Equation of State for Aluminum at High Entropies and Internal Energies in Shock Waves

Abstract

The present theoretical work is devoted to the construction of a model of the equation of state for matter, where the specific volume is used as the thermodynamic potential, and the entropy and the thermal part of the internal energy act as thermodynamic variables. Based on the proposed model, called STEC, calculations were carried out for aluminum in the region of high internal energies and entropies. A comparison of the calculated shock adiabats with the available data from shock-wave experiments indicates that the constructed equation of state describes well the thermodynamic properties of aluminum up to a shock compression pressure of about 1 TPa. The proposed STEC equation-of-state model can be used in numerical simulations of various processes under extreme conditions at high energy densities.

1. Introduction

The theoretical description of the behavior of matter across a wide range of changes in thermodynamic variables is of interest for solving a broad range of problems in fundamental and applied physics of intense pulsed processes [1,2,3]. The impact of high mechanical and thermal loads upon a material leads to a change in the parameters of its state from the starting point towards an increase in internal energy. And this can be accompanied by a noticeable increase in pressure, for example, as occurs during a high-velocity collision of bodies [4,5,6,7,8,9,10,11,12,13,14,15] or during high-intensity short laser irradiation of a solid target [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], or a noticeable increase in temperature and entropy, for example, as during an electrical explosion of conductors under the influence of a powerful current pulse [31,32,33,34,35,36,37]. For numerical modeling of such fast-moving processes, it is necessary to know the equation of state for the substance under consideration in the entire range of realized parameters [38,39,40]. Moreover, the correctness of the equation of state will play a decisive role in ensuring that the modeling results are correct [5,6,20,25,32,41,42].

Aluminum is an example of a material that is widely used as an element of structures that are subject to intense impulse effects during operation [4,12,13,15,43]. In particular, aluminum was the main material for the protective shields (screens) of the Vega unmanned spacecraft against impacts from dust particles moving at a velocity of 60 to 80 km/s during the exploration of Halley’s Comet [4]. In developing the optimal design of these screens, the equation of state for aluminum was used, taking into account melting and evaporation [2,4,44], which was important for the correct numerical modeling of the impact of dust particles with the screen at such velocities. For calculating the parameters of aluminum screens for protecting spacecraft from space debris at lower impact velocities (from 3 to 15 km/s), a simpler equation of state (without taking into account phase transformations) was used in a recent study [13]. A review of some of the works related to protecting spacecraft from space debris can be found elsewhere [12].

A numerical simulation of the impact of an aluminum cylinder with a velocity of 6.6 km/s on an aluminum plate was carried out [5] using the equation of state in three different versions, namely, in caloric form for condensed and rarefied matter [45] and in a form based on the thermodynamic potential of the Helmholtz free energy [2] taking into account melting and evaporation, not allowing or allowing the possibility of the formation of metastable states of liquid and vapor phases. Moreover, the results of modeling [5] with the first and third variants of the equation of state turned out to be similar. To simulate the impact of a titanium plate with a velocity of about 10.4 km/s on a static aluminum plate, in addition to the equation of state [2], which takes into account phase transformations and metastable regions, models of the kinetics of the formation of a liquid–vapor mixture and the mechanical fracture of a metastable liquid were taken into account [6].

For simulating the impact of an aluminum ball with velocities ranging from 2 to 6.5 km/s with targets in the form of round plates made of high-density polyethylene, the equation of state for both materials was used in a simple caloric form together with the constitutive relations and the fracture model [46]. A discrepancy was noted between the simulation results and the experimental data at impact velocities above 5.5 km/s, possibly due to the simplified form of the equation of state used without taking into account the effects of liquid phase formation [46].

Models of the dynamics of shock-wave loading processes in aluminum samples were implemented taking into account plasticity and fragmentation through the dynamics and kinetics of microdefects (dislocations and microcracks) [11], as well as in viscoelastic and hydrodynamic approximations [9]. In the first case [11], the model also included the equation of state for aluminum taking into account melting [2], and in the second case [9], a simpler model of the equation of state for the condensed phase of this metal (without taking into account melting) [47] was used.

