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DOE OSTI2023

Gradient nanostructuring via compositional means

Nanocrystalline metals are inherently unstable against thermal and mechanical stimuli, commonly resulting in significant grain growth. Also, while these metals exhibit substantial Hall-Petch strengthening, they tend to suffer from low ductility and fracture toughness. With regard to the grain growth problem, alloying elements have been employed to stabilize the microstructure through kinetic and/or thermodynamic mechanisms. And to address the ductility challenge, spatially-graded grain size distributions have been developed to facilitate heterogeneous deformation modes: high-strength at the surface and plastic deformation in the bulk. In the present work, we combine these two strategies and present a new methodology for the fabrication of gradient nanostructured metals via compositional means. Here, we have demonstrated that annealing a compositionally stepwise Pt-Au film with a homogenous microstructure results in a film with a spatial microstructural gradient, exhibiting grains which can be twice as wide in the bulk compared to the outer surfaces. Additionally, phase-field modeling was employed for the comparison with experimental results and for further investigation of the competing mechanisms of Au diffusion and thermally induced grain growth. This fabrication method offers an alternative approach for developing the next generation of microstructurally stable gradient nanostructured films.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Implementing contact angle boundary conditions for second-order Phase-Field models of wall-bounded multiphase flows

In the present work, a general formulation is proposed to implement the contact angle boundary conditions for the second-order Phase-Field models, which is applicable to N-phase (N ≥ 2) moving contact line problems. To remedy the issue of mass change due to the contact angle boundary condition, a source term or Lagrange multiplier is added to the original second-order Phase-Field models, which is determined by the consistent and conservative volume distribution algorithm so that the summation of the order parameters and the consistency of reduction are not influenced. To physically couple the proposed formulation to the hydrodynamics, especially for large-density-ratio problems, the consistent formulation is employed. The reduction-consistent conservative Allen-Cahn models are chosen as examples to illustrate the application of the proposed formulation. The numerical scheme that preserves the consistency and conservation of the proposed formulation is employed to demonstrate its effectiveness. Results produced by the proposed formulation are in good agreement with the exact and/or asymptotic solutions. The proposed method captures complex dynamics of moving contact line problems having large density ratios.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Phase-field modeling of dislocation–interstitial interactions

The mechanical behavior of body-centered cubic (BCC) materials can be dramatically affected by the presence of interstitial solute atoms. Here we present a new phase-field dislocation dynamics formulation to include the diffusion of interstitials. Short-range interactions are accounted for by a concentration-dependent lattice energy, and long-range interactions are accounted for by modifications to the elastic energy. The interstitial diffusion law introduces gradients that require methods for minimizing Gibbs oscillations, which is done via a modified Green’s function. The formulation is general to any solute-solvent system and is applied here to Nb-O as a model system, whose interstitial parameters are obtained from ab initio calculations. The effect of O on the core structures of Nb edge and screw dislocations is calculated. The diffusion of O to form interstitial atmospheres around dislocation cores is simulated, as well as the critical stresses required for dislocations to break away or cross slip from these atmospheres. Future applications of the method to simulate complex interstitial embrittlement mechanisms are discussed.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Holistic Microstructure Control Strategies in Photopolymerization‐Induced Phase Separation of Acrylate Systems

Open porous materials, known for their large surface area and interconnected structures, are essential in various applications, including batteries, ion exchange, catalysis, filtration, and electronic waste recycling. A critical aspect of the functionality of porous membranes is the precise control of pore size and morphology. Photopolymerization-induced phase separation (photo-PIPS) offers a convenient and versatile methods for creating porous structures. However, controlling the porous morphology remains challenging due to the complex interplay between thermodynamics, polymerization kinetics, and monomer structures, which makes it difficult to establish the relationship between processing conditions and resulting morphology in photo-PIPS. Herein, a physics-based phase-field model capable of generating and characterizing the microstructures of porous materials based on both average and localized features is developed. Using the phase-field simulations as test bed, the effects of polarity, light intensity, and curing temperature, as well as the previously unexplored roles of chain transfer agents and substrates, on the morphology of the resulting porous microstructure are investigated. Experiments are performed to verify the results predicted by the simulations. This work lays out a comprehensive guide for designing PIPS-derived porous microstructures and offers practical engineering strategies for tailoring microstructure-level topology and size of pores for application-specific needs.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Impact of the plastic deformation microstructure in metals on the kinetics of recrystallization: A phase-field study

