Equation of state of liquid Indium under high pressure

We apply an equation of state of a power law form to liquid Indium to study its thermodynamic properties under high temperature and high pressure. Molar volume of molten indium is calculated along the isothermal line at 710K within good precision as compared with the experimental data in an externally heated diamond anvil cell. Bulk modulus, thermal expansion and internal pressure are obtained for isothermal compression. Other thermodynamic properties are also calculated along the fitted high pressure melting line. While our results suggest that the power law form may be a better choice for the equation of state of liquids, these detailed predictions are yet to be confirmed by further experiment.


I. INTRODUCTION
Thermodynamic properties of materials at high temperature and high pressure are crucial for many fundamental and applied problems.2][3][4] A good EOS can often provide qualitative information of thermodynamic properties of a given system through a limited number of measurements.Various theoretical methods have been developed for EOS, which include quantum and statistical mechanics methods, and used to understand the related physical processes.Many forms of EOS proposed are either based on the theoretical derivation or fitting of experimental data.While the EOS can be derived from theories, the experimental data at high temperature and pressure can serve as reference points for validating the theories and semi-empirical models.One example is the Birch-Murnaghan EOS 1,2 derived by expanding the bulk modulus in a series as a function of pressure.However, all these theories are applicable only in certain limited temperature and pressure ranges.Murnaghan EOS 1 is known to be applicable to the low-pressure region, whereas the Birch-Murnaghan EOS is known to induce unphysical results such as molar volume maximum at high pressure.Logarithmic EOS 3 is obtained from the assumption that the free energy can be expanded into power series of the Hencky strain.Vinet EOS 4,5 is based on the universal relation between binding energy of the solids and intermolecular distance.
7][8][9] This is particularly relevant for liquids where it is generally believed that the EOS models should apply relatively straightforwardly due to the apparent isotropy and uniformity of liquids.For example, the isothermal expansion coefficient may be directly calculated by volume as an explicit function of temperature and pressure from EOS.However, most EOS models fail also.Baonza, et al. [10][11][12][13] have developed an EOS based on a pseudospinodal hypothesis.It has been shown to have reasonable precision for approximating the EOS for a variety of molecular liquids, solids and liquid, i.e. mercury, up to several Mbar.Its unique feature is that the bulk modulus is expressed as a function of pressure in a power law form.This particular functional form allows for coverage of a large span of pressure which is not possible using other widely used EOS such as Birch-Murnaghan.In this work, we shall use the general power-law form of the EOS to estimate and predict the thermodynamic properties of liquid Indium (In).Our purpose is to see if certain suitable and simple forms of EOS, such as the power law form, for liquids can be validated that have not been obvious and easy to obtain so far.For this reason, we shall not pay too much attention to the specific physical meaning of certain parameters appearing in the original model; [10][11][12][13] instead we treat them as merely the fitting parameters for the best EOS.
In this study, we shall use the power law form of EOS to fit the data on molar volume of In melts measured in a diamond anvil cell. 14,15Indium has relatively low melting temperatures and thus, its properties can be measured relatively easily close to room temperature.Using this particular equation of state with power law form fitted with available experimental values, we calculated the molar volume of molten In along the isothermal line at 710K within good precision as compared with the known data from experiments.Bulk modulus, thermal expansion and internal pressure are also obtained for isothermal compression.Thermodynamic properties are also calculated along the fitted high pressure melting line.In addition, we investigated the variation of the internal pressure 16 with the changes of temperature at ambient condition.We hope that these predicted properties could be compared with further experiment measurements and will help us to judge and select more suitable EOS for liquids.

