Diffusion Transformations in solid (a) Precipitation ' Homogeneous Nucleation Heterogeneous Nucleation G V G A VG N hom 0 V Gm G* C exp exp kt kt (b) Eutectoid Transformation S G V ( G G ) A G het V S d 적합한위치는격자결함 ( 핵생성이격자결함제거역할 ) Composition of product phases differs from that of a parent phase. long-range diffusion 1
5. Diffusion Transformations in solid (c) Order-Disorder Transformation ' (d) Massive Transformation : 조성변화없이결정구조가다른단상또는다상으로분해 (e) Polymorphic Transformation : 온도범위에따라서로다른결정구조가안정
Homogeneous Nucleation in Solids Free Energy Change Associated with the Nucleation Negative and Positive Contributions to G? VG 1) Volume Free Energy : V ) Interface Energy : A 3) Misfit Strain Energy : VGS G V G A VG V S for spherical nucleation 4 3 G r ( GV GS) 4 r 3 Plot of G vs r? r* =? G* =? 3
Homogeneous Nucleation in Solids r * ( G G ) V S G* 3 16 3( G G ) V S : driving force for nucleation Fig. 5. The variation of ΔG with r for a homogeneous nucleus. There is a activation energy barrier ΔG*. 4
Homogeneous Nucleation in Solids Concentration of Critical Size Nuclei : 단위체적당임계크기를갖는핵의수 C* C exp( G * / kt ) 0 C 0 : number of atoms per unit volume in the phase Nucleation Rate : 각각의핵이단위시간당 f 의빈도로임계크기값보다커진다면, N hom f C * f exp G kt : f 는임계핵이얼마나빈번히모상 α 로부터원자를공급받는가에따라변하는함수 vibration frequency, area of critical nucleus G : activation energy for atomic migration m m N Gm G* C exp exp kt kt hom 0 5
N Gm G* C exp exp kt kt hom 0 : T 에민감하게변함 G* 3 16 3( G G ) V S G* G V ( 화학적구동력 ) 1) For X 0, solution treatment at T 1 Liquid Solid ) For X 0, quenching down to T α' α + β : B 성분이과포화되어 β 석출 6
Total Free Energy Decrease per Mole of Nuclei Driving Force for Precipitate Nucleation G 0 : 변태를위한전체구동력 / 핵생성을위한구동력은아님 G 1 A X A B X B G : 핵의조성 (X Bβ ) 을갖는작은양이제거될때단위몰당자유 E 변화 (P point) A X A : β 상생성시단위몰당자유 E 변화 (Q point) B X B G n G G 1 G V G V m n per unit volume of : driving force for nucleation For dilute solutions, G X where X X X V G V X T : driving force for nucleation 0 e 7
Rate of Homogeneous Nucleation Varies with Undercooling 실제의평형온도 ΔGs 에의해감소 G V X T G* 3 16 3( G G ) V S 임계핵의증가확률 N 구동력 ΔGv 너무작으므로 N Gm G* C exp exp kt kt hom 0 원자이동도 : T 시 확산이너무느리기때문에 8 N
The Effect of ΔT on G* het & G* hom? Plot G* het & G* hom vs ΔT and N vs ΔT. ΔT hom c 임계과냉도 N hom 1 cm -3 1 s Fig. 4.9 (a) Variation of G* with undercooling ( T) for homogeneous and heterogeneous nucleation. (b) The corresponding nucleation rates assuming the same critical value of G* 9
Rate of Homogeneous Nucleation Varies with Undercooling 실제의평형온도 ΔGs 에의해감소 G V X T G* 3 16 3( G G ) V S 임계핵의증가확률 N 구동력 ΔGv 너무작으므로 N Gm G* C exp exp kt kt hom 0 원자이동도 : T 시 확산이너무느리기때문에 10 N
Homogeneous Nucleation in Solids The Effect of Alloy Composition on the Nucleation Rate Compare the two plots of T vs N(1) and T vs N() Dilute 해질수록최적석출온도의감소 Fig. 5.5 The effect of alloy composition on the nucleation rate. The nucleation rate in alloy is always less than in alloy 1. 11
