Multi-phase equation of state for aluminum

Abstract

Results of theoretical calculations and experimental measurements of the equation of state (EOS) at extreme conditions are discussed and applied to aluminum. It is pointed out that the available high pressure and temperature information covers a broad range of the phase diagram, but only irregularly and, as a rule, is not thermodynamically complete; its generalization can be done only in the form of a thermodynamically complete EOS. A multi-phase EOS model is presented, accounting for solid, liquid, gas, and plasma states, as well as two-phase regions of melting and evaporation. The thermodynamic properties of aluminum and its phase diagram are calculated with the use of this model. Theoretical calculations of thermodynamic properties of the solid, liquid, and plasma phases, and of the critical point, are compared with results of static and dynamic experiments. The analysis deals with thermodynamic properties of solid aluminum at T = 0 and 298 K from different band-structure theories, static compression experiments in diamond anvil cells, and the information obtained in isentropic-compression and shock-wave experiments. Thermodynamic data in the liquid state, resulting from traditional thermophysical measurements, “exploding wire” experiments, and evaluations of the critical point are presented. Numerous shock-wave experiments for aluminum have been done to measure shock adiabats of crystal and porous samples, release isentropes, and sound speed in shocked metal. These data are analyzed in a self-consistent manner together with all other available data at high pressure.

The model's results are shown for the principal shock adiabat, the high-pressure melting and evaporation regions and the critical point of aluminum. New experimental and theoretical data helped to improve the description of the high-pressure, high-temperature aluminum liquid. The present EOS describes with high accuracy and reliability the complete set of available information.

1. INTRODUCTION

Equation of state (EOS) describes fundamental thermophysical properties of matter. This information can only be obtained by using sophisticated theoretical models or from experiments (Al'tshuler, 1965; Bushman & Fortov, 1983; Eliezer et al., 1986; Ross, 1985; Chisolm et al., 2003). The EOS is of considerable interest for basic research and has numerous important applications (Bushman et al., 1992, 1993; Fortov & Yakubov, 1999; Hoffmann et al., 2002, 2005; Tahir et al., 2005b, 2007; Temporal et al., 2005), among them also inertial confinement fusion (Peng et al., 2005; Danson et al., 2005; Eliezer et al., 2007). High intensity heavy ion beams as well as high power lasers and pulsed power discharges are developing into an important tool for EOS experiments (Danson et al., 2005; Jungwirth, 2005; Hoffmann et al., 2005; Ray et al., 2006; Sasaki et al., 2006; Desai et al., 2007). States of matter characterized by high-energy-density occupy a broad region of the phase diagram, for example, hot compressed matter, strongly coupled plasmas, hot expanded liquid and quasi-ideal plasmas. Our knowledge of these states is limited, because theoretical modeling is complicated and experiments are difficult to perform.

Aluminum remains one of the most important metals for mankind, being of wide use in industry and science. Its thermodynamic properties have been investigated in numerous static and dynamic experiments and explored with the use of modern theories as well.

Nevertheless, the EOS for aluminum is still a “hot” problem in the physics of high energy densities. This fact is explained by a long–time intrigue of the structural phase transition at ≃ 200 GPa, inferred from the non-monotonic behavior of the shock adiabat. Other reasons are that properties of liquid aluminum at high pressures and temperatures are not well-known and multi-phase EOSs for aluminum have been developed in 1980 (Holian, 1986; Kerley, 1987; Bushman et al., 1993).

Fortunately, recent progress achieved in both experimental and theoretical areas, i.e., Z-machine, diamond anvils, and quantum molecular dynamic calculations, significantly clarified these old problems. These advances motivated the present reassessment and refinement of the multi-phase EOS for aluminum.

It this work, an advanced multi-phase EOS for aluminum is presented. The thermodynamically complete, temperature EOS for metals is defined by the potential of free energy describing the elastic contribution at T = 0 K and the thermal contribution by atoms and electrons. The EOS provides for a correct description of phase boundaries, melting, and evaporation, as well as the effects of the first and second ionization. The atomic thermal contribution in the model is different in the solid and liquid states, while the electron thermal contribution is identical.

To construct the EOS, the following information was used at high pressure and high temperature: measurements of isothermal compressibility in diamond anvil cells, isentropic–compression experiments, data on sound velocity and liquid metal density at atmospheric pressure, isobaric–expansion measurements, data on the shock compressibility of solid and porous samples in incident and reflected shock waves, impedance measurements of shock compressibility obtained by an underground nuclear explosion, data on the isentropic expansion of shocked metals, calculations by the Thomas–Fermi model and by quantum molecular dynamics methods, numerous evaluations of the critical point.

New experimental and theoretical data helped to improve the description of the high-pressure, high-temperature aluminum liquid. The present EOS describes with high accuracy and reliability the complete set of available information.

2. EOS PROBLEM

EOS is a fundamental property of matter defining its thermodynamic characteristics in a functional form like f(x, y, z) = 0, where x, y, z can be, for example, volume V, pressure P, temperature T, or in the form of graphs or tables. Well-known functional for EOSs include, for instance, the Mie–Grüneisen EOS Grüneisen (1912)

(1)

where the index c indicates the component at T = 0 K; the Birch (1968) potential

(2)

in which σ = V ₀/V, V ₀—specific volume at normal conditions (P = 1 bar, T = 298 K), B _T = −(∂ P/∂ ln V)_T—isothermal bulk compression modulus, B _P = ∂B _T/∂P—its pressure derivative; and the Carnahan–Starling's approximation for the free energy of a system of “hard” spheres (Carnahan & Starling, 1969)

(3)

where N–amount of particles, k–Boltzmann's constant, and η–packing density. One can find examples of EOSs in graphic or tabular form in compendia of shock wave data (van Thiel, 1977; Marsh, 1980; Zhernokletov et al., 1996; Trunin et al., 2001); these are EOSs of shock adiabats.

