Empirical Potential Research Articles

Fitting the charged-optimized many-body potential for the Al-O-Se-Zn system

A charge-optimized many-body (COMB3) potential has been developed for the Al-O-Se-Zn system using first-principles data. This work updates COMB3 for zinc and extends it to include selenium as well as other pairwise and three-body interactions that were not developed previously. We tested the empirical potential for known crystalline phases and find reasonable agreement between the calculated structural, elastic, and vibrational properties and experimental data. We use the anisotropic coefficients of the thermal expansion method to calculate the temperature-dependent properties of a few selected materials. We then compare the total energy of the system and its first- and second-order positional derivatives, as well as the second-order strain derivative with respect to the atomic potential energy surface to thermal properties obtained from molecular dynamics simulations using the fitted COMB3 potential. The results of our temperature-dependent calculations show reasonable agreement with previous work for temperatures below the system’s melting point. With this fitting, we could utilize a versatile, charge-dependent empirical potential to model the interface between ZnSe and Al2O3 by molecular dynamics simulations.

Interaction of pristine and novel graphene allotropes with copper nanoparticles: Coupled density functional and molecular dynamics study

We combined the density functional theory and classical molecular dynamics to study the time evolution and thermal stability of copper nanoparticles wrapped in graphene flakes. We observed a strong attraction between nanoparticles and flakes, which was underestimated in many previous simulations. We found that the two-parameter Lennard-Jones potential with parameters ε = 0.074 eV and r0 = 3.310 Å reproduces DFT data better than other empirical potentials. We confirmed that a nanoparticle could be held reliable inside a graphene flake. The graphene-coated copper system remains stable over the temperature range of 300–1000 K. In addition to pristine graphene, we considered several strained allotropes containing pentagons, as well as heptagons, octagons, and nanometer-sized pores. Strained allotropes interact with copper nanoparticles approximately twice more strongly as compared to pristine graphene. Molecular dynamics revealed nanoparticle flattening due to strong interaction with graphene allotropes at elevated temperatures. The wettability of graphene with respect to copper strongly depends on the sheet structure and can vary significantly for different allotropes. The results may be useful for further research on copper-graphene composites, which are suitable for catalytic and biomedical applications.

Association of Empirical Dietary Inflammatory Potential with Mortality: Results from the Third National Nutrition Examination Survey.

The empirical dietary inflammatory potential (EDIP) score is designed to assess the inflammatory potential of a diet based on the pro- and anti-inflammatory properties of its various components. This study examined the association of EDIP with all-cause mortality in a large, community-based, multiracial sample of the United States population. A prospective cohort study. This analysis included 13155 participants (44.6±18.4 years, 54.21% women, and 40.33% White) without prior history of cardiovascular disease (CVD) from the Third National Health and Nutrition Examination (NHANES III) Survey. A 24-hour dietary recall information was used to calculate EDIP. The National Death Index was employed to identify the date and cause of death. Cox proportional hazard analysis was utilized to evaluate the association between the tertiles of EDIP and all-cause mortality over a median follow-up of 26.6 years. In a model adjusted for demographics and CVD risk factors, a higher EDIP tertile, compared with the lowest tertile, was associated with an increased risk of all-cause mortality (hazard ratio [HR]=1.10; 95% CI: 1.02, 1.19). A standard-deviation increase in EDIP (0.27 units) was related to a 4% increased risk of mortality (HR=1.04; 95% CI: 1.01, 1.08). This association was stronger in older participants compared to younger ones (HR=1.09; 95% CI: 0.98, 1.21 vs. HR=0.89; 95% CI: 0.80, 1.01), respectively, interaction P=0.030)]. Pro-inflammatory diet is associated with an increased risk of mortality, especially in the older population. Dietary changes that reduce inflammation may have the potential to reduce the risk of poor outcomes.

