Ingo Steinbach. Thermal dendrites: Ryo Kobayashi transformation with thermo-mechanical interaction. Grain boundary structure

Transcription

1 Thermal dendrtes: Ryo Kobayash 994 Phase-feld smulaton of sold state transformaton wth thermo-mechancal nteracton Ingo Stenbach f (, T ) L T T c p γ precptates n N-base superalloys: Y. Wang, L.Q. Chen, A.G. Khachaturyan 995 Gran boundary structure TEM Pcture <0> tlt gran boundary n ZrO Y. Ikuhara, N. Shbata, T. Watanabe, F. Oba, T. Yamamoto and T. Sakuma, Ann. Chm. Sc. Mat., 00,7 Suppl :S- 30 nm nm nm N-base super alloy: allgned cubods n crystallograpc drectons 3 Gran boundares n ferrte 4

2 Curvature drven flow Phenomenologcal pcture v Curvature drven flow Mechanstc pcture: The atoms at gran boundares jump between lattce stes of the adjacent crystals How does an atom close on one sde of a gran boundary know ts mean curvature? Characterstc functon Ф ndcates the belongng of one atom to one crystal respectvely The belongng to one crystal s favourable G. Gottsten, Physcal Foundaton of Materals Scence, Sprnger ( )(0.5 ) The balance between both actons leads to a stable nterface 5 6 Outlne Dffuse nterface wth steep concentraton profle Bascs of phase-feld models Elastc nteractons Examples: Pearlte Stran stablzed precptaton Concentraton profle through the nterface 7 8

3 Phase-feld model n the Gbbs-Thomson lmt The phase-feld functonal Order parameter, the Phase-Feld varable 3 F d x m f0 3 d x 7 Gc,c G ~ c c c F 0 v Cagnalp et Ffe 986, Wheeler et al. 99 G σ: nterface energy μ: nterface moblty η: nterface thckness G αβ : Gbbs energy dfference : dffuson ~ potental 9 "A functonal s a unequvocal mappng of a space of functons (x) on to real numbers" 0 F U - TS t 0 Background of phase-feld": The common model phlosophy Thermodynamc prncple Materal specfc Model of the Free energy functonal Mathematcal rgorous dervaton of the "thn nterface lmt" Numercal sound soluton F ~ F dvf,c,t v G j 8 x µ: nterface moblty; : nterface energy; : nterface thckness : nterface curvature; G: thermodynamc undercoolng; x: numercal dscretzaton Mult-phase-feld model coupled to solute dffuson (n,n ) F d K(,,, ), n F F m m n k F k k c M D c k k c k Stenbach MSMSE 009 f c 3

4 Free boundary problem of crystallzaton: the Stephan problem Outlne.) +) ) 3) 3 ) temperatur c p T T L 0 energy ballance L0 v lq T lq sol T sol Gbbs-Thomson equaton v T eq T ntf ( ) (T T ) eq nt f sold lqud v : velocty of the nterface : curvature : heat conductvty c p : specfc heat L 0 : latent heat : Gbbs-Thomson coeffzent Bascs of phase-feld models Elastc nteractons Examples: Pearlte Stran stablzed precptaton 3 4 Motvaton I: Phase dstrbuton n sold state Motvaton II: Processng of multphase steels Pearlte, courtesy Benteler N-base super alloy Hot Rollng Controlled Transformaton Ferrte Temperature M s Bante TRIP steel Bante NT precptates, courtesy Eggeler Tme 5 6 4

5 Mechansms of Mcrostructure Formaton mcrostructure formaton Homogenzaton of stress/stran n the nterface regon thermodynamcs (energy scale) mechancal equlbrum (energy scale) = transport / dffuson (tme scale) nterfaces (length scale) f el * * C dscontnuty of stran or dscontnuty of stress σ σ WBM 994: c L = c S = c Mult-phase-feld: c = L c L + S c S Taden 969 c L = kc S TQ nterface from TCS stress/stran solver dffuson solver phase feld method Km 999 µ L (c L ) = µ S (c S ) OpenPhase 7 8 Reuss lmt of contnuous stress f el * * C! el contnuty of stress between the phases dscontnuty of stran! f * ˆ * f el C Reuss lmt of contnuous stress f el * * Ĉ el el * * f G ˆ C C C C * ˆ * ĈC * * Ĉ * * ; C ; C Hooks matrx s a homogenous mxture between phases el G 9 0 5

6 Verfcaton: growth of a sngle partcle n a matrx Test aganst analytcal soluton (Eshelby) Colour codng: б yy Elastcty modul: C = 80GPa, C = 0GPa, Egenstran = % Thermodynamc data of Fe-C 0.463at% -Mn 0.496at%, TCFe3 database Isothermal transformaton: Growth of a sngle ferrte precptate wthout stress Evoluton of the drvng forces durng transformaton rbuton carbon dstr wth stress 090K 085 K 3 4 6