The current state of the problem of a theoretical description of thermodynamic properties of matter at high pressures and high temperatures is given in a set of publications (Zeldovich & Raizer, 1966; Eliezer et al., 1986; Fortov & Yakubov, 1999; Bushman et al., 1993; Bushman & Fortov, 1983; Ross, 1985, and references therein). In spite of the significant progress achieved in predicting EOS information accurately in solid, liquid, and plasma states with the use of the most sophisticated “ab initio” computational approaches (classic and quantum methods of self-consisted field, diagram technique, Monte-Carlo, and molecular dynamics methods), the disadvantage of these theories is their regional character (Eliezer et al., 1986; Fortov & Yakubov, 1999; Bushman et al., 1993; Bushman & Fortov, 1983; Ross, 1985). The range of applicability of each method is local and, rigorously speaking, no single one of them provides for a correct theoretical calculation of thermodynamic properties of matter on the whole phase plane from the cold crystal to the liquid and hot plasmas (Bushman & Fortov, 1983; Eliezer et al., 1986; Fortov & Yakubov, 1999). The principal problem here is the need to account correctly for the strong collective interparticle interaction in disordered media, which presents special difficulties in the region occupied by dense, disordered, nonideal plasmas (Bushman & Fortov, 1983; Ross, 1985; Eliezer et al., 1986; Fortov & Yakubov, 1999; Bushman et al., 1993).

In this case, experimental data at high pressures and high temperatures are of particular significance, because they serve as reference points for theories and semi-empirical models. Data obtained with the use of dynamic methods (see Zeldovich & Raizer, 1966; Bushman et al., 1993; Fortov & Yakubov, 1999; McQueen et al., 1970; Avrorin et al., 1990; Al'tshuler 1965; Duvall & Graham, 1977, and references therein) are of the importance from the practical point of view. Shock-wave methods permit to study a broad range of the phase diagram from the compressed hot condensed states to dense strongly coupled plasma and quasi-gas states. Detailed presentations of shock-wave methods to investigate high dynamic pressures can be found in the literature (Zeldovich & Raizer, 1966; Bushman et al., 1993; Fortov & Yakubov, 1999; Avrorin et al., 1990, and reviews, Al'tshuler, 1965; Duvall & Graham, 1977).

Available experimental data on the shock compression of solid and porous metals, as well as isentropic expansion, covers nine orders of magnitude in pressure and four in density. The first published shock-wave data were obtained for metals in the megabar (1 Mbar = 100 GPa) (Walsh et al., 1957) and multimegabar (Al'tshuler et al., 1958a, 1958b) pressure range using explosive drivers. Higher pressures of 10 Mbar have been accessed by spherical cumulative systems (Kormer et al., 1962; Al'tshuler et al., 1981) and by underground nuclear explosions (Al'tshuler et al., 1968; Trunin et al., 1969). Maximum pressures of 400 TPa (Vladimirov et al., 1984) were also reported for aluminum through the use of nuclear explosion. Note that data obtained by impedance-matching techniques require the knowledge of the EOS for a standard material. Previously a monotonic approximation of the shock adiabat of lead between the traditional region of pressures ≤10 Mbar to Thomas–Fermi calculations (Kalitkin & Kuzmina, 1975) was used for the standard (Al'tshuler et al., 1968; Trunin et al., 1969). It seems that iron, for which absolute Hugoniot measurements have been reported to pressures of 100 Mbar (Trunin et al., 1992, 1993), is now the best etalon material. Figure 1 shows modern progress achieved in measurements of absolute shock compressibility with the use of traditional explosive techniques and investigations of impedance in experiments with concentrated energy fluxes like lasers and underground nuclear explosions.

Fig. 1. The investigated pressure scale for the elements. Shown are maximum pressures achieved using traditional explosive (gray region), lasers, diamond-anvil-cell static measurements (black), and underground nuclear explosions (points).

The extension of the phase diagram data to greater relative volumes, in comparison with the principal Hugoniot, is achieved with shock compression of porous samples (Zeldovich & Raizer, 1966). Nevertheless, difficulties in fabricating highly porous targets and non-uniform material response of the porous sample to shock loading imposes a practical limit to the minimum density of a specimen. The method of isentropic expansion of shocked matter, depending on the magnitude of the shock pressure and, consequently, the entropy provided, produces in one experiment transitions from a hot metallic liquid shocked state, to a strongly coupled plasma, then to a two-phase liquid-gas region, and to a Boltzmann's weakly ionized plasma, and finally to a nearly ideal gas (Zeldovich & Raizer, 1966; Bushman et al., 1993; Fortov & Yakubov, 1999).

