Zeno Line Research Articles

Using the Zeno line to assess and refine molecular models.

The Zeno line is the locus of points on the temperature-density plane where the compressibility factor of the fluid is equal to one. It has been observed to be straight for a broad variety of real fluids, although the underlying reasons for this are still unclear. In this work, a detailed study of the Zeno line and its relation to the vapor-liquid coexistence curve is performed for two simple model pair-potential fluids: attractive square-well fluids with varying well-widths λ and Mie n-6 fluids with different repulsive exponents n. Interestingly, the Zeno lines of these fluids are curved, regardless of the value of λ or n. We find that for square-well fluids, λ ≈ 1.8 presents a Zeno line, which is the most linear over the largest temperature range. For Mie n-6 fluids, we find that the straightest Zeno line occurs for n between 8 and 10. Additionally, the square-well and Mie fluids with the straightest Zeno line showed the closest quantitative agreement with the vapor-liquid coexistence curve for experimental fluids that follow the principle of corresponding states (e.g., argon, xenon, krypton, methane, nitrogen, and oxygen). These results suggest that the Zeno line can provide a useful additional feature, in complement to other properties, such as the phase envelope, to evaluate molecular models.

Mesoscopic metastable liquid in congruent vapor-liquid diagram of argon from about zero up to boyle`s temperature (review of FT-model)

Such paradigms of the coupled classical metastability and nonclassical criticality as the existence of a unified EOS (common for both gas and liquid phases) with its mean-field (mf), so-called Andrews-van der Waals’ critical point (CP) should be questioned to recognize the realistic stratified structure of a mesoscopic liquid phase. It exists supposedly in the wide range of temperatures located between about zero , K and up to the singular first Boyle’s point . Its opposite, also singular second Boyle’s point corresponds to the alternative origin for the crossover continuous bounds separating the specific structural strata of a mesoscopic liquid. The region of a heterogeneous l-phase spanning the whole temperature range can be termed the non-Gibbsian phase (due to its discrete cluster-like structure) without any appeals to the concept of a spinodal decomposition. The respective metastable liquid stratum is formed by three segments of supercritical , subcritical and sublimation metastable states of a formally incompressible liquid constrained by the pair of fixed extensive parameters (N,V). Its location on the CVL-diagram is restricted by the new introduced here ml-bound and by the known Zeno-line (ZL) bound. Thus, all above-mentioned strata belong to the region of soft fluid with the dominance of interparticle attraction. The remaining parts of CVL-diagram are spanned either by the real gas state-points and solid state-points (crystalline and/or amorphous) or by the region of hard fluid in the classification proposed by Ben-Amotz and Herschbach.

The ideal lines on the phase diagrams of liquids in 2D space

We use the numerical simulations for several model systems to investigate the behavior of the ideal contours and their correlations with the liquid-gas coexistence curve in 2D space. These contours are the line of the unit compressibility factor (Zeno line) and the line of ideal enthalpy (H line). Earlier, in 3D space several similarity relations were found for these lines. For various 3D real and model systems they are the straight lines along all density-temperature plane. The deviations from the linear form occur in cases when the interaction potentials are relatively too short-ranged or too long-ranged. However, in all cases, independently of its form, the Zeno line is the tangent to the liquid binodal branch at relatively low temperatures. These properties were confirmed in 2D cases as well. But one more relation connecting the parameters of the critical point and the Zeno line parameters appears to be different in 2D and 3D cases.

Experimental Investigation and Theoretical Modelling of a High-Pressure Pneumatic Catapult Considering Dynamic Leakage and Convection.

A high-pressure pneumatic catapult works under extreme boundaries such as high-pressure and rapid change of pressure and temperature, with the features of nonlinearity and gas-solid convection. In the thermodynamics processes, the pressure is much larger than the critical pressure, and the compressibility factor can deviate from the Zeno line significantly. Therefore, the pneumatic performance and thermo-physical properties need to be described with the real gas hypothesis instead of the ideal gas one. It is found that the analytical results based on the ideal gas model overestimate the performance of the catapult, in comparison to the test data. To obtain a theoretical model with dynamic leakage compensation, leakage tests are carried out, and the relationship among the leakage rate, pressure and stroke is fitted. The compressibility factor library of the equation of state for compressed air is established and evaluated by referring it to the Nelson-Obert generalized compressibility charts. Based on the Peng–Robinson equation, a theoretical model of the high-pressure pneumatic catapult is developed, in which the effects of dynamic leakage and the forced convective heat transfer between the gas and the metal wall are taken into account. The results from the theoretical model are consistent with the data from ejection tests. This research presents an approach to study the performance of a high-pressure pneumatic catapult with high precision.

