Article pubs.acs.org/JPCB
The Zeno Line for Al, Cu, and U E. M. Apfelbaum*,† and V. S. Vorob’ev†,‡ †
Joint Institute for High Temperatures of Russian Academy of Sciences, Izhorskaya Street 13 Building 2, Moscow 125412, Russia National University of Science and Technology “MISIS”, Leninskiy prospekt 4, Moscow 119049, Russia
‡
ABSTRACT: 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.
NIST database12 there are about 70 different substances for which the Zeno line is straight. The exclusions7,11 among them are only the quantum liquids (hydrogen, helium) and water (see also ref 6). So it was interesting to apply considered similarities to metals, which occupy most of the periodic table. However, the situation with the metals is not so satisfactory in comparison with nonmetallic substances. And there are evident reasons for that. The metals are crystals under ambient conditions, while their liquid and gaseous phases are located at higher temperatures and pressures. However, the maximal temperature T that can be measured reliably in up-to date experiments is approximately 5000 K or even lower,13,14 especially in the case of coexistence curve measurements. The states with supposedly higher temperature can be obtained in dynamical experiments but without direct measurements of the temperature.15 So there are too little data to extract the Zeno-line position from the measurements for most metals. There are only two metals, Hg and Cs, for which there are enough accurate experimental data.16−20 This data allows one to construct the Zeno-line (see ref 18) as well as many other characteristics including the coexistence liquid−gas curve. For other alkali metals there are also analogous measurements but they are less exact and less consistent with each other (compare, for instance, the critical points presented in refs 21 and 22). For other metals, there are no similar measurements. All information that we have is the data about the saturated pressure as a function of temperature for the gaseous phase up to approximately 1 atm23 and some measurements along isobars in the liquid phase up to the temperatures ∼5 kK mentioned above.13 These experimental data are not enough to check the considered similarities for
I. INTRODUCTION Starting from the second half of the 19th century, it was found that many fluids demonstrate some regular and scalable volumetric features, which are general and independent of their interparticle potential. These features are known as the similarity laws.1 Their well-known examples are the corresponding states principle and law of rectilinear diameter. Some of the similarity relations have theoretical grounds. For instance, the corresponding states principle can be directly derived from the van der Waals equation. However, many others relations have no rigorous theoretical substantiations. Nevertheless the search and investigation of various similarity laws are a part of modern statistical physics.2 One more set of regularities concerns of the contour on the density temperature plane known as the Zeno line. The Zeno line is defined as the line where the compressibility factor Z of a liquid is unity: Z = P/(nT) = 1, i.e., it coincides with that one of an ideal gas. (Here P is the pressure, T is the temperature, measured in the units of energy, n is the particle density.) This line has been considered in many studies.2−11 Below we will refer to it as the Z-line as well. The key property of this contour is that it is appeared to be the straight line over all density temperature plane n−T. The number of substances for which this property can be confirmed experimentally are wider than for many other regularities, although there are exclusions.10,11 Another useful property of the Z-line is that, at T → 0, it is tangent to the liquid binodal branch.2,7 In refs 7−10, we have taken into account these properties to construct the coexistence liquid−gas curve and to find possible relation between the critical point coordinates and the Zeno-line parameters. The latter values (Boyle temperature TB and Boyle density nB) are defined by the points where the Zeno line is crossing with the axes on the density−temperature plane. Due to wealth of experimental data for many gases and liquids, it was possible to check the considered regularities directly. For instance, in the © 2016 American Chemical Society
Received: April 7, 2016 Revised: May 5, 2016 Published: May 9, 2016 4828
DOI: 10.1021/acs.jpcb.6b03561 J. Phys. Chem. B 2016, 120, 4828−4833
Article
The Journal of Physical Chemistry B
approach has its limitation when the temperature rises and density decreases (see below); nevertheless, it can be used in liquid metals at relatively low temperatures.31−34 So, in this area we have the advantage to use classical MC/MD technique. We have performed Monte Carlo simulations with EAM potentials, which have shown that the Zeno-lines are really straight in the liquid state of considered metals too, like for Hg and Cs. This property allows us to use the approach developed in refs 9 and 10 to construct the coexistence curve for U and to check the Zeno-line regularity. To our knowledge, no embedded atom methods were applied to the calculations of the Zeno-line previously. The Article is organized as follows. In the next section we will describe the similarity relation mentioned above. The details of EAM will be given in the third section. The results of the Zeno-line calculations for Al, Cu, and U will be considered in the fourth section. The conclusions will complete the Article.