The available experimental and theoretical information is shown in Figure 2 on a three-dimensional (3D), relative volume-temperature-pressure surface calculated by a semi-empirical multi-phase EOS (Bushman et al., 1993). It is well illustrated that besides the shock compressibility, measurements of release isentropes of shocked materials are of especial importance. Such results traverse states in the intermediate region between the solid state and gas, occupied by a hot dense metallic liquid and strongly coupled plasma (Bushman et al., 1993; Fortov & Yakubov, 1999), which is a region poorly described by theory. Experimentally studied release isentropes for copper have as initial high energy states solid, and melted, and compressed liquid metal. The range of thermodynamic parameters covered in the adiabatic expansion process for these states is extremely wide (Fig. 2), covering five orders of magnitude in pressure and two orders of magnitude in density. It extends from a highly compressed metallic liquid, characterized by a disordered arrangement of ions, and degenerate electrons, to a quasi-nonideal Boltzmann plasma and a rarefied metallic vapor. Upon expansion of the system, the degree of degeneracy of the electronic subsystem is decreased and a marked rearrangement of the energy spectrum of atoms and ions occurs. A partial recombination of the dense plasma also takes place. In the disordered electron system, a “metal-insulator” transition takes place and a nonideal (with respect to different forms of interparticle interactions) plasma is formed in the vicinity of the liquid-vapor equilibrium curve and the critical point. Where the isentropes enter the two-phase liquid-vapor region evaporation occurs; on the gas-side condensation occurs (Avrorin et al., 1990; Bushman et al., 1993; Fortov & Yakubov, 1999).

Fig. 2. Generalized 3D volume-temperature-pressure surface for copper in the investigated region of the phase diagram. M–melting region; H ₁ and H _p–principal and porous Hugoniots; DAC–diamond-anvil-cells data; IEX–isobaric expansion data; S–release isentropes; R–boundary of two-phase liquid-gas region with the critical point CP. Phase states of the metal are also shown.

Note that typical shock-wave measurements allow determination of only caloric properties of matter, viz. the dependence of the relative internal energy on pressure and volume as E = E(P, V). The potential E(P, V) is not complete in the thermodynamic sense and a knowledge of temperature T or entropy S is required for completing the thermodynamic equations and calculating first and second derivatives, such as the heat capacity, the sound velocity and others (Fortov & Yakubov, 1999).

Only a few temperature measurements in shocked metals are available (Yoo et al., 1993), as well as analogous measurements in release isentropic waves (Avrorin et al., 1990). This information is of great importance in view of a limitation of purely theoretical calculation methods. From this point of view, thermodynamically complete measurements obtained with the use of the isobaric expansion (IEX) technique (Gathers, 1986) are of a special significance. In this method, metal is rapidly heated by a powerful pulsed current, then expands into an atmosphere of an inertial gas maintained at constant pressure. This data range in density form solid to the critical point and intersect, therefore, the release isentrope data for metals (see Fig. 2).

The region between principal shock adiabat and isotherm can be accessed with use of the isentropic compression technique. This method allows one to obtain simultaneously high pressure and high densities in the material under study. In practice, the sample is loaded by a magnetically driven impactor or by a sequence of reverberating shock waves in a multi-step compression process.

The final conclusion is that shock-wave techniques allow one to investigate material properties in very wide region of the phase diagram—from compressed solid to hot dense liquid, plasma, liquid-vapor, and quasi-gas states. Though the resulting high pressure, high and temperature information covers a broad range of the phase diagram, it has a heterogeneous character and, as a rule, is not complete from the thermodynamic point of view. Its generalization can be done only in the form of a thermodynamically complete EOS.

3. EOS MODEL

Wide-range EOS models for aluminum have been developed since the 1960s (see, for example, Kormer et al., 1962; Holian, 1986; Kerley, 1987; Bushman et al., 1992, 1993; Young & Corey, 1995). The model described below presents a modification of an existing EOS of Bushman et al. (1993). Changes are in the thermal electrons and the cold compression curve, which are from Bushman et al. (1992).

The EOS model is given by a thermodynamically complete potential of free energy F in the traditional form

(4)

describing the elastic contribution at T = 0 K (F _c), and the heat contribution by atoms (F _a) and electrons (F _e).

3.1. Elastic Curve

The elastic energy for the solid phase is given in the form of a series expansion of r _c⁻¹ ~ σ_c^1/3 (Kormer et al., 1962; Bushman et al., 1992, 1993)

(5)

where σ_c = V _0c/V, V _0c–specific volume at P = 0. Note that Eq. (5) automatically provides for the normalizing condition F _c^(s)(V _0c) = 0.

The conditions at σ_c = 1 for the cold pressure P _c(V) = −dF _c(V)/dV, bulk compression modulus B _c(V) = −VdP _c(V)/dV, and its pressure derivative B _p(V) = dB _c/dP _c define the cold curve at moderate compressions, which together with Eq. (5) leads to the formulas

(6)

(7)

(8)

Here B _0c and B _p0, together with the thermal contribution, provide for tabular values of the isentropic bulk compression modulus and its pressure derivative at normal conditions. By minimizing the mean-square deviation from the Thomas-Fermi cold pressure P _c^TFC (Kalitkin & Kuzmina, 1975) over N points in the interval of σ_c from 25 to 500, together with the bounding equations (6)–(8), one can define the Lagrangian problem of determining the minimum of the functional form depending on χ = a _i, λ, μ, ν

(9)

Taking derivatives of Eq. (9) on a _i, λ, μ, ν one can obtain the algebraic system of eight linear equations for eight values, whose solution defines the coefficients a _i.

The cold energy for the liquid in the compression region (σ_c ≥ 1) is given by Eq. (5), while in the rarefaction region (σ_c < 1) it is represented as

(10)

The normalizing condition F _c^(l)(σ_c = 0) = E _sub, where E _sub is the tabular value of the cohesion energy, leads to the formula

(11)

The system of Eqs. (6)–(8) for a cold energy in the form of Eq. (10) at σ_c=1 results in

(12)

(13)

(14)

Here parameter m is usually ≈ 1 (i.e., van der Waals); l is a fitting parameter, defined from the best description of the density and sound velocity data in the liquid phase. Other parameters are found from Eqs. (11)–(14).

