On the relation between virial coefficients and the close-packing of hard disks and hard spheres

Maestre, Miguel Ángel G.; Santos, Andrés; Robles, Miguel; Haro, M. López de

doi:10.1063/1.3558779

Cited by 21 publications

(10 citation statements)

References 75 publications

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“…18 The availability of differently shaped colloidal model systems like spindles, 19,20 plates, 21,22 hypercubes, 23 and polyhedrons 24 has driven the interest in the phase behavior of unusually shaped hard particles. [25][26][27][28][29][30] The virial coefficients of hard ellipsoids up to the fifth order were calculated by Vega. 31 The aim of this contribution is to provide analytical expressions for the second virial coefficients of different geometries in dependence on their aspect ratio.…”

Section: Introductionmentioning

confidence: 99%

Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry

Herold

Hellmann

Wagner

2017

The Journal of Chemical Physics

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We provide analytical expressions for the second virial coefficients of differently shaped hard solids of revolution in dependence on their aspect ratio. The second virial coefficients of convex hard solids, which are the orientational averages of the mutual excluded volume, are derived from volume, surface, and mean radii of curvature employing the Isihara-Hadwiger theorem. Virial coefficients of both prolate and oblate hard solids of revolution are investigated in dependence on their aspect ratio. The influence of one- and two-dimensional removable singularities of the surface curvature to the mutual excluded volume is analyzed. The virial coefficients of infinitely thin oblate and infinitely long prolate particles are compared, and analytical expressions for their ratios are derived. Beyond their dependence on the aspect ratio, the second virial coefficients are influenced by the detailed geometry of the particles.

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Section: Introductionmentioning

confidence: 99%

Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry

Herold

Hellmann

Wagner

2017

The Journal of Chemical Physics

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“…For example, it has been a frequent practice to choose approximants containing a singularity when forming an analytic continuation of the virial series for hard‐sphere fluids, as it is believed that either a physical or nonphysical density‐singularity could set the series' radius of convergence (see Maestre et al and references therein). These approximants are used to either predict or enforce the singularity .…”

Section: Introductionmentioning

confidence: 99%

Critical isotherms from virial series using asymptotically consistent approximants

et al. 2014

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The low-density equation of state of a fluid along its critical isotherm is considered. An asymptotically consistent approximant is formed having the correct leading-order scaling behavior near the vapor-liquid critical point, while retaining the correct low-density behavior as expressed by the virial equation of state. The formulation is demonstrated for the Lennard-Jones fluid, and models for helium, water, and n-alkanes. The ability of the approximant to augment virial series predictions of critical properties is explored, both in conjunction with and in the absence of criticalproperty data obtained by other means. Given estimates of the critical point from molecular simulation or experiment, the approximant can refine the critical pressure or density by ensuring that the critical isotherm remains well-behaved from low density to the critical region. Alternatively, when applied in the absence of other data, the approximant remedies a consistent underestimation of the critical density when computed from the virial series alone. V C 2014 American Institute of Chemical Engineers AIChE J, 60: 3336-3349, 2014 Keywords: virial equation of state, critical phenomena, crossover, approximant, Lennard-Jones P c . When solved using a sufficiently high-order virial series, these predictions for T c and P c are often close to simulation data and/or experiments, but those for q c are consistently about 10% too low in such a comparison. 4,8 Underestimation of the critical density by the virial series is connected to the nonclassical behavior of real fluids at the critical point, for which critical scaling laws assert that q c is a branch-point singularity of the function P(q) along the critical isotherm. In the context of critical phenomena, the molar In this section, critical isotherms are constructed using given values of the critical density q c , critical pressure P c , Values are given in Lennard-Jones units. Virial coefficients are from Schultz et al. 5 q c,AJ is the corrected critical density given by the smallest positive root of (7), which takes the following as inputs: d 5 4.789; P c , q c , and T c as predicted by the virial series at each order J (given above); and the corresponding virial coefficients (taken from an interpolation). Numbers in parentheses specify the 68% uncertainty on the last digit, propagated from uncertainty in the virial coefficients (with the exception of the simulation values). a Predicted using (7) with P c,sim , T c,sim , and the first seven virial coefficients taken at T c,sim (from an interpolation) as inputs.

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“…Other choices of an approximant that matches known virial coefficients are possible and have been proposed [30,44,58,[60][61][62][63][64][65][66][67][68][69]. The one suggested here is given just to make a point about gauging the precision required of the B n , and the observations we make regarding it should to some degree apply to other approximants as well.…”

Section: B Virial Coefficientsmentioning

confidence: 91%

Fifth to eleventh virial coefficients of hard spheres

Schultz

Kofke

2014

Phys. Rev. E

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Virial coefficients B(n) of three-dimensional hard spheres are reported for n=5 to 11, with precision exceeding that presently available in the literature. Calculations are performed using the recursive method due to Wheatley, and a binning approach is proposed to allow more flexibility in where computational effort is directed in the calculations. We highlight the difficulty as a general measure that quantifies performance of an algorithm that computes a stochastic average and show how it can be used as the basis for optimizing such calculations.

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On the relation between virial coefficients and the close-packing of hard disks and hard spheres

Cited by 21 publications

References 75 publications

Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry

Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry

Critical isotherms from virial series using asymptotically consistent approximants

Fifth to eleventh virial coefficients of hard spheres

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