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Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state
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Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state M. Goyal, B.R.K. Gupta∗ Department of Physics, GLA University, Mathura-281406, U.P., India
a r t i c l e
i n f o
Article history: Received 28 May 2016 Revised 1 August 2016 Accepted 29 September 2016 Available online xxx Keywords: Anderson Gruneisen parameter Thermal expansivity Thermal expansion coefficient Equation of state Nanoforesterite
a b s t r a c t A new potential independent equation of state is used in the present work to analyze the thermo-elastic properties of nanocrystalline Foresterite (nc- Fo) under varying temperature and pressure conditions. The newly developed EOS is found to be valid for explaining the elastic behavior of nanocrystalline forsterite satisfactorily over the temperature range from 300 K to 1573 K with a pressure variation from 0 to 9.6 GPa. The values calculated for the volume compression under varying temperature- pressure conditions are compared with the available experimental data and also with those obtained by using different approaches. It is found that the results obtained in the present study are more close to the experimental data in comparison to those reported earlier. The same model is further extended to study the variation in the bulk modulus and thermal expansion coefficient of Foresterite nano-mineral ver the temperature range from 300 K to 1573 K. The results show the same trend of expansion as observed in single nanocrystals at high temperature. © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
1. Introduction The study of the thermo elastic behavior of minerals, ionic solids, metallic solids, alloys etc. under different pressure and temperature conditions is important for understanding the earth’s deep interior and its evolution. It also provides vital information about the dynamics of the earth’s lower mantle. A variation of the temperature-pressure conditions affects the atomic structure, stability and atomic interactions and thus modifies the physical properties, like the compressibility, electrical conductivity, elasticity, and thermal expansivity [1,2] of the material. The physical and rheological properties of a material are greatly influenced by its size and shape, which in turns affects the geo-physical processes [3]. Despite of the limitation of the nanocrystalline forms of minerals in the earth’s crust and mantle, its importance in geophysics cannot be neglected [4–6]. The effect of grain size on the elastic properties of nanocrystalline metal, alloys, ceramics, and oxides has been studied both experimentally [7,8] and theoretically [9,10] during the past years. Some of the experimental studies [11–13] on nanocrystalline materials have concluded that the elastic moduli of nanocrystalline materials decrease with a decrease in grain size. Though the olivine group of minerals are usually studied in bulk form [14–18], yet the nanostructured minerals in nano-form have been studied less under high pressure and high temperature. The olivine group of minerals comprises of Foresterite and Fayalite. Foresterite (Mg2 SiO4 )’ commonly abbreviated as Fo, is the member of the olivine solid solution series rich in magnesium whereas Fayalite (Fe2 SiO4 ) is the iron rich member. Fayalite has a high refractive index and is heavier than Foresterite. Foresterite is found to be the most abundant mineral in ∗
Corresponding author. E-mail address: [email protected] (B.R.K. Gupta).
http://dx.doi.org/10.1016/j.cjph.2016.09.009 0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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the earth’s mantle under the depth of 400 km [19,20]. Foresterite is found mainly in igneous rocks having rich magnesium and iron content and also in meteorites [21–23]. It has exceptionally high toughness and superior mechanical properties. It has been found as a potential material for bone implants and also has wide biomedical applications [24]. So, it is important to study more about the thermo elastic behavior of nanocrystalline Foresterite. Various potential dependent [25,26] and potential independent [27–30] model theories have been evolved and used to investigate the elastic properties of bulk materials and nanomaterials during the past decades. However, the results reported by these investigators show large variations, as compared to the experimental values. So, it becomes necessary to provide a model theory which could determine the thermo elastic properties of nanomaterials so as to explain the experimental results successfully. The main motive of the present study is to formulate a simple pressure and temperature dependent equation of state for studying the thermo elastic properties of nanocrystalline minerals. The method of formulation of the potential independent equation of state and its application is described in Section 2. The results obtained for Foresterite nanocrystal are discussed in Section 3. 2. Formulations and analysis Under high pressure, the product of the volume thermal expansion coefficient (α T ) and the bulk modulus (BT ) can be considered as constant at constant tessmperature [31,32], i.e.
αoBo = αT BT
(1)
Differentiation of Eq. (1) w.r.t. volume V leads to the relation:
α
dB dV
+B T
dα dV
=0
(2)
T
The Anderson-Gruneisen parameter δT at constant temperature T is given by
δT =
V
α
dα dV
(3) T
Using Eqs. (2) and (3), δT can be defined as
δT =
V
α
dα dV
=− T
V B
dB dV
(4) T
Considering δT to be independent of V, it becomes
dB dP
δT =
= B O .
