Phase Transformation of Materials

Transcription

1 2009 fall Phase Transformation of Materials Eun Soo Park Office: Telephone: Office hours: by an appointment 1

2 Contents for previous class Interstitial iffusion / Substitution iffusion 1. Self diffusion in pure material 2. Vacancy diffusion 3. iffusion in substitutional alloys tomic Mobility Tracer iffusion in inary lloys High-iffusivity Paths 1. iffusion along Grain oundaries and Free Surface 2. iffusion long islocation iffusion in Multiphase inary Systems 2

3 iffusion in substitutional alloys C t = ~ C J ~ = X + X t t = 0, J = J + J v = J + vc ~ = C J J V J V J J J t t = t 1, J V J J = J + J v = J + vc ~ = C 농도구배에의한속도 + 격자면이동에의한속도 J = J 3

4 tomic mobility The problem of atom migration can be solved by considering the thermodynamic condition for equilibrium; namely that the chemical potential of an atom must be the same everywhere. In general the flux of atoms at any point in the lattice is proportional to the chemical potential gradient: diffusion occurs down the slope of the chemical potential. J = v c diffusion flux is a combined quantity of a drift velocity and random jumping motion. v = M μ μ : chemical force causing atom to migrate M : mobility of atoms J = M C μ How the mobility of an atom is related to its diffusion coefficient? 4

5 Relationship between M and J = M X RT lnγ X J = M (1 + ) V X ln X = M C m μ C RTF = C μ μ = G 0 = ( G + RT lnγ X ) lnγ ln X = RT ( + ) lnγ ln X = RT (1 + ) ln X n + RT ln a = G + RT C = = = V ( n + n ) Vm n X V m lnγ X = M RTF F = (1 + d lnγ d ln X ) For ideal or dilute solutions, near X 0, γ = const. with respect to X For non-ideal concentrated solutions, thermodynamic factor must be included. F = 1 = M RT Related to the curvature of the molar 5 free energy-composition curve.

6 Contents for today s class Interstitial iffusion / Substitution iffusion tomic Mobility Tracer iffusion in inary lloys High-iffusivity Paths 1. iffusion along Grain oundaries and Free Surface 2. iffusion long islocation iffusion in Multiphase inary Systems 6

7 2.5 Tracer diffusion in binary alloys 1) u* in u 2) u* in u-ni It is possible to use radioactive tracers ( * u ) to determine the intrinsic diffusion coefficients ( u ) of the components in an alloy. * u = u (self diffusivity) How does differ from? * u * u gives the rate at which u* (or u) atoms diffuse in a chemically homogeneous alloy, whereas u gives the diffusion rate of u when concentration gradient is present. u If concentration gradient exhibit, Δ H > 0 <, < * * mix u u Ni Ni the rate of homogenization will there fore be slower. 7

8 Since the chemical potential gradient is the driving force for diffusion in both types of experiment, it is reasonable to suppose that the atomic mobility are not affected by the concentration gradient. What would be the relation between the intrinsic chemical diffusivities and tracer diffusivities *? In the tracer diffusion experiment the tracer essentially forms a dilute solution in the alloy. = M RT = M RT 1 * * = F F * * % d lnγ = MRT + = FMRT dln X F : Thermodynamic Factor = ( * * = X ) + X = F X + X dditional Thermodynamic Relationships for inary Solution: Variation of chemical potential (dμ) by change of alloy compositions (dx) Eq.(1.71) dg dlnγ dlnγ 2 X X = RT RT 2 1+ = 1+ dx d ln X d ln X F 2 dlnγ dlnγ XX dg = 1+ = 1+ = 2 dln X dln X RT dx 8

9 2.5 Tracer diffusion in binary alloys % = X + X * * ( ) % = F X + X * u > * Ni 9

10 2.6 iffusion in ternary alloys: dditional Effects Example) Fe-Si-C system (Fe-3.8%Si-0.48%C) vs. (Fe-0.44%C) at Si raises the μ C in solution. (chemical potential of carbon) at t = 0 2 M Si (sub.) M C (int.), (M : mobility) C 10