7 Dependence of equlbrum fracton on nternal stress / Transformaton Phase dstrbuton van Mses stress ncludng stress no stress [MPa ] 5 6 Outlne Pearlte Bascs of phase-feld models Elastc nteractons Examples: Pearlte Stran stablzed precptaton Temper rature(k) Eutectod steel Fe-C system Fe 3 C ~50K C (wt%) 8 7

8 Pearlte Pearlte The "Zener-Hllert-Tller-Jackson-Hunt Model Pearltc: T= const. Eutectc: v= const. v (a.u.) large ΔT medum ΔT small ΔT ΔT = T eq. -T nt. (a.u.) large V medum V small V Phase-feld calculaton of dffuson controlled growth of pearlte wth dfferent spacngs (Nakajma et al. 006) spacng λ(a.u.) a v = T/λ -a /λ spacng λ(a.u.) T = a vλ + a /λ 30 Transformaton stran and concentraton dependent stran Temperature (K) Lattce constants: austente a =.30 Å/atom ferrte a =.98 Å/atom cementte a =.98 Å/atom; b =.3 Å/atom; c =.953 Å/atom 000 Experment Rdley 0.8C 980 Frye 0.78C Model: Dffuson n phase nodule velocty (mm/s) 3 Phase-feld model coupled to elastc stran f f f ch f * * ( x, t) c ( x, t) C (..) c f el ( x, t) ( x, t) (..) n c f f n f M c m k I f M c c 0 f * C c el G chem G el 3 8

9 The effect of transformaton stran on growth The effect of transformaton stran on growth Phase dstrbuton Hydrostatc stress Carbon concentraton The effect of transformaton stran on growth The effect of transformaton stran on growth Experment Rdley 0.8C Frye 0.78C Temperature (K) Dffuson n Dffuson n + Phase Feld Calculaton Dffuson n phase Dffuson n and phase + Dffuson due to Stress (extrapolated) + transformaton stran staggered growth mechansm of pearlte quenched pearlte front (Darken,Fsher 960) nodule velocty (mm/s) + faceted ansotropy of cementte Model Dffuson n phase

10 Mats Hllert IS John Ågren Outlne Bascs of phase-feld models Elastc nteractons Examples: Pearlte Stran stablzed precptaton Stokholm Precptaton n N-T shape memory alloys: expermental observatons Martenste Start Temperature Experment: Yang, Schryvers, Intern. Journ. Of Appl. Electromagn. And Mech. 3 (006) p Otsuka, Ren Progress n Materal Scence 50 (005), p

11 N4-T3 precptate n NT N4-T3 precptate n NT 4 4 Intal condtons The phase dagram N 4 T 3 precptates n the N oversaturated B matrx C B,0 = 5.0 at.% C = 57. at.% N 4T3 Dffuson controlled precptate growth Dffuson only n the matrx, no dffuson n the stochometrc phase Lattce msmatch Ansotropc egenstrans: 43 44

12 Results: Stran n the matrx Smulaton: Guo et al., Acta Mater. 59 (0) p. 387 Results concentraton n the mattrx Smulaton: Stran (zz) Stran (zz) Experment: Trry, Schryvers; Nature Mater 8 (009) 75 Phase feld N Concentraton Concentraton Dependent Elastc Coeffcents Elastc coeffcents: Pure elements and the alloy Larche, Cahn 98: The effect of self-stress on dffuson n solds c M k f c M M k k f c f c chem chem f c mech c f mech 47 48

13 Elastc Propertes of B N-T wth Defects Perform full relaxaton of B NT supercells wth vacances and substtutons to compare changes n elastc energy Obtan DFT elastc propertes as a functon of N-concentraton Use DFT elastc propertes and trends as nputs for phase-feld (PF) models ) Obtan DFT C j as functon of Nconcentraton ) Import concentraton dependent C j nto PF-model 3) Incorporate nto PF mechancal energy term N vacancy Concentraton Dependent Elastc Coeffcents Lnear couplng between the elastc coeffcents and the N concentraton: Elastc contrbuton to the fluxes: = The role of hydrostatc stress Smulatons wth concentraton dependent elastcty (K=e-3) Stran (zz) = 0 N Concentraton 5 5 3

14 Growth knetcs Growth knetcs Concentraton profle Autocatalytc nucleaton: drvng force for nucleaton chemcal drvng force elastc drvng force chemcal + elastc Favourable poston for new nucleaton

15 Autocatalytc nucleaton Concluson Sold state transformatons cannot be well understood wthout the consderaton of stress-stran effects. The phase-feld method provdes a thermodynamc consstent framework for the nvestgaton of phase transformatons subject to elastc effects. Examples were presented whch demonstrate the predctve capablty of the approach. Experments Eggeler, Bochum Precptaton of γ n a N-Al alloy 59 5