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Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories for equilibrium properties. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression proposed by Yuste et al. (J. Chem. Phys. 99, 2020,1993), which approximates the 2D RDF with an interpolation of the RDFs for hard-rod (1D) and hard-sphere (3D) fluids and provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on the approach proposed by Trokhymchuk et al. for the 3D RDF (J. Chem. Phys., 123, 024501, 2005). The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure constraint and the isothermal compressibility constraint are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results., Comment: 15 pages, 11 Figures

In this work the interface system of the van der Waals fluid is investigated by using the density gradient theory incorporated with the mean-field theory. Based on the mean-field dividing interface generated by the Maxwell construction, we propose a highly accurate density profile model for the density gradient theory, which facilitates reliable predictions of various properties for the interface region. It is found that the local intrinsic Helmholtz free energy peaks at the interface and that the maximum difference of the normal and tangential components of the pressure tensor corresponds to the maximum of the intrinsic Gibbs free energy. It is found that the entire phase space is divided into gas-like and liquid-like regions by the single line composed of the mean-field interface and the Widom line. The two-fluid feature of the supercritical fluid is hence inherited from the coexistence region. Phase diagrams extended into the coexistence region in all the temperature-pressure-volume planes are thus completed with the solutions to the vapor-liquid equilibrium problem by the van der Waals equation of state., Comment: 19 pages, 6+ Figures. arXiv admin note: substantial text overlap with arXiv:2110.07838

We define a mean-field crossover generated by the Maxwell construction as the dividing interface for the vapor-liquid interface area and a highly accurate density-profile equation is thus derived. By using a mean-field equation of sate for the Lennard-Jones fluid incorporated with the density gradient theory, we show that the intrinsic free energy peaks and the isobaric heat capacity exhibits local maxima at the interface. We demonstrate that the mean-field interface is the natural extension of the Widom line into the coexistence region, hence the entire space is coherently divided into liquid-like and gas-like regions in all three (temperature-pressure-volume) planes. Finally, the mean-field theory is found holding all the information for composing the phase diagrams over the entire phase space., Comment: 8 Pages, 5 Figures. arXiv admin note: substantial text overlap with arXiv:2110.07838

In this work we define a mean-field crossover generated by the Maxwell construction as the dividing interface for the vapor-liquid interface area. A highly accurate density-profile equation is thus derived, which is physically favorable and leads to reliable predictions of interfacial properties. By using the density gradient theory and a mean-field equation of sate for the Lennard-Jones fluid, we are able to extensively explore the interface system in terms of the Gibbs free energy, the Helmholtz free energy and heat capacity. The results show that the mean-field dividing interface is the natural extension of the Widom line into the coexistence region. Hence the entire phase space is coherently divided into liquid-like and gas-like regions in all three (temperature-pressure-volume) planes. Some unconventional behaviors are observed for the intrinsic heat capacity, being positive in low temperature region while negative in high temperature region. Finally, a complete picture of the mean-field equation of state is unfolded: all three solutions to a vapor-liquid equilibrium problem have their respective significances., Comment: 23 pages; 19 figures

This paper presents a study on the relationship between transport properties and geometric free volume for hard sphere (HS) system in dense fluid region. Firstly, a generic free volume distribution function is proposed based on recent simulation results for the HS geometric free volume by Maiti et al. [1,2] Combining the new distribution function with a local particle transportation model, we obtain a power law for the HS transport properties. Then a relation between the geometric free volume and thermodynamic free volume is established, which makes it possible to use well-developed equations of state (EoS) for the expressions of the geometric free volume. The new power law models are tested with molecular dynamic (MD) simulation results for HS viscosity, diffusivity and thermal conductivity, respectively and the results are very satisfactory. Using the power law we are able to reproduce several equations obtained from different approaches, such as the entropy scaling laws [3], mode coupling theory [4] or empirical correlations [5]. In particular, A long-standing controversy regarding the well known Cohen-Turnbull-Doolittle free volume model [6,7] is resolved by using the power law combined with an EoS., Comment: 12 Pages, 10 Figures

Since a transcendental equation is involved in vapor liquid equilibrium (VLE) calculations with a cubic equation of state (EoS), any exact solution has to be carried out numerically with an iterative approach [1,2]. This causes significant wastes of repetitive human efforts and computing resources. Based on a recent study [3] on the Maxwell construction [4] and the van der Waals EoS [5], here we propose a procedure for developing analytically approximate solutions to the VLE calculation with the Soave-Redlich-Kwong (SRK) EoS [6] for the entire coexistence curve. This procedure can be applied to any cubic EoS and thus opens a new area for the EoS study. For industrial applications, a simple databank can be built containing only the coefficients of a newly defined function and other thermodynamic properties will be obtained with analytical forms. For each system there is only a one-time effort, and therefore, the wastes caused by the repetitive efforts can be avoided. By the way, we also show that for exact solutions, the VLE problem with any cubic EoS can be reduced to solving a transcendental equation with one unknown, which can significantly simplify the methods currently employed [2,7]., Comment: 14 Pages; 7 Figures

