Monatomic Crystals Research Articles

Mechanism-informed constitutive modeling of molecular crystal cyclotetramethylene tetranitramine (β-HMX)

Molecular crystals are of importance in many applications including energetic materials and pharmaceuticals. Here we focus on the energetic material cyclotetramethylene tetranitramine (β-HMX) whose plastic deformation plays an important role in reaction initiation and detonation. While constitutive modeling of monatomic crystals is well established, the development of models for molecular crystals is less advanced. We present a mechanism-based constitutive formulation for β-HMX crystals, and an equivalent Reuss averaged homogenized model, which are based on a wealth of information about the physics of plastic deformation collected from the literature and developed in this work. We use atomistic models to evaluate the dislocation velocity vs. resolved shear stress relation in multiple slip systems and at various pressures up to 20 GPa, while accounting for slip asymmetry. We use this data to evaluate the strain rate sensitivity on the crystal scale and calibrate all constitutive parameters of the thermally activated and dislocation drag regimes. The resulting crystal plasticity model is then Reuss averaged to produce a homogenized version. The model is perfectly plastic, strongly pressure-dependent and rate sensitive. Further, we observe that strain localization takes place at large enough strain rates and propose a criterion for the occurrence of localization. The formulation can be used to advance the state of the art of continuum modeling of microstructure-dependent initiation in HMX.

Isotope effects on phonon modes in polyatomic crystals

The effect of isotopic impurities on the phonon modes of polyatomic crystals is discussed theoretically. The diagonalized mass-independent force matrix can be considered as force, and the square of the phonon frequency as acceleration, so that the remaining coefficient, which is only related to the atomic mass, can be regarded as effective mass. The effective mass can be obtained by solving the phonon frequencies, and the creation and destruction operators of this system are obtained. Since the first-order perturbation of phonon frequencies in monatomic crystals is zero, we always omit the changes. However, in polyatomic crystals, we find that the first-order perturbation is not zero, which may need to be considered. Finally, we re-derived the formula for the phonon lifetimes limited by isotope scattering, which is independent of the eigenvectors, so that the eigenvectors are not required for calculating phonon lifetimes limited by isotope scattering.

Quantum crystal equation of state

A new and simple equation of state of a monatomic quantum crystal is constructed using the continuum approximation for interacting atoms and the Mie-Gruneisen theory. The equation makes it possible to describe the phase diagram both in the region of compressive pressures and in the vicinity of the critical point. Expressions for critical volume, pressure and temperature are found analytically. Comparison of numerical calculations with experimental data for helium shows satisfactory agreement. Keywords: quantum crystal, continuum approximation, Mie-Gruneisen model, Debye model, equation of state, critical parameters.

Уравнение состояния квантового кристалла

A new and simple equation of state of a monatomic quantum crystal is constructed using the continuum approximation for interacting atoms and the Mie-Gruneisen theory. The equation makes it possible to describe the phase diagram both in the region of compressive pressures and in the vicinity of the critical point. Expressions for critical volume, pressure and temperature are found analytically. Comparison of numerical calculations with experimental data for helium shows satisfactory agreement.

Influence of the Crystal Structure and Type of Interatomic Bond on the Elastic Properties of Monatomic and Diatomic Cubic Crystals

Influence of the Crystal Structure and Type of Interatomic Bond on the Elastic Properties of Monatomic and Diatomic Cubic Crystals

Reformulation of expressions for thermoelastic properties of crystals using harmonic mapping

We introduce alternative statistical mechanics expressions for thermoelastic properties of monatomic crystals, in the classical canonical ensemble, using the mapped-averaging framework. Two reference models are examined as a basis for the coordinate mapping: ideal gas (IG) and quasiharmonic (QH) crystal. The former prescribes affine mapping and yields the conventional expressions for the properties, while the latter has, in addition, a nonaffine component and yields alternative formulas, denoted as ``harmonically mapped averaging'' (HMA). While the (strain-dependent) affine mapping is universal and does not require system-specific information, the HMA method uses a system- (and strain-) dependent matrix to specify the mapping; this mapping matrix is derived from the force-constant matrix of the QH reference and its variation with strains. In practice, implementing HMA (and affine) schemes depends on the availability of analytical (Lagrangian) derivatives of potential energy with respect to strains (e.g., Born term) and coordinates (e.g., Hessian matrix); however, when these derivatives are unavailable, numerical derivatives can be computed via finite difference by moving atoms according the mapping matrices, while straining the crystal. In no circumstances does mapped averaging affect sampling of configurations; it involves only the calculation of averages. Results show that HMA provides CPU speedup ranging from a factor of 4, to over two orders of magnitude, compared to the conventional formulas (especially at low temperature), with the best performance obtained with isotropic-deformation properties. In addition to its application to molecular simulation, HMA can open avenues for developing new theories for thermoelastic properties.