Numerical studies of shock loading processes in initially porous aluminum samples of conical shape, inserted into a solid target (with a conical cavity) made of metal with higher dynamic impedance (steel and lead), made it possible to establish some features of the resulting flow with a converging shock wave [7,8]. In these studies, the equation of state for aluminum (as well as for other metals) was used within the framework of the model [2] that takes into account melting (at the compression stage) and evaporation (at the jet expansion stage, in the target version with an outlet hole [8]).

In the practice of shock-wave experiments, aluminum is frequently used as a standard material with well-studied properties [48]. However, a noticeable discrepancy between the experimental points for the shock compressibility of aluminum occurs at pressures above 0.2 TPa [49,50], possibly due to a discrepancy in the parameters of the shock adiabats of other standard materials available in this region. Calculations using the Thomas–Fermi (with corrections) and Hartree–Fock–Slater models were carried out to refine the equation of state [50,51], and an interpretation of the available data from shock-wave experiments for this metal in the region of higher pressures has been given [51].

Aluminum is widely used as a component of composites (for example, ceramic filler in an aluminum matrix [52], carbon fiber-reinforced plastics between aluminum plates bonded by epoxy adhesive [53] and compositions of layers of elastomeric polyurethane and honeycomb aluminum [54]) and other materials (for example, foams [55,56,57,58]) for structures subject to high dynamic loads.

A numerical model of the ablation process of an aluminum target under the action of an ultrashort (femtosecond) high-intensity laser pulse was proposed [18] taking into account the absorption of radiation energy, the effects of electron thermal conductivity and the exchange of electron energy with phonons or ions, phase transformations (melting and evaporation), and the possibility of forming metastable states of the target material under tensile stresses. The equation of state in such problems [18,20] was used in a variant with a distinction between the temperatures of the electronic and ionic subsystems, and for the ionic subsystem a model approach based on some ideas [1,2] was implemented.

Irradiation of plates made of aluminum and AMg6M alloy with short (picosecond) laser pulses allows, in experiments [19,23], the study of the phenomenon of spalling from the rear side of the target and also, using a hydrodynamic code, the calculation of the tensile mechanical stress at which this spalling occurred. In modeling this process [19,23], the equation of state for aluminum was used in caloric form [45]. The possibility of obtaining additional information about the equation of state for the material under study in such experiments has been considered elsewhere [22]. The use of other variants of the equations of state for aluminum in similar problems is also encountered [59].

The action of shock waves obtained by laser pulses has found application as a tool for surface treatment of metals (in particular, aluminum alloys [60,61,62]) by (laser shock) peening. Other methods of influencing the mechanical properties of metals are also being developed, such as stress relief using ultrasonic vibration [63]. To generalize experimental data in a form convenient for use in numerical modeling, in particular, different models of constitutive relations are used [63].

Aluminum and its alloys are technologically advanced materials for the production of conductive elements for various applications in high-current electronics. For example, a magnetic field created by a high-linear-density current pulse can be used to accelerate a duralumin plate [10], which can serve as an impactor in a shock-wave experiment. Aluminum foils are used in the form of cylindrical liners compressed by a magnetic field to produce hot, dense plasma [31,64]. A variant of the numerical model of the initial stage of this process was proposed in the work [31], in which the equation of state for aluminum was used, taking into account the phase transition of the solid phase into liquid and then the formation of a two-phase liquid–vapor mixture and plasma [2,4,44]. The possible development of instabilities (capable of preventing the achievement of the desired parameters of dense hot plasma) during heating of thin aluminum foil in a vacuum by a high-density current pulse was investigated experimentally in a recent work [64]. Rapid heating of thin aluminum foil in transparent dielectric plates has been studied in experiments [32] and simulations (with different equations of state for aluminum) [32,41,42]; in this case, heating and expansion of the molten metal with the formation of plasma occur at a pressure higher than the critical point of the liquid–vapor phase transition.

Electrical explosions of aluminum wires in water were investigated in experiments [37], and the development of thermoelectric instabilities was discovered after the start of evaporation of the metal. Various methods of accelerating an aluminum disk using electrical explosion of thin foils (made of aluminum or copper) in water have been tested in experiments and simulations [40]. In hydrodynamic modeling [37,40], equations of state for the metals and water in caloric and thermal forms were used.

