Bridging Length Scales Research Articles

Bridging length and time scales in predictive simulations of thermo-mechanical processes

Abstract This work introduces a theoretical formulation and develops numerical methods for finite element implementation of the formulation so as to extend the concurrent atomistic-continuum (CAC) method for modeling and simulation of finite-temperature materials processes. With significantly reduced degrees of freedom, the CAC simulations are shown to reproduce the results of atomically resolved molecular dynamics simulations for phonon density of states, velocity distributions, equilibrium temperature field of the underlying atomistic model, and also the density, type, and structure of dislocations formed during the kinetic processes of heteroepitaxy. This work also demonstrates the need of a mesoscale tool for simulations of heteroepitaxy, as well as the unique advantage of the CAC method in simulation of the defect formation processes during heteroepitaxy.

Gravity-driven flow of liquid bridges between vertical fibres

Liquid bridges are formed when a flowing liquid interacts with multiple parallel fibres, as relevant to heat and mass transfer applications that utilize flow down fibre arrays. We perform a comprehensive experimental study of flowing liquid bridges between two vertical fibres whose spacing is controlled dynamically in our experimental apparatus. The bridge patterns exhibit a regular periodic spacing typical of absolute instability for low flow rates, but become spatially inhomogeneous above a critical flow rate where the base flow is convectively unstable. The shapes of individual bridges and their associated dynamics are measured, as they depend upon the liquid properties, and fibre geometry/spacing. The bridge length scales similarly to static bridges between parallel fibres. The bridge dynamics exhibits a dependence on viscosity and scale with the impedance. A simple energy balance is used to derive a scaling relationship for the bridge velocity that captures the general trend of our experimental data. Finally, we demonstrate that these scalings similarly apply when the fibres are dynamically separated or brought together.

Bridging length scales in hard materials with ultra-small angle X-ray scattering - a critical review.

Owing to their exceptional properties, hard materials such as advanced ceramics, metals and composites have enormous economic and societal value, with applications across numerous industries. Understanding their microstructural characteristics is crucial for enhancing their performance, materials development and unleashing their potential for future innovative applications. However, their microstructures are unambiguously hierarchical and typically span several length scales, from sub-ångstrom to micrometres, posing demanding challenges for their characterization, especially for in situ characterization which is critical to understanding the kinetic processes controlling microstructure formation. This review provides a comprehensive description of the rapidly developing technique of ultra-small angle X-ray scattering (USAXS), a nondestructive method for probing the nano-to-micrometre scale features of hard materials. USAXS and its complementary techniques, when developed for and applied to hard materials, offer valuable insights into their porosity, grain size, phase composition and inhomogeneities. We discuss the fundamental principles, instrumentation, advantages, challenges and global status of USAXS for hard materials. Using selected examples, we demonstrate the potential of this technique for unveiling the microstructural characteristics of hard materials and its relevance to advanced materials development and manufacturing process optimization. We also provide our perspective on the opportunities and challenges for the continued development of USAXS, including multimodal characterization, coherent scattering, time-resolved studies, machine learning and autonomous experiments. Our goal is to stimulate further implementation and exploration of USAXS techniques and inspire their broader adoption across various domains of hard materials science, thereby driving the field toward discoveries and further developments.

Bridging the gap: Connecting pore-scale and continuum-scale simulations for immiscible multiphase flow in porous media

This study aims to bridge length scales in immiscible multiphase flow simulation by connecting two published governing equations at the pore-scale and continuum-scale through a novel validation framework. We employ Niessner and Hassnaizadeh's [“A model for two-phase flow in porous media including fluid-fluid interfacial area,” Water Resour. Res. 44(8), W08439 (2008)] continuum-scale model for multiphase flow in porous media, combined with the geometric equation of state of McClure et al. [“Modeling geometric state for fluids in porous media: Evolution of the Euler characteristic,” Transp. Porous Med. 133(2), 229–250 (2020)]. Pore-scale fluid configurations simulated with the lattice-Boltzmann method are used to validate the continuum-scale results. We propose a mapping from the continuum-scale to pore-scale utilizing a generalized additive model to predict non-wetting phase Euler characteristics during imbibition, effectively bridging the continuum-to-pore length scale gap. Continuum-scale simulated measures of specific interfacial area, saturation, and capillary pressure are directly compared to up-scaled pore-scale simulation results. This research develops a numerical framework capable of capturing multiscale flow equations establishing a connection between pore-scale and continuum-scale simulations.

