Fluid Monte Carlo calculations

In Fig. III-7 we show a molecular dynamics computation for the density profile and pressure difference P - p across the interface of an argonlike system [66] (see also Refs. 67, 68 and citations therein). Similar calculations have been made of 5 in Eq. III-20 [69, 70]. Monte Carlo calculations of the density profile of the vapor-liquid interface of magnesium how stratification penetrating about three atomic diameters into the liquid [71]. Experimental measurement of the transverse structure of the vapor-liquid interface of mercury and gallium showed structures that were indistinguishable from that of the bulk fluids [72, 73]. 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

Fig. 2.8 The pair correlation function g r) for a fluid composed of hard spheres at a packing fraction of = 0.49 calculated as a function of distance, r, using the Ornstein-Zernike equation with the direct correlation function given by equations (2.6.6) and (2.6.10). The data shown as ( ) are from a Monte Carlo calculation.

In the present approach, we apply an accurate and numerically efficient equation of state for the exp-6 fluid based on Zerah and Hansen s hypemetted-mean spherical approximation (HMSA) [111] equations and Monte Carlo calculations to detonation, shocks, and static compression. Thermal effects in the EOS are included through the dependence of the coefficient of thermal expansion on temperature, which can be directly compared to experiment. We find that we can replicate shock Hugoniot and isothermal compression data for a wide variety of solids with this simple form. 

This result is especially important for ionic solution theory for the following reason an ionic solution model at the BO level represents a dense fluid the volumes occupied by the solute and solvent molecules occupy most of the volume of the system. To treat such a model one must use an approximation method that is reliable for dense fluids, such as molecular dynamics or Monte Carlo calculations. On the other hand, in a typical ionic solution with a total ion concentration of less than 2 M, the fraction of the volume occupied by the ions is less than a few tenths. In this range there are many approximation methods that give results accurate enough to be interesting and which therefore may be applied to MM models for solutions because the solvent molecules do not explicitly appear. 

The determination of dense fluid properties from ab initio quantum mechanical calculations still appears to be some time from practical completion. Molecular dynamics and Monte Carlo calculations on rigid body motions with simple interacting forces have qualitatively produced all of the essential features of fluid systems and quantitative agreement for the thermodynamic properties of simple pure fluids and their mixtures. These calculations form the basis upon which perturbation methods can be used to obtain properties for polyatomic and polar fluid systems. All this work has provided insight for the development of the principle of corresponding state methods that describe the properties of larger molecules. 

Optimization of Sampling Algorithms in Monte Carlo Calculations on Fluids... 

This is a class of algorithms which makes feasible on contemporary computers an exact Monte Carlo solution of the Schrodinger equation. It is exact in the sense that as the number of steps of the random walk becomes large the computed energy tends toward the ground state energy of a finite system of bosons. It shares with all Monte Carlo calculations the problem of statistical errors and (sometimes) bias. In the simulations of extensive systems, in addition, there is the approximation of a uniform fluid by a finite portion with (say) periodic boundary conditions. The latter approximation appears to be less serious in quantum calculations than in corresponding classical ones. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

For fluids, this is computed by a statistical sampling technique, such as Monte Carlo or molecular dynamics calculations. There are a number of concerns that must be addressed in setting up these calculations, such as... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

To test the results of the chemical potential evaluation, the grand canonical ensemble Monte Carlo simulation of the bulk associating fluid has also been performed. The algorithm of this simulation was identical to that described in Ref. 172. All the calculations have been performed for states far from the liquid-gas coexistence curve [173]. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

Panagiotopoulos, A. Z., Exact calculations of fluid-phase equilibria by Monte Carlo simulation in a new statistical ensemble, Int. J. Thermophys. 1989,10, 447-457... 

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