Heterogeneous Nucleation in Solids Non-equil. Defect 가 nucleation site 로작용 : disl., gb, excessive vacancies 대부분의핵생성이해당, 적합한위치는격자결함 ( 핵생성이격자결함제거역할 ) G V ( G G ) A G het V S d Nucleation on Grain Boundaries 가정 : G S (misfit strain energy)= 0, 즉 incoherent! 입계에서핵생성이일어날때임계핵의크기 (V*) 핵의모양전체계면자유에너지를최소로하는상태 cos / G V G A A V Mold wall 에서불균일핵생성에의한응고와유사 구형모자형태핵의임계반경 r* / GV 불균일핵생성에필요한활성화에너지장벽 G* het V * G* V * hom het hom S( ) 1 S( ) ( cos )(1 cos ) 1
Barrier of Heterogeneous Nucleation Mold wall nucleation 의경우 G V G A A A het S v SL SL SM SM SM ML G * 16 3G 3 SL V 16 S ( ) 3G 3 SL V ( 3cos cos 4 3 ) G* het V * het S( ) G* hom V * Shape factor S(θ) 와의관계설명은? hom S(θ) has a numerical value 1 dependent only on θ (the shape of the nucleus) G S( ) G * * het hom 3 SL 16 r * and G * G 3G V SL V S ( ) V A 3 * * 3cos cos Gsub Ghomo 4 V B VA V V A B 3cos cos 4 3 S( ) 13
Heterogeneous Nucleation in Solids Rate of Heterogeneous Nucleation Decreasing order of G* (Activation Energy Barrier for nucleation) 1) homogeneous sites ) vacancies 3) dislocations 4) stacking faults 5) grain boundaries and interphase boundaries 6) free surfaces : 아래로갈수록핵생성이빨리일어난다. but 변태총속도에상대적중요성은각위치의상대적인양 (C 1 ) 도고려해야한다. Gm G* Nhet C1 exp exp nuclei m s kt kt 3 1 C 1 : concentration of heterogeneous nucleation sites per unit volume Gm G* Nhom C0 exp exp kt kt 14 : 단위체적당임계크기를갖는핵의수
Heterogeneous Nucleation in Solids The Rate of Heterogeneous Nucleation during Precipitation : 매우작은구동력에서도핵생성발생 균일핵생성과불균일핵생성의상대적인크기 Nhet C G * G * N C kt 1 hom het exp hom 0 > > 1 C 1 /C 0 for GB nucleation? C C 1 0 ( GB thickness) D( grain size) For D = 50 m, = 0.5 nm C C 1 0 D 10 5 15
Precipitate Growth 정합, 반정합평면계면부정합곡면초기석출물모양계면자유에너지를최소로하는모양 석출물성장 계면의이동 : 성장하는동안석출물모양각계면의 상대적이동속도에의해좌우됨. If the nucleus consists of semi-coherent and incoherent interfaces, what would be the growth shape? 다른결정구조 Ledge mechanism 얇은평판이나원판 (disc) Origin of the Widmanstätten morphology 16
Overall Transformation Kinetics TTT Diagram The fraction of Transformation as a function of Time and Temperature f (t,t) Plot f vs log t. - isothermal transformation - ( 등온열처리시전이된혹은형성된상의분율 ) - f :~ β 의체적분율, 0~1 Plot the fraction of transformation (1%, 99%) in T-log t coordinate. Fig. 5.3 The percentage transformation versus time for different transformation temperatures. 17
Overall Transformation Kinetics TTT Diagram Three Transformation Types α β or α β+γ Fig. 5.4 (a) Nucleation at a constant rate during the whole transformation. (b) Site saturation all nucleation occurs at the beginning of transformation. (c) A cellular transformation. (a) continuous nucleation 급냉시많은불균일핵생성처존재 Nucleation rate: constant Wide range of size distribution f depends on the nucleation rate and the growth rate. (b) all nuclei present at t = 0 Nucleation rate: rapid depletion ~ Same size f depends on the number of nucleation sites and the growth rate. (c) All of the parent phase is consumed by the transformation product. 일정한속도로성장, 생성상이서로만나충돌 성장속도감소가아님 pearlite, cellular ppt, massive transformation, recrystallization 18
