Fluid properties, integral equations thermodynamic integration

Verlet, L. Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 165 (1967) 98-103. Ryckaert, J.-P., Ciccotti,G., Berendsen, H.J.C. Numerical integration of the cartesian equations of motion of a system with constraints Molecular dynamics of n-alkanes. Comput. Phys. 23 (1977) 327-341. 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

An example drawn from Deitrick s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the thermodynamic goodness of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4, 5]. 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

We have considered the situation of only one phase for any mixture composition this means that there is no surface tension and the fluid behavior is completely characterized by the turbulent flow described by the mass and momentum balance equations. To solve these equations, one needs to model the diffusional mixing of the species present in the system and to identify local values of the thermodynamic and transport properties, as considered in Section 3.2. Here we just point out that once the methods for predicting local values of fluid density and viscosity have been worked out, one should be able to integrate Eqs. (10) and (11). 

Studies of the thermodynamic properties obtained from solutions of the SSOZ equation have been much more extensive but the success has been mixed. The first such calculations were those of Lowden and Chandler who obtained the pressure of hard diatomic fluids from the RISM (SSOZ-PY) equation. They used two routes to the equation of state a compressibility equation of state in which they integrated the bulk modulus calculated from the site-site correlation functions via... 

In the present chapter we have reviewed a numerically efficient and accurate equation of library state for high pressure fluids and solids. Thermodynamic cycle theories allow us to apply this model profitably to the reactions of energetic materials. The equation of state is based on HMSA integral equation theory, with a correction based on extensive Monte Carlo simulations. We have also shown that our equation of state can be used to accurately model the properties of molecular fluids and detonation products. The accuracy of the equation of state of polar fluids is significantly enhanced by using a multi-species or cluster representation of the fluid. 

An alternative, and particularly successful, approach to liquids is provided by the thermodynamic perturbation theories.In this approach, the properties of the fluid of interest are related to those of a reference fluid through a suitable expansion. One attempts to choose a reference system that is in some sense close to the real system, and whose properties are well known (e.g., through computer simulation studies or an integral equation theory). 

Over the p t several years we and our collaborators have pursued a continuous space liquid state approach to developing a computationally convenient microscopic theory of the equilibrium properties of polymeric systems. Integral equations method [5-7], now widely employed to understand structure, thermodynamics and phase transitions in atomic, colloidal, and small molecule fluids, have been generalized to treat macromolecular materials. The purpose of this paper is to provide the first comprehensive review of this work referred to collectively as Polymer Reference Interaction Site Model (PRISM) theory. A few new results on polymer alloys are also presented. Besides providing a unified description of the equilibrium properties of the polymer liquid phase, the integral equation approach can be combined with density functional and/or other methods to treat a variety of inhomogeneous fluid and solid problems. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Table 2. A comparison between Monte Carlo results and the predictions of integral-equation theories of the thermodynamic properties of a Lennard-Jones fluid. MC = Monte Carlo results of Hansen and Verlet (1969) PY and HNC results are from Levesque (1966) and Verlet and Levesque (1967). c = from compressibility equation p = from pressure equation. (Hansen and McDonald, 1986)... 

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