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.
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)
We present the full thermodynamics of a fluid confined by an arbitrary external potential based on the virial expansion of the grand potential. The fluid may be classical or quantum and it is assumed that interatomic interactions are pairwise additive. We indicate how the appropriate "generalized" volume and pressure variables, that replace the usual volume and hydrostatic pressure, emerge for a given confining potential in the thermodynamic limit. A discussion of the physical meaning and of the measurement of these variables is presented. We emphasize that this treatment yields the correct eq…
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.
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…
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.
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.
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.
A simulation technique is described for quantifying the contribution of three-body interactions to the thermodynamical properties of coarse-grained representations of complex fluids. The method is based on comparing the third virial coefficient $B_3$ for a complex fluid with that of an approximate coarse-grained model described by a pair potential. To obtain $B_3$ we introduce a new technique which expresses its value in terms of the measured volume-dependent asymptote of a certain structural function. The strategy is applicable to both Molecular Dynamics and Monte Carlo simulation. Its utilit…
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…
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…
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 …
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…
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…
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…
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…
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.
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-…
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.
Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path neve…
Thermodynamic consistency of the Mean Spherical Approximation as well as the Self-Consistent Ornstein-Zernike Approximation (SCOZA) with the virial route to thermodynamics is analyzed in terms of renormalized gamma-ordering. For continuum fluids this suggests the addition of a short-range contribution to the usual SCOZA direct correlation function, and the shift of the adjustable parameter from the potential term to this new term. The range of this contribution is fixed by imposing consistency with the virial route at the critical point. Comparison of the results of our theory for the hard-cor…
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 …
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…
We discuss the computation of the third virial coefficient in polymer systems, focusing on an additional contribution absent in the case of monoatomic fluids. We determine the interpenetration ratio and several quantities that involve the third virial coefficient for star polymers with 4 and 6 arms in the good-solvent regime, in the limit of a large degree of polymerization.
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,…
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…
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…
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.
We describe a general simulation scheme for assessing the thermodynamic consequences of neglecting many-body effects in coarse-grained models of complex fluids. The method exploits the fact that the asymptote of a simple-to-measure structural function provides direct estimates of virial coefficients. Comparing the virial coefficients of an atomistically detailed system with those of a coarse-grained version described by pair potentials, permits the role of many-body effects to be quantified. The approach is applied to two models: (i) a size-asymmetrical colloid-polymer mixture, and (ii) a solu…
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…
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…
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.
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 …
The recent theoretical prediction by Maimbourg and Kurchan [arXiv:1603.05023] that for regular pair-potential systems the virial potential-energy correlation coefficient increases towards unity as the dimension $d$ goes to infinity is investigated for the standard 12-6 Lennard-Jones fluid. This is done by computer simulations for $d=2,3,4$ going from the critical point along the critical isotherm/isochore to higher density/temperature. In all cases the virial potential-energy correlation coefficient increases significantly. For a given density and temperature relative to the critical point, wi…
A method is proposed to approximate the unlimited subcritical set of Mayer`s reducible cluster integrals (i.e., the power coefficients of the known virial expansions for pressure and density in powers of activity) on the basis of information about several irreducible integrals (virial coefficients) and the saturation activity. For the Lennard-Jones fluid, the first four irreducible integrals are actually sufficient to reproduce gas isotherms (including the flat phase-transition region) with high accuracy at temperatures between the triple and critical points. At low temperatures, even the usag…
We apply a recently proposed density functional for mixtures of parallel hard cylinders, based on Rosenfeld's fundamental measure theory, to study the effect of length-polydispersity on the relative stability between the smectic and columnar liquid crystal phases.To this purpose we derive from this functional an expression for the direct correlation function and use it to perform a bifurcation analysis. We compare the results with those obtained with a second and a third virial approximation of this function. All three approximations lead to the same conclusion: there is a terminal polydispers…
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 …
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…
Using a field-theoretic approach, we derive the first few coefficients of the exact low-density (``virial'') expansion of a binary mixture of positively and negatively charged hard spheres (two-component hard-core plasma, TCPHC). Our calculations are nonperturbative with respect to the diameters $d_+$ and $d_-$ and charge valences $q_+$ and $q_-$ of positive and negative ions. Consequently, our closed-form expressions for the coefficients of the free energy and activity can be used to treat dilute salt solutions, where typically $d_+ \sim d_-$ and $q_+ \sim q_-$, as well as colloidal suspensio…
The properties of a hard-sphere fluid in contact with hard spherical and cylindrical walls are studied. Rosenfeld's density functional theory (DFT) is applied to determine the density profile and surface tension $γ$ for wide ranges of radii of the curved walls and densities of the hard-sphere fluid. Particular attention is paid to investigate the curvature dependence and the possible existence of a contribution to $γ$ that is proportional to the logarithm of the radius of curvature. Moreover, by treating the curved wall as a second component at infinite dilution we provide an analytical expres…
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 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…
We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between $σ$ and $3σ$ ( $σ$ is the side length of the square). The exact equation of state and the nearest neighbour distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers and a solid-like structure where…
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…
The equation of state for five-dimensional hard hyperspheres arising as a weighted average of the Percus-Yevick compressibility (3/5) and virial (2/5) equations of state is considered. This Carnahan-Starling-like equation turns out to be extremely accurate, despite the fact that both Percus-Yevick equations of state are rather poor.
Liquids displaying strong virial-potential energy correlations conform to an approximate density scaling of their structural and dynamical observables. This scaling property does not extend to the entire phase diagram, in general. The validity of the scaling can be quantified by a correlation coefficient. In this work a simple scheme to predict the correlation coefficient and the density-scaling exponent is presented. Although this scheme is exact only in the dilute gas regime or in high dimension d, a comparison with results from molecular dynamics simulations in d = 1 to 4 shows that it repr…