3.2. Thermal Contribution of Atoms

The atomic thermal contribution to the free energy of the solid phase is defined by the high-temperature Debye approximation formula:

(15)

where R is the gas constant. The characteristic temperature θ_c^(s) is given by the empirical relation

(16)

where x = lnσ. Constants B _s and D _s are found from the compression dependence of the Grüneisen gamma γ(V) = dlnθ(V)/dlnσ. This information is obtained from shock-wave and isentropic–compression data in the solid state. The value of γ_0s is the tabulated value of the Grüneisen parameter at ambient conditions, and the normalizing condition for entropy S(V ₀, T = 293 K) = 0 defines the value of θ_0s. Note that at high compression Eq. (16) provides for the correct ideal-gas asymptote θ_c^(s) ~ σ^2/3.

The atomic thermal contribution to the free energy of the liquid phase is represented as

(17)

The first term accounts for anharmonic effects and the second provides for a properly behaved melting curve.

In the liquid phase, the phonon contribution has a form similar to Eq. (16) but with a volume- and temperature-dependent heat capacity c _a and a characteristic temperature θ^(l):

(18)

The heat capacity in the liquid phase is given by the expression

(19)

describing a smooth variation from the value 3R close to the lattice heat capacity to that of an ideal atomic gas, 3R/2. Coefficients σ_a and T _a define the characteristic density and temperature of this transition.

The variation of the characteristic temperature defines the vibrational spectrum and reflects the gradual change of the Grüneisen coefficient of the liquid phase from values γ^(l) ≈ γ^(s), corresponding to condensed states, to the ideal-gas value of 2/3 in the limit of high temperatures and very low densities. Under these assumptions, the characteristic temperature is given by the approximating formula

(20)

where the characteristic temperature in the liquid phase is given in a form analogous to Eq. (16)

(21)

The parameters in Eq. (21), B _l and D _l, are found from shock-wave experiments for solid and porous samples, while the constant θ₀^l is determined by equation θ_c^(l)(0) = T _ca.

The potential term F _m(V, T) provides for correct values of the entropy changes ΔS = ΔS _m0 and volume changes ΔV = ΔV _m0 on melting at ambient pressure, and disappears in the gas phase. The contribution of F _m should also decrease upon compression due to decreasing differences between the properties of the solid and liquid phases. These requirements are satisfied by the relation

(22)

where σ_m = σ/σ_m0 is the relative density of the liquid phase on the melting curve. The constants A _m, B _m, and C _m are uniquely determined by the equilibrium conditions along the melting curve at T = T _m.

3.3. Thermal Contribution of Electrons

The electronic thermal contribution has an identical form for the solid and liquid phases. It is given by

(23)

It includes the generalized analog of the coefficient of the electronic heat capacity B _e:

(24)

the coefficient of the electronic heat capacity β:

(25)

the heat capacity of the electron gas c _ei

(26)

(27)

and the analog of the electronic Grüneisen coefficient γ_e:

(28)

Approximating dependencies are written in this manner to satisfy primarily the asymptotic relations for the electron gas free energy, namely expressions for the degenerate electron gas F _e(V, T) =−β₀T ²σ^−γ₀/2 at moderate temperatures (T ≪ T _Fermi) and expressions for an ideal electron gas F _e(V, T) = 3RZ  ln(σ^2/3T)/2 as T → ∞. Here Z is the atomic number and R is the gas constant. The specific forms given for the separate terms of Eq. (23) were chosen to satisfy these requirements.

Eqs. (23)–(28) are written in a form which correctly represents the primary ionization effects in the plasma region and the behavior of the partially ionized metal. Eq. (27) for τ_i describes a decrease of the ionization potential as the plasma density increases, and the constants σ_z and T _z define, respectively, the characteristic density of the “metal-insulator” transition and the temperature dependence of the transition from a singly ionized gas to a plasma with the ion–charge mean value Z.

3.4. EOS Construction Procedure

The set of Eqs. (4)–(28) fully defines the thermodynamic potential for metals over the entire phase diagram for the region of practical interest. Some coefficients in the EOS, included in the analytical expressions, are constants characteristic for each metal (atomic weight and charge, density at normal conditions and other) and are obtained from tabulated data. The rest serve as fitting parameters and their values are found from the optimum description of the available experimental and theoretical data, while providing for correct asymptotes to calculations based on the Debye–Hückel and Thomas–Fermi theories (Kalitkin & Kuzmina, 1975). It should be emphasized that, even though the number of coefficients in Eqs. (4)–(28) is large, most of them are rigidly defined constants whose values are assigned explicitly or implicitly from the fulfillment of various thermodynamic conditions at specific points on the phase diagram. A few coefficients (about 10) serve to characterize the densities and temperatures of transition from one typical phase-plane region to another and are found empirically. The tables in the appendix lists the parameters of the EOS model given by Eqs. (4)–(28) along with the values for aluminum.

