Density functional theory hard-sphere fluid

Wichert, J. M. Gulati, H. S. Hall, C. K. J. (1996). Binary hard chain mixtures. I. Generalized Flory equations of state. /. Chem. Phys., Vol. 105, 7669-7682 Woodward, C. E. Forsman, J. (2008). Density functional theory for polymer fluids with molecular weight polydispersity. Phys. Rev. Lett., Vol. 100,098301 Woodward, C. E. (1991). A density functional theory for polymers Application to hard chain-hard sphere mixtures in slitlike pores. /. Chem. Phys., Vol. 94,3183-3191 Woodward, C. E. Forsman, J. (2009). Interactions between surfaces in polydisperse semiflexible polymer solutions. MacromoL, Vol. 42, 7563-7570 Woodward, C. E. Yethiraj A. (1994). Density functional theory for inhomogeneous polymer solutions. /, Chem. Phys., Vol. 100,3181-3186... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

Integral equation methods provide another approach, but their use is limited to potential models that are usually too simple for engineering use and are moreover numerically difficult to solve. They are useful in providing equations of state for certain simple reference fluids (e.g., hard spheres, dipolar hard spheres, charged hard spheres) that can then be used in the perturbation theories or density functional theories. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

The choice of the weighting function w depends on the version of density functional theory used. For highly inhomogeneous confined fluids, a smoothed or nonlocal density approximation is introduced, in which the weighting function is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a... 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

D. Gillespie, W. Nonner, and R.S. Eisenberg. Density functional theory of charged, hard-sphere fluids. Phys. Rev. E, 68 031503, 2003. 

It should be noted that the SPT is not a pure molecular theory in the following sense. A molecular theory is supposed to provide, say, the Gibbs free energy as a function of T, P, N as well as of the molecular parameters of the system. Once this function is available, the density of the system can be computed from the relation p = (9/x/9 )t (with pi = G/N). The SPT utilizes the effective diameter of the solvent molecules as the only molecular parameter (which is the case for a hard-sphere fluid) and, in addition to the specification of T and P, the solvent density Pw is also used as input in the theory. The latter being a measurable quantity carries with it implicitly any other molecular properties of the system. The first application of the SPT to calculate the thermodynamics of solvation in liquids was carried out by Pierotti (1963, 1965). 

Specializing to planar walls for the moment, one has the exact relation [45] that p/kT = where is the local density of the adsorbate in contact with the wall when the pressure of the hard sphere fluid is p. For hard sphere mixtures [46-49] n . is the sum of the individual densities for each of the components in the fluid. Thus, the pressure of the fluid can be obtained from estimates of the intercept of the curve of n z) versus z for example. Fig. 1 indicates that palg/kT is between 8 and 9 for this system. This result, taken together with the calculation of F at a given n from Eq. (10), allows one to construct the isotherm TO). Figure 2 shows the adsorption of a hard sphere fluid on a hard wall as a function of the bulk-phase density [44]. The simulation points compare well with results of two theoretical calculations based on the scaled particle theory. 

Despite the fact that real molecules are not hard spheres, the Enskog theory has been used to describe transport properties of real fluids over a wide range of densities and temperatures with a considerable degree of success. To apply the Enskog theory to real systems one must assume that (a) the mechanisms for the transport of energy and momentum in a real system do not differ in any essential way from the mechanisms of transport in a hard-sphere fluid, and (b) the expressions for the transport coefficients of a real fluid at a given temperature and density are identical to those of a hard-sphere fluid at the same density, provided one replaces a and x(ti) in the hard-sphere expressions by quantities d and x(T) where d is an effective hard-sphere diameter of the molecules at temperature T, and x(T) is an effective radial distribution function that takes into account the temperature dependence of the collision frequency in the real fluid. ... 

Rosenfeld Y Free-energy model for the inhomogeneous hard-sphere fluid mixture and density functional theory of freezing, Phys Rev Lett 63(9) 980—983, 1989. 

Forsman, J. Woodward, C. E. (2003). An improved density functional theory of hard sphere polymer fluids at low density. /, Chem. Phys., Vol. 119,1889-1892 Forsman, J. Woodward, C. E. (2005). Prewetting and layering in athermal polymer solutions. Phys. Rev. Letts., Vol. 94,118301... 

The excess free energy of the hard-sphere fluid may be evaluated by means of methods following from different versions of the density functional theory. Numerous approximations can be used ... 

The Kierhk-Rosinberg [264] theory is formally similar to the treatment of Tarazona. They use four weight functions, which are chosen such that this theory exactly reproduced the Percus-Yevick result for the direct correlation function in the uniform hard-sphere fluid. However, these functions are independent of the density. In consequence, the calculation of the weighted density... 

Kalos, Levesque, and Verlet ( ) reported a study on the hard-sphere fluid and crystal. They found the energy about 3-5% deeper than had been obtained varia-tionally (17) and a structure function some 10-20% sharper. xEe authors also developed a perturbation theory connecting hard-sphere and other strongly repulsive potentials. Using this relation they estimated a minimum energy for fluid He-4 with Lennard-Jones potentials (Eq. (10)) as -6.8 0.2°K/atom occur-ing at l.Ot.l of the experimentally observed density, PO ... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

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