Lennard-Jones Potential Function Research Articles

Vibration analysis of two orthogonal slender single-walled carbon nanotubes with a new insight into continuum-based modeling of van der Waals forces

Transverse vibrations of doubly orthogonal slender single-walled carbon nanotubes (SWCNTs) at the vicinity of each other are of interest. The van der Waals (vdW) forces play an important role in dynamic interactions between two adjacent nanotubes. Using Lennard-Jones potential function, such a phenomenon is appropriately modeled by a newly introduced vdW force density function. By employing Hamilton’s principle, the equations of motion are obtained based on the nonlocal Rayleigh beam theory. In fact, these are integro-partial differential equations and seeking an exact or even analytical solution to them is a very difficult job. Therefore, an efficient numerical solution is proposed. The effects of the intertube distance, slenderness ratio, small-scale parameter, aspect ratio, and elastic properties of the surrounding medium on the free vibration of the nanosystem are addressed. The obtained results could be regarded as a pivotal step for better realizing of dynamic behaviors of more complex systems consist of multiple orthogonal networks of nanotubes.

Monte Carlo simulation study of the effect of chain tacticity on demixing of polyethylene/polypropylene blends

The molecular origin of the demixing behavior for 50: 50 (wt/wt) polyethylene/polypropylene (PE/PP) with different tacticity of PP at the melts (473 K) was investigated by Monte Carlo simulation of coarse-grained polymer model. Isotactic (iPP), atactic (aPP) and syndiotactic (sPP) polypropylenes were used for blending with PE. Coarse-graining polymer chains were represented by 50 beads, corresponding to C100H202 and C150H302 for PE and PP, respectively. The simulation was performed on a high coordination lattice incorporating short-range intramolecular interactions from the Rotational Isomeric State (RIS) model and long-range intermolecular interactions Lennard-Jones (LJ) potential function of ethane and propane units. Chain dimensions, the characteristic ratio (C n ) and self-diffusion coefficient (D) of PE in the blends are sensitive to the stereochemistry of PP chains. Compared with neat PE melts, PE dimension was relatively unchanged in PE/iPP and PE/aPP blends but slightly decreased in PE/sPP blends. PP dimension was increased in PE/iPP and PE/aPP mixture but decreased in PE/sPP blend in comparison with neat PP melts. In addition, diffusion of PE and PP chains in PE/PP mixture was decreased and increased, respectively, compared to the pure melts. Interchain pair correlation functions were used to detect the immiscibility of the blends. The tendency of demixing of PE/aPP and PE/iPP blends were weaker than that of PE/sPP blend.

Molecular Dynamics Simulation on Transformation-Induced Plastic Deformation Using a Lennard-Jones Model

Molecular dynamics simulations were carried out to clarify the atomistic mechanism of transformation plasticity. As the first step for the purpose, a simple thin-film model consisting of 8640 atoms was prepared. Phase transformation was assumed to be expressed by switching the material parameters of Lennard-Jones potential function. As a preliminary calculation, phase transformation was forced to occur homogeneously in the whole region of the model, resulting in no extra strain except volumetric transformation dilatation. In that case, perfect single crystal structure was maintained in the new phase. Simulations were carried out under external load, but specific strain was not generated. On the contrary, when the transformation region was set partially in the model and the region was expanded with time, a large deformation was observed. In the middle process of the phase transformation, slip-like deformation behavior and the change in crystal orientation occurred, indicating that extra plastic strain was induced during phase transformation. The strain was observed even when external load is not applied, and hence it was concluded that not only external load but also local stress distribution may cause the transformation plasticity.