Overall Transformation Kinetics TTT Diagram Johnson-Mehl-Avrami Equation : 변태속도비교 Assumption: - reaction produces by N + G - nucleation rate constant: N - growth as a sphere at constant rate: v - reaction product grows radially until impingement define volume fraction transformed f Vol. of new phase Vol. of specimen t = 0 에형성된핵성장크기 t = τ 에형성된핵성장크기 Number of nuclei at d = N d / unit vol. 19
Overall Transformation Kinetics TTT Diagram f f J-M-A Eq. f f 0 x dfˆ 3 4 3 N v 3 N v t 4 3 1 exp N v t 3 4 1 3 t 0 ( t ) 3 d do not consider impingement & repeated nucleation only true for f 1 임의로배열된생성상의충돌효과를고려하면, 1 exp( z) Z ( z 1) exp ( k t k : sensitive to temp. (N, v) n : 1 ~ 4 Diffusion controlled, sphere: n=3/ Interface controlled, sphere: n=3 i.e. 50% transform Exp (-0.7) = 0.5 kt n ) n 0.5 0.7 시간이지날수록충돌에의한변태속도감소 0.7 k Example above. 0.9 N v t 0.5 1/ n 0.5 1/ 4 3/ 4 0 K가클때, 즉핵생성속도와성장속도가클때변태속도가빠름 t
< 고체에서나타나는여러종류의개별변태 > Precipitation in Age-Hardening Alloys Precipitation in Aluminum-Copper Alloys Al-4 wt%cu (1.7 at %) 0 Quenching + Isothermal 1 +GP zones + 3 + 4 + CuAl ) quenching isothermal Fig. 5.5 Al-Cu phase diagram showing the metastable GP zone. θ and θ solvuses. 1
GP Zones The zones minimize their strain energy by choosing a discshape perpendicular to the elastically soft <100> directions in the fcc matrix. 두께 : 1~ 개의원자층, 지름은대략 5 개의원자직경거리 정합상태의 Cu 농축지역 Section through a GP zone parallel to the (00) plane. : 이러한응집체는완전한석출입자로볼수없으며, 때때로석출대 (zone) 로명명함.
GP zones of Al-Cu alloys x 70,000 Fully coherent, about atomic layers thick and 10 nm in diameter with a spacing of ~ 10 nm : 현미경관찰불가, 사진에나타난명암은 GP 대에수직인방향으로의정합불일치변형때문 3
Transition phases 0 1 +GP zone + 3 + 4 + CuAl ) 같은결정구조 Fig. 5.7 A schematic molar free energy diagram for the Al-Cu system. 4
0 1 +GP zone + 3 + 4 + CuAl ) Low Activation Energy of Transition Phases 천이상의결정구조가모상과평형상의중간상태를갖기때문 천이상 : 정합정도가높고 ΔG* 에기여하는계면에너지는작은값가짐. 평형상 : 기지와격자를잘일치시킬수없는복잡한결정구조 = 고계면 E 5
The Crystal Structures of, and Cu 와 Al 원자가 (001) 면에규칙적으로놓여있는변형된 fcc 구조 CuAl 조성, 복잡한 bct 구조 Fig. 5.9 Structure and morphology of θ, θ and θ in Al-Cu ( Al. Cu). 6
of Al-Cu alloys x 63,000 Tetragonal unit cell, essentially a distorted fcc in which Cu and Al atoms are ordered on (001) planes, fully-coherent plate-like ppt with {001} habit plane. ~ 10 nm thick and 100 nm in diameter. : 석출판상에수직인방향으로의정합불일치변형때문에관찰가능 7
of Al-Cu alloys x 18,000 has (001) planes that are identical with {001} and forms as plates on {001} with the same orientation relationship as. 그러나 (100), (010) incoherent, ~ 1 m in diameter. 8 : 판이성장함에따라판의모서리는부정합이거나복잡한반정합구조의계면을갖게됨.