(5)
T
δ +1
The Anderson-Gruneisen parameter δT and η are related as follows [33,34]:( Tη ) = A, where
η = V/Vo
(6)
where A is a constant. Using Eq. (6) in Eq. (4), and upon integrating it, we get
V B V = exp A 1 − Bo Vo Vo
(7)
where B is the bulk modulus and is defined as
B = −V
dP dV
(8) T
In view of Eq. (8), the expression of the bulk modulus B can be expressed as follows:
V B exp A 1 − dV = −dP, Bo Vo
(9)
where Bo , V0 are the bulk modulus and volume at zero pressure. On integrating Eq. (9), the equation for pressure (P) can be written as
P=
Bo V exp A 1 − A Vo
−1 ,
(10)
where
A = Bo + 1
(11)
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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3
Substituting A from Eq. (11) into Eq. (10), the isothermal usual Tait’s EOS can be expressed as follows:
P=
Bo
exp Bo + 1 (Bo + 1 )
1−
V Vo
−1
(12)
The expression of the bulk modulus corresponding to Eqs. (12) and (8) is obtained as follows [32,35]:
V BT V = exp Bo + 1 1 − Bo Vo Vo
(13)
To incorporate the temperature effect, the thermal pressure (PT h ) term given by Anderson [35] can be added into Eq. (12),
PT h = α0 B0 (T − T0 ).
(14)
On including the thermal pressure (PT h ) term in Eq. (12), the P-V-T relation obtained is given below:
P=
Bo
exp Bo + 1 (Bo + 1 )
1−
V Vo
− 1 + α0 B0 (T − T0 )
(15)
Taking the log of Eq. (15) to express it in terms of volume compression, the inverted form of the usual Tait’s equation so obtained including the thermal effect can now be expressed as
1 log 1 + (Bo + 1 )
V (P, To )/V (o, To ) = 1 −
Bo + 1 (P − α0 B0 (T − T0 ) ) Bo
(16)
The temperature dependence of δ T in accordance with the relation in the empirical form [27,36] is given as follows:
δT = δT0 X k ,
(17)
where X = T /T0 Here T0 refers to the room temperature, and δT0 represents the value of the Anderson-Gruneisen parameter at room temperature. In accordance with Eq. (5), δT0 is equal to Bo at room temperature and zero pressure. k is the thermo- elastic parameter, and its value is determined from the slope of the graph plotted between log (δ T ) versus log (T/T0 ) . Now considering the temperature dependence of the Anderson-Gruneisen parameter (δ T ) and Eq. (17), the new pressurevolume-temperature equation of state obtained is as follows:
P=
Bo
exp
δT0 X k + 1
V − 1 + α0 B0 (T − T0 ) δT0 X k + 1 1 −
(18)
Vo
The inverted form of the above equation including the thermal effect is as follows:
V (P, To )/V (P0 , To ) = 1 −
1
log 1 + δT0 X k + 1
δT0 X k + 1 Bo
(P − α0 B0 (T − T0 ) ) ,
(19)
where δT0 = Bo The expression of the bulk modulus corresponding to Eq. (19) is obtained as
BT = 1− Bo
1
δT0 X k + 1
log 1 +
δT0 X k + 1
Bo
0 k
δT X + 1 (P − α0 B0 (T − T0 ) ) ∗ 1 + (P − α0 B0 (T − T0 ) ) , Bo
(20)
and therefore, the volume thermal expansion coefficient can be expressed as follows:
BT = 1− Bo
1
δT0 X k + 1
log 1 +
δT0 X k + 1 Bo
−1 0 k
δT X + 1 ∗ 1+ (P − α0 B0 (T − T0 ) ) (P − α0 B0 (T − T0 ) ) , Bo
(21)
which is a reciprocal form of Eq. (20), in view of Eq. (1). The inverted form of the EOS formulated by Kholiya et al. [37] for nanomaterials including the thermal effect in terms of the volume compression is as follows:
V −1 V0
⎡
B − 2 ⎣1 + = o Bo − 1
1−
Bo − 1
(Bo − 2 )2
⎤ 1/2 ⎦.
