on providing a fundamental understanding of modeling prospects to young researchers in the field of crystal growth processes, semiconductor devices, NLO, and piezoelectric devices. The workshop aimed to bring together researchers from India and Japan to share ideas and recent developments in photovoltaic technologies. The events featured presentations from experts from various countries and institutions. A total of 24 papers from the symposium have been published in a special issue of Crystal Research and Technology. [Extracted from the article]

We investigate bulk thermodynamic and microscopic structural properties of amorphous solids in the framework of the phase-field crystal (PFC) model. These are metastable states with a non-uniform density distribution, having no long-range order. From extensive numerical simulations, we determine the distribution of free energy density values in varying size amorphous systems and also the point-to-set correlation length, which is the radius of the largest volume of amorphous one can take while still having the particle arrangements within the volume determined by the particle ordering at the surface of the chosen volume. We find that in the thermodynamic limit, the free energy density of the amorphous tends toward a value that has a slight dependence on the initial state from which it was formed—i.e., it has a formation history dependence. The amorphous phase is observed to form on both sides of the liquid linear-stability limit, showing that the liquid to amorphous transition is first order, with an associated finite free energy barrier when the liquid is metastable. In our simulations, this is demonstrated when the noise in the initial density distribution is used to induce nucleation events from the metastable liquid. Depending on the strength of the initial noise, we observe a variety of nucleation pathways, in agreement with previous results for the PFC model, which show that amorphous precursor mediated multi-step crystal nucleation can occur in colloidal systems. [ABSTRACT FROM AUTHOR]

This article focuses on computational studies evaluating the influence of crystallinity, residual stresses, and out-of-plane (OOP) deterministic switching on Terfenol-D nano/microstructures. The computational models use both coupled and uncoupled Landau–Liftshitz–Gilbert equations with elastodynamics to study strain-induced magnetization reorientation. A Voronoi tessellation approach models the crystal distribution in the microstructures subjected to residual stresses with good agreement to experimental data including large changes in coercivity values, i.e., from 100 to 3000 Oe. Parametric studies show how the coercivity is manipulated with residual stresses, including a magnetoelastically induced perpendicular-magnetic-anisotropy (PMA), important for memory applications. Additional parametric studies focus on epitaxially deposited micro-disks, revealing that residual stresses can create magnetoelastically dominant easy axes along the ⟨ 110 ⟩ directions, which are energetically favorable relative to the intrinsic ⟨ 111 ⟩ magnetocrystalline easy axes. Modification of the global easy axis is used to design a strain-mediated multiferroic composite consisting of a 20 nm epitaxially deposited Terfenol-D memory bit with PMA grown on a PZT substrate. The multiferroic disk achieves OOP deterministic clocking with an applied voltage. [ABSTRACT FROM AUTHOR]

Zigzag strip bundles are combinatorial models realizing the crystals B (∞) and the highest weight crystals B (λ) over quantum affine algebras U q (g) , and they are closely related with Nakajima monomials and Kashiwara embeddings of specific type. In this paper, we introduce new zigzag strip bundles for U q (g) , which we call reverse zigzag strip bundles in order to distinguish from the previous ones, and we give explicit 1-1 correspondences with the set of Nakajima monomials and the image of Kashiwara embedding which are different from those appeared in the study of the zigzag strip bundles. Another main result is to give different points of view about zigzag strip bundles and reverse zigzag strip bundles. Indeed, the previous 1-1 correspondences between the sets of zigzag strip bundles or reverse zigzag strip bundles, and the sets of Nakajima monomials or images of Kashiwara embeddings were determined based on the number of blocks in the columns of these strip bundles. In our work, we use the number of blocks in the rows of zigzag strip bundles and reverse zigzag strip bundles to provide characterizations of Nakajima monomials and images of Kashiwara embeddings that differ from the previous ones. That is, we give characterizations of Nakajima monomials and Kashiwara embeddings of 4 types. Further, we discuss the connection between zigzag strip bundles and reverse zigzag strip bundles. [ABSTRACT FROM AUTHOR]

We study the effects of additive and multiplicative noise on the solution landscape of nematic liquid crystals confined to a square domain within the Landau-de Gennes framework, as well as the impact of additive noise on the symmetric radial hedgehog solution for nematic droplets. The introduction of random noise can be used to capture material uncertainties and imperfections, which are always present in physical systems. We implement random noise in our framework by introducing a $Q$ Q -Wiener stochastic process to the governing differential equations. On the square, the solution landscape for the deterministic problem is well understood, enabling us to compare and contrast the deterministic predictions and the stochastic predictions, while we demonstrate that the symmetry of the radial hedgehog solution can be violated by noise. This approach of introducing noise to deterministic equations can be used to test the robustness and validity of predictions from deterministic liquid crystal models, which essentially capture idealised situations. [ABSTRACT FROM AUTHOR]