vdW Coefficients Research Articles

Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis

Abstract In this paper, nonlinear vibration and frequency response analysis of double walled piezoelectric nanoshell (DWPENS) is investigated using nonclassical approach of the Gurtin–Murdoch surface/interface (GMSIT) theory. The piezoelectric nanoshell is simultaneously subjected to visco-Pasternak medium, the nonlinear van der Waals and electrostatic forces. Hamilton’s principles, the assumed mode method combined with Lagrange–Euler’s are used for the governing equations and boundary conditions. Complex averaging method combined with Arc-length continuation is used to achieve the nonlinear frequency response and stability analysis of the DWPENS. It is found that the electrostatic and piezoelectric voltages, the length to radius ratio, the nanoshell gap width, van der Waals (vdW) coefficients and other parameters can effectively change the flexural rigidity of the system which in turn affects the nonlinear frequency response. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system’s instability, nonlinear behavior, and bandwidth.

Mechanical Buckling of Circular Orthotropic Bilayer Nanoplate Embedded in an Elastic Matrix under Radial Compressive Loading

This article investigates the buckling behavior of orthotropic annular/circular bilayer graphene sheet embedded in Winkler–Pasternak elastic medium under mechanical loading. Using the nonlocal elasticity theory, the bilayer graphene sheet is modeled as a nonlocal orthotropic plate which contains small scale effect and van der Waals interaction forces. Differential Quadrature Method (DQM) is employed to solve the governing equations for various combinations of simply supported or clamped boundary conditions. The results show that small scale parameter does not have any effect on critical buckling load of cases without elastic medium in simply supported boundary condition. Also, increase of vdW coefficient leads to increase of critical buckling load smoothly then it has no impact on critical buckling load after a certain value.

Thermal vibration of MoS2/Black phosphorus Bi-layered heterostructure

Thermal vibration is extremely crucial to nanostructure-based nanoresonators. In this paper, a laminated plate model (LPM) with the van der Waals (vdW) interactions between MoS2 and black phosphorus (BP) taken into consideration is proposed to explore the thermally induced vibration of a MoS2/BP heterostructure. The vdW coefficient between single-layered MoS2 and single-layered BP is calculated. The natural frequencies and the root-mean-squared (RMS) amplitudes of the MoS2/BP heterostructure are obtained from molecular dynamics (MD) simulations and the LPM. The natural frequencies and the RMS amplitudes of the MoS2/BP heterostructure calculated by the LPM and those obtained from the MD simulations coincide well. The LPM has in-phase vibrational modes and anti-phase vibrational modes. The natural frequencies of the in-phase vibrational modes are much lower than the frequencies of the anti-phase vibrational modes of the same order. In the thermal vibration of the MoS2/BP heterostructure, the RMS amplitudes of MoS2 and BP are obviously different. Compared with the natural frequencies and RMS amplitudes of the MoS2/BP heterostructure obtained from the MD simulations, the LPM can provide an accurate prediction of the thermal vibration of the MoS2/BP heterostructure.

Origin of the size-dependence of the equilibrium van der Waals binding between nanostructures.

Nanostructures can be bound together at equilibrium by the van der Waals (vdW) effect, a small but ubiquitous many-body attraction that presents challenges to density functional theory. How does the binding energy depend upon the size or number of atoms in one of a pair of identical nanostructures? To answer this question, we treat each nanostructure as a whole object, not as a collection of atoms. Our calculations start from an accurate static dipole polarizability for each considered nanostructure, and an accurate equilibrium center-to-center distance for the pair (the latter from experiment or from the vdW-DF-cx functional). We consider the competition in each term -C2k/d2k (k = 3, 4, 5) of the long-range vdW series for the interaction energy, between the size dependence of the vdW coefficient C2k and that of the 2kth power of the center-to-center distance d. The damping of these vdW terms can be negligible, but in any case, it does not affect the size dependence for a given term in the absence of non-vdW binding. To our surprise, the vdW energy can be size-independent for quasi-spherical nanoclusters bound to one another by vdW interaction, even with strong nonadditivity of the vdW coefficient, as demonstrated for fullerenes. We also show that, for low-dimensional systems, the vdW interaction yields the strongest size-dependence, in stark contrast to that of fullerenes. We illustrate this with parallel planar polycyclic aromatic hydrocarbons. The size dependences of other morphologies or bonding types lie between, as shown by sodium clusters.

