Generalized Lennard-Jones Potential Research Articles

Modeling the degeneration of the collagen architecture in a microstructural model of the human cornea

We propose an enriched micromechanical model of the collagenous reinforcement of the eye stromal tissue. As a departure from an over-simplified model proposed a few years back, where collagen and chemical bonds were modeled as linear-elastic trusses, here we describe the chemical bonds by means of a more realistic generalized Lennard-Jones potential. In keeping with the original model, we disregard the multi-layer nature of the cornea and the continuum nature of the filling elastin matrix. The under-constrained locally orthogonal network of collagen fibrils is stabilized by crosslinks that provide the rigidity of the system and confer the ability to sustain the action of the intraocular pressure. In Ariza-Gracia et al., it has been shown that the weakening and the bulging of the cornea due to ectasia can be ascribed to the reduction of the density of the chemical bonds. The introduction of a pseudo-chemical potential supplies a more realistic model: any mechanical, enzymatic, or chemical cause of the degradation of the tissue observed in ectasia can be effectively introduced in a multi-physic potential, disregarding the adoption of phenomenological models. In numerical calculations, the high non-linearity of the model is suitably controlled by adopting a robust explicit solver based on dynamic relaxation.

Two-dimensional Wigner crystals of classical Lennard-Jones particles

The ground-state of two-dimensional (2D) systems of classical particles interacting pairwisely by the generalized Lennard-Jones potential is studied. Taking the surface area per particle A as a free parameter and restricting oneself to periodic Bravais lattices with one particle per unit cell, Bétermin (2018 Nonlinearity 31 3973) proved that the hexagonal, rhombic, square and rectangular structures minimize successively the interaction energy per particle as A increases. We show here that the second-order transitions between the rhombic/square and square/rectangular phases are of mean-field type. The aim of this paper is to extend Bétermin’s analysis to periodic 2D lattices with more than one particle per elementary cell. Being motivated by previous works dealing with other kinds of models, we propose as the ground-state the extensions of the 2D rectangular (one-chain) lattice, namely the ‘zig-zag’ (two-chain), three-chain, four-chain, etc structures possessing 2, 3, 4, etc particles per unit cell, respectively. By using a recent technique of lattice summation we find for the standard Lennard-Jones potential that their ground-state energy per particle approaches systematically as the number of particles per unit cell increases to the one of a phase separated state (the optimal hexagonal lattice). We analyze analytically the low-density limit and the limiting hard-core case of the generalized Lennard-Jones potential.

Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

In this paper we start with proving that the Schr\"odinger equation (SE) with the classical $12-6$ Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation. We study the $10-6$ potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form $w(r)\propto 1/r^5$. We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so called De Boer principle of corresponding states. A comparison of the second virial coefficient $B(T)$ for both potentials shows a good agreement for low temperatures. As a consequence of these results we propose the $10-6$ potential as an integrable alternative to be applied in further studies instead of the original $12-6$ L-J potential. Finally we study through DGT and SUSYQM the integrability of the SE with a generalized $(2\nu-2)-\nu$ L-J potential. This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.

Calculations of Total Classical Cross Sections for a Central Field

In order to find the total collision cross-section a direct method of the effective potential (EPM) in the framework of classical mechanics was proposed. EPM allows to over come both the direct scattering problem (calculation of the total collision cross-section) and the inverse scattering problem (reconstruction of the scattering potential) quickly and effectively. A general analytical expression was proposed for the generalized Lennard-Jones potentials: (6–3), (9–3), (12–3), (6–4), (8–4), (12–4), (8–6), (12–6), (18–6). The values for the scattering potential of the total cross section for pairs such as electron–N $$_{2}$$ , N–N, and O–O $$_{2}$$ were obtained in a good approximation.

