Variation of Bulk Modulus, Its First Pressure Derivative, and Thermal Expansion Coefficient with Applied High Hydrostatic Pressure

Mahammed, Manaf A.; Mohammed, Hamsa B.

doi:https://doi.org/10.1155/2023/9518475

Advances in Condensed Matter Physics

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9518475 | https://doi.org/10.1155/2023/9518475

Variation of Bulk Modulus, Its First Pressure Derivative, and Thermal Expansion Coefficient with Applied High Hydrostatic Pressure

Manaf A. Mahammed¹and Hamsa B. Mohammed²

Academic Editor: Sergio Ulloa

Received26 Dec 2022

Revised27 Jul 2023

Accepted31 Aug 2023

Published12 Oct 2023

Abstract

Throughout this work, the equations of variation of the isothermal bulk modulus, its first pressure derivative, and the volumetric thermal expansion coefficient as a function of pressure were derived based on the Birch–Murnaghan equation of state (B–M EOS). The bulk modulus and its first derivative at ambient temperature for nine elements were extracted by fitting the published experimental pressure–volume data to B–M EOS, and the results were compared with other published researches, and there was a good agreement. Moreover, those extracted values were used to study the variation of the isothermal bulk modulus, its first pressure derivative, and the isothermal coefficient of thermal expansion as a function of the applied hydrostatic pressure using the equations that were derived from this work.

1. Introduction

Researches concerning materials under high pressure helps to understand the thermoelastic properties of solid materials and find widespread applications in high-pressure physics, material sciences, geoscience, planetary, and astrophysics since the application of high pressure is a very efficient tool to make a unique modification in structure, behavior, as well as properties of the material. In addition, it helps to understand the nature of solids that have been subjected to high pressure and temperature [1, 2], relatively high hydrostatic pressure. The compression of solids under relatively high hydrostatic pressure can make significant effects on some of their properties [3–5]. Those properties include the interatomic lattice structure and thermal, electronic, mechanical, and optical properties. Velisavljevic et al. [6] studied the structural and electrical properties of Be under hydrostatic pressure up to 66 GPa using X-ray diffraction (XRD), and there was no evidence of transition from hcp up to volume compression V/V₀ = 0.708. They fitted their data to adapted polynomial second order equation of state (AP2-EOS), and they got values of bulk modulus B₀ = 106.5 GPa and pressure derivative of bulk modulus = 3.541 [6]. Evans et al. [7] measured the Raman phonon and lattice constants of Be up to 80 and 180 GPa hydrostatic pressure, respectively, and they determined the values B₀ = 109.88 GPa and = 3.584, respectively. Also, they noticed the increasing of c/a ratio with pressure reaching 1.6 when the pressure increased to 180 GPa. Nakano et al. [8] performed XRD experiments on Be element under high pressure to 171 GPa, and they determined the EOS of hcp–Be by fitting their diffraction data to Birch–Murnaghan (B–M EOS) and extracted the values B₀ = 97.2 ± 2.5 GPa and = 3.61 ± 0.07. Their data also revealed the stability of hcp phase of Be under compression up to 171 GPa. Potassium K belongs to alkali metals at ambient condition, crystalizes in bcc structure, and converts to fcc structure at 11.6 GPa [9, 10]. The phase transition sequence of K has been studied by Lundegaard et al. [11] and Degtyareva [10], using a combination of X-ray powder and single crystal diffraction. Fujihisa and Takemura [12] performed diffraction experiments on the hcp–Co under pressure to 79 GPa by using diamond anvil cell (DAC) and imaging plate. The determined value of B₀ = 199 ± 6 GPa and that of = 3.6 ± 0.2. The use of fcc phase of Al in the dual role of pressure standard and solid pressure-transmitting medium (PTM) up to 200 GPa has been discussed by Singh [13] by using XRD patterns with (DAC), and the results showed that Al has all the properties required for a PTM and pressure standard. Molybdenum (Mo) is a bcc 4d transition metal with wide engineering and technology applications for its thermal and mechanical strength. It is also widely used as a pressure standard in DAC [14–16]. The P–V–T EOS of bcc-Mo have been investigated at pressure 31 GPa using XRD measurements by Litasov et al. [16], and data analysis was performed using B–M EOS and MGD approaches combined with Vinet EOS and obtained similar results for both of two equations at pressure below 1 Mbar while significant deviation may occur at higher pressure. Ko et al. [17] used XRD to measure the EOS and mechanical properties of copper (Cu) under high pressure. Their measured value for the bulk modulus of pure crystalline Cu was 15% less than the previously published values. They attributed the large values of the bulk modulus of the Cu that were published previously to the existence of defects or imperfections such as vacancy, interstitial, substitutional defects, and dislocations that make the materials harder than materials without it [17]. Tantalum (Ta) is a bcc with high ductility and thermal, mechanical, and chemical stability, so it is important for using in jet engines and electronic devices and could be a good candidate for an internal pressure standard for static high-pressure researchers [18]. Cynn and Yoo [18] investigated the volume compression of Ta metal by using angle-resolved synchrotron XRD with (DAC) and confirmed that the bcc phase of Ta is stable to 174 GPa. Gold (Au) is usually used as an internal pressure calibrant because of its stable fcc structure at extremely high pressure and temperature [19, 20]. Güler and Güler [5] used embedded atom method (EAM)-based geometry optimization calculation to study effects of pressure on elastic constants, bulk modulus, shear modulus, and young modulus of Au at 1,000 GPa, and the results showed that all the elastic constants increased almost linearly with increasing pressure. Tungsten (W) is a 5d transition metal that is characterized by its highest melting point, and it crystallizes in bcc structure and it is also stable up to extremely high pressure at room temperature. Jiang et al. [21] studied the structural, mechanical, electronic, and thermodynamic properties of W under different pressures by using the first-principles method and showed that W metal is stable up to 100 GPa.

