2010 Crystl Structure nd Formtion Energy of ε-crbide Using First Principles Clcultions Je Hoon Jng, In Gee Kim, Dong Woo Suh nd H. K. D. H. hdeshi Computtionl Metllurgy Lbortory Grdute Institute of Ferrous Technology Pohng University of Science nd Technology
Introduction Mrtensite (α ) ε-crbide η-crbide χ-crbide Cementite (θ) Fe 2.4 C Fe 2 C Fe 2.5 C Fe 3 C Silicon promotes the formtion of ε-crbide below 520 K. S. S. Nyk et.l, Mterils Science nd Engineering. 498, pp.442-456(2008) 900, 100s 250, 30 s 1.0wt% Si : No ε-crbide 1.7wt% Si : ε-crbide M s = 302(1.0 wt%si), 293 (1.7 wt%si) 200, 20s ε-crbide forms without redistribution of Si. S. J. rnrd, G. D. W. Smith, Proceedings of the solid-solid phse trnsformtion., pp.881(1981) No initil prtitioning of Si between ε, θ nd mrtensite S. S. bu, K. Hono, T. Skuri, Metl Mter. Trns. 25 (1994) p. 499
FLPW method E. Wimmer, H. Krkuer, M. Weinert, nd. J. Freemn, Phys. Rev. 28, 864 (1981) nd references therein. M. Weinert, E. Wimmer, nd. J. Freemn, Phys. Rev. 26, 4571 (1982). Muffin-Tin (MT) Sphere Region Interstitil Region $ v G ( r) = c # G( K, r, Wve Function Expnsion! k v k + G " Kmx k, ) ) G( k, r) i( k+ G) ( r "% e = $ v v [ ul r ul "# ' r Y lm lm ( ) + & lm ( )]& lm, r! Interstitil (*,), r! MT sphere v
Clcultion Prmeters E. Wimmer, H. Krkuer, M. Weinert, nd. J. Freemn, Phys. Rev. 28, 864 (1981) nd references therein. M. Weinert, E. Wimmer, nd. J. Freemn, Phys. Rev. 26, 4571 (1982). Perdew, J. P., urke, K., Ernzerhof, M., Phys. Rev. Let. 77,3865 (1996) ll-electron Full-potentil LPW method Generlized Grdient pproximtion for Exchnge-Correltion Potentil Plne-Wve Cutoff : 21 Ry Str-Function Cutoff : 340 Ry k-points : 88 Fe 2.4 C, 365 (Fe 11 M)C 5 Muffin-tin Sphere : Fe, Si, l, Mn (2.04.u.), C (1.30.u.) Mixing Method : royden
Epsilon Crbide C 2 C 3 h C 2 C 3 Formul Unit Structure Spce group c c/ Fe 2.4 C(Fe 2 C~Fe 3 C) hexgonl P6 3 22 or P6 3 /mmc 4.767 Å 4.354 Å 0.913 C 3 C 2 S. Nkgur, J. Phys. Soc. Jpn, 14 (1959) 186.
Fe 3 C nd Fe 2 C (ε-crbide( -crbide) 4.661 Å ( 2.3%) 4.785 Å (+0.3%) c 4.294 Å ( 1.4%) c 4.321 Å ( 0.8%) c/ 0.9213 c/ 0.903 x 0.3190 x 0.3302 # E = E( Fe C2) " 6! E(Fe) " 2! E(C) 8 6 = 5.09kJ/mol # E = E Fe C ) " 6! E(Fe) " 3! E(C) 9 ( 6 3 = 7.00kJ/mol
Fe 2.4 C (ε-crbide( -crbide) Fe 2.4 c 4.740 Å ( 0.6%) 8.631 Å ( 0.9%) # E = E( Fe C5 ) " 12! E(Fe) " 5! E(C) 17 12 = 6.24kJ/mol
Results 0.0-0.5-1.0-1.5-2.0-2.5 Lttice Chnge(%) 0.5 Fe3C Fe2.4C Fe2C c System (Å) c (Å) c h / h Mesured, ε 4.767 8.708 1.582 Fe 3 C 4.661( 2.3%) 8.588 ( 1.4%) 1.596 Fe 2.4 C 4.740( 0.6%) 8.631( 0.9%) 1.577 Fe 2 C 4.785(+0.3%) 8.642( 0.8%) 1.564
Si,, l nd Mn Substitution Si l Mn 4.7303Å ( 0.2%) 4.742Å (+0.0%) 4.738 Å ( 0.1%) c 8.5901 Å ( 0.5%) c 8.685 Å (+0.6%) c 8.664 Å (+0.4%)!E = 9.08kJ/mol!E = 4.98kJ/mol!E = 4.40kJ/mol
Results Lttice Chnge(%) 0.6 0.4 0.2 0.0-0.2-0.4 pure Si l Mn c System (Å) c (Å) c h / h Mesured, ε 4.767 8.708 1.582 Clculted, ε 4.740 8.631 1.577 Si substituted 4.730( 0.2%) 8.590( 0.5%) 1.573 l substituted 4.742(+0.0%) 8.685(+0.6%) 1.586 Mn substituted 4.738( 0.1%) 8.664(+0.4%) 1.584