Comparison of different alpha functions, α(Tr), applied in the prediction of supercritical properties of different polar and nonpolar fluids at Boyle temperature

The Boyle temperature is the temperature at which the second virial coefficient becomes zero. In this work, capability of different alpha functions, α(Tr), in prediction of supercritical properties of different polar and nonpolar fluids at Boyle temperature is investigated. In this direction, eight different alpha functions, α(Tr), including Peng – Robinson original form (1976), Coquelet et al. (2004), Haghtalab et al. (2011), Saffari – Zahedi (2013), Soave – Redlich – Kwong original form (1972), Ozokwelu – Erbar (1987), Soave (1993) and Nasrifar – Bolland (2004) are coupled with different cubic equations of states including Peng – Robinson (PR), Soave – Redlich – Kwong (SRK), volume – translated Peng – Robinson (VTPR), volume translated Soave Redlich-Kwong (VTSRK) and Patel ‒ Teja ‒ Valderrama (PTV). The obtained results have been compared with the most popular relations, such as Tsonopoulos and Meng – Duan correlations. To understand better, the second virial coefficient and Zeno line have been also depicted and compared with the available experimental data. In addition, the residual properties (HR, SR, UR, and GR) and the second order thermodynamic derivatives (Cp, Cv, us, μJT) at the Boyle condition have been also investigated. Furthermore, the trend of the inversion temperature (Tinv) versus the Boyle temperature is studied for n-alkanes, n-alkenes, amines and noble gases.

The Zeno line and binodal for Ga

A line of unit compressibility factor (Zeno line) was calculated for liquid gallium by means of Monte-Carlo simulations with the embedded atom potential. It is shown that the obtained Zeno line is the straight line on the density-temperature plane like for many other nonmetallic substances. The similarity relations between the critical and Zeno line parameters have appeared to be valid for Ga as well. The liquid-gas coexistence curve (binodal) for gallium has been constructed on the basis of the Zeno line similarities and the low-temperature measurements data.

Similarity law and critical properties in ionic systems.

Using molecular simulations, we determine the locus of ideal compressibility, or Zeno line, for a series of ionic compounds. We find that the shape of this thermodynamic contour follows a linear law, leading to the determination of the Boyle parameters. We also show that a similarity law, based on the Boyle parameters, yields accurate critical data when compared to the experiment. Furthermore, we show that the Boyle density scales linearly with the size-asymmetry, providing a direct route to establish a correspondence between the thermodynamic properties of different ionic compounds.

Prediction of the Boyle temperature, second virial coefficient and Zeno line using the cubic and volume – translated cubic equations of state

In this study, the Boyle temperature of 174 substances are predicted using the cubic and volume – translated cubic equations of state. In this regard, the Peng - Robinson (PR), Soave – Redlich – Kwong (SRK), volume – translated Peng – Robinson (VTPR), volume – translated Soave – Redlich – Kwong (VTSRK) and Patel – Teja – Valderrama (PTV) equations of state are used. The Boyle temperature is calculated for different polar, non – polar and associating substances including alkanes, cycloalkanes, alkenes, aromatics, ethers, esters, ketones, refrigerants, halogens, amines and water. The obtained results are compared with the results of other known correlations published so far, such as Tsonopoulos, Meng – Duan and Estrada – Torres et al. equations. For better understanding, the second virial coefficient and the Zeno line (unit compressibility) are also calculated and compared with the available experimental data. The results show that PR, VTPR and PTV have lower average temperature difference between EoSs and Tsonopoulos or Meng – Duan correlations. The dependency of the reduced Boyle temperature (TB/TC) as functions of the Pitzer's acentric factor (ω), hydrocarbon chain length (NC) and the reduced dipole moment (μR) has been also studied.

Thermodynamic behavior of charged Lennard-Jones fluids

In the present paper one of the simplest models for the intermolecular interaction of ionic substances is used. The phase equilibrium and pVT properties in the single-phase area for the fluid with intermolecular interaction, which is described by the Lennard-Jones and Coulomb potentials, is calculated by molecular dynamics simulation. Influence of the charge on position of the Zeno line as well as on a position of the vapor-liquid equilibrium line on phase diagram is determined. It is shown that this system can be described by generalized scaling laws proposed in a series of articles Apfelbaum E.M., Vorob'ev V.S.//J. Phys. Chem. B. 112. (2008) 13064. and Apfelbaum E.M., Vorob'ev V.S.//J. Chem. Phys. 130. (2009) 214111. The expression for the dependence of properties at the critical point, the Boyle temperature and density on the charge value are proposed.

Can the Boyle's and critical parameters be unambiguously correlated for polar and associating fluids, liquid metals, ionic liquids?