The numerous experimental and theoretical data characterizing the thermodynamic properties of metals for a wide range of parameters, was used in determining the numerical values of coefficients in the EOS, This procedure was carried out with the aid of a specially developed computer program using Eqs. (4)–(28) for thermodynamic calculations. At the preliminary stage of calculations, some thermodynamic constants known for each substance (such as normal density, changes in density and entropy at the melting point under normal pressure, cohesion energy, and the like) were used by the program automatically for finding a number of other uniquely defined coefficients (parameters of the cold curve, melting curve, and so on). This further enables one, by way of calculating algebraic or integral relations valid for self-similar hydrodynamic flows, to perform calculations of the kinematic characteristics measured experimentally at high and ultrahigh pressures, namely, the incident and reflected wave velocities and velocities in adiabatic expansion waves (release isentropes), as well as to allow for melting and evaporation effects. The range of action of each fitting parameter that remains free is very localized, as a result of which its value may be selected independently from a comparison of the calculations with available experimental data.

The EOS was constructed using the following high pressure, high temperature information: measurements of isothermal compressibility in diamond anvil cells, data on sound velocity and density in liquid metals at atmospheric pressure, IEX measurements, data on the shock compressibility of solid and porous samples in incident and reflected shock waves, impedance measurements of shock compressibility using an underground nuclear explosion, data on isentropic expansion of shocked metals, calculations by quantum molecular dynamics (QMD), Debye–Hückel, and Thomas–Fermi models, and evaluations of the critical point.

All of the experimental and theoretical data have a given history of accuracy and reliability. The major problem in EOS construction is to provide for a self-consistent and non-contradictory description of the many kinds of available data.

4. THERMODYNAMIC PROPERTIES OF ALUMINUM

This section presents the calculation results of the thermodynamic properties and the phase diagram of aluminum. This is done with the use of Eqs. (4)–(28). The resulting dependencies are compared with the experimental data and theoretical calculations that are most significant at high pressures and temperatures.

All data are presenting in graphs. Earlier results of Al'tshuler et al. (1958a, 1958b, 1960a; McQueen et al., 1970; van Thiel, 1977) have been revised several times. The effects of impactor heating and an attenuation of the shock in the window and specimen have been analyzed (Al'tshuler & Chekin, 1984), and the corrections have been applied based on new precise shock adiabats of reference materials (Marsh, 1980; Zhernokletov et al., 1996; Trunin et al., 2001). The experimental shock wave data in the graphs correspond to the most recent reported values, for which accounting have been made for all of these experimental factors.

The present EOS accounts for the theoretical and experimental data published up to the middle of 2007. The parameters of the liquid state have been changed strongly, in comparison with the predecessor aluminum EOS of Busman et al. (1993), to satisfy new high pressure Hugoniot data, isothermal (Akahama et al., 2006) and isentropic compression measurements (Davis, 2006) and the results of quantum molecular dynamic calculations (Desjarlais, personal communication) in the critical point region.

4.1. Crystal

Aluminum has an face-centered cubic (fcc) structure at room pressure and temperature. The structural fcc hexagonal close packed (hcp) phase transition at T = 0 K has been predicted by different theories (Voropinov et al., 1970; McMahan & Moriarty, 1983; Lam & Cohen, 1983; Wentzcovich & Law, 1991; Boettger & Trickey, 1996) to occur at pressures of 120–360 GPa. According to (Greene et al., 1994) it remains a simple solid to a pressure of 220 GPa under isothermal compression in a diamond anvil cell, but in analogous experiment (Akahama et al., 2006) at 217 GPa aluminum transforms to hcp phase with a volume reduction of 1%. Novel high-pressure data on isentropic compression of aluminum (Davis, 2006), as well as the results of the diamond anvil cell (DAC) experiment of (Greene et al., 1994; Akahama et al., 2006), which are in the solid state region, do not demonstrate dramatic changes in the thermodynamic parameters. So the present EOS provides for a monotonic compression curve.

A comparison of the cold curve, found using Eqs. (5)–(9), with the results of band–structure calculations is shown in Figure 3. It is seen that the elastic curve agrees with different variants of Thomas–Fermi theory (Kalitkin & Kuzmina, 1975; Perrot, 1979), the augmented–plane wave (APW) (McMahan & Ross, 1979), the self–consistent cell model (Liberman, 1979), and the Hartree–Fock–Slater method (Nikiforov et al., 1989), as well in all ranges of densities to 100-fold compression.

Fig. 3. Pressure in aluminum at T = 0  K. Nomenclature: line–EOS; points–theories, 1–Thomas–Fermi model with corrections Kalitkin & Kuzmina (1975), 2–self-consistent cell model Liberman, (1979), 3–APW McMahan & Ross, (1979), 4–Thomas–Fermi model with gradient correction Perrot (1979), 5–modified Hartree–Fock–Slater model Nikiforov et al. (1989).

The room temperature isotherm to 12 GPa (Syassen & Holzaphel, 1978), 220 GPa (Greene et al., 1994) and 333 GPa (Akahama et al., 2006) is drawn in Figure 4, along with isentropic compression data (Davis, 2006) obtained to 300 GPa. Figure 5 also contains theoretical isotherms calculated with the use of the pseudopotential approach (Nellis et al., 1988) and the full-potential linearized augmented plane wave method (Wang et al., 2000). The compression curve for this EOS, which is an isentrope, is also shown. Note that the difference between the compression isentrope and isotherms, with T = 0 and 298 K, in this range of pressures of up to 340 GPa is negligible. The overall agreement between this semi-empirical compression curve and theory and experiment is good and within the error bars of the experimental data.

Fig. 4. Pressure in solid aluminum. Nomenclature: line–EOS; points–experiment, 1–DAC Syassen & Holzaphel (1978), 2–DAC Greene et al. (1994), 4–isentropic compression Davis (2006), 6,7–DAC fcc and hcp Akahama et al. (2006), and theory, 3–Nellis et al. (1988), 5–Wang et al. (2000).

