Lattice-based Monte Carlo Simulations Research Articles

Machine learning-augmented modeling on the formation of Si-dominated Non-β″ early-stage precipitates in Al-Si-Mg alloys with Si supersaturation induced by non-equilibrium solidification

Recent experiments have shown that Al-Si-Mg alloys solidified under high cooling rates may lead to the nucleation of Si-enriched clusters that are remarkably different from the conventional Mg-Si co-clusters (e.g. β″ particles), and yet the responsible mechanism remains to be elucidated. Here we tackle the problem using a multiscale modeling framework that integrates atomistic modeling, energy landscape sampling, and lattice-based kinetic Monte Carlo (kMC) simulation. The migration energy barriers for vacancy-mediated diffusion amid complex local chemical environments are predicted on-the-fly using a surrogate machine learning model. We discover that the actual vacancy-Si migration barriers are much lower than those assumed in the classic linear interpolation approximation. Such a strong deviation from conventional wisdom, in conjunction with differing Si solute composition, can lead to a great variety in the nucleated early-stage precipitates. More specifically, a high-level supersaturation of Si solute (i.e.xSi/(xSi+xMg)>0.75) would lead to an unexpectedly high enrichment of Si in the nucleated clusters with the Si:Mg ratio up to 5∼6; while at a lower-level supply of Si solute the Mg-Si co-clusters (i.e. Si:Mg ratio around 1∼2) are nucleated instead. These findings provide a viable explanation for the diverse types of early-stage precipitates observed in various experiments, from Si-enriched precipitates in high-pressure die cast Al alloys to β″ particles in conventional casting and/or heat-treated alloys. The implications of our findings are also discussed.

Direct computations of viscoelastic moduli of biomolecular condensates.

Biomolecular condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model that leverages information regarding intra- and inter-chain contacts, which we extract from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations of phase separation. The key ingredient of the generalized Rouse model is a graph Laplacian that we compute from equilibrium MMC simulations. We compute two flavors of graph Laplacians, one based on a single-chain graph that accounts only for intra-chain contacts, and the other referred to as a collective graph that accounts for inter-chain interactions. Calculations based on the single-chain graph systematically overestimate the storage and loss moduli, whereas calculations based on the collective graph reproduce the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two graphs proves to be most accurate. In line with the theory of Rouse and contrary to recent assertions, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic vs dominantly viscous behaviors does not imply a single relaxation time. Instead, it is influenced by the totality of the relaxation modes. Hence, our analysis affirms that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain the relaxation time spectra that underlie the dynamics within condensates. This is of practical importance given advancements in passive and active microrheology measurements of condensate viscoelasticity.

Direct computations of viscoelastic moduli of biomolecular condensates.

In vitro facsimiles of biomolecular condensates are formed by different types of intrinsically disordered proteins including prion-like low complexity domains (PLCDs). PLCD condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model and information regarding intra- and inter-chain contacts that is extracted from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations. The key ingredient of the generalized Rouse model is the Zimm matrix that we compute from equilibrium MMC simulations. We compute two flavors of Zimm matrices, one referred to as the single-chain model that accounts only for intra-chain contacts, and the other referred to as a collective model, that accounts for inter-chain interactions. The single-chain model systematically overestimates the storage and loss moduli, whereas the collective model reproduces the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two models proves to be most accurate. In line with the theory of Rouse, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic versus dominantly viscous behaviors is influenced by the totality of the relaxation modes. Hence, our analysis suggests that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain distributions of relaxation times that underlie the dynamics within condensates.

Mean field model parameterization to recover coverage-dependent kinetics

Lateral interactions between adsorbates introduce coverage dependence into adsorption energies and activation barriers of surface reactions. Lattice-based kinetic Monte Carlo (kMC) simulations can capture these interactions quantitatively but are laborious to parameterize and solve. Mean field models are more tractable, but protocols to construct and parameterize them are unclear. Here we explore the ability of a coverage-aware mean-field model to map to a lattice-kMC model of a generic two-step reaction network, including quasi-equilibrated adsorption and rate-limiting dissociation steps. We derive expressions for mean-field and coverage-dependent adsorption energies and dissociation barriers and parameterize against lattice-kMC predictions. We show that the parameterized mean-field rates correlate with ground truth lattice-kMC results across a wide range of reaction conditions and identify regions where the mean field fails. The mean field model similarly captures kMC-derived rate-order, Arrhenius and Sabatier plots at a greatly reduced computational cost. The results provide guidance for parameterizing mean-field models, benchmarked against explicit lattice-based approaches for incorporating the influence of coverage effects.