At absolute 0 k as well as at finite temperature and due to thermal fluctuations, the crystalline solids atoms make small vibrations about their equilibrium positions. The crystal binding energy can be expressed as an expansion of power series in terms of the displacements of those atoms from their equilibrium positions. When the terms beyond the quadratic ones are neglected, then this will be referred to as harmonic approximation, and the vibration is known as harmonic lattice vibration [22]. Among the consequences of harmonic approximation, no thermal expansion can take place, the elastic constants will no longer depend upon pressure and temperature as well as the heat capacity tends to be constant at high temperatures [23]. While for real crystals, none of the abovementioned consequences is satisfied exactly. This happens when terms of lattice binding energy higher than the quadratic ones are taken into consideration, and this is referred to as anharmonic approximation and the lattice vibration is known as anharmonic lattice vibration [24, 25]. Kazanc and Ozgen [26] investigated the changes of acoustic phonon frequencies with hydrostatic pressure for Cu, Ni, Al, Ag, and Au transition metals using molecular dynamics MD simulations based on EAM. Their results show that the pressure causes an increment in phonon frequency and that this increment does not vary linearly with the increased pressure. Jalal et al. [27] used Suzuki and B–M EOSs to study the effect of hydrostatic pressure on two Si nanoparticles. They showed that the volume compression ratio of the smaller nanoparticle was greater, and the shift in its phonon frequency spectrum was more significant. They attributed that to the higher bulk modulus of the smaller silicon nanoparticle [27]. AL-Faris et al. [28] made theoretical calculations to study the thermodynamic properties of gallium nitride (GaN) under high pressure. They calculated the bulk modulus, compression ratio, lattice constant, and phonon frequency spectrum of GaN under high pressure to 40 GPa using three B–M and modified Lennard–Jones EOSs. The agreements of their results for the phonon frequency spectrum that were obtained using B–M EOS with experimental data were better than those obtained from modified Lennard–Jones EOS [28].

