Structural Order Parameter Research Articles

{101¯4} twin boundary in calcite: Structure and physical properties

Calcite, a CaCO3 polymorph, is one of the most significant materials in nature. In addition to its fundamental role in the global carbon cycle, it fills an unparalleled range of applications, both as a natural biomineral and more directly engineered materials. One of the most notable characteristics of calcite is its tendency to form twins, the boundaries of which are expected to determine key attributes of the host mineral at mesoscopic scales. Using a classical molecular dynamics simulation performed with a thermodynamically consistent rigid-ion force field, we show that the walls between the two twin domains aligned along the hexagonal (101¯4) plane of calcite have some unexpected properties: (1) rather than monoatomic planes, these twin walls are composed of two layers with fundamentally different distortions, (2) atomic shifts within the twin walls create strong polarity while the bulk remains centrosymmetric, and (3) the temperature evolution of the structural order parameter is mostly, but not completely, related to the tilt and rotation of CO32− molecules. Molecular dynamics simulations using thermodynamically accurate interatomic potentials demonstrate that the temperature evolution of twin walls is different from that of the bulk material. We show that kinks in twin walls have a low energy and generate high structural disorder near the kink position. Our methods are applicable to other molecular systems and emphasize the dominance of twin walls in materials. Published by the American Physical Society 2024

Structural origin of relaxation in dense colloidal suspensions

Amorphous solids relax via slow molecular rearrangement induced by thermal fluctuations or applied stress. Microscopic structural signatures predicting these structural relaxations have been long searched for but have so far only been found in dynamic quantities such as vibrational quasi-localized soft modes or with structurally trained neural networks. A physically meaningful structural quantity remains elusive. Here, we introduce a structural order parameter derived from the mean-field caging potential experienced by the particles due to their neighbors, which reliably predicts the occurrence of structural relaxations. The structural parameter, derived from density functional theory, provides a measure of susceptibility to particle rearrangements that can effectively identify weak or defect-like regions in disordered systems. Using experiments on dense colloidal suspensions, we demonstrate a strong correlation between this order parameter and the structural relaxations of the amorphous solid. In quiescent suspensions, this correlation increases with density, when particle rearrangements become rarer and more localized. In sheared suspensions, the order parameter reliably pinpoints shear transformations; the applied shear weakens the caging potential due to shear-induced structural distortions, causing the proliferation of plastic deformation at structurally weak regions. Our work paves the way to a structural understanding of the relaxation of a wide range of amorphous solids, from suspensions to metallic glasses.

Classification of solid and liquid structures using a deep neural network unveils origin of dynamical heterogeneities in supercooled liquids

As the temperature decreases, the dynamics of supercooled liquids significantly slow down and become increasingly heterogeneous in space. Many previous studies have found that static structures also become heterogeneous and are spatially correlated with the dynamical heterogeneity. However, there are still debates on whether the dynamical heterogeneity is controlled by the structures, and which structural order parameters should be used to describe the structural heterogeneities (if exist) in amorphous systems. The appropriate order parameter depends on the specific details of the system and needs to be determined for each system. To address this difficulty, here, we use a machine-learning-based method that was trained solely by the static structures. This method combines convolutional neural networks and gradient-weighted class activation mapping, providing interpretable characteristic structures, which can quantify the degrees of liquid-like and solid-like structures in every local part of the system. We apply this method to a canonical glass-forming system and demonstrate that particles in the liquid-like structures are mobile, while those in the solid-like structures are immobile. The present work develops a novel approach to accurately characterize amorphous structures, which will be particularly useful for systems where appropriate structural order parameters have not yet been identified.

Exploring protein-mediated compaction of DNA by coarse-grained simulations and unsupervised learning