The sensitivity of recrystallization kinetics in metals to the heterogeneity of microstructure and deformation history is a widely accepted experimental fact. However, most of the available recrystallization models employ either a mean field approach or use grain-averaged parameters, and thus neglecting the mesoscopic heterogeneity induced by prior deformation. Here in the present study, we investigate the impact of deformation-induced dislocation (subgrain) structure on the kinetics of recrystallization in metals using the phase-field approach. The primary focus here is upon the role of dislocation cell boundaries. The free energy formulation of the phase-field model accounts for the heterogeneity of the microstructure by assigning localized energy to the resulting dislocation microstructure realizations generated from experimental data. These microstructure realizations are created using the universal scaling laws for the spacing and the misorientation angles of both the geometrically necessary and incidental dislocation boundaries. The resulting free energy is used into an Allen-Cahn based model of recrystallization kinetics, which are solved using the finite element method. The solutions thus obtained shed light on the critical role of the spatial heterogeneity of deformation in the non-smooth growth of recrystallization nuclei and on the final grain structure. The results showed that, in agreement with experiment, the morphology of recrystallization front exhibits protrusions and retrusions. By resolving the subgrain structure, the presented algorithm paves the way for developing predictive kinetic models that fully account for the deformed state of recrystallizing metals.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Surrogate Model Integration with MOOSE XFEM for Creep Crack Growth

Ferritic-martensitic steels are key structural materials for advanced reactors but experience time-dependent deformation and damage under prolonged high temperature and irradiation, leading to creep-driven crack initiation and growth. High-fidelity models—crystal plasticity with irradiation mechanisms, phase-field for microstructural evolution, and continuum-damage viscoplasticity—capture the underlying physics but are too computationally intensive for broad design-space exploration and uncertainty quantification. This milestone advances a scalable alternative by integrating a microstructure-sensitive surrogate creep model into the Multiphysics Object-Oriented Simulation Environment (MOOSE) finite element framework and extending it to fracture via the extended finite element method (XFEM). The surrogate model, developed with collaborators at Sandia and Los Alamos National Laboratories, maps relevant microstructural descriptors to the viscoplastic response of HT9. We embed this surrogate within a coupled deformation-damage workflow in MOOSE/XFEM to simulate creep-driven crack initiation and propagation. Implementation enhancements include updates to the material interface, a plastic correction phase involving microstructure evolution, and fracture criteria to ensure numerical robustness and compatibility with the surrogate structure. Demonstrations on canonical creep benchmarks spanning uniaxial and multiaxial states show that the surrogate reproduces key trends of high-fidelity models while substantially reducing computational cost. The resulting capability bridges physics fidelity and performance, providing a practical path to a predictive, microstructure-aware assessment of creep and fracture in reactor materials.

36 - MATERIALS SCIENCE↗

DOE OSTI2023

Emergence of high piezoelectricity from competing local polar order-disorder in relaxor ferroelectrics

Relaxor ferroelectrics are known for outstanding piezoelectric properties, finding a broad range of applications in advanced electromechanical devices. Decoding the origins of the enhanced properties, however, have long been complicated by the heterogeneous local structures. Here, we employ the advanced big-box refinement method by fitting neutron-, X-ray-based total scattering, and X-ray absorption spectrum simultaneously, to extract local atomic polar displacements and construct 3D polar configurations in the classical relaxor ferroelectric Pb(Mg 1/3 Nb 2/3 )O 3 –PbTiO 3 . Our results demonstrate that prevailing order-disorder character accompanied by the continuous rotation of local polar displacements commands the composition-driven global structure evolution. The omnidirectional local polar disordering appears as an indication of macroscopic relaxor characteristics. Combined with phase-field simulations, it demonstrates that the competing local polar order-disorder between different states with balanced local polar length and direction randomness leads to a flattening free-energy profile over a wide polar length, thus giving rise to high piezoelectricity. Our work clarifies that the critical structural feature required for high piezoelectricity is the competition states of local polar rather than relaxor.