Novel effects of grain size and ion implantation on grain boundary segregation in ion irradiated austenitic steel

Nuclear reactor environments are extreme in nature due to the combination of exposure to corrosive coolant, mechanical stresses, and neutron irradiation damage. This combination of environments can lead to irradiation induced/assisted precipitation, segregation, and embrittlement coupled with enhanced creep and stress corrosion cracking. While nanostructured materials are of interest in a variety of applications due to their unique properties including high mechanical strength, for nuclear applications nanostructured materials have the added benefit of irradiation resistance due to the high volume density of sinks for irradiation-induced defects. In this study, we show unique radiation-induced segregation (RIS) behavior in nanocrystalline 304 stainless steel (SS304). Typical RIS in austenitic steels includes Ni enrichment and Cr depletion within the grain boundaries. In nanostructured SS304 this RIS is reduced. Furthermore, interestingly, some grain boundaries in the nanocrystalline SS304 in this study were found to be enriched in Cr after Fe2+ ion irradiation at 500 °C up to 50 displacements per atom (dpa). To understand this unique observation, lattice-based atomic kinetic Monte Carlo simulations were performed to investigate the effects of grain size, thermal segregation, as well as injected interstitials (introduced as Fe self-ions). The results indicate that the Cr enrichment at grain boundaries may be caused by combined effects of grain size, self-ion injection and preferential Cr diffusion via the interstitial mechanism. The findings also suggest a possible difference in RIS between ion and neutron irradiations.

Effects of orientational order on modulated cylindrical interfaces.

Cylindrical interfaces occur in sheared or deformed emulsions and as biological or technological lipid monolayer or bilayer tubules. Like the corresponding spherical droplets and vesicles, these cylinderlike surfaces may host orientational order with n-fold rotational symmetry, for example in the positions of lipid molecules or of spherical nanoparticles. We examine how that order interacts with and induces shape modulations of cylindrical interfaces. While on spherical droplets 2n topological defects necessarily exist and can induce icosahedral droplet shapes, the cylindrical topology is compatible with a defect-free patterning. Nevertheless, once a modulation is introduced by a mechanism such as spontaneous curvature, nontrivial patterns of order, including ones with excess defects, emerge and have nonlinear effects on the shape of the tube. By examining the equilibrium energetics of the system analytically and with a lattice-based Markov chain Monte Carlo simulation, we predict low-temperature morphologies of modulated cylindrical interfaces hosting orientational order. A shape modulation induces a banded pattern of alternatingly isotropic and ordered interfacial material. Furthermore, cylindrical systems can be divided into type I, without defects, and type II, which go through a spectrum of defect states with up to 4n excess defects. The character of the curvature-induced shape transition from unmodulated to modulated cylinders is continuous or discontinuous accordingly.

The Regulatory Roles of Intrinsically Disordered Linker in VRN1-DNA Phase Separation.

Biomacromolecules often form condensates to function in cells. VRN1 is a transcriptional repressor that plays a key role in plant vernalization. Containing two DNA-binding domains connected by an intrinsically disordered linker (IDL), VRN1 was shown to undergo liquid-like phase separation with DNA, and the length and charge pattern of IDL play major regulatory roles. However, the underlying mechanism remains elusive. Using a polymer chain model and lattice-based Monte-Carlo simulations, we comprehensively investigated how the IDL regulates VRN1 and DNA phase separation. Using a worm-like chain model, we showed that the IDL controls the binding affinity of VRN1 to DNA, by modulating the effective local concentration of the VRN1 DNA-binding domains. The predicted binding affinities, under different IDL lengths, were in good agreement with previously reported experimental results. Our simulation of the phase diagrams of the VRN1 variants with neutral IDLs and DNA revealed that the ability of phase separation first increased and then decreased, along with the increase in the linker length. The strongest phase separation ability was achieved when the linker length was between 40 and 80 residues long. Adding charged patches to the IDL resulted in robust phase separation that changed little with IDL length variations. Our study provides mechanism insights on how IDL regulates VRN1 and DNA phase separation, and why naturally occurring VRN1-like proteins evolve to contain the charge segregated IDL sequences, which may also shed light on the molecular mechanisms of other IDL-regulated phase separation processes in living cells.

Lattice-based Monte Carlo simulation of the effects of nutrient concentration and magnetic field exposure on yeast colony growth and morphology.

