Equation of state and thermal expansion of metals with FCC structure: Application to Cu, Al and Ni

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In present paper "Equation of state and thermal expansion of metals with FCC structure: Application to Cu, Al and Ni", by the statistical moment method (SMM) we obtain the equation of state for metals with cubic structure at temperature 0 K and pressure P and from that obtain the expressions of lattice parameter and thermal expansion coefficient for these metals at temperature T and pressure P. Applying to metals Cu, Al and Ni, we derive the equation of state, simple analytic expressions of lattice parameter and thermal expansion coefficient for each metal and our numerical results are in good agreement with the experimental data in the interval of temperature from 0 K to near the melting temperature.

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Nội dung Text: Equation of state and thermal expansion of metals with FCC structure: Application to Cu, Al and Ni

October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 1–6 Modern Physics Letters B Vol. 28, No. 26 (2014) 1450209 (6 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0217984914502091 Equation of state and thermal expansion of metals with FCC structure: Application to Cu, Al and Ni Pham Dinh Tam Le Qui Don University of Technology, 100 Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam Nguyen Quang Hoc∗ and Bui Duc Tinh Department of Physics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam ∗ hocnq@hnue.edu.vn Nguyen Duc Hien Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province, Vietnam Received 5 March 2014 Revised 5 August 2014 Accepted 16 September 2014 Published 8 October 2014 The equation of state, the expressions of lattice parameter and thermal expansion co- eﬃcient in general form are obtained by the statistical moment method. Applying to Cu, Al and Ni metals, we determine these properties in simple analytic form for each metal. Numerical results for the thermal expansion coeﬃcient of these metals in diﬀerent temperatures and pressures are in good agreement with experiments. Keywords: The statistical moment method, modiﬁed Lennard-Jones potential. 1. Introduction The equation of state of metal gives us many information on thermodynamic prop- erties such as the lattice parameter and the thermal expansion coeﬃcient at temper- ature T and pressure P , etc. Finding the equation of state for metals is a complex problem. There are some equations of state which approximately describe states of metals and can be applied to study the thermal expansion. Examples of these equations are the Mie–Gruneisen equation of state1 (approximately in low temperatures) and the ∗ Corresponding author. 1450209-1
October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 2–6 P. D. Tam et al. Bradbam–Furth equation of state.2 Many equations of state are calculated directly from approximate methods such as the Debye approximation,3 the Einstein–Debye approximation4,6 and X-ray approximation.5 Each of these methods is suitable for the study of some metals and several of their thermodynamic properties. In the X-ray method, the interpolated expressions for thermal expansion coeﬃcient have simple analytic form, but only agree with experiments in not large interval of temperature. In present paper, by the statistical moment method (SMM) we obtain the equa- tion of state for metals with cubic structure at temperature 0 K and pressure P and from that obtain the expressions of lattice parameter and thermal expansion coeﬃ- cient for these metals at temperature T and pressure P . Applying to metals Cu, Al and Ni, we derive the equation of state, simple analytic expressions of lattice pa- rameter and thermal expansion coeﬃcient for each metal and our numerical results are in good agreement with the experimental data in the interval of temperature from 0 K to near the melting temperature. 