Bridging length scales in organic mixed ionic-electronic conductors through internal strain and mesoscale dynamics.

Understanding the structural and dynamic properties of disordered systems at the mesoscale is crucial. This is particularly important in organic mixed ionic-electronic conductors (OMIECs), which undergo significant and complex structural changes when operated in an electrolyte. In this study, we investigate the mesoscale strain, reversibility and dynamics of a model OMIEC material under external electrochemical potential using operando X-ray photon correlation spectroscopy. Our results reveal that strain and structural hysteresis depend on the sample's cycling history, establishing a comprehensive kinetic sequence bridging the macroscopic and microscopic behaviours of OMIECs. Furthermore, we uncover the equilibrium and non-equilibrium dynamics of charge carriers and material-doping states, highlighting the unexpected coupling between charge carrier dynamics and mesoscale order. These findings advance our understanding of the structure-dynamics-function relationships in OMIECs, opening pathways for designing and engineering materials with improved performance and functionality in non-equilibrium states during device operation.

Deconstructing 3D Structured Materials by Modern Ultramicrotomy for Multimodal Imaging and Volume Analysis across Length Scales

AbstractBased on the rapid advances in additive manufacturing, micro‐patterned heterostructures of soft materials have become available that need to be characterized down to the nanoscale. Advanced function‐structure relationships are designed by direct 3D structuring of the object and – in the future – fine control over material functionality in 3D will produce complex functional objects. To control their design, fabrication and final structure, morphological and spectroscopical imaging in 3D at nanometer resolution are critically required. With examples of carbon‐based objects, it is demonstrated how serial ultramicrotomy, that is, cutting a large number of successive ultrathin sections, can be utilized to gain access to the interior of 3D objects. Array tomography, hierarchical imaging and correlative light and electron microscopy can bridge length scales over several orders of magnitude and provide multimodal information of the sample's inner structure. Morphology data derived from scanning electron microscopy are correlated with spectroscopy in analytical transmission electron microscopy and probe microscopy at nanometer resolution, using TEM‐electron energy loss spectroscopy and infrared‐scanning‐near‐field microscopy. The correlation of different imaging modalities and spectroscopy of carbon‐based materials in 3D provides a powerful toolbox of complementary techniques for understanding emerging functions from nanoscopic structuring.

Strain-rate dependent mode I cohesive traction laws for glass fiber-epoxy interphase using molecular dynamics simulations

Using molecular dynamics (MD) simulations, we develop strain-rate dependent Mode-I cohesive traction laws for the glass fiber-epoxy interphase as a function of interphase structure. Interphase models with mono-layer glycidoxypropyltrimethoxy silane are prepared varying silane number density from 0.0 nm−2 to 3.9 nm−2. Interphase is formed through epoxy-amine diffusion in the silane layer followed by curing reaction. Traction laws are developed over a full range of strain rates from quasi-static to super-high strain rate (∼1e16/s) where a theoretical plateau strength limit is predicted. A stress-relaxation methodology is introduced to construct quasi-static traction-separation responses from high strain rate loading. Simulation results reveal that the interphase traction-separation responses are strain rate and silane number density dependent. Variations of peak traction and energy with strain rates show a characteristic S-shape pattern in a semi-log plot with a gradual increase in properties up to 1e12/s where a steep transition occurs between 1e13/s to 1e14/s followed by a strain rate independent plateau. Interphase properties (especially peak traction) increase almost linearly with silane number density for all strain rates. Mathematical correlations for the MD predicted traction versus strain rate and energy versus strain rate data are established to bridge length scales providing input micromechanics models for the prediction of fiber-matrix debonding in composites.