Frequency-dependent dielectric function of semiconductors with application to physisorption

The dielectric function is one of the most important quantities that describes the electrical and optical properties of solids. Accurate modeling of the frequency-dependent dielectric function has great significance in the study of the long-range van der Waals (vdW) interaction for solids and adsorption. In this work, we calculate the frequency-dependent dielectric functions of semiconductors and insulators using the $GW$ method with and without exciton effects, as well as efficient semilocal density functional theory (DFT), and compare these calculations with a model frequency-dependent dielectric function. We find that for semiconductors with moderate band gaps, the model dielectric functions, $GW$ values, and DFT calculations all agree well with each other. However, for insulators with strong exciton effects, the model dielectric functions have a better agreement with accurate $GW$ values than the DFT calculations, particularly in high-frequency region. To understand this, we repeat the DFT calculations with scissors correction, by shifting DFT Kohn-Sham energy gap to match the experimental band gap. We find that scissors correction only moderately improves the DFT dielectric function in low-frequency region. Based on the dielectric functions calculated with different methods, we make a comparative study by applying these dielectric functions to calculate the vdW coefficients ($C_3$ and $C_5$) for adsorption of rare-gas atoms on a variety of surfaces. We find that the vdW coefficients obtained with the nearly-free electron gas-based model dielectric function agree quite well with those obtained from the $GW$ dielectric function, in particular for adsorption on semiconductors, leading to an overall error of less than 7% for $C_3$ and 5% for $C_5$. This demonstrates the reliability of the model dielectric function for the study of physisorption.

High-Pressure Structural Phase Transition in Mg<sub>x</sub>Cd<sub>1-x</sub>O Compounds

The high-pressure technique is useful to understand physical properties because the technique can directly control bond length and phase transition. As a general trend, the pressure-induced phase transition causes an increase of coordination number with a drastic change of their physical properties. Here, we attempt to explore the pressure-induced phase transitions from the sixfold-coordinated NaCl structure (B1) to the eightfold-coordinated CsCl structure (B2) in MgxCd1−xO by applying an effective interionic interaction potential, which includes the long range Coulomb, van der Waals (vdW) interaction and the short-range repulsive interaction upto second-neighbor ions within the Hafemeister and Flygare approach. Assuming that both the ions are polarizable, the Slater-Kirkwood variational method is employed to estimate the vdW coefficients for parent compounds. The estimated values of the phase transition pressure (Pt) increase with Mg concentration. The vast volume discontinuity in pressure volume phase diagram identifies the structural phase transition from B1 to B2 structure. The results obtain from the present calculations requires the complete understanding of many physical interactions that are essential to ternary oxides, containing elements with size and chemical mismatch, will lead to a consistent explanation of the documented structural properties.

Accurate van der Waals coefficients between fullerenes and fullerene-alkali atoms and clusters: Modified single-frequency approximation

Long-range van der Waals (vdW) interaction is critically important for intermolecular interactions in molecular complexes and solids. However, accurate modeling of vdW coefficients presents a great challenge for nanostructures, in particular for fullerene clusters, which have huge vdW coefficients but also display very strong nonadditivity. In this work, we calculate the coefficients between fullerenes, fullerene and sodium clusters, and fullerene and alkali atoms with the hollow-sphere model within the modified single-frequency approximation (MSFA). In the MSFA, we assume that the electron density is uniform in a molecule and that only valence electrons in the outmost subshell of atoms contribute. The input to the model is the static multipole polarizability, which provides a sharp cutoff for the plasmon contribution outside the effective vdW radius. We find that the model can generate ${C}_{6}$ in excellent agreement with expensive wave-function-based ab initio calculations, with a mean absolute relative error of only $3%$, without suffering size-dependent error. We show that the nonadditivities of the coefficients ${C}_{6}$ between fullerenes and ${\mathrm{C}}_{60}$ and sodium clusters ${\mathrm{Na}}_{n}$ revealed by the model agree remarkably well with those based on the accurate reference values. The great flexibility, simplicity, and high accuracy make the model particularly suitable for the study of the nonadditivity of vdW coefficients between nanostructures, advancing the development of better vdW corrections to conventional density functional theory.