Yeasts exist in communities that expand over space and time to form complex structures and patterns. We developed a lattice-based framework to perform spatial-temporal Monte Carlo simulations of budding yeast colonies exposed to different nutrient and magnetic field conditions. The budding patterns of haploid and diploid yeast cells were incorporated into the framework, as well as the filamentous growth that occurs in yeast colonies under nutrient limiting conditions. Simulation of the framework predicted that magnetic fields decrease colony growth rate, solidity, and roundness. Magnetic field simulations further predicted that colony elongation and boundary fluctuations increase in a nutrient- and ploidy-dependent manner. These in-silico predictions are an important step towards understanding the effects of the physico-chemical environment on microbial colonies and for informing bioelectromagnetic experiments on yeast colony biofilms and fungal pathogens.

Consequences of adsorbate-adsorbate interactions for apparent kinetics of surface catalytic reactions

Lateral adsorbate interactions at catalyst surfaces are known to influence adsorption energies and reaction rates. Lattice-based kinetic Monte Carlo (kMC) simulations are able to capture these influences, but such models are typically parameterized for a specific reaction network and catalyst surface. Here we report kMC simulations to probe the influence of lateral interactions on simulated rates, rate orders, apparent activation energies, and Sabatier plots. We construct a simple, two-step reaction network involving a single adsorbate and rate-limiting diatomic dissociation, employ a generic repulsive lateral interaction model consistent with known adsorbate–adsorbate interactions on metal surfaces, and a rate model consistent with known Brønsted-Evans-Polyani relationships for diatomic dissocations. We juxtapose reaction kinetics over a wide range of reaction conditions and catalyst binding energies, as a function of interaction strength. We find that at a given zero-coverage binding energy and external conditions, adsorbate coverage decreases monotonically with increasing interaction strength, but absolute rates can vary linearly or nonlinearly. Interactions flatten the Sabatier volcano and shift the maximum towards stronger binding. Influences on apparent rate orders and activation energies are modest and are sensitive to interaction-induced adsorbate ordering. Model predictions are sensitive to lattice size effects at high coverages. The results, modelled for a simple reaction system, highlight the generic consequences of lateral interactions and guidance for identifying their signatures in observed kinetics.

Modelling Organic Gel Growth in Three Dimensions: Textural and Fractal Properties of Resorcinol-Formaldehyde Gels.

Tailoring the properties of porous organic materials, such as resorcinol–formaldehyde gels, for use in various applications has been a central focus for many studies in recent years. In order to achieve effective optimisation for each application, this work aims to assess the impact of the various synthesis parameters on the final textural properties of the gel. Here, the formation of porous organic gels is modelled using a three-dimensional lattice-based Monte Carlo simulation. We model growth from monomer species into the interconnected primary clusters of a gel, and account for varying catalyst concentration and solids content, two parameters proven to control gel properties in experimental work. In addition to analysing the textural properties of the simulated materials, we also explore their fractal properties through correlation dimension and Hurst exponent calculations. The correlation dimension shows that while fractal properties are not typically observed in scattering experiments, they are possible to achieve with sufficiently low solids content and catalyst concentration. Furthermore, fractal properties are also apparent from the analysis of the diffusion path of guest species through the gel’s porous network. This model, therefore, provides insight into how porous organic gels can be manufactured with their textural and fractal properties computationally tailored according to the intended application.

Microphase Separation in Oil-Water Mixtures Containing Hydrophilic and Hydrophobic Ions.

We develop a lattice-based MonteCarlo simulation method for charged mixtures capable of treating dielectric heterogeneities. Using this method, we study oil-water mixtures containing an antagonistic salt, with hydrophilic cations and hydrophobic anions. Our simulations reveal several phases with a spatially modulated solvent composition, in which the ions partition between water-rich and water-poor regions according to their affinity. In addition to the recently observed lamellar phase, we find tubular and droplet phases, reminiscent of those found in block copolymers and surfactant systems. Interestingly, these structures stem from ion-mediated interactions, which allows for tuning of the phase behavior via the concentrations, the ionic properties, and the temperature.