Computational study of Na diffusion and conduction in P2- and O3-Na2x[NixTi1-x]O2 materials with machine-learning interatomic potentials

Sodium-ion batteries (SIBs) provide a feasible solution for large-scale energy storage applications of sustainable energy resources like wind and solar energy. Layered sodium nickel titanates (Na2x[NixTi1-x]O2) (SNTL) are considered a promising electrode material for SIBs since they can function as either a positive or negative electrode due to the coexistence of high redox potential element Ni (V = 3.7 V vs. Na/Na+) and low redox potential element Ti (V = 0.7 V vs. Na/Na+). There have been both experimental and computational studies on the family of Na2x[NixTi1-x]O2 materials, investigating their structures, electrochemical performance, electronic and ionic conductivity, etc. However, the current understanding of the Na migration mechanism is still very limited at an atomic level because of the restrictions of simulation methods: density-functional theory (DFT) simulation is restricted to a small length and time scale, whereas the conventional empirical interatomic potential (IP) is restricted by a limited accuracy. To solve this efficiency-versus-accuracy dilemma, herein we developed a neural network (NN)-based machine-learning interatomic potential (MLIP), which is trained using DFT data. Our MLIP demonstrates a DFT-approaching accuracy in terms of atomic forces. Moreover, the extracted Na self-diffusivity, ionic conductivity, and activation energy values from MLIP-MD simulation agree well with experimental data. In general, P2-SNTL demonstrates a higher Na self-diffusivity and ionic conductivity than O3-SNTL, mainly due to the larger bottleneck size through which Na ions migrate. We also employed the incoherent density correlation function to study the Na migration mechanism. The Singwi-Sjölander (SS) model and Chudley-Elliot (CE) model were used to fit the jump-diffusion behavior in P2- and O3-SNTL, respectively. The obtained jump distance values match well with the typical distance between the nearest Na sites in P2- and O3-SNTL, confirming that the hopping events happen mostly between neighboring Na sites in SNTL. The decrease of residence time and jump distance values with temperature is a result of higher ion mobility and level of delocalization, as validated by the nuclear density maps. In addition, our study also confirmed that both the “straight” and “curved” Na migration paths exist in O3-SNTL, in contrast to previous studies which only mention the “curved” migration path. In the end, our calculated Haven ratios have a value <1, indicating that it is more likely for the Na hopping events to occur via a concerted migration of several atoms other than the jump of an isolated atom.

Structural and electronic properties of U4O9 from ab initio and empirical potential calculations

We address here the description of the different phases of ${\mathrm{U}}_{4}{\mathrm{O}}_{9}$ with ab initio density functional theory and molecular dynamics empirical potential calculations. These phases are built from the $\mathrm{U}{\mathrm{O}}_{2}$ fluorite by the addition of oxygen interstitials which assemble in clusters named cuboctahedra. Due to the unit cell size and complexity, their simulation represents a formidable task for numerical approaches. With $\mathrm{DFT}+U$, we study in detail two potential structures for $\ensuremath{\alpha}\ensuremath{-}{\mathrm{U}}_{4}{\mathrm{O}}_{9}$, however, these are just two representatives among the numerous coexisting structures we found. The role of the different valence uraniums (${\mathrm{U}}^{4+}, {\mathrm{U}}^{5+}$, and possibly ${\mathrm{U}}^{6+}$) is highlighted thanks to our quantum-mechanical approach. Temperature effects are then appraised with empirical potentials to describe the evolution of the ${\mathrm{U}}_{4}{\mathrm{O}}_{9}$ structure with temperature. We observe a continuous symmetrization of the structure with increasing temperature. Indeed, below 800 K, it gradually turns cubic and the cuboctahedral oxygen clusters tend to symmetrize. Beyond 800 K, the cuboctahedra start to disappear up to 1400 K, where none can be found in our simulated structures.

Plastic deformations of ceria nanocubes under compression: An atomistic simulations study

The mechanical properties and elementary deformation processes of ceria nanoparticles are investigated under uniaxial compression using quasi-static atomistic simulations and a fixed charge empirical potential. Results show a strong size-dependence of both the stress response and the plastic deformation phenomena that are characterized by amorphization and recrystallization processes that initiate from the surfaces in smaller-size samples, while the propagation of {111} nanotwins is observed in larger nanoparticles. In particular, we point out that the recrystallized phase is made of reoriented fluorite that behaves as a structural seed for the nucleation of 16<112>{111} partial dislocations at the origin of the nanotwins. Similarities with plastic deformation processes of FCC metals at the nanoscale are outlined and discussed.

В поисках глобальной памяти: куда ведет транснациональный поворот в Memory studies?

