Lennard-Jones particles/spheres

Figure 2.2. The form of the pair correlation function g Rj at very low densities (p — 0) (a) for hard spheres with

Note that this is the exact distance of closest approach for two hard sphere particles. For Lennard-Jones particles <7ab is defined in (2.145). This is... 

The exact solution of the Percus-Yevick (PY) equation is known for a one-component system of hard spheres (Wertheim 1963 Thiele 1963) and for mixtures of hard spheres (Lebowitz 1964). Numerical solutions of the PY equation (for Lennard-Jones particles) have been carried out by many authors, e.g., Broyles (1960, 1961), Broyles et al. (1962), Throop and Bearman (1966), Baxter (1967), Watts (1968), Mandel et al. (1970), Grundke and Henderson (1972a, b)... 

Computer simulation results for S2 are somewhat sparse and involve the usual uncertainties involved in extrapolating results for a truncated T(r) used in a periodic box to untruncated T(r) in an infinite system." Nevertheless for polarizable hard-sphere and Lennard-Jones particles, it is probably safe to say that the estimates currently available from the combined use of analytic and simulation input are enough to provide a reliable guide to the p and dependence of Sj over the full fluid range of those variables. The most comprehensive studies of have been made by Stell and Rushbrooke" and by Graben, Rushbrooke, and Stell," for the hard-sphere and Lennard-Jones cases, respectively. Both these works utilize the simulation results of Alder, Weis, and Strauss," as well as exact density-expansion results, and numerical results of the Kirkwood superposition approximation... 

HSMC allows one, for example, to obtain the free energies of model systems, characterized by a parametrized set of Hamiltonians, by relating them to that of a reference model for which the free energy has already been investigated [1] (e.g., Lennard-Jones particles by reference to soft spheres ... 

Figure 12. Picture of the model 3rd generation siloxane dendrimer. The ellipsoids represent Gay-Berne particles and the spheres represent Lennard-Jones particles.

Computer simulation techniques have become very important tools in the study of condensed matter systems since their development in the 1950s. Such simulations can study systems ranging from the very simple (hard spheres, Lennard-Jones particles) to the complex (for example, proteins and large biomolecules). The total number of atoms or molecules studied also varies greatly and depends upon the application - for the calculation of the bulk properties of simple systems, a few hundred particles is usually sufficient. Other systems such as interfaces and proteins require much larger simulations. In the case of the crystal-melt interface, simulations have an enhanced role owing to the difficulty of performing experiments that probe the interface. 

A large body of the computer simulation work has been reported on model systems such as hard discs, spheres or Lennard-Jones particles. Here the interparticle potential is known and can be used to rapidly calculate the configurational energy of the system as required for Monte Carlo studies or the configurational force on a particle as required for molecular dynamics. 

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

The central point of the present survey is an attempt to show a complete analogy between the free volume of suspensions and that of molecular systems. It is characteristic that the limiting volume fraction of spherical filler particles leaves in the system another 25-40% of unoccupied volume. Precisely the same unoccupied volume exists in molecular systems if we liken them to a volume filled with spheres whose radii are calculated taking into account the Lennard-Jones potential. 

Nauchitel and Pertsin have studied the melting properties of 13-, 19-, and 55-particle Lennard-Jones clusters.Questioning the validity of results obtained from free-volume simulations of such systems, they have used hard-sphere boundaries to constrain their clusters to finite volumes. The results of Nauchitel and Pertsin are most interesting for the 55-particle cluster. For certain ranges of temperature and mean density, structural evidence for surface melting was obtained projections of the cluster s coordinates, and radial density distribution functions, like those given in Fig. 17, characterize the cluster as a 13-particle icosahedral core surrounded by a fluidlike shell. However, dynamic calculations like those described for other clusters in the previous section have yet to be obtained to determine how fluidlike these outer atoms really are. 

The small spheres are fluid molecules, and the large spheres are immobile silica particles. The top visualizations are for a disordered material and the bottom visualizations are for an ordered material of the same porosity. The visualizations on the left are for the saturated vapor state, and those on the right are for the corresponding saturated liquid state, (b) Simulated adsorption and desorption isotherms for Lennard-Jones methane in a silica xerogel at reduced temperature kT/Sfi = 0.7. The reduced adsorbate density p = pa is plotted vs the relative pressure X/Xo for methane silica/methane methane well depth ratios ejf/Sff = 1- (open circles) and 1.8 (filled circles) [44]. (Reproduced with permission from S. Ramalingam,... 

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