Protein-DNA interactions and protein-mediated DNA compaction play key roles in a range of biological processes. The length scales typically involved in DNA bending, bridging, looping, and compaction (≥1 kbp) are challenging to address experimentally or by all-atom molecular dynamics simulations, making coarse-grained simulations a natural approach. Here, we present a simple and generic coarse-grained model for DNA-protein and protein-protein interactions and investigate the role of the latter in the protein-induced compaction of DNA. Our approach models the DNA as a discrete worm-like chain. The proteins are treated in the grand canonical ensemble, and the protein-DNA binding strength is taken from experimental measurements. Protein-DNA interactions are modeled as an isotropic binding potential with an imposed binding valency without specific assumptions about the binding geometry. To systematically and quantitatively classify DNA-protein complexes, we present an unsupervised machine learning pipeline that receives a large set of structural order parameters as input, reduces the dimensionality via principal-component analysis, and groups the results using a Gaussian mixture model. We apply our method to recent data on the compaction of viral genome-length DNA by HIV integrase and find that protein-protein interactions are critical to the formation of looped intermediate structures seen experimentally. Our methodology is broadly applicable to DNA-binding proteins and protein-induced DNA compaction and provides a systematic and semi-quantitative approach for analyzing their mesoscale complexes.

Thermal conductivity of nonunitary triplet superconductors: application to UTe2

Considerable evidence shows that the heavy fermion material UTe2 is a spin-triplet superconductor, possibly manifesting time-reversal symmetry breaking, as measured by Kerr effect below the critical temperature, in some samples. Such signals can arise due to a chiral orbital state or possible nonunitary pairing. Although experiments at low temperatures appear to be consistent with point nodes in the spectral gap, the detailed form of the order parameter and even the nodal positions are not yet determined. Thermal conductivity measurements can extend to quite low temperatures, and varying the heat current direction should be able to provide information on the order parameter structure. Here, we derive a general expression for the thermal conductivity of a spin-triplet superconductor and use it to compare the low-temperature behavior of various states proposed for UTe2.

Rotation-equivariant graph neural networks for learning glassy liquids representations

The difficult problem of relating the static structure of glassy liquids and their dynamics is a good target for Machine Learning, an approach which excels at finding complex patterns hidden in data. Indeed, this approach is currently a hot topic in the glassy liquids community, where the state of the art consists in Graph Neural Networks (GNNs), which have great expressive power but are heavy models and lack interpretability. Inspired by recent advances in the field of Machine Learning group-equivariant representations, we build a GNN that learns a robust representation of the glass’ static structure by constraining it to preserve the roto-translation (SE(3)) equivariance. We show that this constraint significantly improves the predictive power at comparable or reduced number of parameters but most importantly, improves the ability to generalize to unseen temperatures. While remaining a Deep network, our model has improved interpretability compared to other GNNs, as the action of our basic convolution layer relates directly to well-known rotation-invariant expert features. Through transfer-learning experiments displaying unprecedented performance, we demonstrate that our network learns a robust representation, which allows us to push forward the idea of a learned structural order parameter for glasses.

Non-thermal phonon dynamics and a quenched exciton condensate probed by surface-sensitive electron diffraction

Interactions among and between electrons and phonons steer the energy flow in photo-excited materials and govern the emergence of correlated phases. The strength of electron–phonon interactions, decay channels of strongly coupled modes and the evolution of three-dimensional order are revealed by electron or X-ray pulses tracking non-equilibrium structural dynamics. Despite such capabilities, the growing relevance of inherently anisotropic two-dimensional materials and functional heterostructures still calls for techniques with monolayer sensitivity and, specifically, access to out-of-plane phonon polarizations. Here, we resolve non-equilibrium phonon dynamics and quantify the excitonic contribution to the structural order parameter in 1T-TiSe2. To this end, we introduce ultrafast low-energy electron diffuse scattering and trace strongly momentum- and fluence-dependent phonon populations. Mediated by phonon–phonon scattering, a few-picosecond build-up near the zone boundary precedes a far slower generation of zone-centre acoustic modes. These weakly coupled phonons are shown to substantially delay overall equilibration in layered materials. Moreover, we record the surface structural response to a quench of the material’s widely investigated exciton condensate, identifying an approximate 30:70 ratio of excitonic versus Peierls contributions to the total lattice distortion in the charge density wave phase. The surface-sensitive approach complements the ultrafast structural toolbox and may further elucidate the impact of phonon scattering in numerous other phenomena within two-dimensional materials, such as the formation of interlayer excitons in twisted bilayers.

Effect of the presence of pinned particles on the structural parameters of a liquid and correlation between structure and dynamics at the local level.