The article is devoted to the transnational turn in the research of collective and cultural memory in the 21st century, reflecting the desire of scientists working within Memory studies to go beyond “methodological nationalism” and explore the formation and circulation of historical memory across borders. Based on the English-language literature of the 2000s—2020s, the author analyzes the emerging field of Transnational Memory studies from the point of view of the categorical apparatus, disciplinary features and research approaches. Having documented that discussions about the transcultural dimension of collective memory are present even in the works of the classics of Memory studies, the author shows that the efforts of their modern critics are driven by the attempts to conceptualize the memory of the Holocaust, World War II, colonialism and other events of the “difficult past”. The abundance of closely related concepts elaborated in recent years creates an illusion of competition in this field, but these concepts are often based on fundamentally different ideological and methodological settings, and if some of the concepts possess a certain empirical potential, others are completely normative. The author identifies six main research lenses within the current state of Transnational Memory studies — international, supranational, diasporic, mediatized, digital and, finally, global, associated with the problem of memory synchrony/simultaneity. At the same time, according to his conclusion, getting rid of the national perspective in understanding transnational memory is still far from complete, and the very fact that scientists look beyond borders indicates that such borders still exist.

Atomistic determination of Peierls barriers of dislocation glide in nickel

The Peierls barrier measures the lattice resistance to dislocation glide in crystalline solids. We use the nudged elastic band (NEB) method to calculate the Peierls barriers for screw and edge dislocation glide in a face-centered cubic (FCC) metal of Ni. The minimum energy paths (MEPs) across single or sequential Peierls barriers are determined under shear loading. The NEB results show the decreasing Peierls barrier with increasing shear stress, giving the Peierls stress at which the Peierls barrier vanishes. The effects of boundary condition and system size on Peierls barriers are studied by comparing strain- and stress-controlled NEB results. Furthermore, the free-end NEB methods are applied to determine MEPs with improved computational efficiency. The NEB results are also used to evaluate the energetic driving force of dislocation glide, which is consistent with that determined from the Peach-Koehler force. The accuracy of the present NEB results based on an empirical interatomic potential is assessed by comparison with a machine-learning potential. This work demonstrates the robust and efficient quantification of Peierls barriers to dislocation glide in an FCC metal, and it lays a solid foundation for the atomistic determination of Peierls barriers in compositionally complex alloys with the FCC structure in future studies.

Reliable machine learning potentials based on artificial neural network for graphene

Graphene is one of the most researched two dimensional (2D) material in the past two decades due to its unique combination of mechanical, thermal and electrical properties. Special 2D structure of graphene enables it to exhibit a wide range of peculiar material properties like high Young’s modulus, high specific strength, electrical conductivity etc. which are critical for myriad of applications including lightweight structural materials, multi-functional coating and flexible electronics. As it is quite challenging and costly to experimentally investigate graphene and graphene based nanocomposites, computational simulations such as molecular dynamics (MD) simulations are widely adopted for understanding the microscopic origins of their unique properties. However, disparate results were reported from computational studies, especially MD simulations using various empirical inter-atomic potentials. In this work, an artificial neural network (ANN) based interatomic force field potential has been developed for graphene to represent the potential energy surface based on first principle calculations. The developed machine learning potential (MLP) facilitates high fidelity MD simulations to approach the accuracy of ab initio methods but with a fraction of computational cost, which allows larger simulation size and length, and thereby enables accelerated discovery and design of graphene-based novel materials. Lattice parameter, coefficient of thermal expansion (CTE), Young’s modulus and yield strength are estimated using machine learning accelerated MD simulations (MLMD), which are compared to experimental and first principle calculations from previous literatures. It is demonstrated that MLMD can capture the dominating mechanism governing the CTE of graphene, including effects from lattice parameter and out of plane rippling. The MLMD approach is highly scalable for 2D materials and can help in accelerating the research of novel 2D materials and 2D material hybrids with unique atomic structures.

Atomistic model of the anisotropic response of ortho-Mo2C to indentation

Molybdenum carbide has various applications for which studying the material using classical molecular dynamics simulations would be valuable. Here, we develop an empirical potential within the Tersoff formalism using particle swarm optimization for the orthorhombic phase of Mo2C. The developed potential is shown to predict lattice constants, elastic properties, and equation of state results that are consistent with current and previously reported results from experiments and first principles calculations. We demonstrate the potential with simulations of indentation using multiple indenter sizes that load and unload in three different directions relative to the crystallographic lattice of orthorhombic Mo2C. Direction-dependent force-displacement trends are analyzed and explained in terms of the spatial distributions of stress and strain within the material during indentation. This study reveals the anisotropic elasticity of orthorhombic Mo2C and, more generally, provides researchers with a new empirical potential that can be used to explore the properties and behavior of the material going forward.