Predicting Young's Modulus of Linear Polyurethane and Polyurethane-Polyurea Elastomers: Bridging Length Scales with Physicochemical Modeling and Machine Learning.

Predicting the properties of complex polymeric materials based on monomer chemistry requires modeling physical interactions that bridge molecular, interchain, microstructure, and bulk length scales. For polyurethanes, a polymer class with global commercial and industrial significance, these multiscale challenges are intrinsic due to the thermodynamic incompatibility of the urethane and polyol-rich domains, resulting in heterogeneities from molecular to microstructural length scales. Machine learning can model patterns in data to establish a relationship between the monomer chemistry and bulk material properties, but this is made difficult by small data sets and a diverse set of monomers. Using a data set of 63 industrially relevant and complex elastomers, we demonstrate that accurate machine learning predictions are possible when monomer chemistry is used to estimate interactions at interchain length scales. Here, these features were used to accurately (r2 = 0.91) predict the Young's modulus of polyurethane and polyurethane-urea elastomers. Furthermore, by a query of the trained model for compositions that yield a target modulus within the range of accessible values, the capabilities of using this methodology as a design tool are demonstrated. The presented methodology could become increasingly useful in building models for materials with small data sets and may guide the interpretation of the underlying physicochemical forces.

Impact damage modeling in woven composites with two-level Parametrically-Upscaled Continuum Damage Mechanics Models (PUCDM)

This paper develops two-level parametrically-upscaled continuum damage mechanics (PUCDM) models for woven composites. The PUCDM models are thermodynamically consistent, micro-/mesostructure-integrated constitutive models that bridge length scales through explicit incorporation of lower-scale descriptors into coefficients. The level-1 PUCDM model features homogenized intra-yarn constitutive response of unidirectional composite SERVEs, while the level-2 PUCDM model reflects the upscaled effect of mesostructural RVE response. The functional forms of the coefficients are established through machine learning algorithms operating on a micro-/mesostructural response database. The highly efficient two-level PUCDM models are validated using ballistic impact experiments on S-2 glass/SC-15 epoxy woven composite plates with good agreement.

Bridging length scales from micro to mesoscale through rate-dependent traction-separation law predictions

Mesoscale models are useful for modeling dynamic impact on fiber reinforced polymer composites . Mesoscale damage can be modeled using cohesive zones , but determining the governing traction-separation laws is a challenge. We developed finite element models with a stochastic fiber-matrix microstructure embedded within a larger mesoscale continuum to study transverse tension and shear loading of unidirectional S-glass/epoxy composite. We included rate-dependent matrix constitutive behavior and failure and rate-dependent fiber-matrix interface debonding, which are important microscale damage mechanisms that comprise mesoscale fracture. With these multiscale models , we simulated mode I and II crack initiation and propagation. We determined mesoscale, mode I and II, rate-dependent cohesive laws from the models using a domain J-integral approach to calculate the fracture energy as a function of crack opening displacement. We explored the effects of loading rate, matrix failure strain, and fiber-matrix interface debonding properties on the traction laws and crack morphologies. Finally, consistent bridging between length scales is demonstrated by comparing energy dissipation of microstructurally-resolved fracture to mesoscale continuum cohesive zone fracture. This work provides a methodology for determining rate-dependent traction-separation laws, derived from accurate micro-mechanics and experimental input, for bridging lengths scales and use in mesoscale models to evaluate ballistic performance of composites.

Bridging length scales from molecules to the whole organism by cryoCLEM and cryoET.

Resolving atomic structures of isolated proteins has uncovered mechanisms and fundamental processes in biology. However, many functions can only be tested in the context of intact cells and tissues that are many orders of magnitude larger than the macromolecules on which they depend. Therefore, methods that interrogate macromolecular structure in situ provide a means of directly relating structure to function across length scales. Here, we developed several workflows using cryogenic correlated light and electron microscopy (cryoCLEM) and electron tomography (cryoET) that can bridge this gap to reveal the molecular infrastructure that underlies higher order functions within cells and tissues. We also describe experimental design considerations, including cryoCLEM labelling, sample preparation, and quality control, for determining the in situ molecular architectures within native, hydrated cells and tissues.