2P (V, T ) Bo − + 2α0 (T − T0 ) − 3 B0
(22)
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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M. Goyal, B.R.K. Gupta / Chinese Journal of Physics 000 (2017) 1–9 Table 1 Calculated values of the volume compression using the present EOS (Eq. (19)); Kholiya’s EOS [37]; Tait’s EOS [39] and the experimental values [38]. T(K)
P(GPa)
Usual Tait EOS V/V0 [39]
Present EOS V/V0(Eq. (19))
Kholiya EOS V/V0(Eq. (23)) [37]
V/V0(Exp.) [38]
300
0 2.3 3.7 5.1 6.5 2.8 4.2 5.5 6.2 6.9 7.4 3.3 4.8 6.1 6.9 7.6 8 3.7 5.3 6.6 7.4 8.1 8.6 4 5.8 7.1 8 8.8 9.5 5.9 7.4 8.4 9.3 9.6
1 0.9821471 0.9719957 0.9623235 0.9530874 0.982349 0.972187 0.963183 0.958493 0.953909 0.950696 0.983164 0.972254 0.963246 0.957894 0.953323 0.950755 0.984734 0.97303 0.963989 0.958617 0.95403 0.950815 0.991769 0.978204 0.968938 0.962759 0.957419 0.952858 0.984228 0.973262 0.966265 0.960165 0.958172
1 0.9821471 0.9719957 0.9623235 0.9530874 0.9823754 0.9722552 0.9633012 0.9586429 0.9540921 0.9512815 0.9832067 0.9723736 0.963456 0.9581675 0.9536568 0.9515311 0.9847801 0.9731822 0.9642603 0.9589738 0.9544674 0.9517469 0.9917808 0.9783389 0.9692147 0.9631564 0.9579359 0.9539684 0.9843041 0.9734982 0.9666419 0.9606888 0.9587478
1 0.9821 0.972 0.9623 0.9531 0.9823 0.9722 0.9632 0.9585 0.9539 0.9507 0.9832 0.9723 0.9632 0.9579 0.9533 0.9508 0.9847 0.973 0.964 0.9586 0.954 0.9508 0.9918 0.9782 0.9689 0.9628 0.9574 0.9529 0.9842 0.9733 0.9663 0.9602 0.9582
1 0.9824 0.9717 0.9624 0.9562 0.9824 0.9724 0.9652 0.958 0.9535 0.9528 0.9831 0.9738 0.9662 0.9614 0.9531 0.9542 0.9855 0.9745 0.9673 0.9597 0.9549 0.9545 0.9917 0.9773 0.9693 0.9624 0.9576 0.9562 0.9841 0.9728 0.9666 0.96 0.9607
473
673
873
1273
1573
The bulk modulus corresponding to Eq. (22) is expressed as follows:
⎤ 2 ⎡ 1/2 Bo − 1 2P (V, T ) ⎣1 + 1 − ⎦ Bo − + 2α0 (T − T0 ) − 3 B0 (Bo − 2 )2 1/2 Bo − 1 2P (V, T ) × 1− Bo − + 2α0 (T − T0 ) − 3 . B0 (Bo − 2 )2
Bo − 2 BT = Bo (Bo − 1 )
(23)
The expression for the coefficient of volume thermal expansion corresponding to Eq. (23) is as follows:
−1/2 Bo − 1 Bo − 1 2P (V, T ) αT = 1− Bo − + 2α0 (T − T0 ) − 3 αo B0 (Bo − 2 )2 (Bo − 2 )2 ⎡ 1/2 ⎤−1 Bo − 1 2P (V, T ) ⎦ . × ⎣1 + 1 − Bo − + 2α0 (T − T0 ) − 3 B0 (Bo − 2 )2
(24)
3. Results and discussions The equation of state developed in the present study consists of only two important input parameters, i.e. the measured value of the bulk modulus (B0 ) and its first derivative (B0 ) at P = 0 and T = T0 (reference temperature) . The measured value of B0 differs significantly for nanomaterials than that of its counterpart bulk material, and its magnitude depends upon the size of the material, whereas B0 remains more or less same or differs by a negligible amount. For example, for nanocrystalline Foresterite, B0 = 123.3 GPa [38] and for bulk Foresterite B0 = 129.6 GPa [38]. Thus, B0 is the only size dependent parameter which affects the thermodynamic properties of nanomaterials and shows somewhat different behavior as compared to their counterpart bulk material. Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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Fig. 1. Volume compression versus pressure at T = 300 K in nc-Fo.