An excluded volume theory of lattice fluids part II-incorporating specific models of attractive and repulsive interaction energy

The excluded volume theory of lattice fluids, introduced in Fluid Phase Equilibrium 372 (2014) 126–140, is enhanced by specifying models for both the attractive and repulsive interaction energy. The result is an interaction energy function exhibiting a potential well reminiscent of a generalized Lennard-Jones potential. However here the potential well is seen in the Gibbs free energy function at the macro level. The attractive component of the interaction potential, thou based on a lattice model, can be cast as arising from a radial distribution function and as such the model is seen as a bridge between lattice fluid models and models based on the radial distribution method. The success of this theory is based on the following ideas: (1) Intramolecular and intermolecular bonds can be treated as uncorrelated when modeling the partition function. The method used to separate the bonds, which is based on an application of Bayes' theorem, makes a chain molecule appear to have only one segment in a lattice in which the intramolecular bonds are excluded. (2) The attractive forces active between molecules are thought of as being resolved along the generating lines of an assumed lattice. The energy in an interaction bond is assumed to be inversely proportional to some power of the separation distance between molecules along the lattice line connecting the two molecules. This results in an interaction potential expressed as a polylogarithm with an argument given by a function of the ratio of density to maximum density. (3) An entropic based repulsive force is introduced that opposes the attractive forces. The repulsive force is assumed to be inversely related to the number of configurations available to a molecule in the lattice. Increasing the available configurations decreases the repulsive component of the interaction energy. (4) Strong energetic effects can be modeled based on a random occupation of lattice sites. For example if the attractive force between molecules is strong enough to cause clustering the molecular cluster is assumed to randomly occupy lattice sites. The improved model is successfully applied to the vapor liquid equilibrium of ethane, ammonia and water. And to the binary mixtures of propane-butane and R152a-butane.

Corresponding states law for a generalized Lennard-Jones potential.

It was recently shown that vapor-liquid coexistence densities derived from Mie and Yukawa models collapse to define a single master curve when represented against the difference between the reduced second virial coefficient at the corresponding temperature and that at the critical point. In this work, we further test this proposal for another generalization of the Lennard-Jones pair potential. This is carried out for vapor-liquid coexistence densities, surface tension, and vapor pressure, along a temperature window set below the critical point. For this purpose, we perform molecular dynamics simulations by varying the potential softness parameter to produce from very short to intermediate attractive ranges. We observed all properties to collapse and yield master curves. Moreover, the vapor-liquid curve is found to share the exact shape of the Mie and attractive Yukawa. Furthermore, the surface tension and the logarithm of the vapor pressure are linear functions of this difference of reduced second virial coefficients.

Second virial coefficient of a generalized Lennard-Jones potential.

We present an exact analytical solution for the second virial coefficient of a generalized Lennard-Jones type of pair potential model. The potential can be reduced to the Lennard-Jones, hard-sphere, and sticky hard-sphere models by tuning the potential parameters corresponding to the width and depth of the well. Thus, the second virial solution can also regain the aforementioned cases. Moreover, the obtained expression strongly resembles the one corresponding to the Kihara potential. In fact, the Fk functions are the same. Furthermore, for these functions, the complete expansions at low and high temperature are given. Additionally, we propose an alternative stickiness parameter based on the obtained second virial coefficient.

Aggregation in Colloidal Suspensions: Evaluation of the Role of Hydrodynamic Interactions by Means of Numerical Simulations

Numerical simulations constitute a precious tool for understanding the role of key parameters influencing the colloidal arrangement in suspensions, which is crucial for many applications. The present paper investigates numerically the role of hydrodynamic interactions on the aggregation processes in colloidal suspensions. Three simulation techniques are used: Brownian dynamics without hydrodynamic interactions, Brownian dynamics including some of the hydrodynamic interactions, using the Yamakawa-Rotne-Prager tensor, and stochastic rotation dynamics coupled with molecular dynamics. A system of monodisperse colloids strongly interacting through a generalized Lennard-Jones potential is studied for a colloid volume fraction ranging from 2.5 to 20%. Interestingly, effects of the hydrodynamic interactions are shown in the details of the aggregation processes. It is observed that the hydrodynamic interactions slow down the aggregation kinetics in the initial nucleation stage, while they speed up the next cluster coalescence stage. It is shown that the latter is due to an enhanced cluster diffusion in the simulations including hydrodynamic interactions. The higher the colloid volume fraction, the more pronounced the effects on the aggregation kinetics. It is also observed that hydrodynamic interactions slow down the reorganization kinetics. It turns out that the Brownian dynamics technique using the Yamakawa-Rotne-Prager tensor tends to overestimate the effects on cluster diffusion and cluster reorganization, even if it can be a method of choice for very dilute suspensions.