2. Theory By the SMM, we obtain the free energy of metal in the following form:7 u0 ω ψ = 3N + θ ln 1 − exp − , (1) 6 2θ where N is the number of atoms in metal, θ = kB T , kB is the Boltzmann constant, h = 2π , h is the Planck constant, T is the absolute temperature, u0 is the interaction energy corresponding with one atom in metal in equilibrium state and ω is the frequency (coinciding with the Einstein characteristic frequency). The frequency ω is determined from 1 ∂ 2 φ0i k= ≡ M ω2 , (2) 2 i ∂u2ix eq where φ0i is the interaction potential between the 0th atom and ith atom, uix is the displacement of ith atom in the direction x and M is the mass of atom. The equation of state for metal is derived from the thermodynamic relation4,8 ∂ψ a ∂ψ p=− =− . (3) ∂V T 3V ∂a T Substituting (1) into (3) at 0 K, we obtain the equation of state at 0 K and pressure P P v0 1 ∂u0 ∂k0 − = + √ , (4) a0 6 ∂a0 4 M k0 ∂a0 where a0 and v0 are respectively the lattice parameter and the volume of unit cell in crystal lattice of metal at 0 K and pressure P and k0 is the parameter k in (2) calculated at 0 K. 1450209-2
October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 3–6 Equation of state and thermal expansion of metals If the potential φ0i is known, from Eqs. (1)–(4) we determine the equation of state for metal at 0 K and pressure P . The lattice parameter of metal is counted from the formula a = a0 + y , (5) where y is the mean displacement of atom from equilibrium position at 0 K. The mean displacement y is determined according to the formula7,8 2 2γ0 θ2 x γ02 θ2 13 47x y = 1+ + 4 + + Δ1 , (6) 3k 3 2 k0 3 6 where X = xcth, x = θ M k , Δ1 depends on temperature and has small value in comparison with unit and γ0 is the parameter calculated at 0 K. The parameter γ is calculated by the expression7 4 4 1 ∂ φ0i ∂ φ0i γ= +6 . (7) 12 i ∂u4ix eq ∂u2ix ∂u2iy eq If the potential φ0i is known, from Eqs. (2), (5)–(7) we can ﬁnd the lattice parameter a at temperature T and pressure P . The thermal expansion coeﬃcient (linear expansion) of metal is determined by 1 da 1 dy αT = = . (8) a0 dT a0 dT The thermal expansion coeﬃcient of metals and alloys is measured by many diﬀerent methods such as the interferometer, the densitometer, the optical lever and the X-ray photography. The content of these methods and the experimental data on the thermal expansion coeﬃcient of metals and alloys are fully given in Ref. 1. 3. Numerical Results and Discussion We will apply the results obtained in Sec. 2 in order to ﬁnd the equation of state, the lattice parameter and the thermal expansion coeﬃcient of metals Cu, Al and Ni. Using the n − m modiﬁed Lennard-Jones potential for the interaction between atoms in metal as follows:9 n m Dnm r0 r0 φ(a) = m −n , (9) n−m a a where D, r0 , n and m are the parameters of potential and have concrete values depending on each metal. For metals Cu, Al and Ni, values of potential parameters are given in Table 1. From Eqs. (2)–(9) and Table 1, we obtain the equation of state, the lattice parameter and the thermal expansion coeﬃcient of metals Cu, Al and Ni in the following analytic form. 1450209-3
October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 4–6 P. D. Tam et al. Table 1. Parameters D, r0 , n and m for metals Cu, Al and Ni. Metals D/kB (K) r0 (˚ A) n m Cu 3401.0 2.5487 9.0 5.5 Al 2995.6 2.8541 12.5 4.5 Ni 4782.0 2.4780 8.5 5.5 The equations of state for Cu, Al, Ni at 0 K and pressure P , respectively have the following forms: 5.124.10−6P a12 0 − 0.97.10 −4 10.5 a0 + 0.016a70 + 17.11a3.5 0 − 446.21 = 0 , (10) 5.124.10−6P a15.5 0 − 11.3.10−9a21.25 0 + 69.7.10−5a13.25 0 + 5.22a80 − 10.56a5.25 0 − 20929.3 = 0 , (11) 5.124.10−6P a11.5 0 − 18.1.10−4a7.75 0 + 0.166a4.25 0 + 23.55a30 − 3.26a1.25 0 − 342.3 = 0 . (12) The lattice parameters for Cu, Al, Ni at temperature T and pressure P are respectively determined by a = a0 + 2.2.10−9(a10 13.5 0 + 0.026a0 ) × [T + 0.967.10−14a18 3.5 0 (1 + 0.06a0 − 2.04.10 −3 7 a0 )T 3 ] , (13) a = a0 + 0.029.10−9(a13.5 0 + 9.10−5 a21.5 0 ) × [T + 1.53.10−18a25 −4 8 0 (1 + 2.36.10 a0 − 4.47.10 −7 14 a0 )T 3 ] , (14) a = a0 + 3.3.10−9(a9.5 12.5 0 + 0.047a0 ) × [T + 6.33.10−14a17 3 0 (1 + 0.11a0 − 7.55.10 −3 6 a0 )T 3 ] . (15) The thermal expansion coeﬃcients for Cu, Al, Ni are respectively derived from αT = 2.2.10−9(a90 + 0.026a12.5 0 ) × [1 + 2.9.10−14 a18 3.5 0 (1 + 0.06a0 − 2.04.10 −3 7 a0 )T 2 ] , (16) αT = 0.029.10−9(a12.5 0 + 9.10−5 a20.5 0 ) × [1 + 4.6.10−18 a25 0 (1 + 2.36.10 −4 8 a0 − 4.47.10−7a14 2 0 )T ] , (17) αT = 3.3.10−9(a8.5 11.5 0 + 0.047a0 ) × [1 + 19.10−14 a17 3 0 (1 + 0.11a0 − 7.55.10 −3 6 a0 )T 2 ] . (18) Numerical results for the lattice parameter a and the thermal expansion coef- ﬁcient αT for Cu, Al, Ni in the interval of temperature from 0 K to 800 K and at pressures P = 0, 50 kbar and 100 kbar are respectively given in Tables 2–4. 1450209-4