Fig. 2. Volume compression versus pressure at T = 473 K in nc-Fo.
To test the validity and applicability of the present model theory we have used Eqs. (19)–(21) to determine the volume compression (V/V0 ), volume thermal expansion coefficient and bulk modulus of nanocrystalline Foresterite in the temperature range 300 K–1573 K under pressure varying conditions from 0 to 9.6 GPa. The input data needed for the calculation at room temperature and zero pressure are taken as [38]: B0 = 123.3 GPa;B0 = 3.88; α0 = 2.471 × 10−5 K −1 The value of the thermo-elastic parameter (k) calculated from Eq. (17) is determined to be 0.103 for nanoforesterite. The volume compression (V/V0 ) at different temperature-pressure conditions is also calculated using Tait’s equation of state (Eq. 16) [39] and Kholiya’s EOS (Eq. (22)) [37]. It is noted that the results calculated from Eq. (19) are found to be very close to the experimental values [38], as compared to those obtained from Tait’s equation and Kholiya’s equation of state including the thermal effect in them. The reason for obtaining the better results may be ascribed to the fact that the present EOS (Eq. (19)) contains a temperature dependent Anderson-Gruseinen parameter, whereas in Tait’s and Kholiya’s equation of state this parameter is considered as a temperature independent constant parameter. The values of the volume compression V/V0 at different temperatures under varying pressure conditions calculated from the present EOS, Tait’s EOS [39] and Kholiya’s EOS [37] are shown in Table 1. In the present study, we have calculated values for the thermo elastic compression from all three different equations of state and compared the results with the experimental values [38]. It is noted that the values obtained from the new formulation are very close to the experimental values. Figs. 1–7 shows a comparison between the values calculated from the three different equations of state with the experimental data available. Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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Fig. 3. Volume compression versus pressure at T = 673 K in nc-Fo.
Fig. 4. Volume compression versus pressure at T = 873 K in nc-Fo.
Fig. 5. Volume compression versus pressure at T = 1073 K in nc-Fo.
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Fig. 6. Volume compression versus pressure at T = 1273 K in nc-Fo.
Fig. 7. Volume compression versus pressure at T = 1573 K in nc-Fo.
Fig. 8. Variation of the bulk modulus at different pressures in nc-Fo.
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Fig. 9. Variation of the volume thermal expansion coefficient at different pressures in nc-Fo.
The newly formulated EOS is further applied to study the variation of the bulk modulus and thermal expansion coefficient over the temperature range 300 K–1573 K under pressures varying from 0 to 9.6 GPa. Eqs. (20) and (23) of Kholiya [37] are used to determine the variation in the bulk modulus and Eqs. (21) and (24) of Kholiya [37] are used to study the volume thermal expansion coefficient of nanocrystalline Foresterite. The variation of the bulk modulus with increasing pressure values at different temperatures is shown in Fig. 8. It is clear from the figure that the bulk modulus in nc-Fo increases with an increase in pressure across the nanocrystal. The variation of the thermal expansion coefficient under increasing temperature at different pressures is shown in Fig. 9. The variation of the thermal expansion coefficient is found to be the reverse of that of the bulk modulus, which justifies the quasi -harmonic approximation [39], as considered in the present study. The trends of the variation in B versus P and α versus P at different temperatures follow the general trend, which confirms the validity of the new approach. 4. Conclusions In the present theoretical study the authors have studied the thermoelastic properties of a nanoforesterite mineral sample which was collected using two different experimental set ups, diamond anvil cell (DAC) and multi anvil press (MAP) [38], though the occurrence of the traces of such materials also exist under the earth mantle [5,6]. With this motive, the authors have formulated a new equation of state which can provide the predications of the variation in the thermoelastic properties of nanoforesterite mineral considered under high temperature and high pressure, as shown in Figs. 1–9. The same trend, therefore, may be true for minerals present under the earth’s mantle. On the basis of the overall achievement and description, it may be emphasized here that the new EOS formulated is able to explain the thermoelastic properties of nanominerals and nanostructured crystals successfully. The present methodology may also be useful for the study of other mechanical properties, such as the Young modulus, shear modulus, elastic constants and anisotropy of nanocrystals with different morphologies. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state M. Goyal, B.R.K. Gupta∗ Department of Physics, GLA University, Mathura-281406, U.P., India
a r t i c l e
i n f o