A. EOS for liquid In
At relatively low pressure, isothermal bulk modulus is a linear function of pressure in many liquids and solids, as shown in Murnaghan EOS. 1 So isothermal bulk modulus could be approximated by a polynomial function of pressure as B T (T, P) = B T 0 (T) + B ′ T 0 P + ) T at zero pressure.Second order approximation, however, is not applicable at high pressure which can induce unreasonable abnormal results. 2 To consider the higher order term, more parameters are needed although very limited ways are available to obtain the corresponding information from experiments and simulation.Alternatively, one can express the isothermal bulk modulus B T (T, P) = B T (P) at the fixed temperature T in a power law form as [10][11][12][13] B T (P) = 1 where β is a constant, B T and P are isothermal bulk modulus and pressure, respectively, and κ * and P sp are constants along the thermal line.In the original formulation, [10][11][12][13] P sp is identified with the pressure along the pseudospinodal curve but we shall not follow this definition strictly here.][12][13] The molar volume as a function of pressure is derived by integration of Eq. ( 1) in terms of zero pressure quantities, V 0 , the molar volume and B T 0 , the isothermal bulk modulus, and B ′ T 0 , its isothermal pressure derivative, where P sp , κ * and V sp are characteristic parameters related to the zero-pressure quantities through the following relations at the temperature T. Without considering the intrinsic temperature effect 17 the pressure derivative of isothermal bulk modulus can be approximated as To correlate the melting point of solids with high pressure, we use Simon-Glatzel two-parameter equation, 18,19 with T 0 m being the melting point at the reference pressure P 0 , and a and c two empirical parameters.For solid In, we use the following values, T 0 m = 429.76Kat P 0 = 0, a = 3.58GPa and c = 2.3.The above relations furnish the EOS for liquid In.In the following, we shall obtain the free parameters using experimental data.Shen, et al. 14,20 applied radiographic method to measure molar volume of molten In in an isothermal compression at 710K up to the solidification pressure of 8.5GPa in an externally heated diamond anvil cell.The isothermal compressibility K T and density ρ of liquid In at ambient pressure condition have been obtained from speed-of-sound measurements 21 as follows and We fit their data with Eqs. ( 2), ( 4) and ( 5) by using the obtained linear equations for isothermal compressibility K T and density ρ from the speed-of-sound measurements at ambient pressure. 21he relations give V 0= 16.85cm −3 which is in excellent agreement with V 0 = 16.80cm−3 by the sessile-drop method. 22For low pressure (P < 1.0GPa), the fitting from Eq. ( 2) is very close to the experiment data (see Fig. 1(a)).With the increasing pressure, however, the fitted data begin to deviate from the measured compression data with 4% for B ′ T 0 = 5.21 (and 3% for B ′ T 0 = 4.0 1,2 ) at T = 710 K.
Next, we look at the thermal expansion of liquid In from the relations obtained above.Many materials expand upon heating due to the increase of interatomic distances.Using Eq. ( 2), the pressure variation of thermal expansion coefficient can be obtained as the following, To obtain the pressure dependence of the thermal expansion coefficient and its relation with entropy change, we use the Maxwell relation, And we have In Fig. 1(b) and Fig. 1(c), the results show that the calculated thermal expansion decreases and the isothermal bulk modulus increases with the rising pressure at 710K.

