Rigorous derivation of a binary-ternary Boltzmann equation for a dense gas of hard spheres
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This paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.Keywords:
Hard spheres
Momentum (technical analysis)
Lattice Boltzmann methods
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Poisson–Boltzmann equation
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Based on the existing expressions for virial series, molecular-dynamics data, and the expected asymptotic behavior, a new improved equation of state for dense rigid-sphere gases is developed. Comparisons with the existing equations and computer simulation data are made. It is shown that the developed equation of state is in excellent agreement with the simulation results. In particular, the equation is well suited for describing the state of dense-sphere gases at extreme high concentration limits as long as the transition to an order state does not occur.
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Operator (biology)
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We demonstrate that the main features of DPD may be obtained using molecular dynamics employing a deterministic thermostat.This apparent isomorphism holds as long as the MD pair potentials are sufficiently smooth and short ranged, which gives rise to a quadratic equation of state (pressure as a function of density).This is advantageous because it avoids the need to use stochastic forces, enabling a wider choice of integration algorithms, involves fully time reversible motion equations and offers a simpler algorithm to achieve the same objective.The isomorphism is explored and shown to hold in 2 and 3 physical dimensions as well as for binary and ternary systems for two different choices of pair potential.The mapping between DPD and Hildebrand's regular solution theory (a consequence of the quadratic equation of state) is extended to multicomponent mixtures.The procedure for parametrization of MD (identical to that of DPD) is outlined and illustrated for a equimolar binary mixture of SnI4 and isooctane (2,2,4-trimethylpentane).
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In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.
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This paper describes the application of the area method iterative technique to the phase equilibrium problem of pure component, binary and ternary systems. The equations derived from the objective function of the area method for the solution of this important problem are shown to be consistent with basic thermodynamic criteria. The essential soundness of the technique is illustrated using a variety of systems taken from the literature that exhibit two- and three-phase behaviour under different conditions.
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Helmholtz free energy
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We investigate in detail a thermodynamically self-consistent method to calculate the thermodynamics and structure of a binary mixture of simple liquids, introduced recently by one of us [Y. Rosenfeld, J. Chem. Phys. 98, 8126 (1993); Phys. Rev. Lett. 72, 3831 (1994); J. Phys. Chem. 99, 2857 (1995); Phys. Rev. E 54, 2827 (1996)]. This approximation is based on the universality hypothesis of bridge functionals and leads to a modified hypernetted-chain-type closure to the Ornstein-Zernike equations. We employ the fundamental-measure bridge functional of hard spheres. The bridge functions are calculated from this functional by inserting the appropriate structure functions of the actual system and of a suitably chosen hard-sphere reference system. An iterative procedure is repeated until numerical self-consistency is obtained. We demonstrate the reliability and wide applicability of this method by comparing numerical results with computer simulation data for a large variety of systems. Finally, we show for the example of the classical inversion problem of liquid state theory that our method can indeed replace computer simulations in more complex procedures without loss of numerical accuracy. \textcopyright{} 1996 The American Physical Society.
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