Article history: Received 28 May 2016 Revised 1 August 2016 Accepted 29 September 2016 Available online xxx Keywords: Anderson Gruneisen parameter Thermal expansivity Thermal expansion coefficient Equation of state Nanoforesterite
a b s t r a c t A new potential independent equation of state is used in the present work to analyze the thermo-elastic properties of nanocrystalline Foresterite (nc- Fo) under varying temperature and pressure conditions. The newly developed EOS is found to be valid for explaining the elastic behavior of nanocrystalline forsterite satisfactorily over the temperature range from 300 K to 1573 K with a pressure variation from 0 to 9.6 GPa. The values calculated for the volume compression under varying temperature- pressure conditions are compared with the available experimental data and also with those obtained by using different approaches. It is found that the results obtained in the present study are more close to the experimental data in comparison to those reported earlier. The same model is further extended to study the variation in the bulk modulus and thermal expansion coefficient of Foresterite nano-mineral ver the temperature range from 300 K to 1573 K. The results show the same trend of expansion as observed in single nanocrystals at high temperature. © 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
1. Introduction The study of the thermo elastic behavior of minerals, ionic solids, metallic solids, alloys etc. under different pressure and temperature conditions is important for understanding the earth’s deep interior and its evolution. It also provides vital information about the dynamics of the earth’s lower mantle. A variation of the temperature-pressure conditions affects the atomic structure, stability and atomic interactions and thus modifies the physical properties, like the compressibility, electrical conductivity, elasticity, and thermal expansivity [1,2] of the material. The physical and rheological properties of a material are greatly influenced by its size and shape, which in turns affects the geo-physical processes [3]. Despite of the limitation of the nanocrystalline forms of minerals in the earth’s crust and mantle, its importance in geophysics cannot be neglected [4–6]. The effect of grain size on the elastic properties of nanocrystalline metal, alloys, ceramics, and oxides has been studied both experimentally [7,8] and theoretically [9,10] during the past years. Some of the experimental studies [11–13] on nanocrystalline materials have concluded that the elastic moduli of nanocrystalline materials decrease with a decrease in grain size. Though the olivine group of minerals are usually studied in bulk form [14–18], yet the nanostructured minerals in nano-form have been studied less under high pressure and high temperature. The olivine group of minerals comprises of Foresterite and Fayalite. Foresterite (Mg2 SiO4 )’ commonly abbreviated as Fo, is the member of the olivine solid solution series rich in magnesium whereas Fayalite (Fe2 SiO4 ) is the iron rich member. Fayalite has a high refractive index and is heavier than Foresterite. Foresterite is found to be the most abundant mineral in ∗
Corresponding author. E-mail address: [email protected] (B.R.K. Gupta).
http://dx.doi.org/10.1016/j.cjph.2016.09.009 0577-9073/© 2017 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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the earth’s mantle under the depth of 400 km [19,20]. Foresterite is found mainly in igneous rocks having rich magnesium and iron content and also in meteorites [21–23]. It has exceptionally high toughness and superior mechanical properties. It has been found as a potential material for bone implants and also has wide biomedical applications [24]. So, it is important to study more about the thermo elastic behavior of nanocrystalline Foresterite. Various potential dependent [25,26] and potential independent [27–30] model theories have been evolved and used to investigate the elastic properties of bulk materials and nanomaterials during the past decades. However, the results reported by these investigators show large variations, as compared to the experimental values. So, it becomes necessary to provide a model theory which could determine the thermo elastic properties of nanomaterials so as to explain the experimental results successfully. The main motive of the present study is to formulate a simple pressure and temperature dependent equation of state for studying the thermo elastic properties of nanocrystalline minerals. The method of formulation of the potential independent equation of state and its application is described in Section 2. The results obtained for Foresterite nanocrystal are discussed in Section 3. 2. Formulations and analysis Under high pressure, the product of the volume thermal expansion coefficient (α T ) and the bulk modulus (BT ) can be considered as constant at constant tessmperature [31,32], i.e.
αoBo = αT BT
(1)
Differentiation of Eq. (1) w.r.t. volume V leads to the relation:
α
dB dV
+B T
dα dV
=0
(2)
T
The Anderson-Gruneisen parameter δT at constant temperature T is given by
δT =
V
α
dα dV
(3) T
Using Eqs. (2) and (3), δT can be defined as
δT =
V
α
dα dV
=− T
V B
dB dV
(4) T
Considering δT to be independent of V, it becomes
dB dP
δT =
= B O .
