Recent values for virial coefficients up to B12, when expressed in powers of density relative to maximum close packing,lead to a closed equation-of-state for the equilibrium fluid. The series obtained converges for all densities;it becomes negative and diverges to a negative pole at maximum packing. MD data for 64000 spheres in the metastable region shows the virial pressure begins to deviate at the fluid freezing density.
We evaluate the virial coefficients B_k for k<=10 for hard spheres in dimensions D=2,...,8. Virial coefficients with k even are found to be negative when D>=5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D>=5. Further analysis provides evidence that negative virial coefficients will be seen for some k>10 for D=4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D=3.
Several methods of extrapolating the virial coefficients, including those proposed in this work, are discussed. The methods are demonstrated on predicting higher virial coefficients of one-component hard spheres. Estimated values of the eleventh to fifteenth virial coefficients are suggested. It has been speculated that the virial coefficients, B_n, beyond B_{14} may decrease with increasing n, and may reach negative values at large n. The extrapolation techniques may be utilized in other fields of science where the art of extrapolation plays a role.
Exact results are given for the fourth virial coefficient of hard spheres in even dimensions up through 12. The fifth and sixth virial coefficients are numerically computed for dimensions 2 through 50 and it is found that the sixth virial coefficient is negative for D >= 6. Numerical studies are made of the contributing Ree Hoover diagrams up to order 17. It is found for D >= 3 that for large order a class of diagrams we call "loose packed" dominates and the rate of growth of these diagrams is used to study bounds on the radius of convergence.
The fourth virial coefficient is calculated exactly for a fluid of hard spheres in even dimensions. For this purpose the complete star cluster integral is expressed as the sum of two three-folded integrals only involving spherical angular coordinates. These integrals are solved anallytically for any even dimension d and working with existing expressions for the other terms of the fourth cluster integral we obtain an expression for the fourth virial coefficient B4(d) for even d that sums a finite number of simple terms, with the number of terms increasing with d.
We study the virial coefficients B_k of hard spheres in D dimensions by means of Monte-Carlo integration. We find that B_5 is positive in all dimensions but that B_6 is negative for all D >= 6. For 7<=k<=17 we compute sets of Ree-Hoover diagrams and find that either for large D or large k the dominant diagrams are "loose packed". We use these results to study the radius of convergence and the validity of the many approximations used for the equations of state for hard spheres.
We evidence via a computation in the reciprocal space the asymptotic behaviour of the high order virial coefficients for a hard sphere system. These coefficients, if their order is high enough, are those of a geometric series. We thus are able to give an explicit expression of the equation of states of the hard sphere system at high density when the fluid phase is no longer the stable one; in the disordered phase this equation of states exhibits a simple pole at the random close packing density. We can then estimate the packing densities of the freezing point of the disordered phase and also o…
In this paper, we generally expressed the virial expansion of ideal quantum gases by the heat kernel coefficients for the corresponding Laplace type operator. As examples, we give the virial coefficients for quantum gases in $d$-dimensional confined space and spheres, respectively. Our results show that, the relative correction from the boundary to the second virial coefficient is independent of the dimension and it always enhances the quantum exchange interaction. In $d$-dimensional spheres, however, the influence of the curvature enhances the quantum exchange interaction in two dimensions, b…
The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral $b_{2}(V)$ related with the second virial coefficient for the Hard Disc system…
Here we present new results obtained for the equation of state with induced surface and curvature tensions. The explicit formulas for the first five virial coefficients of system pressure and for the induced surface and curvature tension coefficients are derived and their possible applications are briefly discussed.
We propose a new semi empirical expression of the virial coefficients for a hard sphere fluid which is valid in the disordered phase over the whole density range. The results are in good agreement with the numerical data and better than those of the well-known Carnahan & Starling equation of state in the domain of validity of the later; moreover this new expression accounts for the singularity which is required for the equation of state at the close random packing limit.