Equations of state EOSs serve as tests for the calculation of the electronic structure of different properties of metals as well as they are used as scales for high pressure [29]. Most of the experiments of pressure–volume (PV) measurements on solids under high pressure were executed using almost small pressure ranges. As it is known, all EOSs can be fitted to experimental PV data taking into consideration the elastic constants, i.e., the isothermal bulk modulus (B₀), and its pressure derivatives () [30, 31]. There are many methods to determine those elastic constants, like the PV relations and measurement of the velocity of the acoustic waves, besides the dispersion curves of the branches of acoustic phonon [32]. When those experimental data are fitted with any EOS such as B–M, then the isothermal bulk modulus and its first pressure derivative at atmospheric pressure and ambient temperature can be extracted. Those parameters will be useful in the EOS to obtain different important parameters concerning the solid materials under pressure, and they could be helpful for the calculations that can be extrapolated toward higher pressure ranges when theoretical aspects are taken into consideration [29, 33].

The aim of this work is to study the variation of the isothermal bulk modulus, the pressure derivative of bulk modulus, and the volumetric thermal expansion coefficient as a function of high hydrostatic pressure for nine metallic elements of different atomic weights. It is worth to mention that all data concerning the pressure versus relative volume of all nine elements used in this work were taken from the experimental works published by other researchers. Those elements were namely, potassium (K) [34], aluminum (Al) [35], beryllium (Be) [8], gold (Au) [36], copper (Cu) [37], tantalum (Ta) [18], cobalt (Co) [12], molybdenum (Mo) [38], and tungsten (W) [36]. The data of those researchers were either taken from tables or as graphics points of the figures mentioned within their researches. The experimental data taken from the graphs were done with the aid of GRABIT software that works with MATLAB.

2. Theoretical Background

Any equation that relates the pressure (P), specific volume (V), and temperature (T) of a material in the thermal equilibrium state is called the equation of state EOS. The simplest EOS is that of an ideal gas. There are many EOSs describing the state of solid materials under high pressure such as Murnaghan, Birch, Birch–Murnaghan, Vinet, modified Leanard–Jones EOSs, and others.

2.1. Birch–Murnaghan (B–M) EOS

The third order B–M EOS is given as follows [39]:where is the isothermal bulk modulus at atmospheric pressure and ambient temperature, is the first pressure derivative of the bulk modulus at ambient temperature, V₀ and V are the volumes at atmospheric pressure and pressure (P), respectively. It is worth to mention that the above equation was derived on the basis of Eulerian finite strain theory which is useful to describe solids under high pressure applications [40]. This equation is the most widely used by researchers [41].

2.2. Isothermal Bulk Modulus and Its Derivatives

Bulk modulus can be defined as the ability of a material to bear the change in its volume when pressure is applied to all sides. It can be considered one of the fundamental mechanical properties of structural solids which can be deduced through the measurement of the unit cell volume [41]. The isothermal bulk modulus B_T can be calculated from the following equation:

Equation (2) implies that the isothermal bulk modulus is given by the ratio of the increase in the applied pressure divided by the relative deformation. Before deriving the pressure in Equation (1) with respect to the volume and substituting it in Equation (2), it is necessary to simplify Equation (1) to be easier for differentiation. So, the pressure (P) in Equation (1) can be rewritten in the following simple form:

By keeping the temperature constant and differentiating the pressure in Equation (3) with respect to the volume, then,

Substituting Equation (4) into Equation (2), the isothermal bulk modulus under high pressure is expressed as follows:

Most of the EOSs, including B–M finite strain EOS have been widely used for the investigation of high-pressure properties of solid materials. Equations (1) and (5) show that both of the bulk modulus and pressure increase as the relative volume (V/V₀) decreases and that at extreme compression (V = 0) both of P and B_T become infinite [42]. Values of the first pressure derivative of the bulk modulus at a given temperature and atmospheric pressure have been found to decrease with the increase of pressure and then become almost constant at extreme compression (V = 0). The pressure derivative of the bulk modulus can be found from the derivation of the isothermal bulk modulus with respect to the pressure as follows:which can be written as follows:

Now the must be found first,and by substituting for into Equation (5), the pressure derivative of the bulk modulus with respect to the pressure can be written as follows:

The derivative of the bulk modulus with respect to pressure plays an important role in determining thermoplastic parameters of solids at high compression [2]. Making use of Equation (9) together with Equations (1) and (4), the derivative of the bulk modulus with respect to pressure can be obtained as a function of the pressure (P).