Strain Gradient Finite Element Analysis on the Vibration of Double-Layered Graphene Sheets

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

Communication: Accurate higher-order van der Waals coefficients between molecules from a model dynamic multipole polarizability.

Due to the absence of the long-range van der Waals (vdW) interaction, conventional density functional theory (DFT) often fails in the description of molecular complexes and solids. In recent years, considerable progress has been made in the development of the vdW correction. However, the vdW correction based on the leading-order coefficient C6 alone can only achieve limited accuracy, while accurate modeling of higher-order coefficients remains a formidable task, due to the strong non-additivity effect. Here, we apply a model dynamic multipole polarizability within a modified single-frequency approximation to calculate C8 and C10 between small molecules. We find that the higher-order vdW coefficients from this model can achieve remarkable accuracy, with mean absolute relative deviations of 5% for C8 and 7% for C10. Inclusion of accurate higher-order contributions in the vdW correction will effectively enhance the predictive power of DFT in condensed matter physics and quantum chemistry.

Van der Waals coefficients beyond the classical shell model.

Van der Waals (vdW) coefficients can be accurately generated and understood by modelling the dynamic multipole polarizability of each interacting object. Accurate static polarizabilities are the key to accurate dynamic polarizabilities and vdW coefficients. In this work, we present and study in detail a hollow-sphere model for the dynamic multipole polarizability proposed recently by two of the present authors (JT and JPP) to simulate the vdW coefficients for inhomogeneous systems that allow for a cavity. The inputs to this model are the accurate static multipole polarizabilities and the electron density. A simplification of the full hollow-sphere model, the single-frequency approximation (SFA), circumvents the need for a detailed electron density and for a double numerical integration over space. We find that the hollow-sphere model in SFA is not only accurate for nanoclusters and cage molecules (e.g., fullerenes) but also yields vdW coefficients among atoms, fullerenes, and small clusters in good agreement with expensive time-dependent density functional calculations. However, the classical shell model (CSM), which inputs the static dipole polarizabilities and estimates the static higher-order multipole polarizabilities therefrom, is accurate for the higher-order vdW coefficients only when the interacting objects are large. For the lowest-order vdW coefficient C6, SFA and CSM are exactly the same. The higher-order (C8 and C10) terms of the vdW expansion can be almost as important as the C6 term in molecular crystals. Application to a variety of clusters shows that there is strong non-additivity of the long-range vdW interactions between nanoclusters.

Communication: Non-additivity of van der Waals interactions between nanostructures.

Due to size-dependent non-additivity, the van der Waals interaction (vdW) between nanostructures remains elusive. Here we first develop a model dynamic multipole polarizability for an inhomogeneous system that allows for a cavity. The model recovers the exact zero- and high-frequency limits and respects the paradigms of condensed matter physics (slowly varying density) and quantum chemistry (one- and two-electron densities). We find that the model can generate accurate vdW coefficients for both spherical and non-spherical clusters, with an overall mean absolute relative error of 4%, without any fitting. Based on this model, we study the non-additivity of vdW interactions. We find that there is strong non-additivity of vdW interactions between nanostructures, arising from electron delocalization, inequivalent contributions of atoms, and non-additive many-body interactions. Furthermore, we find that the non-additivity can have increasing size dependence as well as decreasing size dependence with cluster size.

Physical Adsorption: Theory of van der Waals Interactions between Particles and Clean Surfaces

van der Waals (vdW) interactions between particles and surfaces are critical for the study of physical adsorption. In this work, we develop a method to calculate the leading- and higher-order coefficients, describing the dependence of vdW interaction on height above the surface. We find that the proposed method can produce the vdW coefficients for atoms on surfaces of metals and semiconductors, with a mean absolute relative deviation of about 5%. As an important application, we study the adsorption energies for rare-gas atoms on noble-metal surfaces by combining the present method, which accounts for the long-range part, with semilocal density functional theory (DFT), which accounts for the short-range part. This combined DFT+vdW approach yields adsorption energies in excellent agreement (5%) with experiments. This suggests that the present method may serve as a useful dispersion correction to density functional approximations.