(5)
T
δ +1
The Anderson-Gruneisen parameter δT and η are related as follows [33,34]:( Tη ) = A, where
η = V/Vo
(6)
where A is a constant. Using Eq. (6) in Eq. (4), and upon integrating it, we get
V B V = exp A 1 − Bo Vo Vo
(7)
where B is the bulk modulus and is defined as
B = −V
dP dV
(8) T
In view of Eq. (8), the expression of the bulk modulus B can be expressed as follows:
V B exp A 1 − dV = −dP, Bo Vo
(9)
where Bo , V0 are the bulk modulus and volume at zero pressure. On integrating Eq. (9), the equation for pressure (P) can be written as
P=
Bo V exp A 1 − A Vo
−1 ,
(10)
where
A = Bo + 1
(11)
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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3
Substituting A from Eq. (11) into Eq. (10), the isothermal usual Tait’s EOS can be expressed as follows:
P=
Bo
exp Bo + 1 (Bo + 1 )
1−
V Vo
−1
(12)
The expression of the bulk modulus corresponding to Eqs. (12) and (8) is obtained as follows [32,35]:
V BT V = exp Bo + 1 1 − Bo Vo Vo
(13)
To incorporate the temperature effect, the thermal pressure (PT h ) term given by Anderson [35] can be added into Eq. (12),
PT h = α0 B0 (T − T0 ).
(14)
On including the thermal pressure (PT h ) term in Eq. (12), the P-V-T relation obtained is given below:
P=
Bo
exp Bo + 1 (Bo + 1 )
1−
V Vo
− 1 + α0 B0 (T − T0 )
(15)
Taking the log of Eq. (15) to express it in terms of volume compression, the inverted form of the usual Tait’s equation so obtained including the thermal effect can now be expressed as
1 log 1 + (Bo + 1 )
V (P, To )/V (o, To ) = 1 −
Bo + 1 (P − α0 B0 (T − T0 ) ) Bo
(16)
The temperature dependence of δ T in accordance with the relation in the empirical form [27,36] is given as follows:
δT = δT0 X k ,
(17)
where X = T /T0 Here T0 refers to the room temperature, and δT0 represents the value of the Anderson-Gruneisen parameter at room temperature. In accordance with Eq. (5), δT0 is equal to Bo at room temperature and zero pressure. k is the thermo- elastic parameter, and its value is determined from the slope of the graph plotted between log (δ T ) versus log (T/T0 ) . Now considering the temperature dependence of the Anderson-Gruneisen parameter (δ T ) and Eq. (17), the new pressurevolume-temperature equation of state obtained is as follows:
P=
Bo
exp
δT0 X k + 1
V − 1 + α0 B0 (T − T0 ) δT0 X k + 1 1 −
(18)
Vo
The inverted form of the above equation including the thermal effect is as follows:
V (P, To )/V (P0 , To ) = 1 −
1
log 1 + δT0 X k + 1
δT0 X k + 1 Bo
(P − α0 B0 (T − T0 ) ) ,
(19)
where δT0 = Bo The expression of the bulk modulus corresponding to Eq. (19) is obtained as
BT = 1− Bo
1
δT0 X k + 1
log 1 +
δT0 X k + 1
Bo
0 k
δT X + 1 (P − α0 B0 (T − T0 ) ) ∗ 1 + (P − α0 B0 (T − T0 ) ) , Bo
(20)
and therefore, the volume thermal expansion coefficient can be expressed as follows:
BT = 1− Bo
1
δT0 X k + 1
log 1 +
δT0 X k + 1 Bo
−1 0 k
δT X + 1 ∗ 1+ (P − α0 B0 (T − T0 ) ) (P − α0 B0 (T − T0 ) ) , Bo
(21)
which is a reciprocal form of Eq. (20), in view of Eq. (1). The inverted form of the EOS formulated by Kholiya et al. [37] for nanomaterials including the thermal effect in terms of the volume compression is as follows:
V −1 V0
⎡
B − 2 ⎣1 + = o Bo − 1
1−
Bo − 1
(Bo − 2 )2
⎤ 1/2 ⎦.