Computing the surface electronic states on the (100), (110) and (111) surfaces of FCC monatomic crystals

In this study, we carry out a simulation of the surface band structures for face-centered cubic (fcc) leads that end up in (100), (110) and (111) surfaces. The surface Hamiltonian matrix is constructed from tight-binding approach and the secular equations of the surface eigenvalue problem. The solution of the problem is performed by integrating the Landauer–Büttiker formalism (LBF) in the phase field matching approach (PFMA). The LBF provides the quantum scattering properties and the PFMA connects the bulk modes to those of the surface based on the quantum scattering coefficients. The combination of these methods allows calculating the electronic bands in the three directions mentioned above. We report the results of ordered slabs for Ag, described as [Formula: see text]-like orbital and Ni given as [Formula: see text]-type orbitals. To show the impact of expanding the crystal wavefunction, we reveal the calculation of the localized states for Rh, Cu, Pt given as [Formula: see text]-type orbitals as first calculation then [Formula: see text]-coupling orbitals as second calculation. The results of the nonordered slabs are applied to Pd and Ir. Cutting the crystals affects the internal energy of the surface atoms, which will be subject to a relaxation effect until equilibrium is achieved.

Effective real-space correlations of crystal lattice vibrations

Effective phonon fields coupled to a temperature-dependent permeability tensor are introduced to model empirical thermodynamic functions of crystal lattices and the temperature variation of Debye-Waller factors inferred by X-ray, γ-ray or neutron diffraction. The permeabilities generate a varying Debye temperature and a temperature-dependent spectral cutoff in the partition function as well as an effective temperature-dependent oscillator mass, to be calculated from diffraction and heat capacity data. The zero-point internal energy of the phonon field is extracted from low-temperature Debye-Waller B-factor measurements. The varying spectral cutoff, Debye temperature and oscillator mass determine the temperature evolution of real-space correlation functions. Closed integral representations are derived for the effective two-point function correlating isotropic lattice vibrations in monatomic cubic crystals as well as for the reduced four-point function correlating fluctuations around the mean-squared atomic displacement. The correlations are long-range with power-law tails and become oscillatory at low temperature. The formalism is illustrated with the correlation functions of copper.

Systematic first-principles study of the on-site spin-orbit coupling in crystals

Spin-orbit coupling (SOC) provides a lot of interesting phenomena in condensed-matter physics, and the strength of the SOC is one of the key factors to control such physical properties. In this paper, we present a systematic first-principles study of the strength of SOC in crystals by using the on-site SOC, namely, the atomic SOC defined by Wannier functions of the corresponding atomic orbitals. By calculating the on-site SOC in isolated atoms, monatomic crystals, and binary compounds, we find that the valence dependence of the atomic SOC in isolated atoms cannot explain the strength of the on-site SOC in crystals. Instead, it is shown that the spread of the Wannier function well describes the material dependence and pressure dependence of the on-site SOC in crystals. The material dependence is found to be substantially larger than the valence dependence for isolated atoms. Particularly, for Bi, the strength of the on-site SOC for the Bi $6p$ orbitals changes from 1.0 eV (${\mathrm{Cs}}_{3}\mathrm{Bi}$) to 2.5 eV (${\mathrm{BiF}}_{3}$).

Anisotropic Fracture Dynamics Due to Local Lattice Distortions.

A brittle material under loading fails by the nucleation and propagation of a sharp crack. In monatomic crystals, such as silicon, the lattice geometries front to the crack-tip changes the way of propagation even with the same cleavage surface. In general, however, crystals have multiple kinds of atoms and how the deformation of each atom affects the failure is still elusive. Here, we show that local atomic distortions from the different types of atoms causes a propagation anisotropy in suspended WS2 monolayers by combining annular dark-field scanning transmission electron microscopy and empirical molecular dynamics that are validated by first-principles calculations. Conventional conditions for brittle failure such as surface energy, elasticity, and crack geometry cannot account for this anisotropy. Further simulations predict the enhancement of the strengths and fracture toughness of the materials by designing void shapes and edge structures.

