MatCalc approach for the modelling of the vacancy concentration evolution

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1 MatCalc approach for the modelling of the vacancy concentration evolution (MatCalc ) P. Warczok

2 Outline Vacancy concentration Vacancies in solids - What is it all about? Influence on diffusion and precipitate nucleation - Why do we care about it? Vacancy concentration evolution models - How does it evolve? Vacancy trapping - Why does it (sometimes) take so long to evolve? Page 2

3 What is it all about? Real materials are not perfect crystals... Page 3

4 Equilibrium vacancy concentration Equilibrium vacancy concentration thermodynamics make them appear yy VVVV,eeee = eeeeee GG ff VVVV RRRR GG ff VVVV = HH ff VVVV Page 4 ff TTSS VVVV

5 Equilibrium vacancy concentration Equilibrium vacancy concentration thermodynamics make them appear yy VVVV,eeee = eeeeee GG ff VVVV RRRR - Vacancy formation entropies are currently not used in MatCalc - Vacancy formation enthalpies come from thermodynamic database. Feel free to change it, GG ff VVVV = HH ff VVVV ff TTSS VVVV if you need it (Variables& function) Page 5

6 Vacancy concentration vs. temperature Page 6

7 Va concentration vs. relative temperature Page 7

8 Relative Va concentration vs. relative temp. Page 8

9 Relative Va concentration vs. relative temp. Page 9

10 What is it all about? How to change the vacancy concentration? Obtain new equilibrium state Change temperature Transform to a new phase Put the system out of equilibrium Deform material Irradiate material Page 10

11 Why do we care? Nucleation Vacancies Precipitation kinetics Diffusion Page 11

12 Why do we care? Vacancies Nucleation Precipitation kinetics dd VVVV ccc = RRRRRRRR yy VVVV yy VVVV,eeee GG = 3 ΔGG ccccccc vv αα 16ππγγ 3 dd VVVV ccc vv αα Δvv + ΔGG eeee vv αα υυ 4EE αα VVVV dd ccc vv αα 2 2 JJ~eeeeee GG RRRR Chemical part dd VVVV ccc - Vacancy contribution to driving force RR Gas constant TT Temperature yy VVVV - Vacancy site fraction on subst. lattice (current) yy VVVV,eeee - Vacancy site fraction on subst. lattice (equil.) GGPage - Critical 12 nucleation energy Elastic part γγ Interfacial energy preciptate/matrix vv αα - Molar volume Δvv volumetric misfit of precipitate υυ Poisson s ratio EE αα - Young s modulus JJ Nucleation rate

13 Why do we care? Vacancies Nucleation Precipitation kinetics dd VVVV ccc = RRRRRRRR yy VVVV yy VVVV,eeee GG = 3 ΔGG ccccccc vv αα 16ππγγ 3 dd VVVV ccc vv αα Δvv + ΔGG eeee vv αα υυ 4EE αα VVVV dd ccc vv αα 2 2 JJ~eeeeee GG RRRR Chemical part Elastic part - Generation/anihilation of the excess vacancies results in a strain of the matrix (last term) - Volumetric misfit of the precipitate may augment/decay the previous effect (second term) Page 13

14 Why do we care? dd VVVV ccc = RRRRRRRR yy VVVV yy VVVV,eeee GG = 3 1 vv αα 16ππγγ 3 ΔGG ccccccc + ΔGG eeee dd VVVV ccc Δvv υυ 4EE αα vv αα VVVV dd 2 ccc 2 Page 14

15 Why do we care? GG = 16ππγγ 3 3 ΔGG ccccccc + ΔGG eeee dd ccc VVVV vv αα Δvv υυ 4EE αα VVVV dd ccc vv αα 2 2 Page 15

16 Why do we care? Vacancies Diffusion Precipitation kinetics DD ii - Diffusion coefficent DD 0,ii - Pre-exponential factor QQ ii - Activation energy of subst. diff. Page 16

17 Why do we care? Vacancies Diffusion Precipitation kinetics DD ii = DD 0,ii eeeeee QQ ii RRRR yy VVVV yy VVVV,eeee Page 17

18 Why do we care? Vacancies Diffusion DD ii = DD 0,ii eeeeee QQ ii RRRR yy VVVV yy VVVV,eeee Page 18

19 How does it evolve? Sources and sinks Source Material Sink Vacancy flow Equilibrium Generation rate at source = Anihilation rate at sink Page 19

20 How does it evolve? 3 models in MatCalc: Mean diffusion distance FSAK * model Vacancy generation during the deformation (with FSAK only!) * Fischer-Svoboda-Appel-Kozeschnik, Fischer et. al., Acta Mater. 59 (2011) Page 20