Fig. 5. Aluminum melting at 1 bar and at 0.3 GPa. Nomenclature: line–EOS; points–experiment, 1–at 1 bar Hultrgen et al. (1973), 2–at 0.3 GPa Gathers (1983).

4.2. Melting

At room pressure, aluminum melts at T = 933 K. The experimental data (Hultrgen et al., 1973) at 1 bar are compared with EOS calculations in Figure 5. Less dense liquid states, characterized by higher values of enthalpy, have been measured in an IEX experiment at 0.3 GPa (Gathers, 1983). The calculated P = 0.3 GPa isobar is also shown in this figure. Note that these EOS isobars are clearly different, while data at 0.3 GPa (Gathers, 1983) correspond to the linear approximation of 1 bar measurements (Hultrgen et al., 1973). The analysis of Figure 4 demonstrates the reliability of the developed EOS in this region of the phase diagram.

The high-pressure melting curve is plotted in the pressure-temperature diagram of Figure 6. The extrapolation of static DAC measurements (Boehler & Ross, 1997; Hanstrom & Lazor, 2000) is in good agreement with the results of a dynamic experiment (McQueen et al., 1984), according to which aluminum melts under shock at a pressure of 110 GPa. The calculated melting line and shock adiabat show good agreement with available data.

Fig. 6. Aluminum melting at high pressures. Nomenclature: lines–EOS calculations, M–melting, H ₁–shock adiabat; points–experiment, 1–Jayaraman et al. (1963), 2–Hanstrom & Lazor (2000), 3–Boehler & Ross (1997).

4.3. Dense Aluminum at High Pressure

In this section, let us discuss the results of shock-wave measurements. Aluminum has been extensively investigated in the past 50 years. Its principal Hugoniot has been measured with the use of traditional high-explosive drivers to pressures of 200 GPa (Al'tshuler et al., 1960a; McQueen et al., 1970; Al'tshuler et al., 1981; Marsh, 1980). Light-gas guns allowed one to obtain precise data to pressures of 210 GPa (Isbell et al., 1968; Mitchell & Nellis, 1981). Much higher pressures of 400 GPa were accessed with the use multi-layer cumulative explosive systems (Glushak et al., 1989) and pressures of 990 GPa with the use of powerful hemi-spherical cumulative drivers (Skidmore & Morris, 1962; Kormer et al, 1962; Al'tshuler & Chekin, 1984; Trunin, 1986; Trunin et al., 1995a, 1995b). High-precision data up to 480 GPa were measured in Z–pinch experiments (Knudson et al., 2003), in which aluminum flyer plates were magnetically accelerated to high velocities.

Extreme pressures in aluminum were generated with the use of underground nuclear explosions. Maximum pressures of 400 TPa were realized (Vladimirov et al., 1984), along with the shock limit of compression ratio. One should specially note that at higher pressures, the contribution of radiation to the pressure and energy will be more significant than the thermal contributions. So we may conclude that the EOS limit for the aluminum shock adiabat has been achieved. Absolute measurements of aluminum shock compressibility were done at 0.9–3.2 TPa (Simonenko et al., 1985). All other measurements are referred to as nuclear impedance measurements (NIM), which are limited in the accuracy with which a reference material is known. In these works, aluminum's compressibility was studied with respect to different standard materials: quartz to pressures of 0.27–2 TPa (Al'tshuler et al., 1977), molybdenum at 2.2–2.9 TPa (Ragan, 1982, 1984), and iron at 4.3–28.9 TPa (Avrorin et al., 1986), 1.7 TPa (Podurets et al., 1994), and 0.25–1.27 TPa (revision Trunin et al., 2001) of data in Avrorin et al., 1987).

Because the structural fcc–hcp phase transition seemed to have confirmed in shock wave experiments (Al'tshuler & Bakanova, 1968; see data points Al'tshuler et al., 1960a; Al'tshuler & Checkin, 1984; Skidmore & Morris, 1962; Isbell et al., 1968; Trunin, 1986; Fig. 7), for some times, investigators used the “soft” shock adiabat at pressures greater than 200 GPa, taking into account this transition effect (see absolute point Trunin (1986) and NIM data Al'tshuler et al. (1977); Trunin et al. (1995b)). More recent high-pressure experiments (Glushak et al., 1989), the revision Trunin et al. (2001) of data in Avrorin et al. (1987), and, especially, precise measurements (Knudson et al., 2003) and QMD calculations (Desjarlais, personal communication), demonstrate a “stiffer” behavior of the aluminum principal Hugoniot. The present EOS describes reliable shock wave data. At high pressure it has an intermediate position between the semi-empirical EOS of Kerley (1987) and novel QMD results (Desjarlais, personal communication). The calculated shock adiabat is also compared against experimental data and theoretical Hugoniots at extreme pressures in Figure 8. This region corresponds to NIM measurements obtained with a highly reliable iron standard to pressures of 10 TPa. These data served to fit the thermal contribution of electrons to the EOS.¹

Fig. 7. Aluminum shock adiabat. Nomenclature: lines–EOS calculations, 1–this work, 2–EOS Kerley (1987), 3–QMD Desjarlais, (2006); points–experiment, 4–Isbell et al. (1968), 5–Al'tshuler et al. (1960a); Al'tshuler & Chekin (1984), 6–Al'tshuler et al. (1981), 7–Al'tshuler et al. (1977), 8–Mitchell & Nellis (1981), 9–Kormer et al. (1962), 10–Volkov et al. (1981), 11–Simonenko et al. (1985), 12–Glushak et al. (1989), 13–Trunin (1986), 14–revision Trunin et al. (2001) of data Avrorin et al. (1987), 15–Knudson et al. (2003), 16–Skidmore & Morris (1962).