most metals. So it is necessary to address the theoretical methods. The rigorous theory can help in the considered question, as to date there have been several powerful simulation techniques developed,24 like Monte Carlo or Molecular Dynamics approaches (MC/MD). However, they require the knowledge of interparticle interaction potential. Although all real substances consist of positively charged nuclei and electrons, it is possible to consider some of them like a one-component system, when the electrons are localized on the nuclei. For instance, argon can be considered this way in gas, liquid, and fluid regions up to TB and higher, as its atomic ionization potential is much greater than TB. So it represents a system of atoms with corresponding interaction potential (of LennardJones or Buckingham type1). This potential appears to be applicable both in gaseous, liquid, and fluid phases, as well as within the coexistence liquid−gas area. Furthermore, one can use MC/MD techniques to find the Zeno-lines, coexistence lines, as well as other characteristics. However, this is not the case for metals. Any metal changes its chemical composition in the process of transition from gaseous to liquid phase. In the rarefied gaseous phase of a metal (at relatively low temperatures), the electrons are also localized at ion-sites, like in the case of argon. It is also possible to find the potential for a corresponding metallic dimer.25 However, in the liquid phase, the electrons are free, so we again have a two-component medium. For liquid metals it is also possible to construct some effective ion−ion potential within the quantum mechanical perturbation theory.26−28 It allows one to reduce the real twocomponent metal to a somewhat effective one-component system. However, the resulting potential for this effective system, whatever approach is used, is completely different from the dimeric potential for gaseous phase. Moreover, the twocomponent nature of metals makes the Zeno-line definition ambiguous. For most substances, we initially know the mass density ρ as the input data. For a one-component system, the particle density is n = ρ/ma, where ma is the mass of corresponding atom. So, we can rewrite the compressibility factor as Z = P/(nT) = Pma/(ρT). However, for a twocomponent metal, the mass density is ρ = (mene + mini), where subscripts “e” and “i” refer to the electrons and ions, correspondingly. In this case, there is no single particle density n. Thus, the definition of Z becomes ambiguous. To overcome this problem, we can consider a metal as a somewhat effective one-component system, as mentioned above. In this case, it is possible to study the Zeno-line regularity at least in one (liquid) phase. However, the problem of potential definition has appeared again. To find the effective potential in simple metals, it is possible to use the pseudopotential theory,26 which was applied, for instance, to Berillium.26 However, this theory does not describe metals with complex electronic structure, like noble (Cu, Ag, Au) or transient (W, Mo, etc.) ones. That is why in the present study we have applied the embedded atom method29−34 (EAM) to construct the Zeno-line for several metals, like Al, Cu, and U. All these elements were chosen as examples of metals with different electron structure (Al is a simple metal, Cu is a noble metal, while U belongs to weakly radioactive metals with very complex electron structure). The concept of EAM is widely used now to describe various properties of a substance in crystal and liquid phases. Within this approach, all collective effects are included in many-particle interaction potential itself due to its functional form. Evidently this
II. THE ZENO-LINE SIMILARITIES To demonstrate the considered similarity relations let us address the phase diagram of Argon as an example. The NIST database12 allows us to build all lines under consideration for this element. They are shown in Figure 1. It is evident that the
Figure 1. Argon phase diagram at the temperature−density plane. Lines: 1 is the binodal; 2, 3, 4, 5 are the isobars with P = 1, 2, 4.683 (Pc), 8 MPa, respectively; 6 is the Zeno line. Pc is the pressure at the critical point.