of Al-Cu alloys x 8,000 CuAl : complex body centered tetragonal, incoherent or complex semicoherent 9
Nucleation sites in Al-Cu alloys Fig. 5.31 Electron micrographs showing nucleation sites in Al-Cu alloys. (a) θ θ. Θ nucleates at dislocation (x 70,000). (b) θ nucleation on grain boundary (GB) (x 56,000) (c) θ θ. Θ nucleates at θ /matrix interface (x 70,000). 30
The Effect of Ageing Temperature on the Sequence of Precipitates 석출순서에따라미세조직조대화됨. 더안정한석출물이덜안정한석출물을소비하면서성장해나가는기구 Fig. 5.3 (a) Metastable solvus lines in Al-Cu (schematic). (b) Time for start of precipitation at different temperatures for alloy X in (a). 31
Age Hardening 중간상형성시커다란격자변형수반하고, 소성변형시전위의이동을방해함 Hardness vs. Time by Ageing 적정시효시간. Ageing at 130 o C produces higher maximum hardness than ageing at 190 o C. Overaging : 석출물간간격증대로경도감소 At 130 o C, however, it takes too a long time. Fig. 5.37 Hardness vs. time for various Al-Cu alloys at (a) 130 (b) 190 How can you get the high hardness for the relatively short ageing time? Double ageing treatment first below the GP zone solvus fine dispersion of GP zones then ageing at higher T. 미세한석출물의분포얻음 3
Spinodal Decomposition Spinodal mode of transformation has no barrier to nucleation How does it differ between inside and outside the inflection point of Gibbs free energy curve? 1) Within the spinodal dg 0 dx : 조성의작은요동에의해상분리 /up-hill diffusion ) If the alloy lies outside the spinodal, small variation in composition leads to an increase in free energy and the alloy is therefore metastable. Fig. 5.38 Alloys between the spinodal points are unstable and can decompose into two coherent phasees α 1 and α without overcoming an activation energy barrier. Alloys between the coherent miscibility gaps and the spinodal are metastable and can decompose only after nucleation of the other phase. The free energy can only be decreased if nuclei are formed with a composition very different from the matrix. nucleation and growth : down-hill diffusion 33
Spinodal Decomposition up-hill diffusion 1) Composition fluctuations within the spinodal ) Normal down-hill diffusion outside the spinodal down-hill diffusion interdiffusion coefficient D<0 34
5.5.5 Spinodal Decomposition * The Rate of Spinodal decomposition Rate controlled by interdiffusion coefficient D within the spinodal D < 0 & composition fluctuation exp( t / ) /4 D : characteristic time constant 1 차원으로가정했을때, : wavelength of the composition modulations Kinetics depends on. Transf. rate as λ 최소의 λ 값이존재하며, 그값이하에서는스피노달분해가일어나지않는다. 35
Spinodal Decomposition * Free Energy change due to decomposition 1) Decomposition of X 0 into X 0 + X and X 0 - X What would be an additional energy affecting spinodal decomposition? 실제로일어나는조성요동의파장을계산하려면, ) Diffuse interface: interfacial energy 3) Atomic size difference: coherency strain energy 1 dg 1) Decomposition of X 0 into X 0 + X and X 0 - X Gchem X dx f (a) f (a h) f (a) f (a)h h! G ( X0 ) G( X0 X ) G( X0 ) G( X0) X X! G ( X0 ) G( X0 X ) G( X0 ) G( X0 ) X X! G( X0 X ) G( X0 X ) Gchem G( X0 ) G( X0 ) 1 dg 36 X X! dx