In addition to the equations of state [2,4,44,47,50,51,59] mentioned in the above cases for aluminum, there are many examples of using different model approaches to describe the thermodynamic properties of this metal at high energy densities [65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96]. These approaches are constantly being improved, and their adequacy is conveniently tested using the example of aluminum, for which there is a lot of experimental data over a wide range of pressures and internal energies obtained using shock-wave techniques. And these circumstances stimulate an unquenchable interest in the study of this material, despite the feeling that it has already been thoroughly studied.

When solving problems of hydrodynamics, the equation of state closes the system of equations of motion of the medium, which express the laws of conservation of mass, momentum and energy. In this case, a necessary requirement for the equation of state is the expression of the relationship between density ($\rho$) or specific volume ($V={\rho }^{-1}$), pressure (P) and specific internal energy (E). Examples of equations of state in the form

$$E=E(V,P)$$

(1)

or

$$P=P(V,E),$$

(2)

which agree well with data from shock-wave experiments, are known [9,33,45,92,97,98,99,100]. This caloric form of the equation of state is sufficient for modeling adiabatic processes [5,9,23,33,59,96].

Another convenient form of representing the equations of state [47,50,65,66,67,76,77,79,82,86,87,93,94,101,102] are the dependencies

$$E=E(V,T)$$

(3)

and

$$P=P(V,T),$$

(4)

which together also relate the specific volume, pressure and internal energy by means of an additional thermodynamic parameter—the absolute temperature (T) [103]. Knowledge of temperature expands the possibilities of constructing numerical models for processes with heat sources, for example, with the effect of thermal conductivity [104].

As is known [103], the relationship between the derivatives of functions (3) and (4) is established by the first and second laws of thermodynamics, which express the law of conservation of energy in a thermodynamic system during a reversible process:

$$\mathrm{d}E=-P\mathrm{d}V+T\mathrm{d}S,$$

(5)

where S is the specific entropy. This relationship can be transformed to a form with the variables V and T, which connects the derivatives of the functions (3) and (4) [103]. This can become the basis for finding the temperature from the equation of state (1) or (2) if one involves some additional information about the temperature of the substance [1], for example, at the atmospheric pressure isobar [97] or an isochor [98]. Of course, the equation of state can be specified directly in the variables through whose differentials the total differential of the internal energy (5) is expressed, namely, in the canonical form $E=E(V,S)$ [105].

In this paper, based on the main thermodynamic law (5), a model of the equation of state for matter is proposed in which specific entropy and the thermal part of the specific internal energy are used as variables. A description of this model is given in Section 2. The results of calculating the thermodynamic characteristics of aluminum using the new model are given in comparison with the available data from shock-wave experiments at high pressures in Section 3. The main conclusions of this work are drawn in Section 4.

2. Equation-of-State Model

In a number of papers [77,86,87,106,107,108,109], the thermodynamic equilibrium component model, called TEC, has been proposed to describe properties of various materials at high pressures. The expressions of this model, given in the form of the caloric and thermal equations of state (3) and (4), do not formally satisfy the main law (5) [109], but the results of calculations using this model demonstrate good agreement with shock-wave data, in particular, for aluminum [77,86,87]. In this section, the expressions of the TEC model are discussed, the issue of their inconsistency with the main law (5) is examined and a new model, called STEC (S stands for entropy), based on the main law (5) is proposed.

2.1. TEC Model

The pressure and specific internal energy in the TEC model are composed of two terms responsible for the cold ($P_c$ and $E_c$) and thermal ($P_t$ and $E_t$) parts:

$$P(V,T)=P_c\left(V\right)+P_t(V,T)$$

(6)

and

$$E(V,T)=E_c\left(V\right)+E_t\left(T\right).$$

(7)

The cold pressure is given in the form of the Murnaghan equation [110]:

$$P_c\left(V\right)=A({\sigma }^n-1),$$

(8)

where $\sigma =V_0/V$; $V_0$ is the specific volume of the substance at $P_c=0$; and A and n are constants. The cold part of the internal energy is chosen from the conditions $P_c=-\mathrm{d}{E}_c/\mathrm{d}V$ and $E_c\left(V_0\right)=0$:

$$E_c\left(V\right)={\displaystyle \frac{A{V}_0}{n-1}}\left({\displaystyle \frac{{\sigma }^n-1+n}{\sigma }}-n\right).$$