The well-known Maxwell construction[1] (the equal-area rule, EAR) was devised for vapor liquid equilibrium (VLE) calculation with the van der Waals (vdW) equation of state (EoS)[2]. The EAR generates an intermediate volume between the saturated liquid and vapor volumes. The trajectory of the intermediate volume over the coexistence region is defined here as the Maxwell crossover, denoted as the M-line, which is independent of EoS. For the vdW or any cubic[3] EoS, the intermediate volume corresponds to the unphysical root, while other two corresponding to the saturated volumes of vapor and liquid phases, respectively. Due to its unphysical nature, the intermediate volume has always been discarded. Here we show that the M-line, which turns out to be strictly related to the diameter[4] of the coexistence curve, holds the key to solving several major issues. Traditionally the coexistence curve with two branches is considered as the extension of the Widom line[5,6-9]. This assertion causes an inconsistency in three planes of temperature, pressure and volume. It is found that the M-line is the natural extension of the Widom line into the vapor-liquid coexistence region. As a result, the united single line coherently divides the entire phase space, including the coexistence and supercritical fluid regions, into gas-like and liquid-like regimes in all the planes. Moreover, along the M-line the vdW EoS finds a new perspective to access the second-order transition in a way better aligning with observations and modern theory[10]. Lastly, by using the feature of the M-line, we are able to derive a highly accurate and analytical proximate solution to the VLE problem with the vdW EoS., Comment: 18 pages, 11 Figures

The well-known Carnahan-Starling (CS) equation of state (EoS) [1] for the hard sphere (HS) fluid was derived from a quadratic relation between the integer portions of the virial coefficients, Bn, and their orders, n. Here we extend the method to the full virial coefficients Bn for the general D-dimensional case. We assume a polynomial function of (D-1)th order for the virial coefficients starting from n=4 and EoS are derived from it. For the hard rob (D=1) case, the exact solution is obtained. For the stable hard disk fluid (D=2), the most recent virial coefficients up to the 10th [2] and accurate compressibility data[3,4] are employed to construct and test the EoS. For the stable hard sphere (D=3) fluid, a new CS-type EoS is constructed and tested with the most recent virial coefficients [5,2] up to the 11th and with the highly-accurate simulation data for compressibility [6-8]. The simple new EoS turn out to be as accurate as the highest-level Pade approximations based on all available virial coefficients, and significantly improve the CS-type EoS in the hard sphere case. We also shown that as long as the virial coefficients obey a polynomial function any EoS derived from it will diverge at the non-physical packing fraction=1., Comment: 8 pages, 6 figures

The hard disc system plays a fundamental role in the study of two-dimensional matters [1-3]. High-precision compressibility data from computer simulations have been reported for all the phases and phase transition regions [4-15]. In particular, Bernard and Krauth (Phys. Rev. Lett., 107, 155704, 2011) [10] presented a complete and accurate picture of the phase transitions of the hard disc system with simulation results. However, thorough descriptions of the system depend on analytical equations of state (EoS) over the entire density range. While majority of EoS published are for the stable fluid region only [1,16], few attempted the liquid-hexact transition region (Phys. Rev. Lett., 11, 241, 1963 [17]; Phys. Rev. E. 63, 042201, 2001 [18]; 74, 061106, 2006 [19]). All the EoS currently available are incapable of quantitative descriptions of the phase transitions. Here we construct a simple EoS to reproduce high-precision simulation data for all the stable liquid, liquid-hexatic transition region and hexatic phase. A global EoS is then obtained when the new EoS is smoothly united with a revisited EoS for the solid phase. Using this global equation, we are able to accurately identify all the phases and the phase transitions from the stable liquid to hexatic, then to solid phases. The liquid-hexatic transition is found to be of weak first-order, namely discontinuous in density and the Gibbs free energy while continuous in entropy and the Helmholtz free energy. The hexatic-solid transition is a continuous high-order phase transition., Comment: 9 Pages, 9 Figures

The hard sphere system plays a basic role in condensed matter physics and related fields, and equation of state (EoS) is the ultimate solution to its thermodynamic properties (1-3). Dozens of EoSs have been proposed since van der Waals historic work and many reliable EoSs are available for the stable fluid region (3). For the metstable region, all available EoSs are not accurate enough for various applications. It has been considered impossible to develop an analytical EoS for the entire stable and metstable region 4. By virtue of a potential energy landscape analysis combined with the Woodcock type EoS (2,5), here we show that a fairly simple analytical equation can be obtained to reproduce the compressibility of the entire region with high accuracy. Therefore, all four amorphous states of matter, gas, liquid, supercooled liquid and glass, can be represented with a single EoS. Examples are given to show that highly accurate EoS is necessary for applications in thermodynamic property or liquid structure predictions. By using conventional approaches, such as appending an attractive term of van der Waals type (6) or using the equation within the framework of perturbation theory (1,7), it can be extended to an EoS for various real systems, including supercooled liquids and glasses., Comment: 26 pages, 4 figures, 3 tables

Glass transition and random packing in the hard sphere system have attracted great attention due to the important role of the system in the investigation of diverse real systems including liquids, colloidal dispersions, supercooled liquids, glasses and granular materials (1-12) Despite the importance and simplicity of the system, some fundamental questions, such as the existence of an ideal glass transition (5-16), the nature of the glass transition and random packing (or jamming), and the entropy crisis or Kauzmann paradox (17,18), remain open. Based on a very accurate equation of state over the entire stable and metstable region within the potential energy landscape framework, here we report two phase transitions observed in the hard sphere system: the first is the ideal glass transition, indicating the configurational entropy vanishing, and the second, the jamming transition between the random loose packing and the random close packing 1,2 (or maximally random jammed packing (11,12)), indicating inherent structure domination. However, it is suggested that the glass and jamming transitions might not be treated as a thermodynamic phase transition. The unbalanced entropy loss suggests that equilibrium thermodynamics does not work for supercooled liquids and glasses. The results presented here for the hard sphere system will have direct impact on studies related to random packing or jamming and will shed a light on studies of glass transition in real systems.