High-pressure phase transformation and elastic behavior of XC (X = Si, Ge, Sn and Pt) compounds

Pressure-induced structural aspects of ZnS-type (B3) to NaCl-type (B1) structure in XC (X = Si, Ge, Sn and Pt) compounds are presented. An effective interionic interaction potential with long-range Coulomb, van der Waals (vdW) interaction and the short-range repulsive interaction up to second-neighbor ions within the Hafemeister and Flygare approach with modified ionic charge is developed. Particular attention is devoted to evaluate the vdW coefficients following the Slater–Kirkwood variational method, as both the ions are polarizable. Our results on vast volume discontinuity in pressure volume phase diagram identify the structural phase transition from B3 to B1 structure. The estimated value of the phase transition pressure (Pt) and the magnitude of the discontinuity in volume at the transition pressure are consistent as compared to the reported data. The variations of elastic constants with pressure follow a systematic trend identical to that observed in other compounds of ZnS and NaCl-type structure family and the Born relative stability criteria are valid in XC compounds.

Large amplitude vibration of a bilayer graphene embedded in a nonlinear polymer matrix

In the present article, nonlinear free and forced vibration of a bilayer graphene embedded in a polymer medium is studied based on the nonlocal elasticity theory. As a nonlinear function of deflection of graphene sheets, a refined pressure expression is established to describe the van der Waals (vdW) interactions between graphene layers and polymer medium. Assuming the large displacements and anisotropic model for graphene layers, the nonlinear couple partial differential equations of a double layered graphene sheet (DLGS) are obtained. The in-phase and out of phase nonlinear to linear natural frequencies are shown for both zigzag and armchair geometries. The effects of small scale parameter, nonlinear coefficients of vdW between two layers, nonlinear factor of polymer matrix and geometric properties on the nonlinear vibrational behavior of a DLGS are discussed in detail. It is found that the nonlinear vdW coefficient has a significant effect on the out of phase frequencies while the nonlinear polymer coefficient has considerable influence on in-phase frequencies.

Pressure induced structural phase transition of XC (X = Si, Ge, Sn)

An effective interionic interaction potential (EIOP) is developed to investigate the pressure induced phase transitions from ZnS-type (B3) to NaCl-type (B1) in XC [X = Si, Ge, and Sn] compounds. The long range Coulomb, van der Waals (vdW) interaction and the short-range repulsive interaction up to second-neighbor ions within the Hafemeister and Flygare approach with modified ionic charge are properly incorporated in the EIOP. The vdW coefficients are computed following the Slater-Kirkwood variational method, as both the ions are polarizable. The estimated value of the phase transition pressure (Pt) and the magnitude of the discontinuity in volume at the transition pressure are consistent as compared to the reported data. The vast volume discontinuity in pressure volume phase diagram identifies the structural phase transition from B3 to B1 structure.

Accurate van der Waals coefficients from density functional theory

The van der Waals interaction is a weak, long-range correlation, arising from quantum electronic charge fluctuations. This interaction affects many properties of materials. A simple and yet accurate estimate of this effect will facilitate computer simulation of complex molecular materials and drug design. Here we develop a fast approach for accurate evaluation of dynamic multipole polarizabilities and van der Waals (vdW) coefficients of all orders from the electron density and static multipole polarizabilities of each atom or other spherical object, without empirical fitting. Our dynamic polarizabilities (dipole, quadrupole, octupole, etc.) are exact in the zero- and high-frequency limits, and exact at all frequencies for a metallic sphere of uniform density. Our theory predicts dynamic multipole polarizabilities in excellent agreement with more expensive many-body methods, and yields therefrom vdW coefficientsC6,C8,C10for atom pairs with a mean absolute relative error of only 3%.

The van der Waals coefficients between carbon nanostructures and small molecules: A time-dependent density functional theory study

We employ all-electron ab initio time-dependent density functional theory based method to calculate the long-range dipole-dipole dispersion coefficient, namely, the van der Waals (vdW) coefficient (C(6)) between fullerenes and finite-length carbon nanotubes as well as between these structures and different small molecules. Our aim is to accurately estimate the strength of the long-range vdW interaction in terms of the C(6) coefficients between these systems and also compare these values as a function of shape and size. The dispersion coefficients are obtained via Casimir-Polder relation. The calculations are carried out with the asymptotically correct exchange-correlation potential-the statistical average of orbital potential. It is observed from our calculations that the C(6) coefficients of the carbon nanotubes increase nonlinearly with length, which implies a much stronger vdW interaction between the longer carbon nanostructures compared with the shorter ones. Additionally, it is found that the values of C(6) and polarizability are about 40%-50% lower for the carbon cages when compared with the results corresponding to the quasi-one-dimensional nanotubes with equivalent number of atoms. From our calculations of the vdW coefficients between the small molecules and the carbon nanostructures, it is observed that for H(2), the C(6) value is much larger compared with that of He. It is found that the rare gas atoms have very low values of vdW coefficient with the carbon nanostructures. In contrast, it is found that other gas molecules, including the ones that are environmentally important, possess much higher C(6) values. Carbon tetrachloride as well as chlorine molecule show very high C(6) values with themselves as well as with the carbon nanostructures. This is due to the presence of the weakly bound seven electrons in the valence state for the halogen atoms, which makes these compounds much more polarizable compared with the others.