Lattice kinetic Monte Carlo simulation study of the early stages of epitaxial GaN(0001) growth

The early stages of GaN(0001) epitaxial growth are modeled using a lattice-based kinetic Monte Carlo simulation. The simulation explicitly includes both the constituent atoms and is performed on the wurtzite crystal structure of GaN. The deposition flux is chosen to mimic conditions in a molecular beam epitaxy chamber. The surface diffusion barriers used for the growth simulation are obtained from ab initio density functional theory based calculations. The evolution of submonolayer islands from small random clusters to ordered triangular islands is captured in these simulations. Further the submonolayer island density statistics is calculated to illustrate standard scaling behavior. The island density as a function of coverage is used to show the different regimes of submonolayer growth and the transition to multi-layer growth. The island density dependence on deposition flux, Ga:N flux ratio, and temperature are also shown to be reasonable and consistent with experiments. We further highlight the importance of N-rotation, which is a diffusion mechanism for nitrogen adatoms under moderate gallium excess.

State transition and electrocaloric effect of BaZrxTi1−xO3: Simulation and experiment

We present a systematic study on the relation of the electrocaloric effect (ECE) and the relaxor state transition of BaZrxTi1−xO3 (BZT) using a combination of computer simulation and experiment. The results of canonical and microcanonical lattice-based Monte Carlo simulations with a Ginzburg-Landau-type Hamiltonian are compared with measurements of BaZrxTi1−xO3 (x = 0.12 and 0.2) samples. In particular, we study the ECE at various temperatures, domain patterns by piezoresponse force microscopy at room temperature, and the P-E loops at various temperatures. We find three distinct regimes depending on the Zr-concentration. In the compositional range 0≤x≤0.2, ferroelectric domains are visible, but the ECE peak drops considerably with increasing Zr-concentration. In the range 0.3≤x≤0.7, relaxor features become prominent, and the decrease in the ECE with Zr-concentration is moderate. In the range of high concentrations, x≥0.8, the material is almost nonpolar, and there is no ECE peak visible. Our results reveal that BZT with a Zr-concentration around x=0.12∼0.3 exhibits a relatively large ECE in a wide temperature range at rather low temperature.

Multiscale Simulations of Anisotropic Grain Growth Using Wavelet Based Multiresolution Analysis

The present work serves to document the development and findings associated with a wavelet-based multiscale simulation analysis for anisotropic grain growth of a two-dimensional polycrystalline material. In particular, lattice-based Monte Carlo and atomically-based Molecular Dynamics simulations are used to compute the grain boundary energies over their respective spatial domains. Serial coupling is performed utilizing an orthonormal set of Haar wavelet transforms embedded within a corresponding multiresolution analysis. For the Monte Carlo approach, anisotropies in grain boundary energies, caused by differences in grain orientation (texturing), are examined using two distinct methods, while the molecular dynamics simulations, offering inherent anisotropy, are conducted assuming the interatomic Lennard Jones potential. Among other findings, under the present context, the results confirm the viability of the wavelet-based multiresolution analysis (MRA) method for use as a potential coupling agent, and provide substantiation for its use with other applications. The results further offer quantitative comparisons between isotropic and anisotropic modeling results, and demonstrate their range of applicability.

Role of [formula omitted] surface reconstruction on Stranski–Krastanov growth illustrated using a modified solid-on-solid model

We develop a modified solid-on-solid model for studying heteroepitaxial growth of Ge on a 2×1 reconstructed Si(001) surface. The model is implemented using lattice-based kinetic Monte Carlo simulations using an atomistic model of elasticity. Our simulations show that the films are grown via Stranski–Krastanov mode with about 2 film layers wetting the substrate below three-dimensional islands. The island shapes evolve from rectangular-based at low coverage(thin films) to square-based at higher coverage. Their size distribution changes from bimodal to unimodal at higher coverage, and subsequently the surface is covered by uniform-sized islands. For simulations performed at lower deposition flux, the shape transition is observed for lower coverage. Our results are consistent with experiments and emphasize the role of the 2×1 reconstruction of the underlying Si(001) surface on the observed behavior. This is the only atomistic model that has been shown to reproduce features of both submonolayer growth and the Stranski–Krastanov growth transition at higher coverage.

Stochastic nonlinear model predictive control applied to a thin film deposition process under uncertainty

This paper investigates the application of stochastic nonlinear model predictive control (NMPC) to a thin film deposition process in the presence of model-plant mismatch while ensuring constraints at a specific probability limit. To capture the multiscale nature of the process, the evolution of the thin film is modelled using nonlinear partial differential equations (PDEs) embedded with lattice-based kinetic Monte Carlo (KMC) simulations. To provide a computationally tractable closed-form expression for online predictive control applications, model identification is performed using data collected from the multiscale deposition model. The closed-form model predicts the expected value and the variance of the thin film properties based on the substrate temperature during the deposition process. The parameters of the closed-form model are determined offline employing power series expansion (PSE). The closed-form model allows the reformulation of probabilistic constraints into their corresponding deterministic expressions thus enabling the design of a computationally tractable stochastic NMPC. To show the effectiveness of the approach, a shrinking horizon stochastic NMPC framework is devised to minimize the final surface roughness while complying with actuator constraints and a probabilistic constraint on the final film thickness.

