Crystal Structure and Formation Energy of ε-carbide Using First Principles Calculations
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1 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
2 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 (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
3 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
4 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
5 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 P or P6 3 /mmc Å Å C 3 C 2 S. Nkgur, J. Phys. Soc. Jpn, 14 (1959) 186.
6 Fe 3 C nd Fe 2 C (ε-crbide( -crbide) Å ( 2.3%) Å (+0.3%) c Å ( 1.4%) c Å ( 0.8%) c/ c/ x x # 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
7 Fe 2.4 C (ε-crbide( -crbide) Fe 2.4 c Å ( 0.6%) Å ( 0.9%) # E = E( Fe C5 ) " 12! E(Fe) " 5! E(C) = 6.24kJ/mol
8 Results Lttice Chnge(%) 0.5 Fe3C Fe2.4C Fe2C c System (Å) c (Å) c h / h Mesured, ε Fe 3 C 4.661( 2.3%) ( 1.4%) Fe 2.4 C 4.740( 0.6%) 8.631( 0.9%) Fe 2 C 4.785(+0.3%) 8.642( 0.8%) 1.564
9 Si,, l nd Mn Substitution Si l Mn Å ( 0.2%) 4.742Å (+0.0%) Å ( 0.1%) c Å ( 0.5%) c Å (+0.6%) c Å (+0.4%)!E = 9.08kJ/mol!E = 4.98kJ/mol!E = 4.40kJ/mol
10 Results Lttice Chnge(%) pure Si l Mn c System (Å) c (Å) c h / h Mesured, ε Clculted, ε Si substituted 4.730( 0.2%) 8.590( 0.5%) l substituted 4.742(+0.0%) 8.685(+0.6%) Mn substituted 4.738( 0.1%) 8.664(+0.4%) 1.584
11 Results Formtion Energy (kj/mol) epsilon cementite pure Si l Mn E (kj/mol) pure-crbide Si substituted l substituted Mn substituted ε-crbide (+2.84) 4.98( 1.26) 4.40( 1.84) cementite (+2.32) 4.53( 0.85) 5.07( 0.31)
12 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
13 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
14 Thnk You!!
15 Equilibrium Clcultion µ pr with silicon µ equi without silicon Crbon θ, ε-crbide
16 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]
17 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
18 Formtion Energy Formtion energy (kj/mol) G xs =ΩX(1 X) (Fe,Si)3C (Fe,l)3C (Fe,Mn)3C Concentrtion (t %)
19 Ternry Phse Digrm t 773K Equilibrium Mn Si l 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
20 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
21 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
22 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
23 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: (2006)
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