ON THE MECHANICS OF ELLIPSOIDAL FULLERENES INSIDE OPEN CARBON NANOCONES: A NOVEL NUMERICAL APPROACH

In this research, mechanics of concentric ellipsoidal fullerenes inside open carbon nanocones (CNCs) is investigated. To this end, using continuum approximation in conjunction with Lennard-Jones (LJ) potential function, quadruple-integral expressions associated with van der Waals (vdW) potential energy and interaction force are first derived. For determination of these expressions, it is assumed that the fullerene molecule enters the open CNC through the small end or wide end. Thereafter, an efficient approach based on the differential quadrature (DQ) method is proposed to numerically evaluate the obtained quadruple integrals. The proposed method takes advantage of computing multidimensional integrals efficiently with using appropriate number of grid points. By introducing DQ-based operational matrices of differentiation and integration, the quadruple-integral expressions are estimated over their domains. Moreover, new semianalytical expressions are introduced in terms of triple integrals to evaluate vdW interactions. The validity and accuracy of the introduced numerical method are proved by comparing the results obtained through this method with ones achieved via the semianalytical expressions. The ease of implementation and quick answer of the demonstrated numerical solution enable us to comprehensively examine the effects of different geometrical parameters such as small end radius wide end radius and vertex angle of nanocone on the distributions of vdW potential energy and interaction force. The results reveal that the ellipsoidal fullerene undergoes an asymmetrical motion along the axis of open CNC.

Dynamics of polypropylene chains in their binary blends of different stereochemical sequences studied by Monte Carlo simulations

The conformational and dynamic properties of polypropylene (PP) for both pure melts and blends with different chain tacticity were investigated by Monte Carlo simulation of isotactic (iPP), atactic (aPP) and syndiotactic (sPP) polypropylenes. The simulation of coarse-grained PP models was performed on a high coordination lattice incorporating short- and long-range intramolecular interactions from the rotational isomeric state (RIS) model and Lennard-Jones (LJ) potential function of propane pairs, respectively. The dynamics of chains in binary PP/PP mixture were investigated with the composition of C150H302 with different chain taciticity. The diffusion rates of PP with different stereochemistry are generally in the order as: iPP > aPP ≫ sPP. For PP/PP blends with 50:50 wt% binary mixtures, immiscibility was observed when sPP was introduced into the mixtures. The diffusion rate of iPP and aPP became slower after mixing, while sPP diffuses significantly faster in the binary mixtures. The mobility of PP chains depends on both intramolecular (molecular size and chain stiffness) and intermolecular (chain packing) interactions. The effect of intramolecular contribution is greater than that of intermolecular contribution for iPP and aPP chains in binary mixtures. For sPP chain, intermolecular interaction has greater influence on the dynamics than intramolecular contribution.

Nonlocal continuous models for forced vibration analysis of two- and three-dimensional ensembles of single-walled carbon nanotubes

Novel nonlocal discrete and continuous models are proposed for dynamic analysis of two- and three-dimensional ensembles of single-walled carbon nanotubes (SWCNTs). The generated extra van der Waals forces between adjacent SWCNTs due to their lateral motions are evaluated via Lennard-Jones potential function. Using a nonlocal Rayleigh beam model, the discrete and continuous models are developed for both two- and three-dimensional ensembles of SWCNTs acted upon by transverse dynamic loads. The capabilities of the proposed continuous models in capturing the vibration behavior of SWCNTs ensembles are then examined through various numerical simulations. A reasonably good agreement between the results of the continuous models and those of the discrete ones is also reported. The effects of the applied load frequency, intertube spaces, and small-scale parameter on the transverse dynamic responses of both two- and three-dimensional ensembles of SWCNTs are explained. The proposed continuous models would be very useful for dynamic analyses of large populated ensembles of SWCNTs whose discrete models suffer from both computational efforts and labor costs.

Energetic Frustrations in Protein Folding at Residue Resolution: A Homologous Simulation Study of Im9 Proteins

Energetic frustration is becoming an important topic for understanding the mechanisms of protein folding, which is a long-standing big biological problem usually investigated by the free energy landscape theory. Despite the significant advances in probing the effects of folding frustrations on the overall features of protein folding pathways and folding intermediates, detailed characterizations of folding frustrations at an atomic or residue level are still lacking. In addition, how and to what extent folding frustrations interact with protein topology in determining folding mechanisms remains unclear. In this paper, we tried to understand energetic frustrations in the context of protein topology structures or native-contact networks by comparing the energetic frustrations of five homologous Im9 alpha-helix proteins that share very similar topology structures but have a single hydrophilic-to-hydrophobic mutual mutation. The folding simulations were performed using a coarse-grained Gō-like model, while non-native hydrophobic interactions were introduced as energetic frustrations using a Lennard-Jones potential function. Energetic frustrations were then examined at residue level based on φ-value analyses of the transition state ensemble structures and mapped back to native-contact networks. Our calculations show that energetic frustrations have highly heterogeneous influences on the folding of the four helices of the examined structures depending on the local environment of the frustration centers. Also, the closer the introduced frustration is to the center of the native-contact network, the larger the changes in the protein folding. Our findings add a new dimension to the understanding of protein folding the topology determination in that energetic frustrations works closely with native-contact networks to affect the protein folding.