21 How does it evolve? Mean diffusion distance Dislocations are taken as a vacancy source/sink Default mean diffusion distance: aa = 10^ 0.5llllll ρρ aa - Mean diffusion distance ρρ Dislocation density Evolution equation: CC - Constant yy ii - Element subst. site fraction Δyy VVVV Δtt = CC 1 eeeeee Page 21 DDΔtt aa yy VVVV,eeee yy VVVV D = yy VVVV yy VVVV,eeee yy ii DD ii ii

22 How does it evolve? Mean diffusion distance Dislocations are taken as a vacancy source/sink Default mean diffusion distance: aa = 10^ 0.5llllll ρρ Evolution equation: Δyy VVVV Δtt = CC 1 eeeeee Page 22 DDΔtt aa yy VVVV,eeee yy VVVV D = yy VVVV yy VVVV,eeee yy ii DD ii ii

23 How does it evolve? Mean diffusion distance Dislocations are taken as a vacancy source/sink Default mean diffusion distance Evolution equation Δyy VVVV Δtt = CC 1 eeeeee DDΔtt aa yy VVVV,eeee yy VVVV Page 23

24 How does it evolve? FSAK: Generation/anihilation on dislocation jogs Evolution equation: Δyy DDDD VVVV Δtt = 2ππDD ff yy VVVV yy VVVV,eeee llll HH = xx jjjjjj ρρ yy VVVV yy VVVV,eeee bbbb ff Diffusion correlation factor (0.75) bb Burger s vector HH Jog density xx jjjjjj Jog fraction on dislocations Page 24

25 How does it evolve? FSAK: Generation/anihilation on dislocation jogs Evolution equation: Δyy DDDD VVVV Δtt = 2ππDD ff yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee bbbb HH = xx jjjjjj ρρ Page 25

26 How does it evolve? FSAK: Generation/anihilation on dislocation jogs Evolution equation: Δyy DDDD VVVV Δtt = 2ππDD ff HH = xx jjjjjj ρρ yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee bbbb Page 26

27 How does it evolve? FSAK: Generation/anihilation on grain boundaries Evolution equation: Δyy GGGG VVVV Δtt = 15DD ff yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee RR 2 RR Radius of the spherical grain (0.5 * grain size) Page 27

28 How does it evolve? FSAK: Generation/anihilation on grain boundaries Evolution equation: Δyy GGGG VVVV Δtt = 15DD ff yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee RR 2 Page 28

29 How does it evolve? FSAK: Generation/anihilation on Frank loops (test-phase?) Evolution equation: Δyy FFFF VVVV Δtt = 4ππ2 DD ff yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee bb2 γγ LL rr FFFF kk BB TT NN FFFF rr FFFF nn PP nn PP = xx jjjjjj,ffff 1 γγ LL - Effective loop energy rr FFFF - Frank loop radius (defined in the console) NN FFFF - Frank loop density (defined in the console) Page 29 nn PP - Jog spacing on Frank loops (number of atoms) xx jjjjjj,ffff - Jog fraction on Frank loops kk BB - Boltzmann constant

30 How does it evolve? FSAK: Generation/anihilation on Frank loops (test-phase?) Evolution equation: Δyy FFFF VVVV Δtt = 4ππ2 DD ff yy VVVV yy VVVV,eeee llll yy VVVV yy VVVV,eeee bb2 γγ LL rr FFFF kk BB TT NN FFFF rr FFFF nn PP nn PP = xx jjjjjj,ffff 1 Page 30

31 How does it evolve? Vacancy generation during the deformation Evolution equation: Δyy dddddd VVVV Δtt = ααααεε eeeeee dddddd yy VVVV,eeee + yy VVVV GGbb 3 4ππ 1 υυ kk BB TT bb ρρ αα Constant MM Taylor factor GG Shear modulus εε - Strain rate Page 31

32 How does it evolve? Vacancy generation during the deformation Evolution equation Δyy DDDDDD VVVV Δtt = ααααεε eeeeee GGbb 3 4ππ 1 υυ kk BB TT bb ρρ Page 32

33 Why does it take so long to evolve? The idea: Some vacancies are trapped in the material and do not proceed to the sinks. Equilibrium between the trapped an free-to-move vacancies Needed: The amount of traps The trapping strength Relation between the amounts of trapped and free vacancies Page 33

34 Why does it take so long to evolve? Possible traps Solute atoms Dislocations Grain- & subgrain boundaries Precipitate surface Page 34