Fig. 8. Aluminum shock adiabat at extreme pressures. Nomenclature: lines–EOS calculations, 1–this work, 2–EOS Kerley (1987), 3–QMD Desjarlais (2006); points–experiment, 4–Kormer et al. (1962), 5–Al'tshuler et al. (1960a); Al'tshuler & Chekin (1984), 6–Al'tshuler et al. (1977), 7–Mitchell & Nellis (1981), 8–Ragan (1982), 9–Ragan (1984), 10–Volkov et al. (1981), 11–Simonenko et al. (1985), 12–Avrorin et al. (1986), 13–Glushak et al. (1989), 14–Podurets et al. (1994), 15–Trunin et al. (1995a), 16–Knudson et al. (2003).

Figure 9 illustrates the phase diagram of aluminum in a pressure–density. States of lower density in comparison with the principal Hugoniot have been investigated by shock compression of porous aluminum samples in the megabar range of pressures (Kormer et al., 1962; Bakanova et al., 1974; van Thiel, 1977; Trunin et al., 2001). The presence of these data allows an accurate fit of the thermal contribution of atoms to the EOS. The analysis of Figure 9 demonstrates the reliability of the present EOS in regions of the phase diagram which are far from the principal Hugoniot. Note that the sound speed in shocked metal is also described with high accuracy, see Figure 9a. Analogous agreement has also been obtained after comparison with megabar–pressure data on double and triple compression of aluminum in reflected shock waves (Al'tshuler & Petrunin, 1961; Neal, 1976; Nellis et al., 1988, 2003).

Fig. 9. Phase diagram of aluminum at high pressures. Nomenclature: lines–EOS calculations, T–isotherms, M–melting region, m–shock adiabats of porous samples (m = ρ₀/ρ₀₀–porosity); points–experimental data, 1–Al'tshuler et al. (1981), 2–Kormer et al. (1962), 3–van Thiel (1977); 4–Al'tshuler et al. (1960a); Al'tshuler & Chekin (1984), 5–Bakanova et al. (1974), 6–Mitchell & Nellis (1981), 7–Simonenko et al. (1985), 8–revision Trunin et al. (2001) of original data Avrorin et al. (1987), 9–Trunin (1986), 10–Glushak et al. (1989), 11–Trunin et al. (1995a), 12–Trunin et al. (2001), 13–Knudson et al. (2003). a) Sound speed in shocked aluminum. Line–EOS, points-experiment, 1–Neal (1975), 2–Al'tshuler et al. (1960b), 3–McQueen et al. (1984), arrows indicate the melting region.

4.4. Aluminum at Lower Densities

The value of shock pressure or, more strictly, of entropy in shocked materials, influences how far a material expands with passage of a release wave. EOS calculations show that nothing special happens with aluminum in adiabatic–expansion experiments done in the range of shock pressures from 8–200 GPa (Bakonova et al., 1983; Zhernokletov et al., 1995). These data are shown in Figure 10. Here initial Hugoniot states correspond to solid or melted aluminum. Final states of the release isentropes have been measured under the condition of expansion into air.

Fig. 10. Release isentropes of solid and melted aluminum. Nomenclature: lines–EOS calculations, H ₁–principal Hugoniot, s _i–release isentropes; points–experiment, 1–Bakanova et al. (1983), 2–Zhernokletov et al. (1995).

Figure 11 illustrates the expansion process in which, according to the present EOS, shocked aluminum is liquid. Again, EOS calculations are compared against experimental isentropes (Glushak et al., 1989; Knudson et al., 2005). At higher pressure one can see from Figures 10 and 11 that the calculated isentropes s ₁₅–s ₂₁ deviate from the experimental data for expansion into air (Bakanova et al., 1983; Glushak et al., 1989) in a non–systematic way. In contrast, the agreement with isentropes s ₂₂–s ₃₁ expanded into aerogel (Knudson et al., 2005) is very good, see Figure 11. Like isentropes s ₁–s ₁₆ from Figure 10, all the isentropes s ₁₇–s ₃₁ show a monotonic dependence at all investigated pressures.

Fig. 11. Release isentropes of liquid aluminum. Nomenclature: lines–EOS calculations, H ₁–principal Hugoniot, s _i–release isentropes; points–experiment, 1–expansion into air Glushak et al. (1989), 2–expansion into aerogel Kundson et al. (2005).

All the release isentropes for aluminum s ₁–s ₃₁ from experiments (Bakanova et al., 1983; Zhernokletov et al., 1995; Glushak et al., 1989; Knudson et al., 2005) are presented in Figure 12. It is interesting to note, the positions of these release isentropes in the phase space. In the experiments of Bakonova et al. (1983) and Zhernokletov et al. (1995), isentropes s ₁–s ₇ and s ₁₀–s ₁₂ are in the solid state, isentropes s ₈ and s ₁₃ originate in the solid state and end in the melting region, while the isentrope s ₁₄ starts in the solid and ends in the liquid. Isentropes s ₁₅–s ₁₆ are mainly in the liquid state. Higher shock pressures, and consequently higher entropies, achieved in the experiments of Glushak et al. (1989) and Knudson et al. (2005) result in all isentropes s ₁₇–s ₃₁ occupying a region of the liquid state. The analysis of the position of the two-phase liquid-gas region with respect to the shock adiabat of air, see line R and points in Figure 12, shows that aluminum does not and will not evaporate upon adiabatic expansion from shocked states into air.