Prediction of methane PVT relations at high temperatures by a simplified virial equation of state

In order to meet the demand of describing the supercritical gas under high temperature and medium-high pressure conditions, such as in detonation circumstance, a simplified virial equation of state (EOS), named Han-Long (HL), is presented, which is based on Lennard-Jones potential function. One hundred and twelve sets of theoretical data for methane above 1000 K are calculated using HL EOS. We obtain that the volume average absolute deviation (AAD) is about 1% and the maximum error is 3.28%; this error is far lower than the calculation deviation of DMW (Duan-Moller-Weare) EOS and BS (Belonoshko-Saxena) EOS. The shockwave data of methane is also calculated by HL EOS and the AAD are less than 3%. Results show that HL EOS can well describe the thermodynamic state of CH4 at high temperatures.

Prediction of transport properties of CO2-N2 binary mixtures via the inversion of reduced-viscosity collision integrals

In the present study, the main purpose is to extract information about the effective intermolecular potential energy function for binary mixture of nitrogen and carbon dioxide by the usage of a direct inversion of the experimentally reduced viscosity and second virial coefficient data and then to reproduce the dilute gas transport properties from the inverted potential energy. The Lennard-Jones (12, 6) potential energy function has been employed as the initial potential model required by the inversion method. The MSV potential obtained in this way is in reasonable agreement with the independently known Co2-N2 potential energy function. Using the inverted pair potential energies, the Chapman-Enskog scheme is employed to calculate transport properties of CO2-N2 in a wide composition and temperature range. The close agreement between the predicted values and the literature results of transport properties demonstrate the predictive power of the inversion scheme.

Molecular dynamics simulation of the nano-ejection system with various interfacial wettabilities

Because of the significance of the interfacial phenomenon, this study examines the nanojet ejection systems with various interfacial wettabilities. A modified Lennard–Jones potential function was used to simulate the various solid–liquid interfacials from strongly hydrophilic to strongly hydrophobic. The properties distributions with various hydrophilic/hydrophobic interfaces are shown and discussed. The results show that decreased degree of hydrophilic interaction between solid–liquid molecules will promote the liquid injection through a nozzle. But the ejection systems are similar in the hydrophobic cases. Results that illustrate various features are presented to aid in the comprehension of the nanojet systems.

Macroscopic Expressions of Molecular Adiabatic Compressibility of Methyl and Ethyl Caprate under High Pressure and High Temperature

The molecular compressibility, which is a macroscopic quantity to reveal the microcompressibility by additivity of molecular constitutions, is considered as a fixed value for specific organic liquids. In this study, we introduced two calculated expressions of molecular adiabatic compressibility to demonstrate its pressure and temperature dependency. The first one was developed from Wada’s constant expression based on experimental data of density and sound velocity. Secondly, by introducing the 2D fitting expressions and their partial derivative of pressure and temperature, molecular compressibility dependency was analyzed further, and a 3D fitting expression was obtained from the calculated data of the first one. The third was derived with introducing the pressure and temperature correction factors based on analogy to Lennard-Jones potential function and energy equipartition theorem. In wide range of temperatures(293<T/K<393)and pressures(0.1<P/MPa<210), which represent the typical values used in dynamic injection process for diesel engines, the calculated results consistency of three formulas demonstrated their effectiveness with the maximum 0.5384% OARD; meanwhile, the dependency on pressure and temperature of molecular compressibility was certified.

Offset configurations for single- and double-strand DNA inside single-walled carbon nanotubes

Nanotechnology is a rapidly expanding research area, and it is believed that the unique properties of molecules at the nano-scale will prove to be of substantial benefit to mankind, especially so in medicine and electronics. Here we use applied mathematical modelling exploiting the basic principles of mechanics and the 6-12 Lennard-Jones potential function together with the continuum approximation, which assumes that intermolecular interactions can be approximated by average atomic surface densities. We consider the equilibrium offset positions for both single-strand and double-strand DNA molecules inside a single-walled carbon nanotube, and we predict offset positions with reference to the cross-section of the carbon nanotube. For the double-strand DNA, the potential energy is determined for the general case for any helical phase angle ϕ,but we also consider a special case when ϕ=π,which leads to a substantial simplification in the analytical expression for the energy. As might be expected, our results confirm that the global minimum energy positions for a single-strand DNA molecule and a double-strand DNA molecule will lie off axis and they become closer to the tube wall as the radius of the tube increases.