Pinning particles at the equilibrium configuration of the liquid is expected not to affect the structure and any property that depends on the structure while slowing down the dynamics. This leads to a breakdown of the structure dynamics correlation. Here, we calculate two structural quantities: the pair excess entropy, S2, and the mean field caging potential, the inverse of which is our structural order parameter (SOP). We show that when the pinned particles are treated the same way as the mobile particles, both S2 and SOP of the mobile particles remain the same as those of the unpinned system, and the structure dynamics correlation decreases with an increase in pinning density, "c." However, when we treat the pinned particles as a different species, even if we consider that the structure does not change, the expression of S2 and SOP changes. The microscopic expressions show that the interaction between a pinned particle and a mobile particle affects S2 and SOP more than the interaction between two mobile particles. We show that a similar effect is also present in the calculation of the excess entropy and is the primary reason for the well-known vanishing of the configurational entropy at high temperatures. We further show that, contrary to the common belief, the pinning process does change the structure. When these two effects are considered, both S2 and SOP decrease with an increase in "c," and the correlation between the structural parameters and the dynamics continues even for higher values of "c."

Superconducting Order Parameter Structure in the Nematic Phase of Iron-based Materials

We consider the effect of the nematic order on the formation of the superconducting state in iron pnictides and chalcogenides. Nematic order with the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{B}_{{2g}}}$$\\end{document} symmetry is modelled as the d-type Pomeranchuk instability and treated within the mean-field approach. Calculated nematic order parameter depends on the nematic interaction coefficient and abruptly changes with the coefficient’s increase. The superconducting solution is obtained within the spin-fluctuation pairing theory. We show that the leading solution in the nematic phase has a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} structure. From the critical temperature \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{T}_{c}}$$\\end{document} estimations, we conclude that the nematic superconducting state of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} type is more favorable than the usual \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{ \\pm }}$$\\end{document} and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$$\\end{document} type states appearing in the absence of the nematicity.

Is the Local Ion Density Sufficient to Drive NaCl Nucleation from the Melt and Aqueous Solution?

Even though nucleation is ubiquitous in different science and engineering problems, investigating nucleation is extremely difficult due to the complicated ranges of time and length scales involved. In this work, we simulate NaCl nucleation in both molten and aqueous environments using enhanced sampling of all-atom molecular dynamics with deep-learning-based estimation of reaction coordinates. By incorporating various structural order parameters and learning the reaction coordinate as a function thereof, we achieve significantly improved sampling relative to traditional ad hoc descriptions of what drives nucleation, particularly in an aqueous medium. Our results reveal a one-step nucleation mechanism in both environments, with reaction coordinate analysis highlighting the importance of local ion density in distinguishing solid and liquid states. However, although fluctuations in the local ion density are necessary to drive nucleation, they are not sufficient. Our analysis shows that near the transition states, descriptors such as enthalpy and local structure become crucial. Our protocol proposed here enables robust nucleation analysis and phase sampling and could offer insights into nucleation mechanisms for generic small molecules in different environments.

Molecular dynamics simulation study of the cosine oscillation electric field's effect on methane hydrate growth

Cosine oscillation electric field could be a promising option to fasten methane hydrate formation in pure water. Molecular dynamics simulation was employed to examine the performance of cosine oscillation electric field in the intensity range of (0.5–2.0) V•nm−1 and frequency range of (0.2–1.0) THz. The hydrate growth time and the hydrate growth rate were firstly defined and obtained by calculating the four-body structure order parameter for different systems at different x positions. The results showed that the added electric field with appropriate parameters could promote methane hydrate formation significantly, represented by shorter hydrate growth time and higher hydrate growth rate. In all studied systems, system with 1.5 V•nm−1 intensity and 0.4 THz field intensity was recommended for fast hydrate growth rate, which could be three times higher than that of system without electric field. It was noteworthy that there was an electric field frequency boundary, lower than which the effect of cosine oscillation electric field on methane hydrate formation could be totally different. The electric field frequency boundary for systems with different electric field intensities was identified and fitted with two exponential growth functions.

Direct measurement of the structural change associated with amorphous solidification using static scattering of coherent radiation.