isobars corresponding to the different pressures at low temperatures thicken and practically coincide with the binodal. Straight line 6 is the Zeno-line. It asymptotically tends to the binodal with decreasing temperature. The linear-like behavior of isobars of a liquid as well as their coincidence with the binodal at low temperatures was described many years ago. That is why it was used to estimate the critical point coordinates.35−37 (Below, the values with subscript “c” refer to the critical point.) However, in reality, only the Zenoline is straight (at least in liquid phase). So in our previous investigations, we have offered an algorithm to find the critical point coordinates, which are based on the linearity of the Zenoline.7,9−11 We have found the following relation connecting the critical and the Zeno-line parameters: nc T + c = S1 nB TB
(1)
The value S1 ≈ 0.67 for most of the real substances. Besides eq 1, it is possible to construct an equation for binodal using its asymptotical properties.10 Namely, the liquid branch of the 4829
DOI: 10.1021/acs.jpcb.6b03561 J. Phys. Chem. B 2016, 120, 4828−4833
Article
The Journal of Physical Chemistry B binodal tends to the Zeno-line at T → 0. In the same limit, the Clapeyron−Clausius equation gives rise to exponential dependence of the density on the inverse temperature along the gas binodal branch. Near the critical point, the density along both branches obey the power law according to the scaling theory.1 It is also convenient to present a binodal in symmetrical form relative to its diameter.9,10 As a result, in ref 10 the following expression for the binodal was derived: ρL,G =
Here, Φ(ρi) is the “embedding potential”, which depends on the effective dimensionless electron density ρi, created by all electrons at the position where the ith ion is located. The density ρi is a sum of contributions Ψ(rij). Each Ψ(rij) describes the influence of the jth neighbor. The function φ(rij) presents the usual pairwise potential. All influence of electrons is included in these two terms Φ(ρi) and φ(rij). Thus, giving particular form for three functionsΦ, Ψ, φ one has a description for a substance under study. In the present research, we have considered Al, Cu, and U. For Al and Cu (and a number of other metals), corresponding potentials were constructed in ref 31. The following forms for Φ, Ψ, φ were offered:31
β⎫ ρ2D ⎧ ⎡ ⎪ qτ ⎞⎤ ⎪ ⎛ ⎜− ⎟ ⎬ ⎨ 1 1 exp ± − ⎢⎣ ⎝ 1 − τ ⎠⎥⎦ ⎪ 2 ⎪ ⎭ ⎩
ρ2D = 2ρc + Aτ + Bτ 2β ⎡T ⎛ ⎢ c − 2β ⎜⎜1 − A= 1 − 2β ⎢⎣ TB ⎝ ρc ρB ⎡ ⎢1 − 2 B= − 1 − 2β ⎢⎣ ρB ρB
ρ ⎞⎤ 2 c ⎟⎟⎥ , ρB ⎠⎥⎦ Tc ⎤ ⎥ TB ⎥⎦
Φ(ρi )/ε = −c ρi ,
The potential is measured in electronvolts, the distance is measured in angstroms. For Al: ε = 0.0090144 eV, a = 4.03230 A, c = 54.97923, m = 5, n = 9. For Cu: ε = 0.0057921 eV, a = 3.60300 A, c = 84.843, m = 5, n = 10. We should note that there are many other forms of Φ, Ψ, and φ for Al and Cu (see ref 31 and references therein). However, they are, as a rule, developed for description of the crystal state, while the EAM of ref 31 was used in liquid phase too. In particular, the binodal for Cu was calculated using this potential.32 Consequently, we have chosen this potential to calculate the Zeno-line. For uranium, there is much less EAM potentials in comparison with other metals.34 Most of them were also constructed to describe the solid phase. Only the potential of ref 33 was fitted to reproduce known experimental data in liquid. That is why it was chosen for our purposes as well. Its functions:
(2)
Ψ(rij) = p1 exp(− p2 rij); φ(rij)/ε = exp[− 2α(rij/d − 1)] − 2 exp[− α(rij/d − 1)] (5)
Here p1 = 5.5619, p2 = 1.3850, ε = 0.209 eV, α = 4.1, d = 3.3318 A. The function Φ is given as a piecewise polynomial function, which can be found in ref 33. Here we would like to mark the following peculiarity of EAM potentials. The two-body potentials usually possess several characteristic parameters. For instance, in the case of potential φ of eq 5, they are α, d, and ε. The two latter parameters (characteristic length and energy) can be used farther as the reducing units for all other values. So all the substances, describing by the same two-body potential have superimposing binodals and other curves, when they are written in reduced units. This property is known as the corresponding state principle.1 However, in the case of many-body EAM potential, besides other parameters, there are also 3 functions Φ, ρ, φ. Due to the last fact, it is principally impossible to extract corresponding reducing parameters from EAM potentials, if these functions are different. To calculate the Zeno-line points we have applied the Monte Carlo simulation in the canonical ensemble.24 We have used from N = 500 to N = 2000 particles to achieve the convergence with respect to particle number. The equilibration of the system took 500 configurations (each configuration corresponds to a move of all N particles). Then 2000 subsequent configurations were used for averaging. The periodic boundary conditions
III. THE EAM POTENTIALS The concept of EAM was proposed about 30 years ago (see refs 29 and 30). Within this concept, a metal is described as a onecomponent system. The interaction between N particles with coordinates ri⃗ in such a system is described by the following many-body potential energy: N
∑ Φ(ρi ) + ∑ i=1
φ(rij)
1≤i