Lattice-based Monte Carlo simulations of the electrocaloric effect in ferroelectrics and relaxor ferroelectrics

Canonical and microcanonical Monte Carlo simulations are carried out to study the electrocaloric effect (ECE) in ferroelectrics and relaxor ferroelectrics (RFEs) by direct computation of field-induced temperature variations at the ferroelectric-to-paraelectric phase transition and the nonergodic-to-ergodic state transformation. A lattice-based Hamiltonian is introduced, which includes a thermal energy, a Landau-type term, a dipole-dipole interaction energy, a gradient term representing the domain-wall energy, and an electrostatic energy contribution describing the coupling to external and random fields. The model is first parametrized and studied for the case of ${\mathrm{BaTiO}}_{3}$. Then, the ECE in RFEs is investigated, with particular focus on the influence of random fields and domain-wall energies. If the strength or the density of the random fields increases, the ECE peak shifts to a lower temperature but the temperature variation is reduced. On the contrary, if the domain-wall energy increases, the peak shifts to a higher temperature and the ECE becomes stronger. In RFEs, the ECE is maximum at the freezing temperature where the nonergodic-to-ergodic transition takes place. Our results imply that the presence of random fields reduces the entropy variation in an ECE cycle by pinning local polarization.

A robust nonlinear model predictive controller for a multiscale thin film deposition process

The economics of the microelectronics industries depend on the operating policies adopted in the plant. Non-optimal strategies may result in undesired plant performance leading to economic loss or environmental and safety hazards. Model-plant mismatch, actuator constraints and sporadic sensor data can potentially drive the process far from the true optimum. The focus of the current work is to address these issues for the thin film deposition process. The evolution of the thin film can be modeled employing nonlinear partial differential equations (PDEs) embedded with lattice-based kinetic Monte Carlo (KMC) simulations to describe the multiscale nature of the process. Since the KMC simulations are computationally prohibitive for online applications, in this paper, a closed-form model is developed based on the data collected from the multiscale model. The identified closed-form model can be used for online control and optimization applications in the presence of model-plant mismatch. The robust performance is quantified by estimates of the distribution of the controlled variables employing power series expansion (PSE) under model parameter uncertainty. Uncertainties are accounted for in one parameter of the KMC model and in the control actions. To show the applicability of the identified model, the model is embedded within a nonlinear model predictive control (NMPC) framework to predict estimates of the control actions that comply with the process control objectives. The shrinking horizon NMPC algorithm is adopted to minimize the roughness of the thin film while satisfying the constraints on the applied substrate temperature at each time interval and film thickness at the end of the deposition process.

Colloidal nanoparticles trapped by liquid-crystal defect lines: a lattice Monte Carlo simulation.

Lattice-based Monte Carlo simulations are performed to study a confined liquid crystal system with a topological disclination line entangling a colloidal nanoparticle. In our microscopic study the disclination line is stretched by moving the colloid, as in laser tweezing experiments, which results in a restoring force attempting to minimize the disclination length. From constant-force simulations we extract the corresponding disclination line tension, estimated as ∼50 pN, and observe its decrease with increasing temperature.

Study of Stranski–Krastanov growth using kinetic Monte Carlo simulations with an atomistic model of elasticity

We have analyzed the Stranski–Krastanov growth mode in heteroepitaxial thin films using lattice-based kinetic Monte Carlo simulations with an atomistic model of elasticity. In this growth mode, elastic effects due to the lattice mismatch between the film and the substrate cause a transition from two-dimensional layer-by-layer growth to three-dimensional island growth. In our simulations on a simple cubic lattice model in a 3-dimensional system with nearest neighbor interactions only, we see very little tendency towards islanding. On modifying the anisotropy using next-to-nearest and next-to-next-to-nearest neighbor interactions, the system shows a greater tendency towards islanding. When the calculations are carried out in a 2-dimensional system, islanding is fairly pronounced. To gain insights into the possible reasons for these observations, we evaluate the elastic energy and bond energy of different configurations in a 2-dimensional system. Our calculations show that island growth in 2-dimensions also involves a significant nucleation barrier. This suggests that the barrier for island formation is more easily overcome in a 2-dimensional system as compared to a 3-dimensional system.