11 How do the compositions of and change with time? 11

12 2.7.1 High-diffusivity paths Real materials contain defects. = more open structure fast diffusion path. Grain boundary dislocation surface s > b > l iff. along lattice iff. along grain boundary iff. along free surface ut area fraction lattice > grain boundary > surface l b s = = = lo bo so Ql exp( ) RT Qb exp( ) RT Qs exp( ) RT 12

13 iffusion along grain boundaries toms diffusing along the boundary will be able to penetrate much deeper than atoms which only diffuse through the lattice. In addition, as the concentration of solute builds up in the boundaries, atoms will also diffuse from the boundary into the lattice. 13

14 Combined diffusion of grain boundary and lattice? app = l + b δ d Thus, grain boundary diffusion makes a significant contribution only when b δ > l d. 14

15 The relative magnitudes of b δ and l d are most sensitive to temperature. -Q/2.3R b > l at all temp. ue to Q b < Q l, the curves for l and b δ/d cross in the coordinate system of ln versus 1/T. Therefore, the grain boundary diffusion becomes predominant at temperatures lower than the crossing temperature. (T < 0.75~0.8 T m ) 15

16 2.7.2 iffusion along dislocations app =? hint) g is the cross-sectional area of pipe per unit area of matrix. app = l + g p app l = 1+ g p l 16

17 ex) annealed metal ~ 10 5 disl/mm 2 ; one dislocation( ) accommodates 10 atoms in the cross-section; matrix contains atoms/mm 2. g *10 10 = = = g = cross-sectional area of pipe per unit area of matrix 7 t high temperatures, diffusion through the lattice is rapid and g p / l is very small so that the dislocation contribution to the total flux of atoms is very small. ue to Q p < Q l, the curves for l and g p / l cross in the coordinate system of ln versus 1/T. t low temperatures, (T < ~0.5 T m ) g p / l can become so large that the apparent diffusivity is entirely due to diffusion along dislocation. 17

18 2.8 iffusion in multiple binary system 18

19 2.8 iffusion in multiple binary system diffusion couple made by welding together pure and pure What would be the microstructure evolved after annealing at T 1? a layered structure containing α, β & γ. α β γ raw a phase distribution and composition profile in the plot of distance vs. X after annealing at T 1. β raw a profile of activity of atom, in the plot of distance vs. a after annealing at T 1. or atom easy to jump interface (local equil.) μ α = μ β, μ β = μ γ at interface (a α = a β, a β = a γ ) 19

20 How can we formulate the interface (α/β, β/γ) velocity? If unit area of the interface moves a distance dx, a volume (dx 1) will be converted from α containing C α atoms/m 3 β to β containing C β atoms/m 3. ( α) C C dx shaded area α β β α ( ) C C dx equal to which quantity? 20

21 Local equilibrium is assumed. flux of towards the interface from the β phase J β C = ( β ) flux of away from the interface into the α phase b % α C J = % ( α ) a In a time dt, there will be an accumulation of atoms given by -[J -J ] dt 2 C C C J = = 2 t t b a C C b a % ( β) % ( α) dt = ( C C) dx a dx 1 C C v = = ( ) ( ) b a % α % β dt ( C C ) (velocity of the αβ / interface) b dcdx 21

22 Contents in Phase Transformation 상변태를이해하는데필요한배경 (Ch1) 열역학과상태도 : Thermodynamics (Ch2) 확산론 : Kinetics (Ch3) 결정계면과미세조직 (Ch4) 응고 : Liquid Solid 대표적인상변태 (Ch5) 고체에서의확산변태 : Solid Solid (iffusional) (Ch6) 고체에서의무확산변태 : Solid Solid (iffusionless) 22

23 3. Crystal interfaces and microstructure Types of Interface 1. Free surface (solid/vapor interface) vapor solid 2. Grain boundary (α/ α interfaces) > same composition, same crystal structure > different orientation 3. inter-phase boundary (α/β interfaces) > different composition & crystal structure β α defect energy 23