Spinodal Decomposition ) During the early stages, the interface between A-rich and B-rich region is not sharp but very diffuse. diffuse interface Interfacial Energy (gradient energy) 이종원자사이의결합수가증가하기때문에생성 X G K K: 동종혹은이종 atomic pair 들의 bond energy 에비례하는비례상수 고용체를구성하는원자의크기가서로다르면, 조성차이 ΔX~ 정합변형에너지 ΔGs 를유발 3) Coherency Strain Energy GS E ( da / dx ) / (atomic size difference) δ: misfit, E: Young s modulus, a: lattice parameter : fractional change in lattice parameter per unit composition change X a 1 da GS ( X ) EV m, where, E E /(1 ) a dx ΔGs 는 λ 에무관 조성의요동으로생긴전체자유 E 의변화 1) + ) + 3) d G K ( X ) G dx EV m 37
Spinodal Decomposition d G K ( X ) G dx EV m <0 이어야 spinodal decomposition! * Condition for Spinodal Decomposition 균질한고용체가불안정하게되어스피노달분해를할수있는조건 d G K dx EVm * The Limit for the decomposition 스피노달분해가가능한온도와조성의한계값 Wavelength for coherent spinodal dg EV m dx coherent spinodal (next page) Chemical spinodal 의안쪽에존재 dg EV m dx K The minimum possible wavelength 가존재 : decreases with increasing undercooling below the coherent spinodal. 38
Spinodal Decomposition Coherent Miscibility Gap This is the line defining the equilibrium compositions of the coherent phases that result from spinodal decomposition. 상태도로부터계산된 curve 39 변형에너지효과를극복하기위해서는매우큰과냉도필요
Spinodal Decomposition
전체계면의 G 값이최소가아닐때는 상합금의미세조직은열역학적으로불안정 Particle Coarsening 작은크기의입자는사라지고입자수는감소 Two Adjacent Spherical Precipitates with Different Diameters를고려 < (Gibbs-Thomson effect) 체적확산이속도제어인자라면, 3 3 r r kt where 0 k D X (X e : 매우큰입자와의평형용해도 ) e < dr dt k r - D and X e ~ exp (-Q/RT) - (Ostwald Ripening) 41
Particle Coarsening The Rate of Coarsening with Increasing Time and Temp. 1) low heat-resistant Nimonic alloys based on Ni-Cr ordered fcc Ni 3 (Ti,Al) (=) Ni/ 의계면완전정합관계 (10 ~ 30 mj m - ) 고온에서도미세한조직유지가능 creep 특성향상 ) low X e fine oxide dispersion ( 분산강화 ) ThO (thoria) in W and Ni 금속에서산화물의낮은용해도에기인 ~ 고온재료의기계적특성과밀접한관계 조대화가일어나면강도감소및기계적성질이저하됨. How can you design an alloy with high strength at high T? hint) dr k k D Xe dt r 3) low D Cementite high D of carbon 매우빠르게조대화 a. substitutional alloying element segregated in the carbide b. strong carbide-forming elements (also low X e ) 4
Precipitation of Ferrite from Austenite (Grain boundary and the surface of inclusions) The Iron-Carbon Phase Diagram Microstructure (0.4%C) evolved by slow cooling (air, furnace)? 43
Precipitation of Ferrite from Austenite Diffusional Transformation of Austenite into Ferrite Fe-0.15%C After being austenitized, held at (a) 800 o C for 150 s (b) 750 o C for 40 s (c) 650 o C for 9 s (d) 550 o C for s and then quenched to room T. What would be the microstructures? 44
