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Byeong-Joo Lee www.postech.ac.kr/~calphad. Byeong-Joo Lee www.postech.ac.kr/~calphad “Numerical Treatment of Moving Interface in Diffusional Reactions,”

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Presentation on theme: "Byeong-Joo Lee www.postech.ac.kr/~calphad. Byeong-Joo Lee www.postech.ac.kr/~calphad “Numerical Treatment of Moving Interface in Diffusional Reactions,”"— Presentation transcript:

1 Byeong-Joo Lee www.postech.ac.kr/~calphad

2 Byeong-Joo Lee www.postech.ac.kr/~calphad “Numerical Treatment of Moving Interface in Diffusional Reactions,” Byeong-Joo Lee and Kyu Hwan Oh, Z. Metallkunde 87, 195-204 (1996). “Numerical Procedure for Simulation of Multicomponent and Multi-Layered Phase Diffusion,” Byeong-Joo Lee, Metals and Materials 5, 1-15 (1999). “Numerical Simulation of Diffusional Reactions between Multiphase Alloys with Different Matrix Phases,” Byeong-Joo Lee, Scripta Materialia 40, 573-579 (1999). “Prediction of the Amount of Retained delta-ferrite and Microsegregation in an Austenitic Stainless Steel,” Byeong-Joo Lee, Z. Metallkunde 90, 522-530 (1999). “Evaluation of Off-Diagonal Diffusion Coefficient from Phase Diagram Information,” Byeong-Joo Lee, J. Phase Equilibria 22, 241-246 (2001). “Thermo-Calc & DICTRA, computational tools for materials science,” J.-O. Andersson, Thomas Helander, Lars Höglund, Pingfang Shi and Bo Sundman, CALPHAD 26, 273-312 (2002) References

3 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusional Reactions – binary & multicomponent systems

4 Byeong-Joo Lee www.postech.ac.kr/~calphad Multicomponent Diffusion Fe-3.8Si-C Fe-C Darken’s uphill diffusion Diffusion between multiphase layers A. Engström, Scand. J. Metall. 24, 12 (1995). B.-J. Lee, J. Phase Equilibria 22, 241 (2001).

5 Byeong-Joo Lee www.postech.ac.kr/~calphad Content 1. Introduction ․ Definition ․ Diffusion Mechanism: Vacancy Mechanism, Interstitial Mechanism 2. Diffusional Flux and Application of Fick's law ․ Fick's first law in two component system ․ Fick's second law Application - Steady State Solution 3. Non-Steady State Diffusion ․ Thin Film Source (Thin Layer) ․ Semi-Infinite Source (Diffusion Couple) ․ Laplace/Fourier Transformation ․ Error function ․ Homogenization/Solute penetration ․ Trigonometric-Series Solutions ․ Determination of diffusion coefficient (Grube, Boltzman-Matano method) ․ Other Examples ․ Diffusion along high diffusion paths 4. Diffusion Coefficients ․ Reference Frame of Diffusion ⇒ Darken's Equation ․ Intrinsic, Inter, Self, Trace, Impurity Trace Diffusion Coefficient ․ Reference : Smithells Metals Reference Book, Chap. 13., Reed-Hil 5. Modelling of Multicomponent Diffusion ․ Darken's experiments : Fe-Si-C ․ Mathematical Formalism for Multicomponent Diffusion Coefficient

6 Byeong-Joo Lee www.postech.ac.kr/~calphad Definition Homogenization phenomena by non-convective mass transport due to chemical potential or electrochemical potential difference in a multicomponent single phase

7 Byeong-Joo Lee www.postech.ac.kr/~calphad General Phenomenological Equation

8 Byeong-Joo Lee www.postech.ac.kr/~calphad Fick’s 1 st law

9 Byeong-Joo Lee www.postech.ac.kr/~calphad Fick’s 2 nd law

10 Byeong-Joo Lee www.postech.ac.kr/~calphad As a thermally activated process for interstitial diffusion More about Diffusion Coefficient – Thermal Activation How about for substitutional diffusion?

11 Byeong-Joo Lee www.postech.ac.kr/~calphad Steady State Solution of Diffusion

12 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion

13 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion - Superposition Principle

14 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion - Superposition Principle

15 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Application of Superposition Principle

16 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Leak Test & Error Function

17 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

18 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

19 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

20 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

21 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

22 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Semi-Infinite Source

23 Byeong-Joo Lee www.postech.ac.kr/~calphad Determination of Diffusivity – Grube method

24 Byeong-Joo Lee www.postech.ac.kr/~calphad Determination of Diffusivity – Boltzmann-Matano

25 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Separation of Variable

26 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Separation of Variable

27 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Separation of Variable

28 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Separation of Variable

29 Byeong-Joo Lee www.postech.ac.kr/~calphad Non-Steady State Solution of Diffusion – Separation of Variable

30 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion along High Diffusion Path – Grain Boundary Diffusion Model

31 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Simulation – Finite Difference Method

32 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Simulation – Finite Difference Method

