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 Fe-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 +NBψB + 2NBψA1 + 4NBψA2 − TSC (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, A1, A2 ≡ Fe atom, B ≡ C atom) 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] E = ∂σ ∂ε = 1 v ∂2ψ ∂ε2 (2.2) where ε denotes the strain and σ is the stress. From Equations (2.1) and (2.2) it is easy to derive the results E = EA  (1− 7cB) + cB   ∂2ψB ∂ε2 + 2 ∂2ψA1 ∂ε2 + 4 ∂2ψA2 ∂ε2 ∂2ψA ∂ε2     (2.3) 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 ∂2ψX ∂ε2 = { 1 2 ∂2U0X ∂r21 + 3 4 ~ωX kX [ ∂2kX ∂r21 − 1 2kX ( ∂kX ∂r1 )2]} 4r201 + { 1 2 ∂U0X ∂r1 + 3 2 ~ωX coth xX 1 2kX ∂kX ∂r1 } 2r01, (2.4) with X = A,B,A1 and A2. Here xX = ~ωX 2θ , ωX = √ kX m , cB is the concentration of the interstitial atoms B and θ = kBT , the second order vibrational constant kX is defined by kX = 1 2 ∑ i ( ∂2ϕXi ∂u2iα ) eq (2.5) where uiα(α = x, y, z) is the atomic displacement of i th particle due to thermal lattice vibrations, and eq denotes the equilibrium interatomic distance, U0α denotes the total energy of the Fe-C system [8] U0α = −Aα √∑ j 6=i ρβα (rij) + 1 2 ∑ j 6=i ϕβα (rij) (2.6) 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 Fe-C potential used in this work, assuming units of length in  A, and units of energy in eV , where AFe = 1.8289905eV and AC = 2.9587870eV 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 Fe-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 + k2r + k3r 2 ) , r ≤ rc,ϕ (2.8) 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) + cBβ (p, T )} (2.9) where EA(p, T, cB) has the form analogous to [9] EA (p, T, cB) = kA pi r1 (p, T, cB) [ 1 + 2γ2Aθ 2 k4A ( 1 + x coth x 2 ) (1 + x coth x) ] (2.10) and β (p, T ) = ∂2ψB ∂ε2 + 2 ∂2ψA 1 ∂ε2 + 4 ∂2ψA 2 ∂ε2 ∂2ψA ∂ε2 (2.11) Since the pressure dependence of the Young modulus of metals is linear, we can expand the Young modulus in terms of pressure p as: EB (p, T ) = 1 ν ∂2ψB ∂ε2 (p, T ) ≈ 1 ν ∂2ψB ∂ε2 ∣∣∣∣ p=0 + a1p = EB (0, T ) + a1p EA 1 (p, T ) = 1 ν ∂2ψA 1 ∂ε2 (p, T ) ≈ EA 1 (0, T ) + a2p (2.12) EA 2 (p, T ) = 1 ν ∂2ψA 2 ∂ε2 (p, T ) ≈ EA 2 (0, T ) + a3p EA (p, T ) = 1 ν ∂2ψA ∂ε2 (p, T ) ≈ EA (0, T ) + a4p 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 (a1p << EB(0, T ), a2p << EA1(0, T ), etc). From Equations (2.11) and (2.12), we obtain the approximate expression of β(p, T ): β (p, T ) = β (0, T ) + a1p+ 2a2p+ 4a3p EA (0, T ) ( 1 + a4p EA (0, T ) ) ≈ β (0, T ) EA (0, T ) − β (0, T ) EA (0, T ) a4p EA (0, T ) + a1p+ 2a2p+ 4a3p EA (0, T ) (2.13) where β (0, T ) = EB (0, T ) + 2EA 1 (0, T ) + 4EA 2 (0, T ) (2.14) From Equations (2.9), (2.12) and (2.13), it is easy to find the Young modulus of Fe-C system at temperature T and pressure p E (p, T, cB) ≈ EA (p, T, cB) { (1− 7cB) + cB β (0, T ) EA (0, T ) } (2.15) 3. Results and discussion The SMM caculations of the nearest neighbour distance and Young moduli of Fe-C interstitial alloys are shown in Figures 2 ÷ 7. The concentration dependence of the nearest neighbour distance r1(0, T ) and Young moduli E of Fe-C interstitial alloys at zero pressure and temperature T is presented in Figures 2 and 3. As show in Figure 2 and 3, nearest neighbour distance and Young moduli depend strongly both upon the C concentration and the temperature T . One can see in Figures 5 and 6 that the present result is in agreement with the experiment result [13]. In Figures 4 and 7, we present the temperature and pressure dependences of the Young moduli E of bcc Fe-C interstitial alloys as a function of the temperature T and pressure p. The rapidly decreasing in the moduli E indicates the stronger anharmonicity contribution of the thermal lattice vibration at high temperatures. It can also be seen in Figure 7 that the Young moduli E of Fe-C interstitial alloys increase rapidly with increasing pressure. The lattice constant decrease due to the effect of increasing pressure, therefore the Young moduli E becomes larger. 24 Study of elastic moduli of Fe-C alloys: pressure dependence Figure 2. Concentration depen- dence cB of the nearest neighbor distance r1(0, T ) A of Fe-C inter- stitial alloys at zero pressure and temperature T . Figure 3. Concentration de- pendence cB of the Young mod- uli E of Fe-C interstitial alloys at zero pressure p and temper- ature T . Figure 4. Temperature depen- dence of the Young moduli E of Fe-C interstitial alloys with the interstitial concentration cB=0%; 0.2%; 0.4%; 1%; 3% and 5% at zero pressure. Figure 5. Temperature depen- dence of the SMM Young mod- ulus E (1010Pa) of Fe-C in- terstitial alloy with cB=0.2% and the experimental results of Young modulus E (1010Pa) of Fe-C interstitial alloys with cB ≤ 0.3%. 25 Vu Van Hung and Nguyen Thi Thu Hien Figure 6. Temperature depen- dence of the SMM Young mod- ulus E (1010Pa) of Fe-C intersti- tial alloy with cB = 0.4% and the experimental results of Young modulus E (1010Pa) of Figure 7. Pressure dependence of the Young moduli E of Fe- C interstitial alloys with the interstitial concentration cB = 0%; 0.2%; 1%; 3% and 5% at temperature T = 300(K). Fe-C interstitial alloys with cB ≥ 0.3%. 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(1010Pa) 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. 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