Machine learning interatomic potential for silicon-nitride (Si3N4) by active learning.

Silicon nitride (Si3N4) is an extensively used material in the automotive, aerospace, and semiconductor industries. However, its widespread use is in contrast to the scarce availability of reliable interatomic potentials that can be employed to study various aspects of this material on an atomistic scale, particularly its amorphous phase. In this work, we developed a machine learning interatomic potential, using an efficient active learning technique, combined with the Gaussian approximation potential (GAP) method. Our strategy is based on using an inexpensive empirical potential to generate an initial dataset of atomic configurations, for which energies and forces were recalculated with density functional theory (DFT); thereafter, a GAP was trained on these data and an iterative re-training algorithm was used to improve it by learning on-the-fly. When compared to DFT, our potential yielded a mean absolute error of 8 meV/atom in energy calculations for a variety of liquid and amorphous structures and a speed-up of molecular dynamics simulations by 3-4 orders of magnitude, while achieving a first-rate agreement with experimental results. Our potential is publicly available in an open-access repository.

Comparing transferability in neural network approaches and linear models for machine-learning interaction potentials

Atomic simulations using machine learning interatomic potential (MLIP) have gained a lot of popularity owing to their accuracy in comparison to conventional empirical potentials. However, the transferability of MLIP to systems outside the training set poses a significant challenge. Here, we compare the transferability of three MLIP approaches: (i) neural network potentials (NNP), (ii) physical LassoLars interactions potential (PLIP) and (iii) linear potentials with Belher-Parrinello descriptors, trained over a small but diverse configuration of zinc oxide polymorphs. We compared the obtained models with density functional theory reference results for physical properties including bulk lattice parameters, surface energies, and vibrational density of states and showed the superiority of both NNP and PLIP models. However, the NNP model performed poorly when compared to the other two linear models for the structural optimization of nanoparticles and molecular dynamics simulation of liquid phases, which are systems outside the training set. While providing less accurate prediction for solid Zinc Oxides phases, both linear models appear more transferable than NNP when testing for nanoscale systems and liquid phases. Our results are finally rationalized by a combination of different statistical analysis including spread in force evaluation, information imbalance, convex hull calculation, and density in descriptor space.

Computational Discovery of New Feasible Crystal Structures in Ce3O3N

Oxynitrides of cerium are expected to have many useful properties but have not been synthesized so far. We identified possible modifications of a not-yet-synthesized Ce3O3N compound, combining global search (GS) and data mining (DM) methods. Employing empirical potentials, structure candidates were obtained via global optimization on the energy landscape of Ce3O3N for different pressure values. Furthermore, additional feasible structure candidates were found using data mining of the ICSD database. The most promising structure candidates obtained were locally optimized at the ab initio level, and their E(V) curves were computed. The structure lowest in total energy, Ce3O3N-DM1, was found via local optimization starting from a data mining candidate and should be thermodynamically metastable up to high pressures.

Fast proper orthogonal descriptors for many-body interatomic potentials

The development of differentiable invariant descriptors for accurate representations of atomic environments plays a central role in the success of interatomic potentials for chemistry and materials science. We introduce a method to generate fast proper orthogonal descriptors for the construction of many-body interatomic potentials and discuss its relation to exising empirical and machine learning interatomic potentials. A traditional way of implementing the proper orthogonal descriptors has a computational complexity that scales exponentially with the body order in terms of the number of neighbors. We present an algorithm to compute the proper orthogonal descriptors with a computational complexity that scales linearly with the number of neighbors irrespective of the body order. We show that our method can enable a more efficient implementation for a number of existing potentials and provide a scalable systematic framework to construct new many-body potentials. The new potentials are demonstrated on a data set of density functional theory calculations for Tantalum and compared with other interatomic potentials.