In this paper, we demonstrate that the weak temperature dependence of the structure factor of supercooled liquids, a defining feature of the glass transition, is a consequence of the averaging of the scattering intensity due to angular averaging. We show that the speckle at individual wavevectors, calculated from a simulated glass former, exhibits a Debye-Waller factor with a sufficiently large temperature dependence to represent a structural order parameter capable of distinguishing liquid from glass. We also extract from the speckle intensities a quantity proportional to the variance of the local restraint, i.e., a direct experimental measure of the amplitude of structural heterogeneity.

Mechanisms of Shock Dissipation in Semicrystalline Polyethylene.

Semicrystalline polymers are lightweight, multiphase materials that exhibit attractive shock dissipation characteristics and have potential applications as protective armor for people and equipment. For shocks of 10 GPa or less, we analyzed various mechanisms for the storage and dissipation of shock wave energy in a realistic, united atom (UA) model of semicrystalline polyethylene. Systems characterized by different levels of crystallinity were simulated using equilibrium molecular dynamics with a Hugoniostat to ensure that the resulting states conform to the Rankine-Hugoniot conditions. To determine the role of structural rearrangements, order parameters and configuration time series were collected during the course of the shock simulations. We conclude that the major mechanisms responsible for the storage and dissipation of shock energy in semicrystalline polyethylene are those associated with plastic deformation and melting of the crystalline domain. For this UA model, plastic deformation occurs primarily through fine crystallographic slip and the formation of kink bands, whose long period decreases with increasing shock pressure.

Characterization of the quasi-liquid layer on gas hydrates with molecular dynamics simulations

Understanding the properties of the surficial quasi-liquid layer (QLL) is a prerequisite in proper handling of gas hydrates, a potential alternative future energy resource. Despite of its critical importance, the characterization of QLL has long been known to be highly challenging. In this work, the evolution of QLL during hydrate decomposition was systematically investigated by Molecular Dynamics simulations. Using the instant displacement of water molecules as a measure of the fluidity of the molecular layer adjacent to hydrate surfaces, QLL thickness was accurately measured using the dynamic properties of water molecules for the first time. The QLL thickness obtained by this new molecular dynamics-based approach takes amorphous water molecules close to hydrate surfaces into account, which was often ignored in the previous results based on the static structural order parameters. The variation of QLL thickness throughout the hydrate decomposition process at different temperatures and pressures was then evaluated. The thickness of QLL was found to increase with elevated temperature and pressure, owing to the varied changes of the amorphous and hydrate-like molecular content in this important layer. The results regarding the dynamics of the QLL with molecular resolution improved the current understanding on the stability of gas hydrate and shed new light on the physical fundamentals relevant to future hydrate exploration as well as anti-hydrate materials design.

Orientational Order in Dense Colloidal Liquids and Glasses.

We construct structural order parameters based on local angular and radial distribution functions in dense colloidal suspensions. All the order parameters show significant correlations to local dynamics in the supercooled and glass regime. In particular, the correlations between the orientational order and dynamical heterogeneity are consistently higher than those between the conventional two-body structural entropy and local dynamics. The structure-dynamics correlations can be explained by a excitation model with the energy barrier depending on local structural order. Our results suggest that in dense disordered packings, local orientational order is higher than translational order, and plays a more important role in determining the dynamics in glassy systems.

A comparative study of the correlation between the structure and the dynamics for systems interacting via attractive and repulsive potentials.

We present the study of the structure-dynamics correlation for systems interacting via attractive Lennard-Jones (LJ) and its repulsive counterpart, the Weeks-Chandler-Andersen (WCA) potentials. The structural order parameter (SOP) is related to the microscopic mean-field caging potential. At a particle level, the SOP shows a distribution. Although the two systems have similar pair structures, their average SOP differs. However, this difference alone is insufficient to explain the well known slowing down of the dynamics in the LJ system at low temperatures. The slowing down can be explained in terms of a stronger coupling between the SOP and the dynamics. To understand the origin of this system specific coupling, we study the difference in the microscopic structure between the hard and soft particles. We find that for the LJ system, the structural differences of the hard and soft particles are more significant and have a much stronger temperature dependence compared to the WCA system. Thus, the study suggests that attractive interaction creates more structurally different communities. This broader difference in the structural communities is probably responsible for stronger coupling between the structure and dynamics. Thus, the system specific structure-dynamics correlation, which also leads to a faster slowing down in the dynamics, appears to have a structural origin. A comparison of the predictive power of our SOP with the local energy and two body excess entropy in determining the dynamics shows that in the LJ system, the enthalpy plays a dominant role and in the WCA system, the entropy plays a dominant role, and our SOP can capture both these contributions.