Growth competition between columnar dendritic grains – The role of microstructural length scales

The growth competition between columnar dendritic grains is investigated by both phase-field (PF) and cellular automaton (CA) models in a growth regime where the primary dendrite spacing is much larger than the solutal diffusion length. This growth regime favors the formation of highly branched hierarchical dendritic microstructures prevalent in castings. Previous PF and CA studies have shed light on the complex relationship between GB orientation selection and GB bi-crystallography. They showed that, in the CA model, the orientation is governed by the favorably oriented grain (FOG) criterion and the geometrical limit (GL) in the limit of large and small cell size, respectively. The present study focuses on exploring how to quantitatively bridge length scales between PF simulations that resolve the whole solid-liquid interface dynamics and the CA model that resolves the dynamics of the grain envelope under a certain set of assumptions. For this purpose, we study grain boundary (GB) orientation selection as a function of the imposed temperature gradient G under the frozen-temperature approximation, which allows us to vary G-dependent microstructural length scales with a fixed GB bi-crystallography. PF simulations reveal the existence of a transition from FOG to GL dominance with decreasing G at non-degenerate bi-crystallography. Simulations further reveal that this transition can be quantitatively reproduced by the CA model with a choice of the cell size that corresponds, in PF simulations, to the active secondary dendrite arm spacing of the favorably oriented grain preceding the tertiary branching event that gives birth to a new stable primary dendrite. PF simulations are also used to obtain a detailed quantitative characterization of the dynamics of the grain envelope and its internal length scales, thereby providing a quantitative test of the inherent approximations of the existing CA approach and paving the way for its future development.

Bridging length scales in granular materials using convolutional neural networks

Granular materials are complex systems whose macroscopic mechanics are governed by particles at the grain-scale. The need to understand their grain-scale behavior has motivated significant experimental and modeling efforts. Bridging the grain-scale with the continuum scale is important in order to develop constitutive theories based on grain-scale behavior, as well as for interpreting the results of grain-scale models and experiments from a macroscopic context. In this work, we present a new data-driven framework based on convolutional neural networks to bridge the grain-scale and continuum scale in granular materials. We use this framework to obtain a micromechanical model of stress and demonstrate that spatial correlations at the grain-scale are critical for bridging length scales. Our results suggest that it is possible to learn data-driven relationships between the grain-scale and macroscale even if we have limited knowledge about the physical state of a granular system. We also observed that it is possible to train a model to predict macroscopic stress using only a subset of the contact data for each time step. This points to the discovery of a new pattern in granular systems, whereby any spatially correlated subset of contact data is sufficient to model macroscopic stress, regardless of how much force they may be carrying. Finally, we demonstrated that our framework is robust with potential for generalizability in time.

An enhanced tool for probing the microscopic behavior of granular materials based on X-ray micro-CT and FDEM

We propose an enhanced tool by combining X-ray micro-computed tomography test and hybrid finite and discrete element method to investigate the mechanical behaviors of granular materials. We first conduct a min-triaxial test of Ottawa sand under X-ray micro-CT. Then, spherical harmonic analysis is performed to characterize multi-scale morphological characteristics of particles and used in the particle matching. The particle tracking algorithm ensures the matching accuracy between particle configurations even at large strain intervals. To probe intra-particle contact force, we reconstruct the numerical sample from X-ray image data. Without calibrating material parameters, FDEM simulation quantitatively agrees with the overall response of Ottawa sand recorded in experiment. Moreover, the particle scale dynamics obtained by simulation are remarkably quantitatively consistent with experiment results. The proposed tool sheds new light on bridging length scales from particle to granular system. We find that the granular material deforms plastically through spatially localized zones of large nonaffine displacements, and the spatiotemporal evolution of these zones controls the macroscopic responses of the system. The force chain collapse is relevant to the large induced structural voids formation within the shear transformation zones. Furthermore, we discover a connection between particle stress fluctuations and particle plastic rearrangements in granular materials.