Mechanics of single-walled carbon nanotubes inside open single-walled carbon nanocones

This study investigates the mechanical characteristics of single-walled carbon nanotubes (CNTs) inside open single-walled carbon nanocones (CNCs). New semi-analytical expressions are presented to evaluate van der Waals (vdW) interactions between CNTs and open CNCs. Continuum approximation, along with the the Lennard-Jones (LJ) potential function, is used in this study. The effects of geometrical parameters on alterations in vdW potential energy and the interaction force are extensively examined for the concentric CNT-open CNC configuration. The CNT is assumed to enter the nanocone either through the small end or the wide end of the cone. The preferred position of the CNT with respect to the nanocone axis is fully investigated for various geometrical parameters. The optimum nanotube radius minimizing the total potential energy of the concentric configuration is determined for different radii of the small end of the cone. The examined configuration generates asymmetric oscillation; thus, the system constitutes a nano-oscillator.

DNA adsorption on graphene

Here we use classical applied mathematical modeling to determine surface binding energies between both single-strand and double-strand DNA molecules interacting with a graphene sheet. We adopt basic mechanical principles to exploit the 6–12 Lennard-Jones potential function and the continuum approximation, which assumes that intermolecular interactions can be approximated by average atomic line or surface densities. The minimum binding energy occurs when the single-strand DNA molecule is centred 20.2 A from the surface of the graphene and the double-strand DNA molecule is centred 20.3 A from the surface, noting that these close values apply for the case when the axis of the helix is perpendicular to the surface of graphene. For the case when the axis of the helix is parallel to the surface, the minimum binding energy occurs when the axis of the single-strand molecule is 8.3 A from the surface, and the double-strand molecule has axis 13.3 A from the surface. For arbitrary tilted axis, we determine the optimal angles Ω of the axis of the helix, which give the minimum values of the binding energies, and we observe that the optimal angles tend to occur in the intervals Ω ∈ (π /4,π/2) and Ω ∈ (π /7,π/5) for the single and double-strand DNA molecules, respectively.

All-Atom Force Field for Molecular Dynamics Simulations on Organotransition Metal Solids and Liquids. Application to M(CO)n(M = Cr, Fe, Ni, Mo, Ru, or W) Compounds

A previously developed OPLS-based all-atom force field for organometallic compounds was extended to a series of first-, second-, and third-row transition metals based on the study of M(CO)(n) (M = Cr, Fe, Ni, Mo, Ru, or W) complexes. For materials that are solid at ambient temperature and pressure (M = Cr, Mo, W) the validation of the force field was based on reported structural data and on the standard molar enthalpies of sublimation at 298.15 K, experimentally determined by Calvet-drop microcalorimetry using samples corresponding to a specific and well-characterized crystalline phase: Δ(sub)H(m)° = 72.6 ± 0.3 kJ·mol(–1) for Cr(CO)(6), 73.4 ± 0.3 kJ·mol(–1) for Mo(CO)(6), and 77.8 ± 0.3 kJ·mol(–1) for W(CO)(6). For liquids, where problems of polymorphism or phase mixtures are absent, critically analyzed literature data were used. The force field was able to reproduce the volumetric properties of the test set (density and unit cell volume) with an average deviations smaller than 2% and the experimentally determined enthalpies of sublimation and vaporization with an accuracy better than 2.3 kJ·mol(–1). The Lennard-Jones (12-6) potential function parameters used to calculate the repulsive and dispersion contributions of the metals within the framework of the force field were found to be transferable between chromium, iron, and nickel (first row) and between molybdenum and ruthenium (second row).

Vibration Simulations of Double-Walled Carbon Nanotubes by Finite Element Method

Finite element (FE) method is used to study the vibration behavior of armchair and zigzag double-walled carbon nanotubes (DWCNTs). In the analysis, nonlinear spring elements and the Lennard-Jones potential function are used to simulate the Van der Waals' force between non-bond atoms of different tube layers. We systematically analyze the effects of aspect ratio, double-atom vacancy defects and Van der Waals' force on the vibration behavior of DWCNTs. The simulation results show that the first order natural frequency decreases with the increase of length-to-diameter ratio (aspect ratio), the existence of Van der Waals' force causes the increase of natural frequency, and double-atom vacancy defects results in the decrease of each order natural frequency.