This work is devoted to analyze the relation between the thermodynamic properties of a confined fluid and the shape of its confining vessel. Recently, new insights in this topic were found through the study of cluster integrals for inhomogeneous fluids, that revealed the dependence on the vessel shape of the low density behavior of the system. Here, the statistical mechanics and thermodynamics of fluids confined in wedges or by edges is revisited, focusing on their cluster integrals. In particular, the well known hard sphere fluid, which was not studied in this framework so far, is analyzed un…
The energy route to the equation of state of hard-sphere fluids is ill-defined since the internal energy is just that of an ideal gas and thus it is independent of density. It is shown that this ambiguity can be avoided by considering a square-shoulder interaction and taking the limit of vanishing shoulder width. The resulting hard-sphere equation of state coincides exactly with the one obtained through the virial route. Therefore, the energy and virial routes to the equation of state of hard-sphere fluids can be considered as equivalent.
Virial coefficients for the two-dimensional hard-disk fluid, when expressed in powers of density relative to maximum close packing, lead to an accurate closed equation-of-state for the equilibrium fluid, analogous to that recently found for hard spheres. The 2D series also converges for all densities up to a negative pole at close packing density. The virial pressure begins to deviate from the thermodynamic fluid in the approach to the ordering transition.
The composition-independent virial coefficients of a $d$-dimensional binary mixture of (additive) hard hyperspheres following from a recent proposal for the equation of state of the mixture [Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1] are examined. Good agreement between theoretical estimates and available exact or numerical results is found for $d=2$, 3, 4 and 5, except for mixtures whose components are very disparate in size. A slight modification that remedies this deficiency is introduced and the resummation of the associated virial series is carried out, le…
A closed virial equation-of-state for the low density fluid phase of hard disks is obtained from the known virial coefficients. The equation exhibits 6-figure accuracy for the thermodynamic (MD) pressure up to the reduced number density ~ 0.4 Interpolation of the discrepancy at higher densities indicates a higher-order thermodynamic phase transition at the extensive-intensive free-volume percolation transition previously located by Hoover et al. (JCP 70 1837 1979)
The fourth virial coefficient of additive hard-sphere mixtures, as predicted by the Percus-Yevick (PY) and hypernetted-chain (HNC) theories, is derived via the compressibility, virial, and chemical-potential routes, the outcomes being compared with exact results. Except in the case of the HNC compressibility route, the other five expressions exhibit a common structure involving the first three moments of the size distribution. In both theories the chemical-potential route is slightly better than the virial one and the best behavior is generally presented by the compressibility route. Moreover,…
The question of whether the known virial coefficients are enough to determine the packing fraction $η_\infty$ at which the fluid equation of state of a hard-sphere fluid diverges is addressed. It is found that the information derived from the direct Padé approximants to the compressibility factor constructed with the virial coefficients is inconclusive. An alternative approach is proposed which makes use of the same virial coefficients and of the equation of state in a form where the packing fraction is explicitly given as a function of the pressure. The results of this approach both for hard-…
The problem of demixing in a binary fluid mixture of highly asymmetric additive hard spheres is revisited. A comparison is presented between the results derived previously using truncated virial expansions for three finite size ratios with those that one obtains with the same approach in the extreme case in which one of the components consists of point particles. Since this latter system is known not to exhibit fluid-fluid segregation, the similarity observed for the behavior of the critical constants arising in the truncated series in all instances, while not being conclusive, may cast seriou…
Quantum effects in statistical mechanics are important when the thermal wavelength is of the order of, or greater than, the mean interatomic spacing. This is examined at depth taking the example of a hard-sphere Boltzmann gas. Using the virial expansion for the equation of state, it is shown that the interaction energy of a classical hard-sphere gas is exactly zero. When the (second) virial coefficient of such a gas is obtained quantum mechanically, however, the quantum contribution to the interaction energy is shown to be substantial. The importance of the semiclassical corrections to the int…
Different theoretical approaches for the thermodynamic properties and the equation of state for multicomponent mixtures of nonadditive hard spheres in $d$ dimensions are presented in a unified way. These include the theory by Hamad, our previous formulation, the original MIX1 theory, a recently proposed modified MIX1 theory, as well as a nonlinear extension of the MIX1 theory proposed in this paper. Explicit expressions for the compressibility factor, Helmholtz free energy, and second, third, and fourth virial coefficients are provided. A comparison is carried out with recent Monte Carlo data …
A new method of estimating high-order virial coefficients for fluids composed of equal three-dimensional rigid spheres is proposed. The predicted $B_{11}$ and $B_{12}$ values are in good agreement with reliable estimates previously reported. A new application of the Levin's transformations is developed, as well as a new way of using Levin's transformations is suggested. For the virial series of packing factor powers, this method estimates the $B_{13}$ value near 173.