2P (V, T ) Bo − + 2α0 (T − T0 ) − 3 B0
(22)
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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M. Goyal, B.R.K. Gupta / Chinese Journal of Physics 000 (2017) 1–9 Table 1 Calculated values of the volume compression using the present EOS (Eq. (19)); Kholiya’s EOS [37]; Tait’s EOS [39] and the experimental values [38]. T(K)
P(GPa)
Usual Tait EOS V/V0 [39]
Present EOS V/V0(Eq. (19))
Kholiya EOS V/V0(Eq. (23)) [37]
V/V0(Exp.) [38]
300
0 2.3 3.7 5.1 6.5 2.8 4.2 5.5 6.2 6.9 7.4 3.3 4.8 6.1 6.9 7.6 8 3.7 5.3 6.6 7.4 8.1 8.6 4 5.8 7.1 8 8.8 9.5 5.9 7.4 8.4 9.3 9.6
1 0.9821471 0.9719957 0.9623235 0.9530874 0.982349 0.972187 0.963183 0.958493 0.953909 0.950696 0.983164 0.972254 0.963246 0.957894 0.953323 0.950755 0.984734 0.97303 0.963989 0.958617 0.95403 0.950815 0.991769 0.978204 0.968938 0.962759 0.957419 0.952858 0.984228 0.973262 0.966265 0.960165 0.958172
1 0.9821471 0.9719957 0.9623235 0.9530874 0.9823754 0.9722552 0.9633012 0.9586429 0.9540921 0.9512815 0.9832067 0.9723736 0.963456 0.9581675 0.9536568 0.9515311 0.9847801 0.9731822 0.9642603 0.9589738 0.9544674 0.9517469 0.9917808 0.9783389 0.9692147 0.9631564 0.9579359 0.9539684 0.9843041 0.9734982 0.9666419 0.9606888 0.9587478
1 0.9821 0.972 0.9623 0.9531 0.9823 0.9722 0.9632 0.9585 0.9539 0.9507 0.9832 0.9723 0.9632 0.9579 0.9533 0.9508 0.9847 0.973 0.964 0.9586 0.954 0.9508 0.9918 0.9782 0.9689 0.9628 0.9574 0.9529 0.9842 0.9733 0.9663 0.9602 0.9582
1 0.9824 0.9717 0.9624 0.9562 0.9824 0.9724 0.9652 0.958 0.9535 0.9528 0.9831 0.9738 0.9662 0.9614 0.9531 0.9542 0.9855 0.9745 0.9673 0.9597 0.9549 0.9545 0.9917 0.9773 0.9693 0.9624 0.9576 0.9562 0.9841 0.9728 0.9666 0.96 0.9607
473
673
873
1273
1573
The bulk modulus corresponding to Eq. (22) is expressed as follows:
⎤ 2 ⎡ 1/2 Bo − 1 2P (V, T ) ⎣1 + 1 − ⎦ Bo − + 2α0 (T − T0 ) − 3 B0 (Bo − 2 )2 1/2 Bo − 1 2P (V, T ) × 1− Bo − + 2α0 (T − T0 ) − 3 . B0 (Bo − 2 )2
Bo − 2 BT = Bo (Bo − 1 )
(23)
The expression for the coefficient of volume thermal expansion corresponding to Eq. (23) is as follows:
−1/2 Bo − 1 Bo − 1 2P (V, T ) αT = 1− Bo − + 2α0 (T − T0 ) − 3 αo B0 (Bo − 2 )2 (Bo − 2 )2 ⎡ 1/2 ⎤−1 Bo − 1 2P (V, T ) ⎦ . × ⎣1 + 1 − Bo − + 2α0 (T − T0 ) − 3 B0 (Bo − 2 )2
(24)
3. Results and discussions The equation of state developed in the present study consists of only two important input parameters, i.e. the measured value of the bulk modulus (B0 ) and its first derivative (B0 ) at P = 0 and T = T0 (reference temperature) . The measured value of B0 differs significantly for nanomaterials than that of its counterpart bulk material, and its magnitude depends upon the size of the material, whereas B0 remains more or less same or differs by a negligible amount. For example, for nanocrystalline Foresterite, B0 = 123.3 GPa [38] and for bulk Foresterite B0 = 129.6 GPa [38]. Thus, B0 is the only size dependent parameter which affects the thermodynamic properties of nanomaterials and shows somewhat different behavior as compared to their counterpart bulk material. Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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Fig. 1. Volume compression versus pressure at T = 300 K in nc-Fo.
Fig. 2. Volume compression versus pressure at T = 473 K in nc-Fo.
