Study of elastic moduli of Fe-C alloys pressure dependence

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Temperature and pressure dependence of elastic moduli for interstitial alloys like Fe-C alloys with bcc structure has been investigated by using statistical moment method (SMM). The Young moduli of Fe-C alloys is calculated as a function of the temperature and pressure.

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Nội dung Text: Study of elastic moduli of Fe-C alloys pressure dependence

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 20-27 STUDY OF ELASTIC MODULI OF Fe-C ALLOYS PRESSURE DEPENDENCE Vu Van Hung(∗) and Nguyen Thi Thu Hien Hanoi National University of Education (∗) Email: bangvu57@yahoo.com Abstract. Temperature and pressure dependence of elastic moduli for in- terstitial alloys like Fe-C alloys with bcc structure has been investigated by using statistical moment method (SMM). The Young moduli of Fe-C alloys is calculated as a function of the temperature and pressure. We discusss the temperature and pressure dependence of the elastic moduli of Fe-C alloys with the different interstitial concentrations and compare the SMM elastic moduli calculations with those of the experimental results. Keywords: Elastic moduli, Fe-C alloys, temperature dependence, pressure dependence. 1. Introduction Carbon interstice in metals and alloys is a current subject of considerable ex- perimental and theoretical investigations [1-5]. A small number of theoretical studies have focused on diffusion in pure metals (Ti, Fe, etc.) at high interstitial concentra- tion [2,5]. Some approaches in the form of Green's function technique [6] and first principles calculations [1,3,5,7] have been used to explain the lattice dynamics and predict the lattice parameters, elastic constant and moduli, vibrational properties of Fe-C systems. To understand the natrure of the lattice dynamics and influence of interstitial elements on the mechanical and thermodynamic properties of Fe-C system, knowl- edge of the many - body interaction potential of bcc Fe alloyed with C is required. Recently, empirical potentials [8] have been developed for point defect clusters in Fe-C alloys. The purpose of the present article is to investigate the Young moduli E of interstitial Fe-C alloys using the statistical moment method (SMM) [9-12]. In the study, the influences of the temperature, pressure and C interstitial concentration on the Young modulus E of Fe-C alloys have been studied using the many body interaction potential [8]. We will compare the present calculations with the available experimental results. 20
Study of elastic moduli of F e-C alloys: pressure dependence 2. Theory Let us consider an AB interstitial alloys consisting of NA atoms A and NB atoms B (NA >> NB ) with the bcc structure. The free energy of system has an approximate form: ψ = (NA − 6NB )ψA + NB ψB + 2NB ψA1 + 4NB ψA2 − T SC (2.1) where ψA is the free energy of an atom A of the pure metal A, ψA1 (or ψA2 ) is the free energy of an atom A with an interstitial atom B located at the nearest neighbour lattice site as schematically shown in Figure 1; ψB is the free energy of an interstitial atom B in the interstitial alloy AB and SC the configurational entropy. Figure 1. The lattice site of interstitial atom B of bcc interstitial alloy AB. (A, A , A ≡ Fe atom, B ≡ C atom) 1 2 The temperature dependence of elastic moduli of AB interstitial alloys are calculated by using the general expression of the Helmholtz free energy ψ of Equation (2.1). Young modulus E of AB interstitial alloys is given as [12] ∂σ 1 ∂2ψ E= = (2.2) ∂ε v ∂ε2 where ε denotes the strain and σ is the stress. From Equations (2.1) and (2.2) it is easy to derive the results ∂ 2 ψB ∂ 2 ψA1 ∂ 2 ψA2       +2 +4   ∂ε2 ∂ε 2 ∂ε 2  E = EA (1 − 7cB ) + cB  (2.3) ∂ 2 ψA      ∂ε 2 21