Adsorption of single polymer molecules in shear flow near a planar wall

Adsorption of homopolymers from a dilute solution to a planar wall in the presence of shear flow is studied using a bead-spring dumbbell model. The bead-bead and bead-wall interactions are described by generalized Lennard-Jones potentials. A kinetic theory incorporating bead-wall hydrodynamic interaction is developed in order to obtain an analytical expression for the steady-state dumbbell concentration profile. The concentration profile exhibits an exclusion zone in the immediate vicinity of the wall, is followed by a peak, and finally approaches the bulk concentration far away from the wall. Using the analytical expression, the amount adsorbed and the equivalent film thickness are studied as a function of flow strength and the parameters characterizing the bead-wall interaction potential. Shear flow causes migration of the dumbbells due to bead-wall hydrodynamic interaction, which leads to desorption. On increasing the flow strength, the quantity adsorbed and the film thickness decrease until complete desorption occurs. The dependence of the flow strength required for desorption on the model parameters is also studied and a scaling law is derived for the strong-interaction limit. Brownian dynamics simulations are performed to verify the predictions from the kinetic theory. Although the theory makes a number of simplifying assumptions, it captures many of the key features seen in the simulations.

Scaling of viscous dynamics in simple liquids: theory, simulation and experiment

Supercooled liquids are characterized by relaxation times that increase dramatically by cooling or compression. From a single assumption follows a scaling law according to which the relaxation time is a function of h(ρ) over temperature, where ρ is the density and the function h(ρ) depends on the liquid in question. This scaling is demonstrated to work well for simulations of the Kob–Andersen binary Lennard-Jones mixture and two molecular models, as well as for the experimental results for two van der Waals liquids, dibutyl phthalate and decahydroisoquinoline. The often used power-law density scaling, h(ρ)∝ργ, is an approximation to the more general form of scaling discussed here. A thermodynamic derivation was previously given for an explicit expression for h(ρ) for liquids of particles interacting via the generalized Lennard-Jones potential. Here a statistical mechanics derivation is given, and the prediction is shown to agree very well with simulations over large density changes. Our findings effectively reduce the problem of understanding the viscous slowing down from being a quest for a function of two variables to a search for a single-variable function.

Qualitative analysis of the anisotropic two-body problem with generalized Lennard-Jones potential

This paper continue the study of the generalized Lennard-Jones potential started in Bărbosu et al. (J Math Chem 49(9):1961–1975, 2011) for a more general situation. More precisely we study the two-body problem with generalized Lennard-Jones potential in an anisotropic space. We will show that the set of initial conditions leading to collisions and ejections have positive measure. We also study the capture and escape solutions in the zero-energy case using the infinity manifold. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and the two anisotropic perturbations corresponding to the generalized Lennard-Jones potential is chaotic.

The two-body problem with generalized Lennard-Jones potential

We consider a generalization of the famous Lennard-Jones potential. To study the two-body problem associated to this potential, we use the foliations of the phase space by the invariant sets corresponding to the first integrals of energy and angular momentum. We investigate all possible situations created by the interplay among the constants of integration and the field parameters. We obtain the global flow, and illustrate it in both 3D and 2D. This flow exhibits a great variety of orbits, a homoclinic one included. All phase portraits are interpreted in terms of physical trajectories.

Local Self-Consistent Ornstein−Zernike Integral Equation Theory and Application to a Generalized Lennard-Jones Potential

Local self-consistent Ornstein-Zernike (OZ) integral equation theory (IET) provides a rapid and easy route for obtaining independently thermodynamic and structural information for a single state point. Because of neglect of information of neighboring state points in determining a self-consistent adjustable parameter performance of the local self-consistent OZ IET is somewhat vulnerable and worthy of intensive investigation. For this reason, we have performed Monte Carlo simulations to obtain thermodynamic and structural properties of fluid with a generalized Lennard-Jones potential, and the present simulation results are employed to verify the quality of a local version of a recently developed global self-consistent OZ IET and a local expression for computation of excess chemical potential directly from the structural functions of the state point of interest. Comprehensive comparison and analysis demonstrate the following (i) the present local self-consistent OZ IET performs quite well for calculation of pressure and excess internal energy; (ii) using the same structural functions from the present local self-consistent OZ IET, the previously derived local expression by the present author has by and large the same accuracy in calculating the excess chemical potential as an exact virial formula for the pressure; (iii) although the excellent performance exhibited for the above thermodynamic quantities persists to very low temperature and very short-ranged potential and remains even in the liquid-solid coexistence region, the excess Helmholtz free energy calculated from the pressure and excess chemical potential shows evident inaccuracy for a density-temperature combination deep in the liquid-solid coexistence region, and this makes it necessary to derive a local formulation for the excess free energy.