Competition between phonon-vacancy and four-phonon scattering in cubic boron arsenide by machine learning interatomic potential

Point defects can strongly suppress the thermal conductivity $\ensuremath{\kappa}$ of solid materials, which is crucial for a broad range of applications, such as thermal management of electronic devices and thermoelectrics. Understanding thermal transport in materials containing point defects often relies on atomistic simulations based on density functional theory (DFT) or empirical potentials (EPs). However, modeling thermal transport in defective materials using DFT is very computationally expensive or even prohibitive due to the breaking of crystal symmetry while EPs suffer from low accuracy. Recently, machine learning has been applied to the development of interatomic potentials, offering opportunities to model defective systems accurately and efficiently. Here, we present a Gaussian approximation potential (GAP) developed for crystalline cubic boron arsenide (c-BAs) with vacancies, which can achieve DFT-level accuracy in predicting its $\ensuremath{\kappa}$ and phonon transport properties at four orders of magnitude reduced computational cost, especially for phonon-vacancy and four-phonon scatterings. Particularly, we applied the GAP to investigate the effect of vacancies on the $\ensuremath{\kappa}$ of c-BAs by considering both three-phonon and four-phonon scattering. Special attention was paid to the competition between phonon-vacancy and four-phonon scattering, which tend to decrease and increase the temperature dependence of $\ensuremath{\kappa}$, respectively. Specifically, when the vacancy concentration is much lower than $0.07%$ $(2.56\ifmmode\times\else\texttimes\fi{}{10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3})$, four-phonon scattering plays stronger roles in determining the temperature dependence of $\ensuremath{\kappa}$. As the vacancy concentration increases to $0.07%$, the temperature dependence of $\ensuremath{\kappa}$ becomes close to that considering only three-phonon scattering, indicating the comparable effect of phonon-vacancy and four-phonon scattering. As the vacancy concentration further increases, the phonon-vacancy scattering becomes more dominant and pushes the $\ensuremath{\kappa}$ towards a temperature-independent behavior. Our work deepens the understanding of the phonon scattering landscape in c-BAs with vacancies and will be helpful for tailoring its thermal properties. Atomistic simulations combined with machine learning interatomic potentials are expected to be able to greatly advance the understanding of thermal transport in defective materials.

Relationship between empirical dietary inflammatory potential and myocardial infarction

BackgroundInflammation plays an important role in developing myocardial infarction (MI). This study examined whether a proinflammatory diet is associated with increased risk of myocardial infarction. MethodsThis analysis included 7,134 participants (60.3 ± 13.6 years; 51.8% females) from the third National Health and Nutrition Examination (NHANES-III). The proinflammatory diet was assessed using the empirical dietary inflammatory potential (EDIP) score, calculated from the Food Frequency Questionnaire. MI was defined from electrocardiograms (ECGs) using the Minnesota ECG Classification. The cross-sectional association between levels of EDIP modeled as tertiles and per 1-standard deviation (1-SD) increase in separate models with the risk of MI using multivariable logistic regression analysis. ResultsParticipants with MI (n=230 (3.2%)) had higher levels of EDIP scores compared to those without MI (0.148 ± 0.241 score units vs. 0.106 ± 0.256 score units, respectively; p=0.01). In multivariable-adjusted models, each 1-SD (0.256 score unit) increase in EDIP was associated with 20% increased odds of MI (OR (95% CI); 1.20 (1.05 to 1.38)). Odds of MI increased as the levels of EDIP tertiles increased, indicating a dose-response relationship (OR (95% CI); 1,41 (1.0 to 1.99) and 1.48 (1.05 to 2.09), respectively). These results were consistent among subgroups of the participants stratified by hypertension, obesity, diabetes, and hyperlipidemia, but effect modification by smoking status was observed (interaction p-value=0.04). ConclusionsDietary patterns with high proinflammatory properties are associated with an increased risk of MI. Advocating for low proinflammatory dietary patterns could be an approach for preventing coronary heart disease.

Graph neural networks predict energetic and mechanical properties for models of solid solution metal alloy phases

We developed a PyTorch-based architecture called HydraGNN that implements graph convolutional neural networks (GCNNs) to predict the formation energy and the bulk modulus for models of solid solution alloys for various atomic crystal structures and relaxed volumes. We trained the GCNN surrogate model on a dataset for nickel–niobium (NiNb) generated by the embedded atom model (EAM) empirical interatomic potential for demonstration purposes. The dataset was generated by calculating the formation energy and the bulk modulus as a prototypical elastic property for optimized geometries starting from initial body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal compact packed (HCP) crystal structures, with configurations spanning the possible compositional range for each of the three types of initial crystal structures. Numerical results show that the GCNN model effectively predicts both the formation energy and the bulk modulus as function of the optimized crystal structure, relaxed volume, and configurational entropy of the model structures for solid solution alloys.