2.3. Volumetric Thermal Expansion Coefficient

Since the product of the volumetric thermal expansion coefficient (α) and the isothermal bulk modulus (B_T) is constant and holds for all solids. Thus,

Shanker and Kumar [43] showed the following:where α₀ is the volumetric expansion coefficient at atmosphere pressure (P = 0) and δ_T is the isothermal Anderson parameter. Moreover, they showed the following:from which one can write the following:or

, from which one can write the following:

By combining Equations (4) and (14), the volumetric thermal expansion coefficient can be written as follows:

It is worth to mention here that the equations for the bulk modulus, its first pressure derivative and that of the volumetric bulk thermal expansion coefficient are not expressed directly in terms of pressure, but they were expressed instead in terms of the relative volume (V/V₀). Then, by making use of Equations (1), (4), and (15), and knowing the values of , the volumetric thermal expansion coefficient of the given solid elements can be calculated as a function pressure (P).

3. Results and Discussion

The experimental P–V data of all nine elements of the researchers that were mentioned in the introduction of this work were fitted to the B-EOS given by Equation (1). The experimental method used by those researchers for hcp-beryllium, fcc-gold, hcp-cobalt, bcc-tantalum, bcc-molybdenum, and bcc-tungsten was DAC.

The experimental method used for bcc potassium was a pressure vessel and that used for fcc aluminum was tungsten carbide opposed anvil. Values of and were extracted for each element and they were tabulated in Table 1. The P–V isotherms of those elements were plotted in the figures, i.e., Figures 1–5. The P–V isotherm of potassium is plotted within Figure 1 using Equation (1) after substitution of its and from Table 1. Moreover, the experimental data for potassium done by Kim and Ruoff [34] were plotted in Figure 1. Other experimental data of Lundegaard et al. [11] and Winzenick et al. [9] were plotted with the same potassium isotherm for comparison. Data of Winzenick et al. [9] are generally in good agreement with the plotted potassium isotherm and the data [11] also seem in good agreement with it at low pressures, i.e., up to nearly 2.5 GPa, but as the pressure exceeds 2.5 GPa, differences between them increase gradually.

This is attributed to B–M EOS that was driven on the basis of Eulerian finite strain, and its description of the solid material compression under high pressure is not adequate, as mentioned by Alkammash [67]. The isotherm of aluminum is also plotted in Figure 1, making use of B–M EOS given by Equation (1) with its values of and from Table 1. Moreover, the experimental data of Syassen and Holzapfel [35] were plotted with the isotherm. The values of and extracted in this work were very close to those deduced by Syassen and Holzapfel [35], and the little differences may be due to the errors in picking the points from Figure 1 of Syassen and Holzapfel [35]. Also, the experimental data of Dewaele et al. [36] and Ming and Manghnani [60], as well as those of Thomas [48], were plotted within the same figure for comparison. The data of Dewaele et al. [36] and those of Vaboya and Kennedy [60] seem to be in good agreement with the fitted data, while those of Ming et al. [48] have little deviation from the fitted data.