42 ENGINEERING↗

DOE OSTI2023

Learning heterogeneous reaction kinetics from X-ray videos pixel by pixel

Reaction rates at spatially heterogeneous, unstable interfaces are notoriously difficult to quantify, yet are essential in engineering many chemical systems, such as batteries and electrocatalysts. Experimental characterizations of such materials by operando microscopy produce rich image datasets, but data-driven methods to learn physics from these images are still lacking because of the complex coupling of reaction kinetics, surface chemistry and phase separation. Here we show that heterogeneous reaction kinetics can be learned from in situ scanning transmission X-ray microscopy (STXM) images of carbon-coated lithium iron phosphate (LFP) nanoparticles. Combining a large dataset of STXM images with a thermodynamically consistent electrochemical phase-field model, partial differential equation (PDE)-constrained optimization and uncertainty quantification, we extract the free-energy landscape and reaction kinetics and verify their consistency with theoretical models. We also simultaneously learn the spatial heterogeneity of the reaction rate, which closely matches the carbon-coating thickness profiles obtained through Auger electron microscopy (AEM). Across 180,000 image pixels, the mean discrepancy with the learned model is remarkably small (<7%) and comparable with experimental noise. Our results open the possibility of learning nonequilibrium material properties beyond the reach of traditional experimental methods and offer a new non-destructive technique for characterizing and optimizing heterogeneous reactive surfaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Learning heterogeneous reaction kinetics from X-ray movies pixel-by-pixel

Reaction rates at spatially heterogeneous, unstable interfaces are notoriously difficult to quantify, yet are essential in engineering many chemical systems, such as batteries 1 and electrocatalysts 2. Experimental characterizations of such materials by operando microscopy produce rich image datasets 3–6, but data-driven methods to learn physics from these images are still lacking because of the complex coupling of reaction kinetics, surface chemistry and phase separation 7. Here we show that heterogeneous reaction kinetics can be learned from in situ scanning transmission X-ray microscopy (STXM) images of carbon-coated lithium iron phosphate (LFP) nanoparticles. Combining a large dataset of STXM images with a thermodynamically consistent electrochemical phase-field model, partial differential equation (PDE)-constrained optimization and uncertainty quantification, we extract the free-energy landscape and reaction kinetics and verify their consistency with theoretical models. We also simultaneously learn the spatial heterogeneity of the reaction rate, which closely matches the carbon-coating thickness profiles obtained through Auger electron microscopy (AEM). Across 180,000 image pixels, the mean discrepancy with the learned model is remarkably small (<7%) and comparable with experimental noise. Our results open the possibility of learning nonequilibrium material properties beyond the reach of traditional experimental methods and offer a new non-destructive technique for characterizing and optimizing heterogeneous reactive surfaces.

Chueh, William↗

DOE OSTI2023

Towards inverse microstructure-centered materials design using generative phase-field modeling and deep variational autoencoders

The field of Integrated Computational Materials Engineering (ICME) combines a broad range of methods to study materials’ responses over a spectrum of length scales. A relatively unexplored aspect of microstructure-sensitive materials design is uncertainty propagation and quantification (UP/UQ) of materials’ microstructure, as well as establishing process-structure–property (PSP) relationships for inverse material design. In this study, an efficient UP technique built on the idea of changing probability measures and a deep generative unsupervised representative machine learning method for microstructure-based design of thermal conductivity of materials is proposed. Probability measures are used to represent microstructure space, and Wasserstein metrics are used to test the efficiency of the UP method. By using deep Variational AutoEncoder (VAE), we identify the correlations between the material/process parameters and the thermal conductivity of heterogeneous dual-phase microstructures. Through high-throughput screening, UP, and the deep-generative VAE method, PSP relationships that are too complex can be revealed by exploiting the materials’ design space with an emphasis on microstructures. As a last point, we demonstrate generative machine learning serves as a useful tool for inverse microstructure-centered materials design, and we demonstrate this by examining the inverse design of thermal conductivity in nano-structured materials. Here, the results reveal the effects of morphology, volume fraction, characteristic length scale, and the individual thermal diffusivity of phases on the thermal conductivity of dual-phase alloys. Our findings emphasize the advantages of high-throughput phase-field modeling and generative deep learning for linking PSP and inverse microstructure-centered materials design.