A correlation between maxima in virial coefficients (Bn), and "kissing" numbers for hard hyper-spheres up to dimension D=5, indicates a virial equation and close-packing relationship. Known virial coefficients up to B7, both for hard parallel cubes and squares, indicate that the limiting differences Bn-B(n-1) behave similar to spheres and disks, in the respective expansions relative to maximum close packing. In all cases, the increment Bn-B(n-1) will approach a negative constant with similar functional form in each dimension. This observation enables closed-virial equations-of-state for cubes …
Using the first seven known virial coefficients and forcing it to possess two branch-point singularities, a new equation of state for the hard-sphere fluid is proposed. This equation of state predicts accurate values of the higher virial coefficients, a radius of convergence smaller than the close-packing value, and it is as accurate as the rescaled virial expansion and better than the Padé [3/3] equations of state. Consequences regarding the convergence properties of the virial series and the use of similar equations of state for hard-core fluids in $d$ dimensions are also pointed out.
An equation of state of the hard sphere fluid which is not analytical at the freezing density is proposed and tested. The nonanalytical term is based on the the classical nucleation theory and is able to capture the observed ``anomalous increase'' of pressure at high densities. It is combined with the virial expansion at low densities.
In this note, we propose a simple derivation of the one dimensional hard rod equation of state, with and without a Kac tail (appended long range and weak potential). The case of hard spheres in higher dimension is also addressed and it is shown there that our arguments --which avoid any mathematical complication-- allow to recover the virial form of the equation of state in a direct way.
Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for th…
The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pres…
We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the l…
The internal energy of hard spheres (HS) is the same as that of an ideal gas, so that the energy route to thermodynamics becomes useless. This problem can be avoided by taking an interaction potential that reduces to the HS one in certain limits. In this paper the square-shoulder (SS) potential characterized by a hard-core diameter $σ'$, a soft-core diameter $σ>σ'$ and a shoulder height $ε$ is considered. The SS potential becomes the HS one if (i) $ε\to 0$, or (ii) $ε\to\infty$, or (iii) $σ'\toσ$ or (iv) $σ'\to 0$ and $ε\to\infty$. The energy-route equation of state for the HS fluid is obtaine…
We present new molecular dynamics results for the pressure of the pure hard disk fluid up to the hexatic transition (about reduced density 0.9). The data combined with the known virial coefficients (up to $B_{10}$) are used to build an equation of state, to estimate higher-order virial coefficients, and also to obtain a better value of $B_{10}$. Finite size effects are discussed in detail. The ``van der Waals-like'' loop reported in literature in the vicinity of the fluid/hexatic transition is explained by suppressed density fluctuations in the canonical ensemble. The inflection point on the p…
A method for taking into account the long-range potential of atoms in the framework of the hard-sphere model is proposed. It is shown that thermodynamic quantities can be represented as a sum of three contribution: that of an ideal gas, the interaction of hard sphere, and the long-range potential. In the leading approximation on density, the corrections to the virial coefficient and heat capacity due to the smooth component of the potential are calculated. Attention is drawn to the fact that the effects determined by the long-range part of of the potential can be described in the framework of …
The equation of state for a two-dimensional hard-sphere gas is difficult to calculate by usual methods. In this paper we develop an approach for calculating the equation of state of hard-sphere gases, both for two- and three-dimensional cases. By regarding a hard-sphere gas as an ideal gas confined in a container with a multi-core (excluded sphere) boundary, we treat the hard-sphere interaction in an interacting gas as the boundary effect on an ideal quantum gas; this enables us to treat an interacting gas as an ideal one. We calculate the equation of state for a three-dimensional hard-sphere …