Results Formtion Energy (kj/mol) 10 8 6 4 2 0 epsilon cementite pure Si l Mn E (kj/mol) pure-crbide Si substituted l substituted Mn substituted ε-crbide 6.24 9.08(+2.84) 4.98( 1.26) 4.40( 1.84) cementite 5.38 7.70(+2.32) 4.53( 0.85) 5.07( 0.31)
Orienttion Reltionship H. K. D. H. hdeshi, inite (1992) (101) α (1011) ε (211) α (1010) ε [011] α [0001] ε (111) α (1210) ε C 2 C 3 C 2 C 3 C 3 C 2 C [1210] 1 ε h 4. 7% Mrtensite with 1 wt% C = 2.85 Å c = 2.98 Å [111] α (0001) ε 9.2% (011) α
Summry First Principles Clcultion cn be pplied for hypotheticl crystl structure. Si ddition increses the formtion energy of θ nd ε- crbide. The formtion energy clcultion : ε-crbide θ The role of silicon in trnsition of crbide : Reducing the misfit Mngnese ddition : stble ε-crbide
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Equilibrium Clcultion µ pr with silicon µ equi without silicon Crbon θ, ε-crbide
ppendix - DFT Hohenberg-Kohn Theorem : The ground stte property is functionl of electron density. E [ n] = " V ( r)! n( r) dr + < # T + Vee # >! E! n [ n] ext = V [ n]! EGS, E[ ngs] EGS E = Kohn-Shm Eqution : Introducing the non-intercting fictitious prticle. E 1 = " ext C 2 " r [ n] V ( r)! n( r) dr + T[ n] + V ( r)! n( r) d + Exc[ n] ext ( r)! T +! n [ n] + V C ( r)! Exc +! n [ n] = µ! E! n [ n]! T[ n] = + v eff r! n ( ) = µ ( 1 2 % & * ) + v #! = " eff i i! i ' 2 $ ( r) ( r) ( r) v! ( ) V ( ) VC( ) xc eff r = ext r + r + n N ( r ) = " ( r)! i= 1 i 2 E! n [ n]
First-Principles Clcultion Quntum Mechnics tomic Position Schrödinger Eqution Electron Density Physicl Principles Structure Moleculr Crystl Density Defect Surfce Interfce Thermodynmic Internl Energy Entropy Free Energy Formtion Energy Phse Trnsform Phse digrms Mechnicl - Compressibility Elstic constnts ulk modulus Vibrtions Etc. Electron Ioniztion Energy bnd nd gps Mgnetic
Formtion Energy Formtion energy (kj/mol) 300 250 200 150 100 50 0-50 G xs =ΩX(1 X) (Fe,Si)3C (Fe,l)3C (Fe,Mn)3C 0 10 20 30 40 50 60 70 80 Concentrtion (t %)
Ternry Phse Digrm t 773K Equilibrium Mn Si l 0.2 0.2 0.2 θ+γ θ+γ θ+γ Fe 0.8 Fe 0.8 Fe 0.8 Pr-equilibrium 0.2 Mn θ+γ 0.2 Si γ θ+γ 0.2 l θ+γ Fe 0.8 Fe 0.8 Fe 0.8
Equilibrium Phse Digrm t 723K γ-θ-ε γ-θ-ε γ γ2+ ε γ2+ ε γ+ ε γ+ ε γ γ+ θ+ ε γ+ θ θ+ ε γ+γ2+ ε γ+ θ γ+ θ+ ε
Gibbs Free Energy of Crbide H(ε) H(θ) G(ε) G(θ) G(θ) G(ε) H(θ) H(ε) Fe-C System Mn-C System
Gibbs Free Energy of l-crbide H(ε) H(θ) G(θ) G(ε) H(θ) H(ε) G(ε) G(θ) Si-C System l-c System
Density Functionl Theory full-reltivistic sclr-reltivistic non-reltivisitic 2 [# 1?? + V ( r) + V ( r) + V ( r)]ø ( r) = Ø ( r ) 2 non-periodic periodic symmetrized rel-spce ll electron, full-potentil ll electron, sphericl potentil pseudopotentil (vlence electron s) jellium pproximtion ext C xc! " i! locl density pprox. (LD) generlized grident pprox. (GG) non-spinpolrized spinpolrized, vector-spin density LD+U, OEP hybrid functionls current functionls rel-spce grid plne-wves (PW) non-liner methods - PW: ugmented method - KKR-GF linerized methods - LPW - SW - LMTO liner combintion of t omic orbitls (LCO) - tight-binding - Guss-O - Slter type-o - numericl O lugel, S. ihlmyer, G., Computtionl nnoscience: Do it yourself, 31:85-129 (2006)