Experimental P–V data of Nakano et al. [8] for Be were fitted to the B–M EOS, and those data, together with the fitted curve, were drawn in Figure 2. Be has hcp (phase II) at atmospheric pressure up to 1,523 k; its unit cell has the dimensions of a = 0.2286 nm, c = 0.3584 nm, and c/a = 1.56 [8]. There is no practical evidence for phase transformation of Beryllium up to 171 GPa [8]. The elastic constants that were extracted from the fitting were = 101.428 ± 2.157 GPa, = 3.515 ± 0.055. These are in good agreement with those that were found by other researchers that were tabulated in Table 1. In order to compare the above fitted data with the data published by other authors, the experimental P–V data of Ming and Manghnani [68] and Velisavljevic et al. [6], were plotted in the same figure. It seems that Velisavljevic et al. [6] data are in good agreement with those of Nakano et al. [8], while the data of Ming and Manghnani [68] are in good agreement with Nakano et al. [8] data up to nearly 30 GPa but after that, it is clear that deviation between them began increasing until 170 GPa. This deviation can also be justified by the reason mentioned by Alkammash [67]. Gold isotherm was also plotted within Figure 2 using the extracted and after their substitution in Equation (1). Values of and were in good agreement with those found by Heinz and Jeanloz [69] that are = 167 GPa and = 5.5. The P–V experimental data of Ming et al. [48], Takemura and Dewaele [70], Heinz and Jeanloz [69], and Bridgman [71] for the Gold were also plotted on the same figure. They show good agreement with the fitted data.

The P–V isotherm of Cu was plotted in Figure 3 using Equation (1) and by taking the values of and from Table 1. Experimental data from Mao et al. [37] that were fitted to B–M EOS were plotted with them in Figure 2. The experimental P–V data of Cu of researchers Dewaele et al. [36] and Xu et al. [72] were also plotted on the same figure for comparison. Xu et al. [72] data showed almost good agreement with the fitted data, while Dewaele et al. [36] data showed some deviation from B–M EOS. Also, this deviation increases as the pressure increases. This can also be attributed to the reason that B–M EQS cannot describe adequately the solid under the compression at very high pressures, as mentioned above. At 300 k and 0.1 MPa, Ta has a bcc structure, and its lattice constant and melting point are a = 0.3297 nm and 3,247°C, respectively [73]. According to Söderlind and Moriarty [74], it was shown theoretically that the Ta bcc lattice structure remains stable up to 1 TPa. By fitting the P–V experimental data of Cynn and Yoo [18] for Ta to the B–M EOS, it was shown that = 194.2 GPa and = 3.453 which are in good agreement with those values that were got by ultrasonic measurements. Using theoretical calculations, it was estimated that the value of of Ta to be 203–211 GPa at 0 k [18]. Figure 3 shows the experimental data of Xu et al. [72] and Litasov et al. [61] that were plotted with the Ta-fitted data for comparison. There was good agreement between the experimental data and B–M EOS at low pressure and again deviation between them starts as pressure increases until it exceeds 50 GPa.

For Cobalt (Co) elements, the experimental P–V data were taken from Fujihisa and Takemura [12], as mentioned earlier. Co has hcp and fcc phases. In this work, only the hcp phase data of cobalt were taken into consideration and were fitted to B–M EOS, and the bulk modulus and its first pressure derivative at ambient temperature were found to be = 209.345 ± 5.145 GPa and = 3.244 ± 0.185, respectively. The above hcp phase experimental data, together with co isotherm, were plotted in Figure 4, and the experimental data of Yoo et al. [56] and those of Ono [57] were plotted in the same figure for comparison. From Figure 4, it is obvious that almost good agreement can be seen between the fitted data to B–M, EOS, and Yoo. The elastic parameters of the molybdenum (Mo), = 249.55 ± 2.786 GPa and = 3.86 ± 0.086 were extracted through the fitting of the experimental P–V data of Dewaele et al. [38] to B–M EOS and were used to plot the Mo isotherm in Figure 4. Moreover, the P–V experimental data of Huang et al. [14], Litasov et al. [61], and Sokolova et al. [62] were plotted in the same figure for the sake of comparison. Good agreement was shown between those data with the fitted data of Dewaele et al. [38], as shown in Figure 4.

The P–V experimental data of Dewaele et al. [36] for tungsten were fitted to B–M EOS, and the isothermal bulk modulus at atmospheric pressure and its first pressure derivative was found to be = 279.626 ± 3.132 GPa and = 6.265 ± 0.51. Pressures between 200 and 300 GPa have been used when the Tungsten has been studied under hydrostatic compression, and the crystallographic structures were found to be stable according to the expectation of the electronic transfer of s to d [75]. The experimental data of tungsten of Litasov et al. [61], Dubrovinsky et al. [16], and Wilson [76] were plotted together with the fitted data of a B–M EOS in Figure 3 for comparison. The agreement is almost good, especially at low pressures.