Microscopic and Macroscopic Characterization of Grain Boundary Energy and Strength in Silicon Carbide via Machine-Learning Techniques.

Predicting the properties of grain boundaries poses a challenge because of the complex relationships between structural and chemical attributes both at the atomic and continuum scales. Grain boundary systems are typically characterized by parameters used to classify local atomic arrangements in order to extract features such as grain boundary energy or grain boundary strength. The present work utilizes a combination of high-throughput atomistic simulations, macroscopic and microscopic descriptors, and machine-learning techniques to characterize the energy and strength of silicon carbide grain boundaries. A diverse data set of symmetric tilt and twist grain boundaries are described using macroscopic metrics such as misorientation, the alignment of critical low-index planes, and the Schmid factor, but also in terms of microscopic metrics, by quantifying the local atomic structure and chemistry at the interface. These descriptors are used to create random-forest regression models, allowing for their relative importance to the grain boundary energy and decohesion stress to be better understood. Results show that while the energetics of the grain boundary were best described using the microscopic descriptors, the ability of the macroscopic descriptors to reasonably predict grain boundaries with low energy suggests a link between the crystallographic orientation and the resultant atomic structure that forms at the grain boundary within this regime. For grain boundary strength, neither microscopic nor macroscopic descriptors were able to fully capture the response individually. However, when both descriptor sets were utilized, the decohesion stress of the grain boundary could be accurately predicted. These results highlight the importance of considering both macroscopic and microscopic factors when constructing constitutive models for grain boundary systems, which has significant implications for both understanding the fundamental mechanisms at work and the ability to bridge length scales.

Assessing pH and Voltage Effects on the Binding of CO to Model Copper Catalyst Surfaces Via a DFT-Continuum Theory

The electrochemical conversion of CO2 to commodity products such as hydrocarbons and oxygenates is an attractive approach for generating non-fossil derived chemical and fuel stocks and for the storage of energy in chemical bonds. To date, a number of studies have shown that the activity and selectivity of copper-based catalysts for the CO2 reduction reaction towards valuable higher order carbon products are highly sensitive to the local reaction environment. Furthermore, the local electrolyte concentration and local pH is intimately tied to the applied voltage and the mass transport of reactants and products within the interfacial region. In recognition of this, several recent efforts focused on engineering the design of electrochemical reactors to reduce mass transport limitations indicate a pronounced effect on the apparent selectivity of the copper catalyst. While these efforts have highlighted a new direction for improving the efficiency of the CO2 reduction reaction, the rational design of these reactors remains challenging due to the need to optimize features of the reactor across multiple time and length scales. In this talk, we will discuss recent efforts within our laboratory to address the influence of the local pH on the CO2 reduction reaction at the atomistic level. We focus the discussion on the binding of CO, a key reaction intermediate in the CO2 reduction pathway, whose preferential hydrogenation or dimerization leads to the production of C1 and C2 products, respectively. We additionally demonstrate how the use of hybrid approaches employing density functional theory (DFT) and continuum models can help bridge time and length scales within the context of atomistic CO2 electroreduction modeling.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

P.009 MDK gene expression is correlated with human brain activity during acceptance of monetary rewards

Computationally efficient methods for bridging length scales, from highly resolved micro/meso-scale models that can explicitly model crack growth, to macro-scale continuum models that are more suitable for modeling large parts, have been of interest to researchers for decades. In this work, an improved brittle damage model is presented for the simulation of dynamic fracture in continuum scale quasi-brittle metal components. Crack evolution statistics, including the number, length, and orientation of individual cracks, are extracted from high-fidelity, finite discrete element method (FDEM) simulations and used to generate effective material moduli that reflect the material’s damaged state over time. This strategy allows for the retention of small-scale physical behaviors such as crack growth and coalescence in continuum scale hydrodynamic simulations. However, the high-fidelity simulations required to generate the crack statistics are computationally expensive. Thus, steps were taken to produce a flexible constitutive model to reduce the number of costly high-fidelity simulations needed to produce accurate results. A new stress based degradation criterion is introduced for the degradation of individual material zones. This allows for the development of a heterogeneous damage distribution within the bulk material. Then a flow stress model is added to the hydrodynamic simulation to account for plasticity in quasi-brittle materials. As a result, the effective moduli model can be applied to a larger range of materials. The effective moduli constitutive model is used to simulate beryllium flyer plate experiments. The results from the continuum scale simulations using statistics from a single high-fidelity simulation are found to be in excellent agreement with numerical and experimental velocity interferometer data. The same set of crack statistics are used to extrapolate the results of a higher rate flyer plate case using the effective moduli model. The extension of this model to higher rate cases shows promise for further reducing the number of costly high-fidelity simulations needed to generate crack statistics.