Experimental evidence for thermal generation of interstitials in a metallic crystal near the melting temperature

The only intrinsic point defects of simple crystalline metals known from solid state physics are vacancies and interstitials. It is widely believed that while vacancies play a major role in crystal properties and their concentration reaches relatively big values near the melting temperature Tm, interstitials essentially do not occur in thermodynamic equilibrium and their influence on properties is minor. Here, taking aluminum single crystals as an example, we present compelling experimental evidence for rapid thermoactivated growth of interstitial concentration upon approaching Tm.Using high precision measurements of the shear modulus we found a diaelastic effect of up to near Tm. It is argued that this effect is mostly due to the generation of dumbbell (split) interstitials. The interstitial concentration ci rapidly increases upon approaching Tm and becomes only 2–3 times smaller than that of vacancies just below Tm. The reason for this ci-increase is conditioned by a decrease of the Gibbs free energy with temperature, which in turn originates from the high formation entropy of dumbbell interstitials and a decrease of their formation enthalpy at high ci. Special molecular dynamic simulation confirmed all basic aspects of the proposed interpretation.The results obtained (i) demonstrate the significance of interstitial concentration near Tm that could lead to the revaluation of vacancy concentration at high temperatures, (ii) suggest that dumbbell interstitials play a major role in the melting mechanism of monatomic metallic crystals and (iii) support a new avenue for in-depth understanding of glassy metals.

Electron Doping of Ultrathin Black Phosphorus with Cu Adatoms.

Few-layer black phosphorus is a monatomic two-dimensional crystal with a direct band gap that has high carrier mobility for both holes and electrons. Similarly to other layered atomic crystals, like graphene or layered transition metal dichalcogenides, the transport behavior of few-layer black phosphorus is sensitive to surface impurities, adsorbates, and adatoms. Here we study the effect of Cu adatoms onto few-layer black phosphorus by characterizing few-layer black phosphorus field effect devices and by performing first-principles calculations. We find that the addition of Cu adatoms can be used to controllably n-dope few layer black phosphorus, thereby lowering the threshold voltage for n-type conduction without degrading the transport properties. We demonstrate a scalable 2D material-based complementary inverter which utilizes a boron nitride gate dielectric, a graphite gate, and a single bP crystal for both the p- and n-channels. The inverter operates at matched input and output voltages, exhibits a gain of 46, and does not require different contact metals or local electrostatic gating.

On the intrinsic ripples and negative thermal expansion of graphene

The intrinsic ripples of suspended graphene have attracted intensive attention due to their influence on the electronic transport and other properties. Negative thermal expansion (NTE), another unconventional phenomenon found in graphene, can be utilized to control the intrinsic ripples in a reversible way, thus opening new perspective for application. In this case, understanding the mutual relation and physical origin of the intrinsic ripples and NTE is crucial, especially since they are both widely observed in other 2D materials. Here we clarify through lattice dynamical analysis that at low temperature the two phenomena are both intrinsic for any 2D crystals with a honeycomb structure (or any monatomic 2D crystals). We find that the intrinsic ripples, generally believed to be caused by thermal fluctuation, have another origin that is the appearance of soft ZA modes near long wavelength limit when the lattice constant is shortened. Moreover, the soft ZA modes and NTE have the same physical origin at low temperature. At finite temperature, NTE is dominantly caused by the “vibrational elongation” effect owing to large out-of-plane fluctuation according to our calculation based on self-consistent phonon theory.

From atomistics to continuum: Effects of a free surface and determination of surface elasticity properties

We present an analysis of surface elasticity from the Born–Oppenheimer approximation for monatomic crystals. The analysis shows that the relaxations of crystal planes parallel to a free surface can be sufficiently determined by a low-rank algebraic Riccati equation instead of a full-scale molecular dynamic (MD) simulation, and gives new restrictions on physically reasonable atomistic models and simple criteria for surface reconstructions. In the case of surface relaxations, we calculate surface elasticity properties, i.e., surface tension, residual surface stress and surface elastic stiffness tensor, from atomistic models which are compared with experimental data and prior simulation results. The formulation also proves that surface relaxations always lower surface tension and surface elastic stiffness tensor. Together with the proposed algorithm, the formulation may be useful for investigating a variety of size-dependent phenomena of nano-structures.

Isomorph invariance of the structure and dynamics of classical crystals

This paper shows by computer simulations that some crystalline systems have curves in their thermodynamic phase diagrams, so-called isomorphs, along which structure and dynamics in reduced units are invariant to a good approximation. The crystals are studied in a classical-mechanical framework, which is generally a good description except significantly below melting. The existence of isomorphs for crystals is validated by simulations of particles interacting via the Lennard-Jones pair potential arranged into a face-centered cubic (FCC) crystalline structure; the slow vacancy-jump dynamics of a defective FCC crystal is also shown to be isomorph invariant. In contrast, a NaCl crystal model does not exhibit isomorph invariances. Other systems simulated, though in less detail, are the Wahnstrom binary Lennard-Jones crystal with the ${\rm MgZn_2}$ Laves crystal structure, monatomic FCC crystals of particles interacting via the Buckingham pair potential and via a novel purely repulsive pair potential diverging at a finite separation, an ortho-terphenyl molecular model, and SPC/E hexagonal ice. Except for NaCl and ice, the crystals simulated all have isomorphs. Based on these findings and previous simulations of liquid models, we conjecture that crystalline solids with isomorphs include most or all formed by atoms or molecules interacting via metallic or van der Waals forces, whereas covalently- or hydrogen-bonded crystals are not expected to have isomorphs. Crystals of ions or dipolar molecules constitute a limiting case for which isomorphs are only expected when the Coulomb interactions are relatively weak. We briefly discuss the consequences of the findings for theories of melting and crystallization.