The available virial coefficients for the 2D hard disks model are transformed into a matrix representation of the thermodynamic potentials, which allows for an accurate description of the whole fluid phase, up to the phase transition. We find that the fluid phase terminates at the transition point, supporting the Kosterlitz-Thouless-Halperin-Nelson-Young picture of a second order phase transition into a Hexatic phase. The density and pressure at the transition are calculated from the available first ten virial coefficients, and are found to be in excellent agreement with recent Monte-Carlo cal…
We use the extension of scaled particle theory (ESPT) presented in the accompanying paper [Stillinger et al. J. Chem. Phys. xxx, xxx (2007)] to calculate numerically pair correlation function of the hard sphere fluid over the density range $0\leq ρσ^3\leq 0.96$. Comparison with computer simulation results reveals that the new theory is able to capture accurately the fluid's structure across the entire density range examined. The pressure predicted via the virial route is systematically lower than simulation results, while that obtained using the compressibility route is lower than simulation p…
We consider a system of monodisperse hard spheres immersed in a sheared fluid. We obtain the distortion of the structure factor of the hard spheres at low shear rates, within a Percus-Yevick like framework. The consequent distortion of the pair distribution function is shown to affect the transition of the hard sphere fluid into a jammed state, which is similar to the transition to the state of random close packing in the absence of shear.
When granular systems are modeled by frictionless hard spheres, particle-particle collisions are considered as instantaneous events. This implies that while the velocities change according to the collision rule, the positions of the particles are the same before and after such an event. We show that depending on the material and system parameters, this assumption may fail. For the case of viscoelastic particles we present a universal condition which allows to assess whether the hard-sphere modeling and, thus, event-driven Molecular Dynamics simulations are justified.
I use recent advances in the study of the susceptibility of the Ising model to propose a new mechanism for the freezing transition which is observed in three dimensional hard spheres.
The simplest bounded potential is that of penetrable spheres, which takes a positive finite value $ε$ if the two spheres are overlapped, being 0 otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of $T^*\equiv k_BT/ε$ (where $k_B$ is the Boltzmann constant and $T$ is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Mo…
The fourth virial coefficient of asymmetric nonadditive binary mixtures of hard disks is computed with a standard Monte Carlo method. Wide ranges of size ratio ($0.05\leq q\leq 0.95$) and nonadditivity ($-0.5\leq Δ\leq 0.5$) are covered. A comparison is made between the numerical results and those that follow from some theoretical developments. The possible use of these data in the derivation of new equations of state for these mixtures is illustrated by considering a rescaled virial expansion truncated to fourth order. The numerical results obtained using this equation of state are compared w…
Dynamical behavior of steady granular flow is investigated numerically in the inelastic hard sphere limit of the soft sphere model. We find distinctively different limiting behaviors for the two flow regimes, i.e., the collisional flow and the frictional flow. In the collisional flow, the hard sphere limit is straightforward; the number of collisions per particle per unit time converges to a finite value and the total contact time fraction with other particles goes to zero. For the frictional flow, however, we demonstrate that the collision rate diverges as the power of the particle stiffness …
By Wertheim-method the exact solution of the Percus-Yevick integral equation for a system of particles with the repulsive step potential interacting (collapsing hard spheres) is obtained. On the basis of this solution the state equation for the repulsive step potential is built and determined, that the Percus-Yevick equation does not show phase transition for collapsing hard spheres.