An independent test of thermodynamic consistency applied to a presently widespread methodology of Zeno-line (ZL) has demonstrated its restricted predictive abilities. Unfortunately, a subjective choice of the reference Boyle (B) parameters for the reduced critical temperature Tc/TB and the critical density ρc/ρB as well as the imposed empirical linear constraints on their presumably universal correlation can lead to unadjusted predictions of critical properties for liquid metals, polar and associating fluids, water and many other complicated substances. To study such “difficult” compounds the alternative universal set of correlations between the critical compressibility factor Zc, the Riedel's factor of similarity Ac and both B-parameters ρB,TB is introduced. It follows from the model of fluctuational thermodynamics (FT) and the principle of global fluid asymmetry (GFA) formulated recently as its consequence. This new FT-methodology describes a set of known B-temperatures with an excellent accuracy while the respective B-densities are fixed by the empirical Timmermans correlation. We believe that critical properties for the afore-mentioned “difficult” fluids can be reasonably represented and/or predicted by the proposed universal FT-methodology. The revealed interrelation between the effective potential parameters established in terms of the actual critical properties forces us to disavow the tentative supposition about the violation of PCS (principle of corresponding states) in complex liquids.

Classical and quantum many-body effects on the critical properties and thermodynamic regularities of silicon

Using molecular simulation, we determine the critical properties of Si as well as the loci for several remarkable thermodynamic contours spanning the supercritical region of the phase diagram. We consider a classical three-body potential as well as a quantum (tight-binding) many-body model, and determine the loci for the ideality contours, including the Zeno line and the H line of ideal enthalpy. The two strategies (classical or quantum) lead to strongly asymmetric binodals and to critical properties in good agreement with each other. The Zeno and H lines are found to remain linear over a wide temperature interval, despite the changes in electronic structure undergone by the fluid along these contours. We also show that the classical and quantum model yield markedly different results for the parameters defining the H line, the exponents for the power-laws underlying the line of minima for the isothermal enthalpy and for the density required to achieve ideal behavior, most notably for the enthalpy.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. V. Impact of an electric field on the thermodynamic properties and ideality contours of water.

Using molecular simulation, we assess the impact of an electric field on the properties of water, modeled with the SPC/E potential, over a wide range of states and conditions. Electric fields of the order of 0.1 V/Å and beyond are found to have a significant impact on the grand-canonical partition function of water, resulting in shifts in the chemical potential at the vapor-liquid coexistence of up to 20%. This, in turn, leads to an increase in the critical temperatures by close to 7% for a field of 0.2 V/Å, to lower vapor pressures, and to much larger entropies of vaporization (by up to 35%). We interpret these results in terms of the greater density change at the transition and of the increased structural order resulting from the applied field. The thermodynamics of compressed liquids and of supercritical water are also analyzed over a wide range of pressures, leading to the determination of the Zeno line and of the curve of ideal enthalpy that span the supercritical region of the phase diagram. Rescaling the phase diagrams obtained for the different field strengths by their respective critical properties allows us to draw a correspondence between these systems for fields of up to 0.2 V/Å.

The application of the Zeno line similarities to alkaline earth metals

We show that similarity relations connected with the line of the unit compressibility factor (Zeno line) is valid for the alkaline earth metals Mg, Ca, Sr and Ba. We have applied the Monte Carlo simulations in the canonical ensemble to establish linearity of the Zeno line and to find its parameters. We have also constructed liquid-gas binodals for these metals which describes the available low temperature measurements data. We check the other relations linking the critical and Zeno line parameters and found that they are correct within the range of experimental errors.

Modeling the volumetric properties of metals from heat of vaporization

We have applied the Song and Mason equation of state [Y. Song, E.A. Mason, Phys. Rev. A 42 (1990) 4743–4748] with a simple modification for alkali metals, alkaline earth metals, aluminum, bismuth, lead, copper and mercury. The present equation of state consists of four temperature dependent parameters. These parameters have been calculated as function of reduced temperature with the use of the law of corresponding states based on latent heat of vaporization and the liquid density at the normal boiling point. The predicted densities were compared with those obtained from the experiment. From 396 data points examined for the aforementioned metals, the total average absolute deviation was found to be 0.72%. Moreover, we indicate that the Zeno line regularity can well be predicted by the proposed model for metals.

Scaling Laws and Critical Properties for fcc and hcp Metals

The determination of the critical parameters of metals has remained particularly challenging both experimentally, because of the very large temperatures involved, and theoretically, because of the many-body interactions that take place in metals. Moreover, experiments have shown that these systems exhibit an unusually strong asymmetry of their binodal. Recent theoretical work has led to new similarity laws, based on the calculation of the Zeno line and of the underlying Boyle parameters, which provided results for the critical properties of atomic and molecular systems in excellent agreement with experiments. Using the recently developed expanded Wang-Landau (EWL) simulation method, we evaluate the grand-canonical partition function, over a wide range of conditions, for 11 fcc and hcp metals (Ag, Al, Au, Be, Cu, Ir, Ni, Pb, Pd, Pt, and Rh), modeled with a many-body interaction potential. This allows us to calculate the binodal, Zeno line, and Boyle parameters and, in turn, obtain the critical properties for these systems. We also propose two scaling laws for the enthalpy and entropy of vaporization, and identify critical exponents of 0.4 and 1.22 for these two laws, respectively.