ATP-powered molecular recognition to engineer transient multivalency and self-sorting 4D hierarchical systems

Biological systems organize multiple hierarchical structures in parallel, and create dynamic assemblies and functions by energy dissipation. In contrast, emerging artificial non-equilibrium self-assembling systems have remained relatively simplistic concerning hierarchical design, and non-equilibrium multi-component systems are uncharted territory. Here we report a modular DNA toolbox allowing to program transient non-equilibrium multicomponent systems across hierarchical length scales by introducing chemically fueled molecular recognition orchestrated by reaction networks of concurrent ATP-powered ligation and cleavage of freely programmable DNA building blocks. Going across hierarchical levels, we demonstrate transient side-chain functionalized nucleic acid polymers, and further introduce the concept of transient cooperative multivalency as a key to bridge length scales to pioneer fuel-driven encapsulation, self-assembly of colloids, and non-equilibrium transient narcissistic colloidal self-sorting on a systems level. The fully programmable and functionalizable DNA components pave the way to design chemically fueled 4D (3 space, 1 time) molecular multicomponent systems and autonomous materials.

Modeling of electrochemical deionization across length scales: Recent accomplishments and new opportunities

Theoretical models have recently been used to simulate deionization technology by capturing electrochemical processes at atomistic, electrode, and plant length scales in electrodialysis, capacitive deionization using electric double layers, and Faradaic deionization using intercalation materials and redox-active polymers. We review the salient features of such models, identifying their major accomplishments in quantifying energy consumption and ion removal, analyzing the feasibility of large-scale systems, and discovering new electrode materials and understanding their deionization mechanisms. After summarizing strengths and weaknesses of recent modeling strategies, we identify research directions to expand modeling capabilities that can be used to inform electrode material/microstructure design, to assign energy losses to electrode-scale mechanisms, to bridge length scales, and to capture Faradaic kinetic/diffusion processes.

Assessing pH and Voltage Effects on the Binding of CO to Model Copper Catalyst Surfaces Via a DFT-Continuum Theory

The electrochemical conversion of CO2 to commodity products such as hydrocarbons and oxygenates is an attractive approach for generating non-fossil derived chemical and fuel stocks and for the storage of energy in chemical bonds. To date, a number of studies have shown that the activity and selectivity of copper-based catalysts for the CO2 reduction reaction towards valuable higher order carbon products are highly sensitive to the local reaction environment. Furthermore, the local electrolyte concentration and local pH is intimately tied to the applied voltage and the mass transport of reactants and products within the interfacial region. In recognition of this, several recent efforts focused on engineering the design of electrochemical reactors to reduce mass transport limitations indicate a pronounced effect on the apparent selectivity of the copper catalyst. While these efforts have highlighted a new direction for improving the efficiency of the CO2 reduction reaction, the rational design of these reactors remains challenging due to the need to optimize features of the reactor across multiple time and length scales. In this talk, we will discuss recent efforts within our laboratory to address the influence of the local pH on the CO2 reduction reaction at the atomistic level. We focus the discussion on the binding of CO, a key reaction intermediate in the CO2 reduction pathway, whose preferential hydrogenation or dimerization leads to the production of C1 and C2 products, respectively. We additionally demonstrate how the use of hybrid approaches employing density functional theory (DFT) and continuum models can help bridge time and length scales within the context of atomistic CO2 electroreduction modeling.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.