The Competition Effect of Strain and Interatomic Distance on Ferromagnetic Critical Temperature in Ising Ultra-Thin-Film: Monte Carlo Simulation

This work used Monte Carlo simulation to investigate the effect of lattice strain on critical temperature of the ferromagnetic Ising ultra-thin-film. Using Bethe-Slater interpretation, exchange interaction was assumed Lennard-Jones (LJ) potential function, and becomes functions of interatomic distance. The cluster flip algorithm was used to update the spin configuration, where energy and magnetization were measure to extract critical temperature via the fourth order cumulant of the magnetization. Results show that initial interatomic distance and strain have strong effect on magnetic critical point behavior. The critical point can be enhanced or suppressed depending on how exchange interaction is modified by the values of initial interatomic distance and strain. A scaling function in predicting critical temperatures is also given in this work. Results agree well with previous applicable works, which confirms the reliability of the reported results.

On the van der Waals interaction of carbon nanotubes as electromechanical nanothermometers

Electromechanical carbon nanothermometers are devices that work based on the interactions and relative motions of double-walled carbon nanotubes (DWCNTs). In this paper, the mechanics of carbon nanotubes (CNTs) constituting two well-known configurations for nanothermometer, namely shuttle configuration and telescope configuration are fully investigated. Lennard-Jones (LJ) potential function along with the continuum approximation is employed to investigate van derWaals (vdW) interactions between the interacting entities. Accordingly, semi-analytical expressions in terms of single integrals are obtained for vdW interactions. Acceptance condition and suction energy are studied for the shuttle configuration. In addition, a universal potential energy is presented for the shuttle configuration consisting of two finite CNTs. Also, for the telescope configuration, extensive studies are performed on the distributions of potential energy and interaction force for various radii and lengths of CNTs. It is found that these geometrical parameters have a considerable effect on the potential energy.

Oscillation of C60 Fullerene in Carbon Nanotube Bundles

This paper aims to present a thorough investigation into the mechanics of a C60 fullerene oscillating within the center of a carbon nanotube bundle. To model this nanoscale oscillator, a continuum approximation is used along with a classical Lennard–Jones potential function. Accordingly, new semianalytical expressions are given in terms of single integrals to evaluate van der Waals potential energy and interaction force between the two nanostructures. Neglecting the frictional effects and using the actual van der Waals force distribution, the equation of motion is directly solved. Furthermore, a new semianalytical formula is derived from the energy equation to determine the precise oscillation frequency. This new frequency formula has the advantage of incorporating the effects of initial conditions and geometrical parameters. This enables us to conduct a comprehensive study of the effects of significant system parameters on the oscillatory behavior. Based upon this study, the variation of oscillation frequency with geometrical parameters (length of tubes or number of tubes in bundle) and initial energy (potential energy plus kinetic energy) is shown.

Modified Lennard-Jones Potentials with a Reduced Temperature-Correction Parameter for Calculating Thermodynamic and Transport Properties: Noble Gases and Their Mixtures (He, Ne, Ar, Kr, and Xe)

The three-parameter Lennard-Jones(12-6)potential function is proposed to calculate thermodynamic property (second virial coefficient) and transport properties (viscosity, thermal conductivity, and diffusion coefficient) of noble gases (He, Ne, Ar, Kr, and Xe) and their mixtures at low density. Empirical modification is made by introducing a reduced temperature-correction parameterτto the Lennard-Jones potential function for this purpose. Potential parameters (σ,ε, andτ) are determined individually for each species when the second virial coefficient and viscosity data are fitted together within the experimental uncertainties. Calculated thermodynamic and transport properties are compared with experimental data by using a single set of parameters. The present study yields parameter sets that have more physical significance than those of second virial coefficient methods and is more discriminative than the existing transport property methods in most cases of pure gases and of gas mixtures. In particular, the proposed model is proved with better results than those of the two-parameter Lennard-Jones(12-6)potential, Kihara Potential with group contribution concepts, and other existing methods.