36 MATERIALS SCIENCE↗

DOE OSTI2023

On the convexity of phase-field fracture formulations: Analytical study and comparison of various degradation functions

Efficient and accurate fracture modeling is of great importance in applications where catastrophic outcomes under extreme scenarios are possible. The phase-field (PF) approach to fracture received significant attention over the past decade, due to its capability to capture complicated fracture patterns (e.g., crack merging and branching). Specifically, crack initiation and propagation are modeled via minimization of the total energy functional, which is regularized with the aid of a phase field. Despite the promising results and modeling capabilities of the PF method in many applications, the solution of fracture problems remains computationally challenging mainly due to the non-convexity of the total energy functional with respect to the combined unknown (phase field and displacement) fields. Understanding the effects of their coupling on convexity is crucial in order to address frequently encountered hurdles in fracture modeling (e.g., inefficient solvers and non-physical crack nucleation). In this paper, we develop convexity criteria for a wide class of PF fracture formulations. For this class of formulations, the second variation of the total energy functional is expressed in terms of Hessian matrices (evaluated at individual material points). Depending on the choice of geometric crack functions and degradation functions, we classify the formulations into three categories and analytically study each one separately. To study the sign of the second variation, we derive inequalities which are satisfied at material points when the Hessian matrix is locally positive semi-definite. These inequalities provide objective criteria for comparing degradation functions. Finally, the applicability of the proposed convexity criteria is demonstrated in the context of a one-dimensional problem, solved using a conventional monolithic solver.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Theory of terahertz pulse transmission through ferroelectric nanomembranes

An analytical model is developed to predict the temporal evolution of the lattice polarization in ferroelectric nanomembranes upon the excitation by a terahertz (THz) electromagnetic pulse of an arbitrary waveform and the concurrent transmission of the THz pulse in both linear and nonlinear regimes. It involves the use of the perturbation method to solve the equation of motion for the lattice polarization in both unclamped and strained ferroelectric nanomembranes within the framework of Landau-Ginzburg-Devonshire theory. The model is applicable to perovskite oxides such as BaTiO 3 and SrTiO 3 , wurtzite Al 1−𝑥 ⁢Sc 𝑥 ⁢N, and trigonal LiNbO 3 . Our analytical model provides a theoretical basis for determining the thermodynamic and kinetic parameters of ferroelectric materials through a THz transmission experiment. The calculation results also suggest an approach to reversing the chirality of a circularly polarized THz pulse by harnessing the resonant polarization-photon coupling in ferroelectrics. This capability of chirality reversal, along with the high tunability from a strain applied along any arbitrarily oriented in-plane axis, provides new opportunities for THz wave modulation without relying on complex metasurface designs.

36 MATERIALS SCIENCE↗

DOE OSTI2022

A multi-resolution approach to hydraulic fracture simulation

Abstract We present a multi-resolution approach for constructing model-based simulations of hydraulic fracturing, wherein flow through porous media is coupled with fluid-driven fracture. The approach consists of a hybrid scheme that couples a discrete crack representation in a global domain to a phase-field representation in a local subdomain near the crack tip. The multi-resolution approach addresses issues such as the computational expense of accurate hydraulic fracture simulations and the difficulties associated with reconstructing crack apertures from diffuse fracture representations. In the global domain, a coupled system of equations for displacements and pressures is considered. The crack geometry is assumed to be fixed and the displacement field is enriched with discontinuous functions. Around the crack tips in the local subdomains, phase-field sub-problems are instantiated on the fly to propagate fractures in arbitrary, mesh independent directions. The governing equations and fields in the global and local domains are approximated using a combination of finite-volume and finite element discretizations. The efficacy of the method is illustrated through various benchmark problems in hydraulic fracturing, as well as a new study of fluid-driven crack growth around a stiff inclusion.

58 GEOSCIENCES↗

DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗

DOE OSTI2023

Understanding and control of Zener pinning via phase field and ensemble learning

Zener pinning refers to the dispersion of fine particles which influences grain size distribution via movement of grain boundaries in a polycrystalline material. Grain size distribution in polycrystals has a significant impact on their properties including physical, chemical, mechanical, and optical to name a few. We explore the use of Phase-field modeling and machine-learning techniques to understand and improve the control of grain size distribution via Zener pinning in polycrystalline materials. We develop a machine learning model that determines the relative importance of various parameters to exercise microstructure control via Zener pinning. Our workflow combines high-throughput phase-field simulations and machine learning to address the computational bottlenecks associated with large-scale simulations as well as identify features necessary for microstructure control in polycrystals. A random forest (RF) regression model was developed to predict grain sizes based on five Phase-field model parameters, achieving an average prediction error of 0.72 nm for the training data and 1.44 nm for the test data. The importance of the input parameters is analyzed using the SHapley Additive exPlanations (SHAP) approach which reveals that diffusivity, volume fraction, and particle diameter are the most important parameters in determining the final grain size. These findings will allow us to select the best second-phase particles, optimize grain size distributions and thus design microstructures with the desired properties. The developed method is a highly versatile and generalizable approach that can be used to assess the combined effects of individual features in the presence of multiple variables.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Interaction of extended dislocations with nanovoid clusters