October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 5–6 Equation of state and thermal expansion of metals Table 2. Lattice parameter and thermal expansion coeﬃcient for Cu in diﬀerent temperatures and pressures. P (kbar) T (K) 0 200 400 600 800 a (˚ A) 2.5229 2.5306 2.5387 2.5475 2.5574 0 αT 105 (K−1 ) 1.5152 1.5514 1.6602 1.8415 2.0954 αT 105 (K−1 ) (Ref. 10) — 1.52 1.73 1.86 2.01 a (˚ A) 2.4983 2.5053 2.5125 2.5202 2.5288 50 αT 105 (K−1 ) 1.3694 1.3977 1.4827 1.6244 1.8227 a (˚ A) 2.4781 2.4844 2.4904 2.4978 2.5055 100 αT 105 (K−1 ) 1.2582 1.2812 1.3501 1.4649 1.6256 Table 3. Lattice parameter and thermal expansion coeﬃcient for Al in diﬀerent temperatures and pressures. P (kbar) T (K) 0 200 400 600 800 a (˚ A) 2.8516 2.8630 2.8754 2.8897 2.9069 0 αT 105 (K−1 ) 1.9727 2.0576 2.3122 2.7365 3.3305 αT 105 (K−1 ) (Ref. 10) — 2.02 2.45 2.81 3.26 a (˚ A) 2.8121 2.8213 2.8310 2.8419 2.8546 50 αT 105 (K−1 ) 1.6081 1.6608 1.8188 2.0822 2.4510 a (˚ A) 2.7821 2.7899 2.7980 2.8069 2.8170 100 αT 105 (K−1 ) 1.3762 1.4122 1.5200 1.6997 1.9512 Table 4. Lattice parameter and thermal expansion coeﬃcient for Ni in diﬀerent temperatures and pressures. P (kbar) T (K) 0 200 400 600 800 a (˚ A) 2.4469 2.4483 2.4497 2.4511 2.4526 0 αT 105 (K−1 ) 1.1202 1.1543 1.2566 1.4271 1.6658 αT 105 (K−1 ) (Ref. 10) — 1.14 1.37 1.59 1.67 a (˚ A) 2.4305 2.4317 2.4330 2.4343 2.4357 50 αT 105 (K−1 ) 1.0492 1.0786 1.1667 1.3137 1.5195 a (˚ A) 2.4160 2.4172 2.4184 2.4196 2.4209 100 αT 105 (K−1 ) 0.9901 1.0159 1.0930 1.2216 1.4017 In equations and expressions from (10) to (18), the pressure P and the lattice parameter a0 are taken in kbar and ˚A, respectively. The dependence of thermal expansion coeﬃcient for metals Cu, Al and Ni on temperature at pressure P = 0 is expressed in Fig. 1. Our numerical results of thermal expansion coeﬃcient for metals Cu, Al and Ni at pressure P = 0 (atmospheric pressure) are in good agreement with experiments 1450209-5
October 7, 2014 9:36 WSPC/147-MPLB S0217984914502091 6–6 P. D. Tam et al. Fig. 1. Dependence of thermal expansion coeﬃcient for metals Cu, Al and Ni on temperature at pressure P = 0. in the interval of temperature from 200 K to 800 K. That shows that the thermal expansion coeﬃcient for each of metals Cu, Al and Ni is described rather exactly by the system of two equations in simple analytic form: Equations (10) and (16) describe the thermal expansion of Cu, Eqs. (11) and (17) describe the thermal expansion of Al and Eqs. (12) and (18) describe the thermal expansion of Ni. 4. Conclusion The thermal expansion coeﬃcient for metals Cu, Al and Ni is described by the systems of two equations (10), (16); (11), (17) and (12), (18), respectively. These equations have simple analytic forms and are easy to calculate numerically and the obtained numerical results are in good agreement with experiments. Acknowledgments This work was carried out by the ﬁnancial support from HNUE, the Le Quy Don University of Technology and the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant No. 103.01-2013.20. References 1. M. Delannoy-Coutris and G. Perrin, Phys. Status Solidi B 138(1) (1986) 93. 2. H. D. Pandey and B. Dayan, Phys. Status Solidi B 5(2) (1964) 273. 3. B. M. Askerov and M. Cankurtaran, Phys. Status Solidi B 185(2) (1994) 341. 4. M. Cankurtaran and B. M. Askerov, Phys. Status Solidi B 194(2) (1996) 499. 5. B. N. Dutta and B. Dayal, Phys. Status Solidi B 3(12) (1963) 2253. 6. A. Dewaele, P. Loubevre and M. Mezouar, Phys. Rev. B 70 (2004) 094112. 7. N. Tang and V. V. Hung, Phys. Status Solidi B 149(2) (1988) 511. 8. K. Masuda-Jindo, V. V. Hung and P. D. Tam, Phys. Rev. B 67 (2003) 094301. 9. S. Zhen and G. J. Davies, Phys. Status Solidi A 78(2) (1983) 595. 10. L. N. Laricov, Yu. F. Yurchenko, Thermal Properties of Metals and Alloys (Naukova Dumka, Kiev, 1985) (in Russian). 1450209-6