(9)

The thermal pressure is given in the following form:

$$P_t(V,T)={\displaystyle \frac{\Gamma }{V}}{C}_{V0}(T-T_0),$$

(10)

where $C_{V0}$ and $T_0$ are constants; $\Gamma$ is a coefficient that depends only on temperature,

$$\Gamma \left(T\right)={\Gamma }_{\infty }+{\displaystyle \frac{{\Gamma }_0-{\Gamma }_{\infty }}{1+C_1(T-T_0)}},$$

(11)

${\Gamma }_0$, ${\Gamma }_{\infty }$ and $C_1$ are constants, and

$$C_1={\displaystyle \frac{{\Gamma }_0-{\Gamma }_1}{(T_1-T_0)({\Gamma }_1-{\Gamma }_{\infty })}},$$

(12)

where ${\Gamma }_1$ and $T_1$ are also constants. The thermal part of the specific internal energy is given as

$$E_t\left(T\right)=C_{V0}(T-T_0).$$

(13)

One can easily verify that the expressions (6)–(13) of the TEC model do not satisfy the main law (5). Directly from (5), the expression for the total differential of the entropy with respect to the variables E and V follows:

$$\mathrm{d}S=T^{-1}\mathrm{d}E+P{T}^{-1}\mathrm{d}V.$$

(14)

From (14), the relations ${(\partial S/\partial E)}_V=T^{-1}$ and ${(\partial S/\partial V)}_E=P{T}^{-1}$ follow in turn. By equating the mixed derivatives of entropy with respect to E and V, one can determine the relationship between the derivatives of functions (3) and (4):

$${(\partial E/\partial V)}_T-T{(\partial P/\partial T)}_V+P=0.$$

(15)

This relation for thermodynamically consistent equations of state (3) and (4) is an identity. For expressions (6)–(13) this requirement is not met.

Some illustrations of this thermodynamic inconsistency of the TEC model are given in Appendix A.

It should be noted that according to the definitions of the thermal parts of pressure (10) and internal energy (13), it turns out that the cold parts (8) and (9) correspond to the isotherm $T=T_0$.

2.2. STEC Model

Following the accepted thermodynamic definition of pressure, which succeeds from the main law (5),

$$P=-{(\partial E/\partial V)}_S,$$

(16)

the STEC model assumes that the cold parts of pressure ($P_s$) and specific internal energy ($E_s$) correspond to the isentrope $S=S_0$.

Then the pressure and internal energy are written as functions of specific volume and specific entropy:

$$P(V,S)=P_s\left(V\right)+P_t(V,S)$$

(17)

and

$$E(V,S)=E_s\left(V\right)+E_t(V,S).$$

(18)

The cold parts of pressure and specific internal energy are given similarly to (8) and (9):

$$P_s\left(V\right)=A({\sigma }^n-1)$$

(19)

and

$$E_s\left(V\right)={\displaystyle \frac{A{V}_0}{n-1}}\left({\displaystyle \frac{{\sigma }^n-1+n}{\sigma }}-n\right).$$

(20)

The thermal part of pressure is given similarly to (10) after eliminating temperature using (13):

$$P_t(V,S)={\displaystyle \frac{{\Gamma }_t}{V}}{E}_t(V,S),$$

(21)

where

$${\Gamma }_t\left(E_t\right)={\Gamma }_{\infty }+{\displaystyle \frac{{\Gamma }_0-{\Gamma }_{\infty }}{1+{\mathrm{A}}_1{E}_t}},$$

(22)

${\Gamma }_0$ and ${\Gamma }_{\infty }$ are the same constants as in (11), and

$${\mathrm{A}}_1={\displaystyle \frac{C_1}{C_{V0}}},$$

(23)

where $C_1$ is the same constant as in (12).