Precipitation of Ferrite from Austenite Microstructures of an Fe-0.15%C alloy austenitized (a) 800 o C for 150 s 초석 Ferrite 가판상으로성장 판상 Ferrite 수증가 / 주로입계로부터성장 (c) 650 o C for 9 s 입계타형 GB allotriomorphs (b) 750 o C for 40 s (d) 550 o C for s White area γ α, Gray area: γ M Widmanstätten side plates (b), (c), (d) 과냉도가클수록더미세하게됨 45
Precipitation of Ferrite from Austenite Undercooling 이작으면 GB allotriomorph 형태 Undercooling 이크면 Widmanstatten 형태
Cellular precipitation Austenite to ferrite trans: g.b. allotriomorph or Widmanstätten structure GB precipitation 은항상위와같이일어나지않는다. 그예로 cellular precipitation: g.b. 가 ppt. 의 growing tip 과같이이동 - Very similar to eutectoid transformation - 그러나, α (supersaturated matrix) α + β - 예 : Mg-9wt%Al, β (Mg 17 Al 1 ) 47
Cellular precipitation Growth of cellular ppt -Partitioning of solute to the tips of ppt in contact with the advancing g.b.: 아래의두가지방법으로! -1) diff. through the lattice ahead of the advancing cell ) diff. in the moving boundary matrix 의조성이 cell front 의바로전까지변하지않고있어야가능 일반적인 ppt 는 continuous ppt. Discontinuous ppt.: matrix 의조성이 cell front 에서불연속적으로변함. 48
Cellular precipitation Interlamellar spacing change
Eutectoid transformation - Pearlite reaction in Fe-C alloy - Austenite (~0.8%C) cooled below A1 temp. γ α + Fe 3 C - Liq. 에서 eutectic reaction 과비슷, 그러나 very fine ( 협동성장 ) - Peralite 는 austenite 의 GB 에서 nucleate Growth dir. 50
Eutectoid transformation - If small undercooling below A 1 temp.: hemispheres or spheres - If large undercooling : nucleation rate is much higher, : site saturation occurs (Fig. 5. 61)
Massive transformation Cu-Zn alloy ~38at% Zn >800 β, <500 α, between α + β, cooling rate dependant Slow to moderate cooling: α precipitates like γ α in Fe-C alloy slower cooling rate: equiaxed α ppt. higher cooling rate: Widmanstätten α needle ppt. 5
Massive transformation Rapid quenching β상이 500 이하에서 retained. β α transform massively with same comp. rapid growth / irregular interface Diffusionless civilian transformation: thermally activated migration of int. Recrystallization과비슷, 그러나 very rapid 53
Massive transformation: CCT (continuous cooling transformation) diagram High cooling rate Slow cooling rate Equiaxed α Widmanstätten morphology 54
Massive transformation Zn: 38% 에서 850 에서는상태도에서 β 상이안정상. 700 에서는 α 와 β 의 G 가같아짐 : T 0 curve ( 각조성에서이와같은온도를찾으면!) 600 에서 massive trans 의구동력생김 (pptn 의구동력과비교 ) 500 에서는 massive trans. 와 pptn. 의구동력이같아짐.
Diffusionless transformation Martensitic transformation Martensite: product of diffusionless transformation Martensitic transformation: 각 atom 의이동거리가 one atomic spacing 이내인 transformation Diffusional trans. 가나타나지않도록 cooling rate 를크게해주면원칙적으로모든금속혹은합금에서일어날수있다. 대표적으로철강의중요한강화기구 : Simply, supersaturated solid soln of carbon in α-fe Martensite 형태 : lens shape / grain 을가로질러서형성 Martensite: coherent with surrounding γ matrix Grain 을가로지르는 martensite ~10-7 s 내에형성 : 그래서 athermal process
Diffusionless transformation High cooling rate: martensite 형성 Low cooling rate: Primary α + eutectoid α /Fe 3 C