Lennard-Jones, generally fluid

At variance from the HS system, it had been observed that PY is not as accurate for attractive potentials. Hence, an alternative closure has been derived and consists in a generalization of the MSA closure [49,50], This has been feasible by incorporating the division scheme introduced by Weeks et al. [51] for the (12-6) Lennard-Jones (LJ) fluid composed of particles interacting through the potential... 

Application of DFT as a general methodology to classical systems was introduced by Ebner et al. (1976) in modeling the interfacial properties of a Lennard-Jones (LJ) fluid. The basis of all DFTs is that the Helmholtz free energy of an open system can be expressed as a unique functional of the density distribution of the constituent molecules. The equilibrium density distribution of the molecules is obtained by minimizing the appropriate free energy. 

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 F[p(r)] functional can be separated into an ideal gas term and contributions from the repulsive and attractive forces between the adsorbed molecules (i.e. the fluid-fluid interactions). Hard-sphere repulsion and pairwise Lennard-Jones 12-6 potential are usually assumed and a mean field treatment is generally applied to the long-range attraction. However, the evaluation of the density profile of an inhomogeneous hard-sphere fluid presents a special problem since its free energy is... 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Mryglod, I.M., and Omelyan, I.P. Generalized collective modes for a Lennard-Jones fluid in higher mode approximations. Phys. Lett. A, 1995, 205, p. 401 106. 

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... 

Fig. 7. A random configuration of atoms (black) surrounded by exclusion spheres (gray). The disconnected pockets of space that lie outside of the generally overlapping exclusion spheres are termed cavities (cross-hatched). A natural choice for the effective exclusion radius for the Lennard-Jones fluid is r.v = ct, the Lennard-Jones diameter.

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. 

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]. 

Figure 7. Time dependence of the wavelength of the fastest growing density fluctuation (Affiax) during a molecular dynamics simulation of isothermal liquid-vapor spinodal decomposition in the three-dimensional Lennard-Jones fluid kT/e = 0.8 p

The object of any statistical mechanical theory of polymer systems is ultimately to relate the measurable physical properties of the system to the properties of the constituent monomers and their mutual interactions. It is imperative that the initial statistical mechanical theories of these physical properties of polymer systems not depend on the exact details of a particular polymer. Instead, these theories should reflect those generic properties of polymer systems that are a result of the chainlike structures of polymer molecules. Once the properties of simple, yet general, models of polymers are well understood, it is natural to focus attention upon the particular aspects of a polymer of interest. The initial use of simple models of polymers is not solely dictated by an attempt to obtain those general features of polymer systems. The mathematical simplicity of the model is required so that we avoid the use of uncontrollable mathematical approximations which necessarily arise with the use of more complicated models. When the model is sufficiently simple, yet physically nontrivial, we are able to test different approximation schemes to find those that are useful. Presumably these methods of approximation would also be useful for more complicated models. This emphasis upon mathematical simplicity has its analog in the theory of fluids. First hard-core interactions can be used to test the physical principles associated with various methods of approximation. Once physically sound approximation schemes have been obtained with this model, they may be applied with more realistic potentials, e.g., the Lennard-Jones potential, which require subsequent numerical approximations. Thus we wish to separate approximations of a physical origin from those of purely a numerical nature. This separation... 

The existence of two competing local structures may appear as an essential condition for the occurrence of anomalous phase behaviors in general, and of LLPT in particular. To investigate if this is actually the case, we wish to examine models of CS fluids in which the soft repulsion is made progressively weaker and weaker so as to explore the behavior of systems with features intermediate between CS fluids with two distinct length scales and standard simple fluids with only one length scale (such as, e.g., the Lennard-Jones fluid). 

A second, but less satisfactory, version of the same idea is provided by the use of a prescribed intermolecular pair potential such as the Lennard-Jones (12-6) model (Maitland et al. 1987). This model, while not an exact representation of any known intermolecular force, has the virtues of mathematical simplicity and the generally correct gross features of real intermolecular potentials. The disadvantages are that when used in the manner outlined above to evaluate the appropriate functionals for a particular property of a fluid, it is unable to represent the experimentally observed behavior correctly even in a limited temperature range. As a consequence, the predictions of properties that follow from its use are intrinsically less accurate than those of the corresponding-states procedure. Indeed, this example serves to illustrate a general point that the more constrained the model attached to a rigorous theory, the less reliable the predictions of the procedure based upon it. 

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