Rotational defects in cyclotrimethylene trinitramine (RDX) crystals

Cyclotrimethylene trinitramine (RDX) crystalizes in the orthorhombic α-phase at the ambient pressure and temperature. In principle, the point defects commonly found in monatomic crystals, such as vacancies and interstitials, may exist in RDX as well. However, in molecular crystals one encounters additional point defects associated with the distortion of the molecules. A set of rotational defects are described in this article. These are molecules which are located in the proper positions in the crystal but are rotated relative to the molecules in the perfect crystal, and their ring is slightly puckered. The energetic barriers for defect formation and for their annealing back to the perfect crystal configuration are computed using an atomistic model. It is shown that the formation energy of rotational defects is smaller than the vacancy formation energy. Such defects are identified in the cores of dislocations in RDX and hence their concentration in the crystal is expected to increase during plastic deformation. The importance of such point defects is related to their role in phonon scattering and in dislocation-mediated plastic deformation.

Isotope effects in Raman spectra of crystalline sulfur α-S8

The Raman spectra of 32S, 33S, and 34S powders, as well as naturS powder with the natural isotopic composition, have been measured in the temperature range of 5–300 K. The isotope mass dependences of the vibrational frequencies have been obtained. It has been shown that such a dependence in the region of mixed translational-librational modes differs from the dependences known for monatomic inorganic crystals. The role of zero-point vibrations for low- and high-frequency modes has been revealed throughout the entire temperature range.

Slip asymmetry in the molecular crystal cyclotrimethylenetrinitramine

Slip asymmetry is a common occurrence in some monatomic crystals where it is due to complex core structures or specific packing of slip planes. Here we present another mechanism, based on molecular steric hindrance, which leads to asymmetric dislocation motion in cyclotrimethylenetrinitramine (RDX) molecular crystal. Dislocations move at different critical stresses when shear is applied in the positive and negative directions of the Burgers vector in the slip system that contributes most to plastic deformation.

Melting of Mesoscale Lennard-Jones Crystals with Free Surfaces

It is widely considered that crystalline solids melt heterogeneously, i.e., near the equilibrium melting point (\(T_{\text{m}}\)) a liquid phase appears at the surface and gradually penetrates into the interior. However, via molecular dynamics (MD) simulations we find a quite new scenario of melting of mesoscale monatomic Lennard-Jones crystals with free surfaces. In the early stage of pre-melting, a quasi-liquid surface layer occurs which contains both liquidlike and solidlike atoms. Further heating leads to the homogeneous occurrence of liquidlike atoms in the interior. Liquidlike configuration grows fast with temperature and a pure liquid surface layer appears just in the next stage of pre-melting together with a homogeneous occurrence/growth of liquidlike atoms throughout the interior. Melting proceeds further by two different mechanisms: the heterogeneous one in the surface shell and the homogeneous one in the interior leading to the fast collapse of crystal lattice. Our findings are supported by the...

Soliton microdynamics of heat conduction in plutonium and uranium in the temperature range of martensitic phase transitions

The microdynamics of large-amplitude nonlinear lattice vibrations of plutonium and uranium has been investigated at high reactor temperatures in the ranges of martensitic phase transitions. Solutions of nonlinear dynamic equations have been obtained using the Lennard-Jones interatomic potential for the soliton generation and the energy transfer by solitons between the crystal boundaries in the shells. The synchronism of the soliton trajectories and peaks of energy fluxes demonstrates an analog of the shot effect. The temperature dependences of thermal conductivity coefficients are consistent with the experiment and exhibit local maxima in the ranges of phase transitions. A spectral analysis has revealed that the dominant heat transfer is provided by rarefaction solitons. The martensitic phase transitions are accompanied by a reorganization of the spectral density in the phase plane with a sharp increase in the high-frequency range. The spectral density has maxima of the quasi-biphonon type. The obtained data in dimensionless form, apart from plutonium and uranium, can be used for other monatomic crystals. The specific features of the thermal conductivity and microdynamics of the formation of vacancies and pores in crystals without shells have been discussed.