B. Internal pressure
Internal pressure of liquids is defined as the negative value of the partial derivative of the internal energy of system with respect to the volume,  2), Eq. ( 6), Eq. (1), Eq. ( 9), and Eq. ( 12), respectively, (solid line for B ′ T 0 = 5.21, dotted line for U is the internal energy of liquids that is contributed from the interactions between atoms and P is the external pressure.In general it can be divided into two parts, U = U r + U a , where U r and U a are the energy from repulsive and attractive interaction.Substituting U into Eq.( 9), we have where the repulsive internal pressure, P r int , is positive in sign while attractive internal pressure , P a int , is negative.And from Eq. ( 9) and the cyclic relation with α P = 1 V ∂V ∂T P and B T = −V ∂P ∂V T , we have And where ) .
When the external pressure increases from ambient condition at fixed temperature (Fig. 1(d)), the magnitude of internal pressure of liquid In decreases but with negative sign, which shows the attractive forces play the major role under low-pressure conditions with the increasing influence from the repulsive force.For B ′ T = 4.0, we obtain without considering the intrinsic temperature effect and with the approximation , where δ is some constant.And it does show the internal pressure is a linear function of hydrostatic pressure (Fig. 1(d)).For liquid In at 710K, P int could reach zero near P = 2.30 GPa for B ′ T = 4.0 (P = 2.10 GPa for B ′ T = 5.21) (Fig. 1(d)).When external pressure increases further, the internal pressure of liquid Indium is larger than zero with P r int > P a int .The magnitude of the temperature derivative of internal pressure ( ∂P int ∂T ) P is increasing with the pressure (Fig. 1(e)).At ambient pressure, the temperature dependence of internal pressure of liquid In (Fig. 2(a)) shows that the attractive force component is dominant over the repulsion force within the high temperature region, P r int < P a int .A linear function of internal pressure with temperature variation was suggested in liquefied permanent gases, liquid metals, non-polar organic liquids, and indeed it can be seen that the temperature variation of ( ∂P int ∂T ) P for liquid In (Fig. 2(b)) is nearly linear for temperature range 430 < T < 800 K. 18 In Fig. 2(c), it shows the pressure P sp is nonlinearly increasing with the rising of temperature.However, considering the pressure variation of B ′ T , the temperature dependence of becomes nonlinear for high temperature and could be described by the exponential where a int , β int , and b int are the material parameters (Fig. 2(b)).For liquid In, we have ∂T < 0 and both a int and b int are negative, a int = −3.597MPaK−1 , β int = 0.816, and b int = −1.206MPa K −1 .A solution of Eq. ( 11) is obtained by integration of Eq. ( 15), with the integration constant c int = −0.319GPa at ambient pressure.Using Eq. ( 9) and Eq. ( 11), we could solve the equation P int = 0 to the pressure (Fig. 3), with the temperature variation and then substitute the pressure into Eq.( 2) to have the volume. of liquid indium at ambient condition from Eq. ( 12) and (c) the pressure P sp from Eq. ( 3). with the increasing temperature (dotted line for B ′ T 0 = 5.21 and solid line for B ′ T 0 = 4.0).

C. High pressure melting
The changes of physical properties, such as volume, entropy, electrical and thermal conductivities, are measured in experiments at melting at high pressures. 14,15,18,19The melting of In under high pressure has been studied by measuring x-ray diffraction in an externally heated diamond anvil cell and a sharp change in resistivity in piston cylinder apparatus was observed. 18From Eq. ( 4), we can express the melting temperature as a function of pressure in the following form, T m = ( (P−P 0 ) a + 1 ) 1/c T 0 m .The predicted melting pressure at 710K is 7.77GPa upon heating (Fig. 3), although the corresponding solidification pressure is 8.5GPa from experiments along the isothermal compression line of 710K.As discussed in the Refs.18 and 19, there is some discrepancy on the determination of melting point of liquid indium at high pressure.With this relation, along with Eqs. ( 1), ( 2), ( 6) and ( 11 pressure along the melting line under pressure in Fig. 4. For the above calculation, we also assume the pressure derivative of bulk modulus is constant at ambient condition (B ′ T = 4.0 or B ′ T = 5.21).With the increasing pressure, the specific volume (Fig. 4(a)) and the thermal expansion coefficient (Fig. 4(b)) are decreasing at the corresponding high pressure melting point.
Under ambient pressure condition, the calculated bulk modulus at melting point (Fig. 4(c)) is 34.09GPa, which is larger than the estimated value of 23.96GPa from experimental data fitting, and the magnitude of internal pressure of liquid In at melting temperature is around 1.63Gpa (Fig. 4(d)), smaller than the estimated value 2.86GPa from the linear relationship approximation between internal pressure and temperature. 23Both bulk modulus and internal pressure of liquid states at melting points increase with the rising pressure, and also the internal pressure switches the sign from negative to positive at P = 1.65 GPa for B ′ T = 4.0 (and at P = 1.76 GPa for B ′ T = 5.21) (Fig. 4(b)).The attraction force plays the dominant role along the melting line at low pressure region while the repulsive force may dominate over the high pressure range.

