Lennard-Jones systems

Agrawal R and Kofke D A 1995 Thermodynamio and struotural properties of model systems at solid-fluid ooexistenoe. II. Melting and sublimation of the Lennard-Jones system Mol. Phys. 85 43-59... 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

Errington, J. R., Solid-liquid phase coexistence of the Lennard-Jones system through phase-switch Monte Carlo simulation, J. Chem. Phys. 2004,120, 3130-3141... 

Figure 3 shows an ordering map for this Lennard-Jones system, with the translational order t (of Eq. [2]) plotted against the bond-orientational order Qs (of Eq. [5]). It can be observed that the data, collected over a wide range of temperatures and densities, collapse onto two distinct equilibrium branches... 

Numerical Calculation of the Rate of Crystal Nucleation in a Lennard-Jones System at Moderate Undercooling. 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

This strategy has been applied to the study of a range of coexistence problems, initially focused on lattice models in magnetism [41] and particle physics [42]. Figure 5 [43—4-5] shows the results of an application to liquid-vapor coexistence in a Lennard-Jones system with the particle number density chosen as an order parameter. 

Example A hep and fee Phases of Lennard Jones Systems. In the case of crystalline phases it is natural to choose the reference configurations to represent the states of perfect crystalline order, described by the appropriate sets of lattice vectors... 

The method was validated for Lennard-Jones systems and applied44 to Sin and Sin+. This discovered lower-energy isomers for n> 19. All structures for n = 19-23 are prolate (Fig. 6). Compact geometries first become competitive in energy at n = 24 - 25, exactly as observed in IMS. Our ongoing studies for n> 24 will be presented in future publications. 

The same procedure can be applied to higher linear clusters of, let us say, n molecules. In order to retain the simplicity of the approach we have to assume equal nearest neighbour distances R = RBC = RCD =. This is an approximation for clusters with n > 3, actually a very good one for Lennard-Jones systems. This assumption is not only exact for n = 3 but also for the infinite chain, n = oo. By straightforward calculation we derive... 

Now, we perform the limit n - oo and obtain a measure of the sum of all further neighbour effects in an infinite linear chain of Lennard-Jones systems ... 

Some numerical results are given in Table 10. We observe rather small effects the contraction of the equilibrium distance is 0.3 % only and the energy per nearest neighbour interaction increases in absolute value by 3.5 %. In three dimensions the effects are somewhat larger. The expressions for fn, dn, fx and d are free of parameters and valid for all Lennard-Jones systems, independently of the particular values of Re and De. 

Figure 47. Diffusion coefficient D as obtained from a molecular dynamics simulation study of a binary Lennard-Jones system reaching temperatures below the crossover temperature of mode coupling theory (MCT). Solid line represents interpolation by MCT power law note the large temperature range covered by the power law. (From Ref. 371.)...

III. Computer Simulation of a One-Dimensional Lennard-Jones System.233... 

First (Sections II-V), we shall tackle the problem of translation. The simplest way of doing this is to study one-dimensional chains of particles. Bishop et al. have shown via computer simulation that one-dimensional Lennard-Jones systems exhibit the same dynamic properties as real three-dimensional liquids. This makes our investigations less academic than they seem at a purely intuitive level, as physical intuition would refuse to take as a Uquid sample a chain of particles which cannot bypass each other. 

Fig. 3.11. Molecular dynamic simulation results for the average fracture stress CTf for various disorder concenrations on triangular lattices, (a) For site dilute Lennard-Jones system (Chakrabarti et al 1986), and (b) for bond dilute spring network (Beale and Srolovitz 1988).

Fig. 3.16. Variation with time t (in number of iterations) of the molecular dynamic simulation data for the number nt of broken bonds in a given configuration of site dilute Lennard-Jones system (of linear size L = 21), for different initial concentraions of dilution, when subjected to a stress just greater than the corresponding fracture stress af (Chakrabarti et al 1986).

The JE is used by Mu and Song to calculate the interfacial free energies of different crystal orientations for a Lennard-Jones system, and they find it more efficient than thermodynamic integration. 

Mandell, M.J., McTague, J.P., and Rahman, A. (1976) Crystal Nucleation in a Three-Dimensional Lennard-Jones System A Molecule- Dynamics Study, / Chem. Phys., Vol. 64, pp.3699-3702. 

Nucleation was first observed by Mandell, McTague, and Rahman in simulations of small (128-particle) Lennard-Jones systems. They monitored the nucleation process by following the magnitude of the structure factor, the increase in temperature associated with the release of the heat of fusion, and the apparent absence of diffusion. In subsequent work they examined the effect of an increase in system size, which led to larger undercoolings before crystallization was seea In addition, they determined the structure of the resulting solid to be bcc, as opposed to the thermodynamically stable fee phase. They also introduced a method for locating the critical nucleus at a series of times the velocities of the particles were randomized and it was then determined whether the nucleation had disappeared or whether it still took place. In the former case the intervention time was taken to be precritical, while in the latter it was postcritical. In this way they estimated the critical nucleus to contain 40-70 atoms. 

This study is consistent with the idea that crystal surfaces at temperatures close to melting have some kind of disordered layer or layers, often called liquid-like . Due to the different equilibrium volumes of the liquid and solid phases, this region makes the surface either contract (as in the case of the ice surface) or expand (as it is for Lennard-Jones systems). The positive interfacial excess stress of the ice/water interface therefore makes it similar to liq-uid/vapor interfaces, and the water/vapor interface in particular, for which the excess stress is equal to the interfacial free energy (surface tension). 

M. J. Mandell, J. P. McTaque, and A. Rahman (1976) Crystal nucleation in a 3-dimensional lennard-jones system - molecular-dynamics study. J. Chem. Phys. 64, pp. 3699-3702... 

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