Lennard-Jones, generally liquid

The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

The force fields used in the QM/MM methods are typically adopted from fully classical force fields. While this is in general suitable for the solvent-solvent interactions it is not clear how to model, e.g., the van der Waals interaction between the solute and the solvent. The van der Waals interactions are typically treated as Lennard-Jones (LJ) potentials with parameters for the quantum atoms taken from the classical force field or optimized for the particular QM/MM method for some molecular complexes. However, it is not certain that optimizing the (dispersion and short-range repulsion) parameters on small complexes will improve the results in a QM/MM simulation of liquids [37],... 

We illustrate the behavior for a first order transition between a vapor and a dense liquid in the framework of a simple Lennard-Jones model. The condensation of a vapor into a dense liquid upon cooling is a prototype of a phase transition that is characterized by a single scalar order parameter - the density, p. The thermodynamically conjugated field is the chemical potential, p. The qualitative features, however, are general and carry over to other types of phase coexistence, e.g., Sect. 3.4. 

An optimal choice of weights can be found by measuring the local dif-fusivity of a random walk along the reaction coordinates and applying the feedback method to shift weight towards the bottlenecks in the simulation. This generalized ensemble optimization approach has recently been illustrated for the simulation of dense Lennard-Jones fluids close to the vapor-liquid equilibrium [21]. The interaction between particles in the fluid is described by a... 

Figure 5,1 shows the pair correlation function of a typical Lennard-Jones liquid. Two general features are seen First, the short range stiucture that shows that atoms in liquids arrange themselves about a central atom in a way that reflects their atomic diameters (here expressed by the Lennard-Jones parameter cr), and, second, the relative fast decay of this short-range order, expressed by the rapid approach of g(r) to 1. 

In a recent study, a new model of fluids was described by using the generalized van der Waals theory. Actually, van der Waals over 100 years ago suggested that the structure and thermodynamic properties of simple fluids could be interpreted in terms of neatly separate contributions from intermolecular repulsions and attractions. A simple cubic equation of state was described for the estimation of the surface tension. The fluid was characterized by the Lennard-Jones (12-6) potential. In a recent study the dependence of surface tension of liquids on the curvature of the liquid-vapor interface has been described. ... 

This comment should be borne in mind when the theory is applied to real fluids. In any real liquid, and certainly for water, we need a few molecular parameters to characterize the molecules, say and a in a Lennard-Jones fluid, or in general, a set of molecular parameters a, b, c,. Thus, a proper statistical-mechanical theory of real liquid should provide us with the Gibbs energy as a function of T, P, N and the molecular parameters a,b,c,..., i.e., a function of the form G(T, P, N a, b, c,...). Instead, the SPT makes use of only one molecular parameter, the diameter a. No provision of incorporating other molecular parameters is offered by the theory. This deficiency in the characterization of the molecules is partially compensated for by the use of the measurable density p as an input parameter. 

The first indications that certain systems might violate the phase rule came from computer simulations of small clusters of atoms. A number of studies revealed clearly defined solid-like and liquid-like forms [5-14]. These embraced both molecular dynamics and Monte Carlo simulations, and explored a variety of clusters. These included several based on atomic models with interparticle Lennard-Jones forces, which mimic rare gas clusters rather well. There were also models of alkali halide clusters. Hence, the existence of solid and liquid forms for such small systems seemed not only plausible but general, not restricted to any one kind of system. Shortly after these studies appeared, another, of a 55-atom cluster with Lennard-Jones interparticle forces, showed not only solid and liquid forms but also a form in which the surface of the cluster (with icosahedral structure) is liquid... 

The simulation results for coexistence properties are from Hoover and Ree [24] and the value of L is from the work of Ohnesorge et al. [135], The result from Barker s SCF theory [94] is from the leading term (order a ) in an expansion of the mean square displacement in powers of a = (p p/p) - 1 and may be an underestimate of the true value from that theory, SCF, self-consistent field LJD, Lennard-Jones and Devonshine MWDA, modified weighted-density approximation GELA, generalized effective liquid approximation FMF, fundamental measures functional. 

The parameter A tunes the stiffness of the potential. It is chosen such that the repulsive part of the Lennard-Jones potential makes a crossing of bonds highly improbable (e.g., k=30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Lennard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly form liquid crystalline phases. 

Equations of state derived from statisticai thermodynamics arise from proper con-figurationai partition functions formuiated in the spirit of moiecuiar modeis. A comprehensive review of equations of state, inciuding the historicai aspects, is provided in Chapter 6. Therefore, we touch briefly in oniy a few points. Lennard-Jones and Devonshire [1937] developed the cell model of simple liquids, Prigogine et al. [1957] generalized it to polymer fluids, and Simha and Somcynsky [1969] modified Pri-gogine s cell model, allowing for more disorder in the system by lattice imperfections or holes. Their equations of state have been compared successfully with PVT data on polymers [Rodgers, 1993]. 

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