To find the entropy dependence for the internal energy, it is assumed that the isochoric heat capacity at $V=V_0$ is a constant value, $C_V{|}_{V=V_0}=C_{V0}$. Then the thermal part of the internal energy at $V=V_0$ depends on the entropy as follows:

$$E_v\left(S\right)=C_{V0}{T}_0\left(exp{\displaystyle \frac{S-S_0}{C_{V0}}}-1\right).$$

(24)

By substituting the expressions (17) and (18) into the definition of pressure (16), one can obtain a differential equation for the relationship between $E_t$ and V on an isentrope. By solving this equation taking into account (24), an explicit expression for the specific volume as a function of the thermal part of the specific internal energy and the specific entropy can be obtained:

$$V(E_t,S)=V_{0}exp\left({\displaystyle \frac{1}{{\Gamma }_0}}ln{\displaystyle \frac{E_v\left(S\right)}{E_t}}+{\displaystyle \frac{{\Gamma }_0-{\Gamma }_{\infty }}{{\Gamma }_0{\Gamma }_{\infty }}}ln{\displaystyle \frac{{\Gamma }_0+{\Gamma }_{\infty }{\mathrm{A}}_1{E}_v\left(S\right)}{{\Gamma }_0+{\Gamma }_{\infty }{\mathrm{A}}_1{E}_t}}\right).$$

(25)

This quantity V (25) can now be considered as a thermodynamic potential for which the total differential is expressed through the differentials of the variables $E_t$ and S by transforming the basic law (5) taking into account relations (17) to (20):

$$\mathrm{d}V=-{P_t}^{-1}\mathrm{d}{E}_t+T{P_t}^{-1}\mathrm{d}S.$$

(26)

From this, the relations for the factors of the differentials of the variables on the right side of the expression for the differential $\mathrm{d}V$ (26) follow:

$${P_t}^{-1}=-{(\partial V/\partial E_t)}_S$$

(27)

and

$$T{P_t}^{-1}={(\partial V/\partial S)}_{E_t}.$$

(28)

Applying relation (27) to the thermodynamic potential V (25) results in the expression (21). Applying relation (28) to the potential V (25) together with the formula for $P_t$ (21) allows one to obtain an explicit expression for the temperature as a function of the variables $E_t$ and S:

$$T(E_t,S)=T_0{\displaystyle \frac{{\Gamma }_t{E}_t}{{\Gamma }_v{E}_v\left(S\right)}}exp{\displaystyle \frac{S-S_0}{C_{V0}}},$$

(29)

where

$${\Gamma }_v\left(E_v\right)={\Gamma }_{\infty }+{\displaystyle \frac{{\Gamma }_0-{\Gamma }_{\infty }}{1+{\mathrm{A}}_1{E}_v}}.$$

(30)

Expressions (16)–(30) fully formulate the STEC model for thermodynamically consistent calculations of the properties of matter over a wide range of changes in internal energy and entropy.

3. Thermodynamic Characteristics of Aluminum in Shock Waves

The results of the calculations of shock compressibility of aluminum using the STEC model with constants that were previously proposed [77,86,87] for the TEC model expressions are presented below. The values of the constants used are as follows: ${\rho }_0=2.71$ g/cm³, $A=24.57$ GPa, $n=3.1$, $C_{V0}=1$ J/[g K], ${\Gamma }_0=2.13$, ${\Gamma }_1=1.25$, ${\Gamma }_{\infty }=0.3$, $T_0=0.3$ kK and $T_1=23$ kK. When calculating, it is set that $V_0=1/{\rho }_0$ and $S_0=0$.

The shock compressibility of aluminum has been investigated over a wide range of pressures [51,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134]. In these experiments, various loading means have been used, including traditional planar explosive devices [48,135], layered cumulative systems [124,136] and converging-shock-wave explosive generators [135,137,138], as well as gas guns [121,139,140]. Underground nuclear explosions [49,141,142,143], laser pulses [16,17,21,22,23,24,144] and magnetic fields [10,130,140,145] have also been used to generate shock waves in aluminum samples. Here, the initial porosity of the samples (${\rho }_0/{\rho }_{00}$, where ${\rho }_{00}$ is the initial density of the samples) is varied from 1 to 8.