Consequences of an Equation of State in the Thermodynamic Scaling Regime

An equation of state following from the generalized Lennard-Jones potential has been successfully applied to describe isothermal volumetric data for several glass-forming liquids on the assumption of the thermodynamic scaling validity. The scaling exponents gamma evaluated from the fitting procedure for each tested material have turned out at least 2 times larger than those earlier found from relaxation data. The discrepancy indicates that the appealing idea of thermodynamic scaling, which has been recently intensively explored, requires more efforts to find a satisfying theoretical grounds for it.

Lattice vibrations of armchair carbon nanotubes: phonons, soliton deformations and lattice discreteness effects

Radial solitons are investigated in armchair carbon nanotubes using a generalized Lennard-Jones potential. The radial solitons are found in terms of moving kink defects whose velocity obeys a dispersion relation. Effects of lattice discreteness on the shape of kink defects are examined by estimating the Peierls stress. Results suggest that the typical size for an unpinned kink phase is of the order of a lattice spacing.

Theory and simulation of short-range models of globular protein solutions

We report theoretical and simulation studies of phase coexistence in model globular proteinsolutions, based on short-range, central, pair potential representations of the interactionamong macro-particles. After reviewing our previous investigations of hard-core Yukawaand generalized Lennard-Jones potentials, we report more recent results obtainedwithin a DLVO-like description of lysozyme solutions in water and added salt. Weshow that a one-parameter fit of this model, based on static light scattering andself-interaction chromatography data in the dilute protein regime, yields demixing andcrystallization curves in good agreement with experimental protein-rich–protein-poor andsolubility envelopes. The dependence of cloud and solubility point temperatures ofthe model on the ionic strength is also investigated. Our findings highlight theminimal assumptions on the properties of the microscopic interaction sufficient for asatisfactory reproduction of the phase diagram topology of globular protein solutions.

Low temperature Pb deposits on Cu(0 0 1): Monte Carlo structural studies

Monte Carlo simulations with adapted generalized Lennard-Jones potentials, already used to investigate the room temperature superstructures of Pb/Cu(0 0 1), enable us to derive the superstructures of low temperature ( T=160 K) deposited Pb on Cu(0 0 1). Two stable structures are found to agree with recent observations and are compared to the structures ( 6 1× 6 1)R tan −1(5/6) and (5×5) Rtan −1(3/4) earlier proposed by Vidali et al. On the other hand, a comparative study of the energy per atom of one, two and three Pb monolayers deposited at low temperature over the annealed room temperature dense Pb c (5 2 × 2 )R45° structure on a Cu(0 0 1) substrate shows evidence of the stability of Pb bilayers rather than Pb monolayers, in agreement with recent experiments.

Effective Lennard–Jones potential for cubic metals in the frame of embedded atom model

Generalized Lennard–Jones potentials are proposed in the frame of the embedded-atom method (EAM) of Daw and Baskes for homonuclear cubic metals. The basic functions of the model (the pair interaction potential, electron density function and embedding energy function) are presented in analytical forms. It is shown that N-body Finnis–Sinclair potentials and Daw–Baskes embedded atom potentials are mathematically equivalent. The developed potentials suit very well most of the fcc metals and alkali metals, and they are convenient for computer simulations.

Structural transitions in clusters

Monatomic clusters are studied by Monte Carlo relaxation using generalized Lennard-Jones potentials. A transition from an icosahedral symmetry to a crystalline symmetry with stacking faults is always observed. Bcc-based soft atom clusters are found to have a lower energy than the corresponding hcp and fcc ones below the melting point.

Short-wavelength collective density excitations in monatomic liquids

Computer simulation data for liquids with generalized Lennard-Jones potentials are analysed with the aim of elucidating the key features required of a liquid for it to support a short-wavelength, collective density excitation. The data suggest that, for high densities, both the repulsive and attractive components of the Lennard-Jones potential must be modified substantially for such a mode to exist, and that softening the potential core does not in itself suffice to induce the correlation necessary for a liquid to support a mode. Viscoelastic theory is shown to provide a good description of the dynamical properties and to provide a useful condition for the existence of a collective density excitation.