Transport properties in liquids from first-principles: The case of liquid water and liquid argon.

Shear and bulk viscosities of liquid water and argon are evaluated from first-principles in the density functional theory (DFT) framework, by performing molecular dynamics simulations in the NVE ensemble and using the Kubo-Greenwood equilibrium approach. The standard DFT functional is corrected in such a way to allow for a reasonable description of van der Waals effects. For liquid argon, the thermal conductivity has been also calculated. Concerning liquid water, to our knowledge, this is the first estimate of the bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first-principles. By analyzing our results, we can conclude that our first-principles simulations, performed at a nominal average temperature of 366 to guarantee that the systems are liquid-like, actually describe the basic dynamical properties of liquid water at about 330K. In comparison with liquid water, the normal, monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-viscosity ratio (close to unity) and this feature is well reproduced by our first-principles approach, which predicts a value of the ratio in better agreement with experimental reference data than that obtained using the empirical Lennard-Jones potential. The computed thermal conductivity of liquid argon is also in good agreement with the experimental value.

Fast general two- and three-body interatomic potential

We introduce a new class of machine learning interatomic potentials---fast general two- and three-body potential (GTTP), which is as fast as conventional empirical potentials and require computational time that remains constant with increasing fitting flexibility. GTTP does not contain any assumptions about the functional form of two- and three-body interactions. These interactions can be modeled arbitrarily accurately, potentially by thousands of parameters not affecting resulting computational cost. Time complexity is O(1) per every considered pair or triple of atoms. The fitting procedure is reduced to simple linear regression on ab initio calculated energies and forces and leads to effective two- and three-body potential, reproducing quantum many-body interactions as accurately as possible. Our potential can be made continuously differentiable any number of times at the expense of increased computational time. We made a number of performance tests on one-, two- and three-component systems. The flexibility of the introduced approach makes the potential transferable in terms of size and type of atomic systems as long as they involve the same atomic species. We show that trained on randomly generated structures with just eight atoms in the unit cell, it significantly outperforms common empirical interatomic potentials in the study of large systems, such as grain boundaries in polycrystalline materials.

On-the-fly machine learning potential accelerated accurate prediction of lattice thermal conductivity of metastable silicon crystals

In this paper, we propose a convenient strategy to accelerate the evaluation of lattice thermal conductivity through combining phonon Boltzmann transport equation (PBTE) and on-the-fly machine learning potential (FMLP). The thermal conductivity of diamond silicon ($d\text{\ensuremath{-}}\mathrm{Si}$) is evaluated firstly by density functional theory (DFT), FMLP, and empirical potential with PBTE, respectively. The results demonstrate the proposed strategy integrates the prediction accuracy of DFT and computational speed of empirical potential, breaking the dilemma of traditional thermal conductivity assessment schemes. Based on this, the efficient strategy is applied to predict thermal conductivity of 102 low-energy metastable silicon crystals with energies between $d\text{\ensuremath{-}}\mathrm{Si}$ and experimentally ${\mathrm{Si}}_{24}$. Among them, the $Cmcm\text{\ensuremath{-}}{\mathrm{Si}}_{16}, P6/mmm\text{\ensuremath{-}}{\mathrm{Si}}_{36}$-2, $Pnma\text{\ensuremath{-}}{\mathrm{Si}}_{32}$-2 are predicted to host lowest lattice thermal conductivity in $xx$ (8.213 ${\mathrm{Wm}}^{\ensuremath{-}1}{\mathrm{K}}^{\ensuremath{-}1}$), $yy$ (10.917 ${\mathrm{Wm}}^{\ensuremath{-}1}{\mathrm{K}}^{\ensuremath{-}1}$), and $zz$ (11.807 ${\mathrm{Wm}}^{\ensuremath{-}1}{\mathrm{K}}^{\ensuremath{-}1}$) directions, respectively. Such low lattice thermal conductivity benefits from the combined effect of low phonon group velocity and intense phonon scattering caused by distorted $s{p}^{3}$ hybrid states in these metastable silicon crystals. The findings presented in this work provide new candidates and insights of silicon-based materials with ultra-low thermal conductivity, which will greatly expand the applications in thermoelectric and thermal insulation fields.