To test the validity and applicability of the present model theory we have used Eqs. (19)–(21) to determine the volume compression (V/V0 ), volume thermal expansion coefficient and bulk modulus of nanocrystalline Foresterite in the temperature range 300 K–1573 K under pressure varying conditions from 0 to 9.6 GPa. The input data needed for the calculation at room temperature and zero pressure are taken as [38]: B0 = 123.3 GPa;B0 = 3.88; α0 = 2.471 × 10−5 K −1 The value of the thermo-elastic parameter (k) calculated from Eq. (17) is determined to be 0.103 for nanoforesterite. The volume compression (V/V0 ) at different temperature-pressure conditions is also calculated using Tait’s equation of state (Eq. 16) [39] and Kholiya’s EOS (Eq. (22)) [37]. It is noted that the results calculated from Eq. (19) are found to be very close to the experimental values [38], as compared to those obtained from Tait’s equation and Kholiya’s equation of state including the thermal effect in them. The reason for obtaining the better results may be ascribed to the fact that the present EOS (Eq. (19)) contains a temperature dependent Anderson-Gruseinen parameter, whereas in Tait’s and Kholiya’s equation of state this parameter is considered as a temperature independent constant parameter. The values of the volume compression V/V0 at different temperatures under varying pressure conditions calculated from the present EOS, Tait’s EOS [39] and Kholiya’s EOS [37] are shown in Table 1. In the present study, we have calculated values for the thermo elastic compression from all three different equations of state and compared the results with the experimental values [38]. It is noted that the values obtained from the new formulation are very close to the experimental values. Figs. 1–7 shows a comparison between the values calculated from the three different equations of state with the experimental data available. Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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Fig. 3. Volume compression versus pressure at T = 673 K in nc-Fo.
Fig. 4. Volume compression versus pressure at T = 873 K in nc-Fo.
Fig. 5. Volume compression versus pressure at T = 1073 K in nc-Fo.
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Fig. 6. Volume compression versus pressure at T = 1273 K in nc-Fo.
Fig. 7. Volume compression versus pressure at T = 1573 K in nc-Fo.
Fig. 8. Variation of the bulk modulus at different pressures in nc-Fo.
Please cite this article as: M. Goyal, B.R.K. Gupta, Analysis of the thermoelastic properties of nanocrystalline Foresterite using a thermodynamic equation of state, Chinese Journal of Physics (2017), http://dx.doi.org/10.1016/j.cjph.2016.09.009
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Fig. 9. Variation of the volume thermal expansion coefficient at different pressures in nc-Fo.
The newly formulated EOS is further applied to study the variation of the bulk modulus and thermal expansion coefficient over the temperature range 300 K–1573 K under pressures varying from 0 to 9.6 GPa. Eqs. (20) and (23) of Kholiya [37] are used to determine the variation in the bulk modulus and Eqs. (21) and (24) of Kholiya [37] are used to study the volume thermal expansion coefficient of nanocrystalline Foresterite. The variation of the bulk modulus with increasing pressure values at different temperatures is shown in Fig. 8. It is clear from the figure that the bulk modulus in nc-Fo increases with an increase in pressure across the nanocrystal. The variation of the thermal expansion coefficient under increasing temperature at different pressures is shown in Fig. 9. The variation of the thermal expansion coefficient is found to be the reverse of that of the bulk modulus, which justifies the quasi -harmonic approximation [39], as considered in the present study. The trends of the variation in B versus P and α versus P at different temperatures follow the general trend, which confirms the validity of the new approach. 4. Conclusions In the present theoretical study the authors have studied the thermoelastic properties of a nanoforesterite mineral sample which was collected using two different experimental set ups, diamond anvil cell (DAC) and multi anvil press (MAP) [38], though the occurrence of the traces of such materials also exist under the earth mantle [5,6]. With this motive, the authors have formulated a new equation of state which can provide the predications of the variation in the thermoelastic properties of nanoforesterite mineral considered under high temperature and high pressure, as shown in Figs. 1–9. The same trend, therefore, may be true for minerals present under the earth’s mantle. On the basis of the overall achievement and description, it may be emphasized here that the new EOS formulated is able to explain the thermoelastic properties of nanominerals and nanostructured crystals successfully. The present methodology may also be useful for the study of other mechanical properties, such as the Young modulus, shear modulus, elastic constants and anisotropy of nanocrystals with different morphologies. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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