The Zeno Line for Al, Cu, and U.

We show that the property of linearity for a line of unit compressibility factor (Zeno line) can be confirmed for metals (Al, Cu, and U) in liquid phase. The embedded atom potentials (EAM) have been used to describe the interaction between the particles. The numerical simulations within Monte Carlo (MC) technique with the EAM potential have resulted in the straight Zeno-line for considered metals and have allowed us to define the Zeno line parameters. The similarity relations between the critical and the Zeno line parameters, which were observed previously for nonmetallic substances, have appeared to be valid for Al and Cu as well. For uranium there is a contradiction between the calculated and experimental data, indicating the limitation for these similarities.

New Zeno-Like Liquid States.

A Zeno line is a straight line in a fluid's temperature-density plane where the compressibility factor equals unity and stretches from supercritical to subcritical regimes. A second unexpected linearity also occurs in liquids in their normal liquid range (NLR) where the compressibility factor approaches zero. This Zeno-like line includes liquids as diverse as monatomic and diatomic elements to hydrogen bonding liquids, with water and helium being the only currently known exceptions. Remarkably, this second linearity is also present in polymeric liquids, molten metals, and salts, i.e., it is a generic characteristic of liquids with negligible or low vapor pressures. This observation yields the following new corresponding states principle: Saturated liquid densities are a linear function of temperature in the NLR for molecular, polymeric, inorganic, ionic, and metallic liquids and superpose to form a single master curve. Another Zeno-like linearity in the NLR has also been identified for the configurational energy of many liquids. Extension of this line to zero temperature defines the ground state configurational energy of the hypothetical disordered liquid (glass).

PVT properties of imidazolium-, phosphonium-, pyridinium- and pyrrolidinium-based ionic liquids using critical point constants

ABSTRACTSong and Mason equation of state (EOS) with a simple modification has been extended to modelling PVT properties of ionic liquids (ILs). The considered ILs are [C1mim][MeSO4], [C1mim][CH3OC2H5SO4], [C1mim][(CH3)2PO4], [C2mim][MeSO4], [C2mim][BF4], [C2mim][SCN], [C2eim][NTf2], [C4mim][C(CN)3], [C4mim][CF3SO3], [C4mim][SCN], [C5mim][NTf2], [C8mim][NTf2], [(C6H13)3P(C14H29)][Cl], [(C6H13)3P(C14H29)][NTf2], [(C6H13)3P(C14H29)][Ac], [C3mpyr][NTf2], [C4mpyr][NTf2] and [Py][C2H5OC2H4SO4]. Three temperature-dependent parameters in the proposed EOS have been scaled as functions of reduced temperature with the use of the law of corresponding states. It is shown that the knowledge of just critical temperature and critical density is sufficient to predict the PVT properties of these ILs. The overall average absolute deviation of calculated densities from literature values for 1347 data points of 18 ILs was found to be 0.58%. The predicted density of ILs from proposed EOS has been compared with those obtained by other literature work. Moreover, we indicate that the Zeno line regularity can well be predicted by proposed model for ILs.

Modeling the volumetric properties of some imidazolium and phosphonium based ionic liquids from surface tension

In this work, we have extended Song and Mason equation of state with a simple modification to imidazolium and phosphonium based ionic liquids. Four temperature-dependent parameters in the equation of state have been calculated as functions of reduced temperature with the use of the law of corresponding states. It is shown that the knowledge of surface tension, liquid density at the room temperature and two other constants that are obtained by fitting against experimental data is sufficient to predict the PVT properties of these ILs. Predicted densities are in good agreement with experimental literature data in a wide range of temperatures, 273.15–472.6K, and pressures, 0.1–200.0MPa. The overall average absolute deviation of calculated densities from literature values for 3302 data points of 17 ionic liquids was found to be 0.89%. Moreover, we indicate that the Zeno line regularity can well be predicted by proposed model for ILs. Also, we have demonstrated the density behavior of studied imidazolium based ILs in terms of the chain length of alkyl group of cation using modified SM EOS.

The saturation pressure for different objects in reduced variables and the justification of some empirical relations set from the van der Waals equation

We analyse the saturation pressure (SP) behaviour along the coexistence curve liquid–gas in reduced variables, containing only the Zeno line parameters. Using experimental and simulation data for substances (Ar, N2, Hg, Cs) and two models (van der Waals and Lennard–Jones) we create a global 3D picture of SP dependence on temperature and density. We also show that the well-known empirical Timmermans relation and partly Riedel equation are the consequences of the van der Waals equation and the rectilinear diameter law.