33 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Simulation – Finite Difference Method implicit integer (i-n) implicit double precision (a-h,o-z) dimension U(1000), UF(1000) c write(*,'(a)',advance='NO') ' Length of Simulation (micro-m) ? ' read(*,*) XL write(*,'(a)',advance='NO') ' Initial Composition (U-fraction) ? ' read(*,*) Uini write(*,'(a)',advance='NO') ' Boundary (Left-end) Composition ? ' read(*,*) U0 write(*,'(a)',advance='NO') ' Diffusion Coefficient (cm^2/sec) ? ' read(*,*) D write(*,'(a)',advance='NO') ' Reaction Time (sec) ? ' read(*,*) Tend write(*,'(a)',advance='NO') ' number of grid ? ' read(*,*) n c D = 1.d+08 * D dx = XL / dble(n-1) dt = 0.25d0 * dx * dx / D dtdx = D * dt / dx / dx c xiter = Tend / dt nprnt = idint(xiter/10.d0) c c initial condition c U = Uini UF = Uini time = 0.d0 iter = 0 open(unit=1,file='result.exp',status='unknown') write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddo c c Boundary condition U(1) = U0 UF(1) = U0 U(n+1) = U(n-1) c 1 iter = iter + 1 time = time + dt do i = 2, n uf(i) = u(i) + dtdx * ( u(i+1) - 2.d0*u(i) + u(i-1) ) enddo uf(n+1) = uf(n-1) c u = uf c if(mod(iter,nprnt).eq. 0) then write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddo endif c if(time.lt.tend) goto 1 stop end

34 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Simulation – Finite Difference Method

35 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Inter Diffusion

36 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Inter Diffusion

37 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Self/Tracer Diffusion

38 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Intrinsic Diffusion Coefficient

39 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Inter Diffusion Coefficient

40 Byeong-Joo Lee www.postech.ac.kr/~calphad Inter-diffusion Coefficient in a binary alloy – linked to intrinsic diffusion by the Darken’s relation Intrinsic diffusion Coefficient – composed of mobility term (Tracer Diffusion) and thermodynamic factor Tracer diffusion Coefficient – as a function of composition & temp. : tracer impurity diffusion coefficient : self-diffusion of A in the given structure Diffusion Coefficient – Modeling

41 Byeong-Joo Lee www.postech.ac.kr/~calphad Diffusion Coefficient – Modeling  assuming composition independent D o Linear composition dependence of Q B in a composition range N 1 ~ N 2  Tracer diffusion Coefficient at an intermediate composition is a geometrical mean of those at both ends – from experiments  the same for the D o term  Both Q o & Q are modeled as a linear function of composition

42 Byeong-Joo Lee www.postech.ac.kr/~calphad Moving Boundary Problem – Basic Equation

43 Byeong-Joo Lee www.postech.ac.kr/~calphad Binary Diffusion

44 Byeong-Joo Lee www.postech.ac.kr/~calphad Modeling of Multi-Component Diffusion - Basic Assumption for k  S (substitutional) for k  S

45 Byeong-Joo Lee www.postech.ac.kr/~calphad Modeling of Multi-Component Diffusion - Reference Frame

46 Byeong-Joo Lee www.postech.ac.kr/~calphad Mathematical Formalism of Multi-Component Diffusion Coefficient

47 Byeong-Joo Lee www.postech.ac.kr/~calphad Mathematical Formalism of Multi-Component Diffusion Coefficient

48 Byeong-Joo Lee www.postech.ac.kr/~calphad Mathematical Formalism - Application to Binary and Ternary Solutions

49 Byeong-Joo Lee www.postech.ac.kr/~calphad Mathematical Formalism - Application to Binary and Ternary Solutions

50 Byeong-Joo Lee www.postech.ac.kr/~calphad Smithells Metals Reference Book, 1992 Mathematical Formalism - Application to Binary and Ternary Solutions

51 Byeong-Joo Lee www.postech.ac.kr/~calphad John Ågren, Scripta Metallurgica 20, 1507-10 (1986). Mathematical Formalism - Application to Binary and Ternary Solutions

52 Byeong-Joo Lee www.postech.ac.kr/~calphad Mathematical Formalism - Application to Binary and Ternary Solutions

53 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – for C in Fe-C-M ternary system

54 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Darken’s uphill diffusion Fe-3.8Si-C and Fe-C

55 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Darken’s uphill diffusion

56 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Darken’s uphill diffusion Fe-3.8Si-C and Fe-6.45Mn-C

57 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Darken’s uphill diffusion

58 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – FDM approach for Fe-Si-C

59 Byeong-Joo Lee www.postech.ac.kr/~calphad Moving Boundary Problem – Basic Equation

60 Byeong-Joo Lee www.postech.ac.kr/~calphad Binary Diffusion

61 Byeong-Joo Lee www.postech.ac.kr/~calphad Application to Interfacial Reactions – Ti/Al 2 O 3 Reaction

62 Byeong-Joo Lee www.postech.ac.kr/~calphad Application to Interfacial Reactions – Ti/Al 2 O 3 Reaction

63 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni

64 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni

65 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – between Multi-Phase Layers

66 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – between Multi-Phase Layers

67 Byeong-Joo Lee www.postech.ac.kr/~calphad Multi-Component Diffusion Simulation – between Multi-Phase Layers

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