First principles study on the structural, electronic, and thermophysical properties of BiFeO3

Due to their special aspects and simple structure, perovskite structure materials are of enormous curiosity to scientists. Because of their ferroelectric, ferromagnetic, or ferroelastic properties, multiferroic materials could have interrelated electronic, magnetic, or structural order parameters. We represent the structural, electronic, and thermophysical properties in the ground state space group (221) of BiFeO3 according to Density Functional Theory (DFT). A BiFeO3 is investigated using Local Density Approximation (LDA) and Generalised Gradient Approximation (GGA) functional using ultrasoft pseudopotential. The present work calculated the structural, electronic, and thermophysical properties of BiFeO3 Perovskite. It demonstrates that the estimated structural properties of space group (221) BiFeO3 correspond well with existing research. The electronic attributes include band structure, total and projected density of states, electronic charge density, and Fermi surface topology. The thermophysical properties, like specific heat capacity, thermal expansion (α), Gruneisen parameter (γ), isothermal bulk modulus (B0), and Debye temperature (θD), were calculated in the temperature range of 0 K to 1073 K.

Dynamic heterogeneity in polydisperse systems: A comparative study of the role of local structural order parameter and particle size.

In polydisperse systems, describing the structure and any structural order parameter (SOP) is not trivial as it varies with the number of species we use to describe the system, M. Depending on the degree of polydispersity, there is an optimum value of M = M0 where we show that the mutual information of the system increases. However, surprisingly, the correlation between a recently proposed SOP and the dynamics is highest for M = 1. This effect increases with polydispersity. We find that the SOP at M = 1 is coupled with the particle size, σ, and this coupling increases with polydispersity and decreases with an increase in M. Careful analysis shows that at lower polydispersities, the SOP is a good predictor of the dynamics. However, at higher polydispersity, the dynamics is strongly dependent on σ. Since the coupling between the SOP and σ is higher for M = 1, it appears to be a better predictor of the dynamics. We also study the Vibrality, an order parameter independent of structural information. Compared to SOP, at high polydispersity, we find Vibrality to be a marginally better predictor of the dynamics. However, this high predictive power of Vibrality, which is not there at lower polydispersity, appears to be due to its stronger coupling with σ. Therefore, our study suggests that for systems with high polydispersity, the correlation of any order parameter and σ will affect the correlation between the order parameter and dynamics and need not project a generic predictive power of the order parameter.

Unsupervised machine learning discovery of structural units and transformation pathways from imaging data

We show that unsupervised machine learning can be used to learn chemical transformation pathways from observational Scanning Transmission Electron Microscopy (STEM) data. To enable this analysis, we assumed the existence of atoms, a discreteness of atomic classes, and the presence of an explicit relationship between the observed STEM contrast and the presence of atomic units. With only these postulates, we developed a machine learning method leveraging a rotationally invariant variational autoencoder (VAE) that can identify the existing molecular fragments observed within a material. The approach encodes the information contained in STEM image sequences using a small number of latent variables, allowing the exploration of chemical transformation pathways by tracing the evolution of atoms in the latent space of the system. The results suggest that atomically resolved STEM data can be used to derive fundamental physical and chemical mechanisms involved, by providing encodings of the observed structures that act as bottom-up equivalents of structural order parameters. The approach also demonstrates the potential of variational (i.e., Bayesian) methods in the physical sciences and will stimulate the development of more sophisticated ways to encode physical constraints in the encoder–decoder architectures and generative physical laws and causal relationships in the latent space of VAEs.

Amorphous solidification of a supercooled liquid in the limit of rapid cooling.

We monitor the transformation of a liquid into an amorphous solid in simulations of a glass forming liquid by measuring the variation of a structural order parameter with either changing temperature or potential energy to establish the influence of the cooling rate on amorphous solidification. We show that the latter representation, unlike the former, exhibits no significant dependence on the cooling rate. This independence extends to the limit of instantaneous quenches, which we find can accurately reproduce the solidification observed during slow cooling. We conclude that amorphous solidification is an expression of the topography of the energy landscape and present the relevant topographic measures.