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    6. Fundamental-measure free energy density functional for hard spheres: Dimensional crossover and freezing
    Y. Rosenfeld, M. Schmidt, H. Löwen, and P. Tarazona, Phys. Rev. E 55, 4245 (1997).
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    Abstract. A geometrically based fundamental-measure free-energy density functional unified the scaled-particle and Percus-Yevick theories for the hard-sphere fluid mixture. It has been successfully applied to the description of simple "atomic" three-dimensional 3D fluids in the bulk and in slitlike pores, and has been extended to molecular fluids. However, this functional was unsuitable for fluids in narrow cylindrical pores, and was inadequate for describing the solid. In this work we analyze the reason for these deficiencies, and show that, in fact, the fundamental-measure geometrically based theory provides a free-energy functional for 3D hard spheres with the correct properties of dimensional crossover and freezing. After a simple modification of the functional, as we propose, it retains all the favorable D=3 properties of the original functional, yet gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D drastically. The modified functional is accurate for hard spheres between narrow plates (D=2), and inside narrow cylindrical pores (D=1), and it gives the exact excess free energy in the D=0 limit a cavity that cannot hold more than one particle. It predicts the vanishingly small vacancy concentration of the solid, provides the fcc hard-sphere solid equation of state from closest packing to melting, and predicts the hard-sphere fluid-solid transition, all in excellent agreement with the simulations. [figures]


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    Basic model fluids

    Density functionals were constructed for hard spheres [5] [6], penetrable spheres that interact with a step-function pair potential [7] (see [14] for more discussion), the Asakura-Oosawa-Vrij model of colloid-polymer mixtures [11], the Widom-Rowlinson model [17], and non-additive hard sphere mixtures [53].

    DFT for soft potentials

    Based on the hard sphere case [5] [6], a density functional theory (DFT) for soft (i.e. continuously varying as a function of interparticle distance) potentials was obtained by considering dimensional crossover [8], and subsequently extended to mixtures [10]. Applications concern the liquid structure of model fluids [13], and structure and freezing of star polymer solutions [15] (see also [16].)

    Hard spheres

    The hard sphere system freezes between [2] and [3], and in ~[23] dimensions (tags as contents!), as well as on stripe-patterned substrates [26]. Deep relations to dimensional crossover exist [5] [6]. Hard spheres were immersed in emulsions [9], confined to a flexible container [18], exposed to surfaces of other quenched spheres [37] and of random fiber networks [39], and subject to gravity [51]. Recently, the Rosenfeld functional [5] [6] was generalized to non-additive mixtures [53].

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MS, 20 Apr 2009.