Voids of nanoscale dimensions in irradiated metals can act as obstacles to dislocation motion and cause strengthening. In this work, nanovoid strengthening and the influences of void size, void spacing and material properties, such as stacking fault energies, on dislocation bypass mechanisms are investigated using Phase Field Dislocation Dynamics, a three-dimensional mesoscale model that predicts the minimum energy pathway taken by discrete dislocations. A broad range of face centered cubic metals (copper, nickel, silver, rhodium, and platinum) and nanovoid sizes and spacings are treated, altogether spanning void size–to–dislocation stacking fault width ratios from less than unity to ten. Material γ-surfaces, calculated from ab initio methods, are input directly into the formulation. The analysis reveals that the critical bypass stress scales linearly with the linear void fraction, effective isotropic shear modulus, and ratio of the intrinsic to unstable stacking fault energies. With only a few exceptions, the critical stress is controlled by the stress required for the leading partial to impinge the voids (to move within range of the attractive image stress field of the void). When the void diameter is nearly an order of magnitude greater than the stacking fault width, the mechanism determining critical strength shifts to the stress for the dislocation to breakaway after partially cutting the void. Furthermore, this situation corresponds to that treated by line tension models and is realized here for Pt, with a sub-nanometer stacking fault width.

36 MATERIALS SCIENCE↗

DOE OSTI2023

A probabilistic-phase field model for the fracture of brittle materials

Abstract We develop a computational method to determine the failure probability of brittle materials under general mechanical loading conditions. The method is a combination of two parts: (1) numerical simulations of materials with multiple cracks using phase field theory, where the complete fracture process is viewed as ‘damage percolation’ along critical paths or clusters of cracks, rather than the traditional weak-link failure mechanism of Weibull, and (2) an extension of the Batdorf statistical theory of fracture to finite domains, where it is implemented within the finite element framework. The results of phase-field simulations at the ‘percolation threshold’ are used as failure data in the Batdorf theory to determine the overall probability of failure. The input to this approach is the size distribution of cracks in a pristine material. An example is shown, where alumina samples that were previously tested by Abe and coworkers (Abeet al2003J. Am. Ceram. Soc.861019–21) in four-point loading are compared to the results of our numerical simulations. The approach developed here has the advantage of being extendable to more complex thermomechanical loading.

Materials Science↗

DOE OSTI2026

PyJMAK: An Open-Source Python Toolkit for Modeling Solid-State Metallurgical Phase Transformations

Accurate prediction of metallurgical phase transformations is an essential basis for autonomous optimization and rapid part qualification. Several methods can be used to estimate the evolution of phase fractions such as JMAK kinetics-based models, phase-field models, thermodynamic models, and data-driven machine learning models. Thermodynamic and phase-field-based methodologies solve multiphysics equations requiring numerous calibration parameters and significant computational resources. As a result, the computation domain is limited to a point or on order of micron-meters. The data-driven models rely on large datasets from experiments and simulations. While the JMAK model only provides information about phase fraction evolution, it can predict this evolution in near real-time using thermal history and thermodynamic data without restriction on the domain. JMAK models have been popularly used by researchers to model phase transformations occuring during additive manufacturing or over arbitrary temperature profiles. Commercial proprietary software such as Abaqus and Ansys or closed-source in-house implementations offer the ability to model JMAK based kinetics to predict phase transformation. However, these software packages are not open-source or freely available for use and development in conjunction with manufacturing machines, sensors, and machine learning algorithms. In addition, the use of the model is restricted by a license token. In contrast, given temperature profiles at multiple points in the domain, this Python-based PyJMAK model can compute phase evolution in parallel due to its stand-alone modular, voxel-based structure, and it can be executed on high-performance computing resources without any license restrictions.

Prabhune, Bhagya [Oak Ridge National Laboratory (O↗