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Register a new userEnergy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard--Jones $(m,n)$ Interaction
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It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the step-bunching phenomena for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard--Jones (LJ) type potential. It is characterized by two exponents $m$ and $n$ describing the singular and decaying behaviors of the interacting potential at small and large distances, and henceforth are called generalized LJ $(m,n)$ potential. We provide a systematic analysis of the asymptotic properties of the step configurations and the value of the minimum energy, in particular, their dependence on $m$ and $n$ and an additional parameter $alpha$ indicating the interaction range. Our results show that there is a phase transition between the bunching and non-bunching regimes. Moreover, some of our statements are applicable for any critical points of the energy, not necessarily minimizers. This work extends the technique and results of [Luo et al, SIAM MMS, 2016] which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). As a by-product, our result also leads to the well-known fact that the classical LJ (6,12) potential does not demonstrate step-bunching type phenomena.
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This paper investigates the relation between the density-scaling exponent $gamma$ and the virial potential-energy correlation coefficient $R$ at several thermodynamic state points in three dimensions for the generalized $(2n,n)$ Lennard-Jones (LJ) system for $n=4, 9, 12, 18$, as well as for the standard $n=6$ LJ system in two, three, and four dimensions. The state points studied include many low-density states at which the virial potential-energy correlations are not strong. For these state points we find the roughly linear relation $gammacong 3nR/d$ in $d$ dimensions. This result is discussed in light of the approximate extended inverse power law description of generalized LJ potentials [N. P. Bailey et al., J. Chem. Phys. 129, 184508 (2008)]. In the plot of $gamma$ versus $R$ there is in all cases a transition around $Rapprox 0.9$, above which $gamma$ starts to decrease as $R$ approaches unity. This is consistent with the fact that $gammarightarrow 2n/d$ for $Rrightarrow 1$, a limit that is approached at high densities and/or temperatures at which the repulsive $r^{-2n}$ term dominates the physics.
Liquids displaying strong virial-potential energy correlations conform to an approximate density scaling of their structural and dynamical observables. This scaling property does not extend to the entire phase diagram, in general. The validity of the scaling can be quantified by a correlation coefficient. In this work a simple scheme to predict the correlation coefficient and the density-scaling exponent is presented. Although this scheme is exact only in the dilute gas regime or in high dimension d, a comparison with results from molecular dynamics simulations in d = 1 to 4 shows that it reproduces well the behavior of generalized Lennard-Jones systems in a large portion of the fluid phase.
This paper studies physical aging by computer simulations of a 2:1 Kob-Andersen binary Lennard-Jones mixture, a system that is less prone to crystallization than the standard 4:1 composition. Starting from thermal-equilibrium states, the time evolution of the following four quantities is monitored following up and down jumps in temperature: the potential energy, the virial, the average squared force, and the Laplacian of the potential energy. Despite the fact that significantly larger temperature jumps are studied here than in previous experiments, to a good approximation all four quantities conform to the single-parameter-aging scenario derived and validated for small jumps in experiments [Hecksher et al., J. Chem. Phys. 142, 241103 (2015)]. As a further confirmation of single-parameter aging with a common material time for the different quantities monitored, their relaxing parts are found to be almost identical for all temperature jumps.
We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic FPU-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and highlight in that the asymmetric nature of the latter interaction potential may lead to a form of hybrid bulk-surface localized solutions.
The homogeneous and heterogeneous nucleation of a Lennard-Jones liquid is investigated using the umbrella sampling method. The free energy cost of forming a nucleating droplet is determined as a function of the quench depth, and the saddle point nature of the droplets is verified using an intervention technique. The structure and symmetry of the nucleating droplets is found for a range of temperatures. We find that for deep quenches the nucleating droplets become more anisotropic and diffuse with no well defined core or surface. The environment of the nucleating droplets form randomly stacked hexagonal planes. This behavior is consistent with a spinodal nucleation interpretation. We also find that the free energy barrier for heterogeneous nucleation is a minimum when the lattice spacing of the impurity equals the lattice spacing of the equilibrium crystalline phase. If the lattice spacing of the impurity is different, the crystal grows into the bulk instead of wetting the impurity.
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