For comparison with these experimental data, the parameters of the shock adiabats were calculated by solving a system of equations, including the equation of state and the expression for the law of conservation of energy of matter through the shock-wave front [1]:

$$E=E_0+2^{-1}(P+P_0)(V_{00}-V),$$

(31)

where E, P and V are the specific internal energy, pressure and specific volume of the substance behind the front, and $E_0$, $P_0$ and $V_{00}$ are the same parameters ahead of the front.

The velocity of the shock-wave front ($U_s$, the shock velocity) and the velocity of the substance behind the front ($U_p$, the particle velocity) are determined from the laws of conservation of mass and momentum through the shock-wave front [1] in the following form:

$$U_s=V_{00}\sqrt{{\displaystyle \frac{P-P_0}{V_{00}-V}}},$$

(32)

$$U_p=\sqrt{(P-P_0)(V_{00}-V)}.$$

(33)

In the selected initial state for relations (31)–(33), the pressure is assumed to be $P_0=0.1$ MPa and the internal energy is found from the equation of state, $E_0=E(V_0,S_{\mathrm{in}})$, where the entropy at the initial point $S_{\mathrm{in}}$ is given by the condition $P_0=P(V_0,S_{\mathrm{in}})$ and $V_{00}=1/{\rho }_{00}$.

The results of calculations of the parameters of shock compression of aluminum using the STEC model for samples with ${\rho }_{00}=2.71$, 2.676, 2.607, 1.92, 1.89, 1.816, 1.585, 1.35, 1.3, 0.909, 0.77 and 0.34 g/cm³ are presented in Figure 1, Figure 2 and Figure 3 in comparison with experimental data [51,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134].

One can see that, in the considered pressure range up to 1 TPa, the shock adiabats calculated using the presented STEC model with constants from [77,86,87] are in good agreement with the data from shock-wave experiments [51,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134].

In Figure 1, Figure 2 and Figure 3, shock (Hugoniot) adiabats calculated using the equation of state for aluminum [92] within the framework of the KEOS2M model are also shown. The KEOS2M model [9,25,33,92,99,100] is defined in the form of pressure as a function of volume and internal energy (2). In this case, the parameters of the states on the shock adiabats are found by solving the system of two equations: (31) and (2). In the initial state, the pressure in front of the shock wave is assumed to be $P_0=0.1$ MPa, and the internal energy $E_0$ is found from the condition $P_0=P(V_0,E_0)$.

Analysis of the calculated curves and experimental points drawn in Figure 1, Figure 2 and Figure 3 shows that, in the considered pressure range up to 1 TPa, the both STEC and KEOS2M models give results close to each other and to experiments for non-porous samples of aluminum. With increasing porosity of the samples, the difference becomes noticeable already at pressures up to 100 GPa.

In general, the equations of state within the framework of the STEC and KEOS2M models are in good agreement with the set of available shock-wave data for both solid [51,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131] and porous samples [112,114,115,131,132,133,134] of aluminum; differences appear at pressures higher than those achieved experimentally for a fixed initial density, or above 1 TPa. It can be assumed that this is due to the fact that the Grüneisen coefficient $\gamma =V{(\partial P/\partial E)}_V$ according the STEC model depends only upon the thermal part of the internal energy, whereas this coefficient $\gamma$ according the KEOS2M model explicitly depends upon both the thermal part of the internal energy and the specific volume.

It should be noted that the proposed STEC equation-of-state model does not take into account possible phase transformations of the metal within the range of thermodynamic parameters under consideration. Examples of equations of state for aluminum that take melting and evaporation into account can be found elsewhere.

4. Conclusions

Thus, the STEC model proposed in this work makes it possible to describe the properties of matter in a thermodynamically consistent manner within the framework of an unusual thermodynamic potential of specific volume, for which the variables are the entropy and the thermal part of the internal energy. For aluminum, the present equation of state based on fairly simple expressions of the STEC model with constants previously proposed for the TEC model demonstrates good agreement with the data of shock-wave experiments in the pressure range up to 1 TPa. Taking into account possible phase transformations of the metal in this range of achievable parameters remains outside the scope of the STEC model. The developed STEC model can be used in numerical simulations of hydrodynamics of various physical processes at high entropies and energy densities.