A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E. {\bf 76}, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) $d$ dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients $B_j$ are expressed in terms of the solution of a set of $(d-1)/2$ coupled algebraic equations which become nonlinear for $d \geq 5$. Results have been derived up to $d=13$. A confirmation of the alternating character…
The shear stress autocorrelation function has been studied recently by molecular dynamics simulation using the 1/q^n potential for very large n. The results are analyzed and interpreted here by comparing them to the shear stress response function for hard spheres. It is shown that the hard sphere response function has a singular contribution and that this is reproduced accurately by the simulations for large n. A simple model for the stress autocorrelation function at finite n is proposed, based on the required hard sphere limiting form.
A hard sphere fluid confined by hard, structureless, and parallel walls is investigated using a certain version of weighted density functional theory. The density profile, the excess coverage, the finite size contribution to the free energy, the solvation force, and the total correlation function are determined as function of the slit width $L$ for various bulk number densities $ρ_b$. In quantitative agreement with rigorous results the present version of density functional theory yields a constant and large but finite number density profile for the limiting case that $L$ is reduced to the diam…
An equation of state for a multicomponent mixture of non-additive hard spheres in $d$ dimensions is proposed. It yields a rather simple density dependence and constitutes a natural extension of the equation of state for additive hard spheres proposed by us [A. Santos, S. B. Yuste, and M. López de Haro, Mol. Phys. 96, 1 (1999)]. The proposal relies on the known exact second and third virial coefficients and requires as input the compressibility factor of the one-component system. A comparison is carried out both to another recent theoretical proposal based on a similar philosophy and to the ava…
Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume for a general convex body that applies in any space dimension, including both the rotationally-averaged exclusion volume and with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume $v_{ex}(K)/v(K)$ among all convex bodies, whether randomly oriented or uniformly oriented, for any $d$, where $v(K)$ is the volume of $K$. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any $d…
Model of hard sphere system is important part of modern theories of liquids. Radial distribution function of hard sphere fluid represented in form of explicit analytical expression allows to obtain thermodynamic potentials in analytical form, which is very helpful for further analysis and applications. A new analytical expression of radial distribution function of hard sphere fluid is developed. The density derivative of the radial distribution function and the first two terms of Barker-Henderson perturbation theory are derived as analytical expressions. The obtained results agree well with pu…
We obtain, using extensive Monte Carlo simulations, virial expansion and a high-density perturbation expansion about the fully packed monodispersed phase, the phase diagram of a system of bidispersed hard rods on a square lattice. We show numerically that when the length of the longer rods is $7$, two continuous transitions may exist as the density of the longer rods in increased, keeping the density of shorter rods fixed: first from a low-density isotropic phase to a nematic phase, and second from the nematic to a high-density isotropic phase. The difference between the critical densities of …
In this paper, we present a cluster algorithm for the simulation of hard spheres and related systems. In this algorithm, a copy of the configuration is rotated with respect to a randomly chosen pivot point. The two systems are then superposed, and clusters of overlapping spheres in the joint system are isolated. Each of these clusters can be ``flipped'' independently, a process which generates non-local moves in the original configuration. A generalization of this algorithm (which works perfectly well at small density) can be successfully made to work at densities around the solid-liquid trans…
Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schrödinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary …
Values of the fifth virial coefficient, compressibility factors, and fluid-fluid coexistence curves of binary asymmetric nonadditive mixtures of hard disks are reported. The former correspond to a wide range of size ratios and positive nonadditivities and have been obtained through a standard Monte Carlo method for the computation of the corresponding cluster integrals. The compressibility factors as functions of density, derived from canonical Monte Carlo simulations, have been obtained for two values of the size ratio ($q=0.4$ and $q=0.5$), a value of the nonadditivity parameter $Δ=0.3$), an…
As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus--Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient $B_4$ predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of $B_4$ obtained from the virial route in the HNC theory…
We calculate the second and third virial coefficients of the effective sphere-sphere interaction due to polymer depletion. By utilizing the anisotropy of a typical polymer conformation, we can consider polymers that are roughly the same size as the spherical inclusions. We argue that recent experiments can confirm this anisotropy.