High-pressure induced structural phase transition in alkaline earth CaX (X = S, Se and Te) semiconductors: NaCl-type ( B1) to CsCl-type ( B2)

The present paper addresses the pressure-induced structural aspects of NaCl-type ( B1) to CsCl-type ( B2) structure in alkaline earth chalcogenides (AECs) calcium chalcogenides (CaX; X = S, Se and Te) by formulating an effective interionic interaction potential (EIoIP) with long-range Coulomb interactions and the Hafemeister and Flygare type short-range overlap repulsion extended upto the second neighbor ions and the van der Waals (vdW) interaction. The vdW coefficients are evaluated following the Slater-Kirkwood variational method, as both the ions are polarizable. The present calculations have revealed reasonably good agreement with the available experimental data on structural transition ( B1 to B2 structure), the phase transition pressures ( P t ) of 40 (CaS), 38 (CaSe) and 34 (CaTe) GPa as well the elastic properties. The calculated values of the volume collapses [Δ V( P)/ V(0)] are also closer to their observed data. Further, the variations of the second and third order elastic constants with pressure have followed a systematic trend, which are almost identical to those exhibited by the observed data measured for other semiconducting compounds with rocksalt ( B1) type crystal structure. The Born and relative stability criteria is valid in Ca monochalcogenides.

Structural phase transition (zincblende–rocksalt) and elastic properties in AlY (Y=N, P and As) compounds: Pressure-induced effects

The present paper addresses the pressure-induced structural aspects of ZnS-type (B3) to NaCl-type (B1) structure in AlY (Y=N, P, As). An effective-interionic interaction potential (EIoIP) with long-range Coulomb and three-body interactions and the Hafemeister-and-Flygare-type short-range overlap repulsion extended up to the second-neighbour ions and the van der Waals (vdW) interaction is developed. Emphasis has been given on evaluating the vdW coefficients by the Slater–Kirkwood variational method, as both the ions are polarizable. The lattice model calculations have revealed reasonably good agreement with the available experimental data on the phase-transition pressures ( P t =16, 14, 7.5 GPa) and the elastic properties of AlY (Y=N, P, As). The equation of state curves (plotted between V( P)/ V(0) and pressure) for both the B3 and B1 structures obtained are in fairly good agreement with the experimental results. The calculated values of the volume collapses [Δ V( P)/ V(0)] are also close to their observed data. Further, the variations of the second-order elastic constants with pressure follow a systematic trend that is almost identical to that exhibited by the observed data measured for other semiconducting compounds with B3→B1 structural phase transitions.

High-pressure structural phase transition and elastic properties of yttrium pnictides

Pressure-induced structural aspects of NaCl-type (B1) to CsCl-type (B2) structures in yttrium pnictides (YX; X = N, P and As) are presented. An effective interionic interaction potential with the long-range Coulomb and van der Waals (vdW) interaction and the short-range repulsive interaction up to second-neighbor ions within the Hafemeister and Flygare approach with a modified ionic charge is developed. Particular attention is devoted to evaluate the vdW coefficients following the Slater–Kirkwood variational method, as both the ions are polarizable. Our results on vast volume discontinuity in the pressure–volume phase diagram identify the structural phase transition from B1 to B2 structure. The estimated value of the phase-transition pressure (P t) and the magnitude of the discontinuity in volume at the transition pressure are consistent when compared with the reported data. The variations of elastic constants and their combinations with pressure follow a systematic trend identical to that observed in other compounds of the NaCl-type structure family, and the Born relative stability criteria are valid in yttrium pnictides.