Fig. 12. Pressure–entropy diagram for aluminum. Nomenclature: lines–EOS calculations, H ₁–principal Hugoniot, M–melting region, R–liquid–gas region with the critical point CP, s _i–release isentropes (see Figs. 10, 11); points–experiment, 1–Zhernokletov et al. (1995), 2–Bakanova et al. (1983), 3–Glushak et al. (1989), 4–Knudson et al. (2005).

Density measurements at P = 1 bar are available for solid (Toloukian et al., 1975) and liquid (Lang, 1995) aluminum at lower densities. Thermophysical properties of the liquid aluminum have been studied at P = 0.3 GPa in an isobaric expansion experiment (Gathers, 1983). EOS calculations are compared against these data and different evaluations of the critical point in Figure 13. This figure demonstrates that the present EOS describes the experimental points with outstanding accuracy. The EOS parameters of the critical point, P _c = 0.197 GPa, T _c = 6250 K, and V _c = 1.423 cm³/g, agree with QEOS parameters P _c = 0.168 GPa, T _c = 5520 K (Young & Corey, 1995) and are also very closely in density with the other evaluations. This position of the critical point in the present EOS provides for an accurate description of IEX data (Gathers, 1983), see the 0.3 GPa isobar in Figure 13. The calculated evaporation temperature at room pressure, T _V = 2770 K, coincides with the tabulated value (Hultrgen et al., 1973).

Fig. 13. Phase diagram of aluminum at lower densities. Nomenclature: lines–EOS calculations, M–melting region, R–liquid-gas region with the critical point CP, P–isobars, and L–density of liquid metal at 1 bar Lang (1994–1995); points–experiment, 1–Toloukian et al. (1975), 2–Gathers (1983), and evaluations of the critical points, 3–Gates & Thodos (1960), 4–Morris (1964), 5–Young & Alder (1971), 6–Fortov & Yakubov (1999), 7–Gathers (1986), 8–this work, 9–Likalter (2002).

EOS isotherms are compared with QMD calculations (Desjarlais, personal communication) in Figure 14. The EOS critical isotherm at T = 6250 K describes the QMD results very well; the density of the critical point is also very close to the QMD result. The isotherm at T = 12000 K also shows good agreement with the QMD calculations.

Fig. 14. Pressure–density diagram of aluminum's critical region. Nomenclature: lines–EOS calculations, R–liquid–gas region with the critical point CP, T–isotherms; open circles–QMD calculations Desjarlais (2006).

Experimental isochors of heated aluminum (Renaudin et al., 2003) occupy the super critical domain on the phase diagram. EOS isochors and the isochore 0.1 g/cm³ from Saha plasma model (Gryaznov et al., 1998) are drawn in Figure 15, together with experimental points. The comparison with other advanced theoretical models is also available in the original work (Renaudin et al., 2003).

Fig. 15. Pressure–energy diagram of isochorically heated aluminum. Nomenclature: lines–EOS calculations, points with bars–EPI experiment Renaudin et al. (2003), open stars–Saha model Gryaznov et al. (1998) (numbers near stars indicate ionization ratio).

Finally, the summary of all mentioned above experimental data is given in 3D pressure–volume–temperature surface in Figure 16. It is worth to underline good potential capabilities of intense heavy ion beams for inducing high-energy-density states in expanded aluminum. In this region of practical interest and importance, we have today limited amount of a data for testing and calibrating modern theoretical models.

Fig. 16. Generalized 3D volume-temperature-pressure surface for aluminum. M–melting region; R–boundary of two-phase liquid-gas region with the critical point CP; H ₁ and H _p–principal and porous Hugoniots; H _air and H _aerogel – shock adiabats of air and aerogel; DAC–diamond-anvil-cells data; ICE–isentropic compression experiment; IEX–isobaric expansion data; S–release isentropes; HIHEX–region accessible with use of intense heavy ion beams SIS18 and SIS100 (see for details Hoffman et al. (2002); Tahir et al. (2005a)). Phase states of the metal are also shown.

5. CONCLUSION

New advanced studies of aluminum properties at extreme conditions include experiments and theoretical calculations. Experimental investigations of isothermal, shock and isentropic compression of aluminum at high pressures up to 500 GPa have demonstrated behavior that results in a more “stiff” compression curve and shock adiabat for aluminum than was previously figured. Adiabatic-expansion measurements of shocked aluminum have brought new accurate data for the hot expanded liquid. Theoretical “ab initio” QMD calculations in the vicinity of the critical point have established the reference domain of thermodynamic data.

These data serve as a fundamental basis for the new multi-phase EOS for aluminum presented in this paper. It accounts for the high pressure, high temperature experimental, and theoretical data that was available up to the middle of 2007. The compression curve and the principal Hugoniot in the resulting EOS are fit to new data, as well as the critical point. According to the EOS model, shocked aluminum melts at 113 GPa and the parameters of the critical point are: P _c = 0.197 GPa, T _c = 6250 K, and V _c = 1.423 cm³/g.

The developed EOS describes with high accuracy and reliability a broad range of the phase diagram, from the high-pressure shocked metal to more dense states in reflected shock waves and to regions of the phase diagram with much lower densities accessed in the process of the adiabatic expansion of shocked metal. The high accuracy suits this EOS to be applied in advanced numerical modeling for solving numerous problems in the physics of high energy densities. The developed EOS together with QMD data provides for self-consisted description of both thermodynamic and transport properties of aluminum.