One-Component Fluids

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

This vanishes at the critical-solution point as does (d p/d k) at the one-component fluid critical point. Thus... 

The field-density concept is especially usefiil in recognizing the parallelism of path in different physical situations. The criterion is the number of densities held constant the number of fields is irrelevant. A path to the critical point that holds only fields constant produces a strong divergence a path with one density held constant yields a weak divergence a path with two or more densities held constant is nondivergent. Thus the compressibility Kj,oi a one-component fluid shows a strong divergence, while Cj in the one-component fluid is comparable to (constant pressure and composition) in the two-component fluid and shows a weak... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

To illustrate this theory, we consider a one-component fluid with the interaction between the same species given by Eq. (36). Obviously, the model differs from that described in Sec. (II Bl). In particular, the geometrical constraints, which determine the type of association products in the case of a two-component model, are no longer valid. If we restrict ourselves to the case L < cr/2, only dimers and -mers built up of rigid, regular polygons are possible. 

As in Sec. II, we consider a mixture composed of a dimerizing one-component fluid and a giant hard sphere [21,119]. We begin with the multidensity Ornstein-Zernike equation for the mixture... 

We report here some results for a simple model of a one-component fluid interacting via a slightly modified Lennard-Jones potential, with angular-dependent associative forces. The model is considered in contact with the adsorbing surface. The principal aim of the simulation is to investigate the... 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

Fig. 10.5. Schematic diagram of the mean number of particles, (N), versus chemical potential, /u for a subcritical and a supercritical isotherm of a one-component fluid. The curve for the supercritical isotherm has been shifted up for clarity. Reprinted by permission from [6], 2000 IOP Publishing Ltd...

For a single-component or a pseudo-one-component fluid, the dimensionless parameter co is evaluated at the inlet conditions (subscript 0) from (Leung, 1996)... 

The 6-12 potential is only qualitatively like the realistic potentials that can be derived by calculations at Schroedinger level for, say, Ar-Ar interactions. But it requires careful and detailed study to see how real simple fluids (i.e. one component fluids with monatomic particles) deviate from the behavior calculated from the 6-12 model. Moreover the principal structural features of simple fluids are already quite realistically given by the hard sphere fluid. 

Let us now illustrate some of these ideas more concretely with the simple example of a one-component fluid, say a sample of water. In this case c = p = 1, leading to... 

Near the critical point a fluid is known to behave differently, and many anomalies appear in the static and dynamical properties. The important anomalies in the dynamical properties are the critical slowing down of the thermal diffusivity (Dt) in a one-component fluid and the interdiffusion of two species in a binary mixture and also the divergence of the viscosity in a binary mixture. 

S.E. show that for a simple one-component fluid the equilibrium particle density n r) in the presence of the perturbation F(r) is given exactly by... 

For a one-component fluid confined by a planar wall to which the z-axis is perpendicular, the inhomogeneous OZ equation can be written as [102]... 

Let us consider a one-component fluid confined in a pore of given size and shape which is itself located within a well-defined solid structure. We suppose that the pore is open and that the confined fluid is in thermodynamic equilibrium with the same fluid (gas or liquid) in the bulk state and held at die same temperature. As indicated in Chapter 2, under conditions of equilibrium a uniform chemical potential is established throughout the system. As the bulk fluid is homogeneous, its chemical potential is simply determined by the pressure and temperature. The fluid in the pore is not of constant density, however, since it is subjected to adsorption forces in the vicinity of the pore walls. This inhomogeneous fluid, which is stable only under the influence of the external field, is in effect a layerwise distribution of the adsorbate. The density distribution can be characterized in terms of a density profile, p(r), expressed as a function of distance, r, from the wall across the pore. More precisely, r is the generalized coordinate vector. 

For a one-component fluid, which is under the influence of a spatially varying external potential, the grand potential functional becomes... 

Consider the KS expansion. Fig. 6.3, applied to a one-component fluid and truncated at the second term displayed this will be satisfactory for low density because the subsequent terms have higher-power initial density multipliers. What is the corresponding MM approximate theory for the excess chemical potential Show that this KS approximate theory, expressed for is... 

Consider a one-component fluid of simple molecules in equilibrium with its vapour at given temperature. The interface is flat, gravity is ignored. At the interface the density profile p (z) will adjust itself in such a way as to minimize F at given T and V. 

In many colloidal and micellar systems the asymmetry in size is large enough for the experiment to measure only the macroion-macroion correlation [35], For this reason various approximations, by which macroions are assumed to interact via an effective potential, are often applied. Macroions are assumed to be surrounded by a cloud of an opposite charge and it is assumed that the overlap of two clouds results in the repulsive interaction. In a popular theory, referred to as the one-component fluid (OCF) model, the macroions interact via the repulsive screened Coulomb potential in the form,... 

---