It is mentioned by many researchers that the bulk modulus of solid materials increases linearly with increasing the applied pressure [32, 41]. Equations (1) and (4) were used to plot the variation of bulk modulus values as a function of the applied pressure for the selected nine elements after substitution of the values of B₀ and that were found from the fitting in Figure 6. It can be seen from the figure that the variation of the bulk modulus with the pressure for all nine elements is almost linear. It is worth to mention that the variation of the bulk modulus with the applied pressure is also given by the equation B = + P and that the higher orders of nonlinear terms B₀″, B₀‴ were neglected and this is known as Murnaghan approximation [43]. This is why some of the curves in Figure 6 show a slight tendency toward curvature shape. The differences in the slopes of the lines in Figure 6 were attributed to the differences in the values of the first pressure derivatives at atmospheric pressure of the elements that are shown in Table 1. The highest slope is for tungsten, which has the highest value (6.265), while the lowest slope is for potassium, which has the lowest value is 3.207.

The first pressure derivatives of the bulk modulus Bʹ (P) as a function of pressure for all nine elements were determined from Equations (1) and (9) and plotted versus pressure in Figure 7. It can be seen from this figure that the Bʹ (P) of most of the selected elements decreases slightly with increasing pressure. This can be attributed to the fact that most of those elements, like W, Au, Mo, Al, Co, Ta, Cu, and Be, relatively possess high values of bulk modulus, as shown in Table 1. Therefore, Bʹ (P) seems to decrease slowly with pressure and tends to be constant. Thus, the total variation from the initial values of seems to be not too much. On the other hand, the last selected element in this work, which is potassium, has a very low bulk modulus, as can be seen from Table 1. It is clear from Figure 7 that the Bʹ (P) for potassium decreases rapidly with increasing pressure. This can be interpreted that potassium is a relatively soft material compared to other selected metallic elements and has small stiffness, so the pressure rate of change of its bulk modulus decreases with increasing pressure.

With a few exceptions, all materials expand when heated. However, the extent of expansion per degree change in temperature is different for different materials. Equation (15) is used to determine the volumetric thermal expansion coefficient α (P) of all the nine elements. Values of thermal expansion coefficients at atmospheric pressure α_o shown in Table 2 for those elements together with the first derivative of the bulk modulus at atmospheric pressure were substituted in the equation. Figure 8 shows the variation of α (P) with the applied pressure. It is clear from the figure that α (P) decreases with increasing pressure. Thermal expansion is reduced by the effect of large pressures that cause decoupling heat from mechanical work.

4. Conclusion

The third-order B–M EOS was used in this work, and the isothermal bulk modulus, its first pressure derivative and the volumetric thermal expansion coefficient equations were derived based on this equation. The isothermal P–V diagrams for nine metallic elements was plotted based on the B–M EOS mentioned above by making use of their and values. Values of and were extracted by fitting the experimental P–V data of those metallic elements executed under high hydrostatic compression by different researchers to the B–M EOS and compared with the published values and showed good agreement. The variation of the isothermal bulk modulus with the applied pressure for the nine metallic elements were studied. It was shown that as the applied pressures increase, the isothermal bulk modulus also increases. The first derivative of the bulk modulus with respect to the pressure for the nine metals seems to decrease slowly as the pressure increases. The variation of the volumetric thermal expansion coefficient of those elements with the applied pressure was studied in this research. It was shown that as the applied pressure increases, the volumetric thermal expansion coefficient decreases.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The authors want to acknowledge Prof. Matteo Alvaro, Professor of Mineralogy at the University of Pavia, Italy and Dr. Nawzat S. Saadi from the College of Science, University of Duhok, Iraq for the scientific advisory effort during the completion of this work. This research was supported by the University of Duhok, Iraq and the University of Mosul, Iraq.

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