Vu Van Hung and Nguyen Thi Thu Hien where the second derivatives of the free energies with respect to the strain ε are calculated as ( " 2 #) 2 2 2 3 ~ωX ∂ kX ∂ ψX 1 ∂ U0X 1 ∂kX 2 2 = 2 + 2 − 4r01 ∂ε 2 ∂r1 4 kX ∂r1 2kX ∂r1 (2.4) 1 ∂U0X 3 1 ∂kX + + ~ωX coth xX 2r01 , 2 ∂r1 2 2kX ∂r1 r ~ωX kX with X = A, B, A1 and A2 . Here xX = , ωX = , cB is the concentration of 2θ m the interstitial atoms B and θ = kB T , the second order vibrational constant kX is defined by ∂ 2 ϕXi 1X kX = (2.5) 2 i ∂u2iα eq where uiα (α = x, y, z) is the atomic displacement of ith particle due to thermal lattice vibrations, and eq denotes the equilibrium interatomic distance, U0α denotes the total energy of the Fe-C system [8] 1 X sX U0α = −Aα ρβα (rij ) + ϕβα (rij ) (2.6) j6=i 2 j6=i Here j refers to the nearest neighbours within a cutoff distance from atoms i, α is the element type of atoms i, β is the element type of atom j , Aα is a positive coefficient, ρβα (rij ) refers to the density contribution of j to atom i, and ϕβα (rij ) refers to the pairs of interaction between atom i and its neighbours. Table 1. Constant of the F e-C potential used in this work, assuming units of length in A, and units of energy in e , where V A = 1.8289905eV and A = 2.9587870eV Fe C α β rc,p t1 t2 rc,ϕ k1 k2 k3 Fe Fe 3.569745 1 0.504238 3.40 1.237115 -0.35921 -0.038560 Fe C 2.545937 10.024001 1.638980 2.468801 8.972488 -4.086410 1.483233 C Fe 2.545937 10.482408 3.782595 C C 2.892070 0 -7.329211 2.875598 22.061824 -17.468518 4.812639 22
Study of elastic moduli of F e-C alloys: pressure dependence We used Fe-Fe, Fe-C, and C-C interactions of the Finnis-Sinclair (FS) form [8] as: ρβα (r) = t1 (r − rc,ρ )2 + t2 (r − rc,ρ)3 , r ≤ rc,ρ (2.7) ϕβα (r) = (r − rc,ϕ )2 k1 + k2 r + k3 r 2 , (2.8) r ≤ rc, ϕ where these functions are zero for r ≥ rc . The constants rc , ti , and ki are presented in Table 1. From Equations (2.1) and (2.2) Young modulus of Fe-C system at the tem- perature T and pressure p can be given as: E (p, T, cB ) = EA (p, T, cB ) {(1 − 7cB ) + cB β (p, T )} (2.9) where EA (p, T, cB ) has the form analogous to [9] k EA (p, T, cB ) = 2 2 A (2.10) 2γA θ x coth x π r1 (p, T, cB ) 1 + 4 1+ (1 + x coth x) kA 2 and ∂ 2 ψB ∂ 2 ψA 1 ∂ 2 ψA 2 2 + 2 + 4 β (p, T ) = ∂ε ∂ε2 ∂ε2 (2.11) 2 ∂ ψA ∂ε2 Since the pressure dependence of the Young modulus of metals is linear, we can expand the Young modulus in terms of pressure p as: 1 ∂ 2 ψB 1 ∂ 2 ψB
EB (p, T ) = (p, T ) ≈ + a1 p = EB (0, T ) + a1 p ν ∂ε2 ν ∂ε2
p=0 1 ∂ 2 ψA 1 EA 1 (p, T ) = (p, T ) ≈ EA 1 (0, T ) + a2 p (2.12) ν ∂ε2 1 ∂ 2 ψA 2 EA 2 (p, T ) = (p, T ) ≈ EA 2 (0, T ) + a3 p ν ∂ε2 1 ∂ 2 ψA EA (p, T ) = (p, T ) ≈ EA (0, T ) + a4 p ν ∂ε2 23
Vu Van Hung and Nguyen Thi Thu Hien where a1 , a2 , a3 , and a4 are the constants with respect to pressure. Note that the second terms in Equation (2.12) are small for the wide pressure range (a1 p
Study of elastic moduli of F e-C alloys: pressure dependence Figure 2. Concentration depen- Figure 3. Concentration de- dence c of the nearest neighbor pendence c of the Young mod- distance r (0, T ) A of F e-C inter- uli E of F e-C interstitial alloys B B stitial alloys at zero pressure and at zero pressure p and temper- 1 temperature T . ature T . Figure 4. Temperature depen- Figure 5. Temperature depen- dence of the Young moduli E of dence of the SMM Young mod- F e-C interstitial alloys with the ulus E (10 P a) of F e-C in- 10 interstitial concentration c =0%; terstitial alloy with c =0.2% 0.2%; 0.4%; 1%; 3% and 5% at and the experimental results of B B zero pressure. Young modulus E (10 P a) of10 F e-C interstitial alloys with c ≤ 0.3%. B 25
Vu Van Hung and Nguyen Thi Thu Hien Figure 6. Temperature depen- Figure 7. Pressure dependence dence of the SMM Young mod- of the Young moduli E of F e- ulus E (10 P a) of F e-C intersti- 10 C interstitial alloys with the tial alloy with c = 0.4% and the interstitial concentration c = experimental results of Young 0%; 0.2%; 1%; 3% and 5% at B B modulus E (10 P a) of 10 temperature T = 300(K ). F e-C interstitial alloys with c ≥ 0.3%. B In conclusion, the SMM calculations are performed using the many body in- teraction potentials for the Fe-C interstitial alloys with bcc structure. The nears- est neighbour distance and Young moduli of the Fe-C system are calculated and compared with the available experimental results. Present SMM results of Young modulus E(1010 P a) of Fe-C interstitial alloy are in agreement with the experimental data. Acknowledgements This work is supported by the research project No. 103.01.2609 of NAFOS- TED. REFERENCES [1] Y. Song, Z. X. Guo, R. Yang, Philosophical Magazine A, Vol. 82, No. 7, (2002) 1345. [2] A. V. Nazorov, and A. A. Mikheev, Physica Scripta, T 108, pp. 90-94, (2004) [3] C. Jiang, S. G. Srinivasan, A. Caro, and S . A. Maloy, J. Appl. Phys., 103, (2008) 043502 [4] A. Yu. Moskrichev, I. A. Nechaev, and V. S. Demidenko, Russian physics Journal, 26
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