III. DISCUSSIONS
The pressure derivative of bulk modulus at zero pressure (B ′ T 0 ) is one of the critical parameter for semi-empirical equation of states. 1-4B ′ T 0 can be obtained from direct measurement of sonic velocity in a material with variation of pressure along the isothermal line.From static high pressure experiments, B ′ T 0 can also be determined by polynomial fitting to the experimental P-V isothermal data.For materials obeying Birch relation, a linear sound velocity density relation is suggested in liquid indium 20,21,24,25 and the intrinsic temperature effect may be neglected under the condition α P γT ≪ 1, where γ is the Grüneisen parameter.For liquid In, we have α P γT < 0.2 for T < 800K. 21We may calculate the pressure derivative of isothermal bulk modulus as and have B ′ T 0 = 5.21 at 710K for liquid In, which is larger than B ′ T 0 = 4.0 from the numerical results of fitting Birch-Murnaghan EOS in liquid In under high pressure. 21In general, the molten metal is more compressible than the corresponding solid state -From the compression experiments at ambient condition, 26,27 B ′ T for crystalline indium could be 4.0 ∼ 7.0.Brosh, et al. 28 pointed out that the pseudospinodal assumption is not necessary to derive the Baonza EOS.Based on their explanation of physical meaning of the spinodal condition for liquids, Baonza EOS does not satisfy the spinodal condition and the energy analytic condition.Both isothermal compressibility and volume thermal expansion coefficient are related to the first derivatives of the pressure-volume-temperature surface of equation of state.For positive thermal expansion in compressible liquids, the expression of α P (P,T) is independent of the function form of V sp and κ * .Thermal expansion coefficient α P (P,T) might obey the power law along the isothermal path as α P (P) = α * P − P sp −β ′ with α * a constant but dependent on temperature, β ′ is a material-dependent parameter.If β ′ < β, the thermal expansivity would approach zero, slower than the compressibility.For negative thermal expansion materials, the above assumption of thermal expansion coefficient could no longer be valid.As the starting condition of Baonzo EOS, we use the measured values of isothermal compressibility and density of liquid indium in the linear function of temperature for certain range at ambient condition.However, that may limit the validity of the Baonzo EOS since both isothermal compressibility and density of liquid may be nonlinear function at high temperature.
The internal pressure of liquids plays an important role in the study of liquids and can be used to represent the principle characteristics of liquids. 29Internal pressure may be considered as a measurement of cohesive forces in liquids.At ambient pressure, the average nearest neighbor distance of liquid indium decreases with the macroscopic volume expansion which is revealed by x-ray diffraction experiments. 30The volume per atom is related to both the coordination number and interatomic distances.Local contraction of liquid indium atoms leads to an increase of local atomic density and the corresponding structural change is a result of the increasing attractive component of interactions among atoms.The enhancement of the attractive force is indicated by the increasing magnitude of negative internal pressure shown in Fig. 2(a).The volume contraction under high pressure might cause spatial inhomogeneity in liquid related to the solidification processes.

IV. CONCLUSIONS
In this paper, we apply an equation of state in a power law form to study the thermodynamics of liquid indium under high temperature and high pressure where the bulk modulus and volume are assumed to obey the power law form dependent of pressure at constant temperature.We obtain the specific volume of liquid indium as the function of pressure along the isothermal line.The calculated result is reasonable and compared with the data from the experiments measured in an externally heated diamond anvil cell.We calculate the thermal expansion of liquid indium and found that it decreases with pressure along the isothermal line.We find that bulk modulus of liquid indium increases with pressure at constant temperature.At ambient pressure, we calculate the internal pressure as a function of temperature and found that the attractive force is dominant at low pressure and at constant temperature, the repulsive force will take a dominant role with the increasing pressure.From the internal pressure, we get the line of sign changing from negative to positive that intersects with the high pressure melting line in pressure-temperature plane.We investigate thermodynamics properties as a function of pressure along the high pressure melting line.Bulk modulus and internal pressure increase with pressure and specific volume and thermal expansion coefficient decrease with pressure along the melting line.While our results suggest that the power law form may be a better choice for the equation of state of liquids, these detailed predictions are yet to be confirmed by further experiment.

FIG. 4 .
FIG. 4. Calculated thermodynamic properties of liquid indium along the high pressure melting line of Eq. (4) (solid line for B ′ T 0 = 5.21, and dotted line for B ′ T 0 = 4.0): (a) molar volume V , (b) thermal expansion coefficient α T , (c) isothermal bulk modulus B T , and (d) internal pressure P int .