35 Why does it take so long to evolve? Possible traps Trappingsite Solute atoms Dislocations Fraction of trappingsites xx ii ππrr 2 dddd ρρ NN BB - Number of bulk sites NN GGGG - Number of sites at grain boundary NN SSSSSS - Number of sites at subgrain boundary Grain boundary NN GGGG NN BB Subgrain boundary NN SSSSSS NN BB Precipitate surface 4ππVV aaaa 1 3 ccllllllll NN jj rr jj 2 rr dddd - Dislocation core radius (5*10-19 m) xx ii - mole fraction of component i NN jj - Number of precipitates in class j rr jj - Radius of precipitates in class j VV aaaa - Atomic volume Page 35

36 Why does it take so long to evolve? Trap strength : delta H KK = eeeeee ΔHH RRRR ΔHH - Trapping enthalpy Trap range modifier: coordination number Page 36

37 Why does it take so long to evolve? The amount of traps Amount of lattice sites expressed by molar volumes nn tttttt - Total number of lattice sites nn tttttt = nn LL + nn TT nn ii VV mm,ii = nn tttttt VV mm xx ii = nn ii nn tttttt = VV mm,tttttt VV mm,ii nn LL - Number of sites on which the vacancy is free to move nn SS - Number of sites on the vacancy traps nn ii - Number of i -sites (e.g. nn LL, nn TT ) VV mm - Molar volume (volume for nn tttttt = 1) VV mm,ii - Volume of material containing 1 mole of i -sites xx ii - Mole fraction of i -sites Page 37 Svoboda J., Fischer F.D., Acta Mater. 60 (2012)

38 Why does it take so long to evolve? The amount of traps Amount of lattice sites expressed For clarity: by molar volumes nn tttttt = nn LL + nn TT nn ii VV mm,ii = nn tttttt VV mm xx ii = nn ii nn tttttt = VV mm,tttttt VV mm,ii nn tttttt - Total number of lattice sites - VV mm,ii is the volume of the material under consideration nn LL - Number of sites on which the vacancy is free to move needed to have the amount of 1 mole of i -sites in it. nn SS - Number of sites on which the vacancy is trapped The less i -sites in the material, the more of it is nn ii - Number of i -sites (e.g. nn LL, nn TT ) needed. VV mm - Molar volume (volume for nn tttttt = 1) VV mm,ii - Volume of material containing 1 mole of i -sites - VV mm,ii IS NOT the volume of 1 mole of i -sites! The xx ii - Mole fraction of i -sites material does not consist of i -sites alone Page 38 Svoboda J., Fischer F.D., Acta Mater. 60 (2012)

39 Why does it take so long to evolve? Vacancy balance VV mm,tt xx VVVV,tttttt = yy VVVV,tttttt = xx LL yy VVVV,LL + xx TT yy VVVV,TT = yy VV mm,tt +VV VVVV,LL + mm,ll VV mm,ll VV mm,tt +VV mm,ll yy VVVV,TT yy VVVV,ii - Site fraction of vacancies on i -sites (= fraction of i -sites occupied by the vacancies) VV mm,tt according fraction of trapping sites*coordination number VV mm,ll assumed as VV mm,tttttt (molar volume of system) Page 39 Svoboda J., Fischer F.D., Acta Mater. 60 (2012)

40 Why does it take so long to evolve? Vacancy balance: xx VVVV,tttttt = yy VVVV,tttttt = xx LL yy VVVV,LL + xx TT yy VVVV,TT = yy VV mm,tt +VV VVVV,LL + mm,ll Minimization of free energy: yy TT VVVV = LL yy VVVV VV mm,tt Combine these two above solve for yy VVVV,LL and yy VVVV,TT VV mm,ll VV mm,tt +VV mm,ll yy VVVV,TT LL KK + yy VVVV 1 KK KK = eeeeee ΔHH RRRR The immobile vacancies xx TT yy VVVV,TT xx LL yy VVVV,LL Page 40 Svoboda J., Fischer F.D., Acta Mater. 60 (2012)

41 Examples Examples from MatCalc website (P2-1, P2-2) FSAK model Al-Cu system, 4.3 wt.% Cu Precipitates: TH_DP_GPB- (bulk), THETA_PRIME- (disl.), THETA_AL2CU- (disl.) Al-fcc domain Page 41

42 Examples Al-Cu system (Example P2-1), tempered at 200 C Page 42

43 Examples Al-Cu system (Example P2-1), tempered at 150 C Page 43

44 Examples Al-Cu system (Example P2-1), tempered at 100 C Page 44

45 Examples Al-Cu system (Example P2-1) Page 45

46 Examples Al-Cu system (Example P2-2), influence of cooling rate Page 46

47 Examples Al-Cu system (Example P2-2), influence of dislocation density Page 47

48 Examples System Al-Mg-Si Mg = 0,4 wt.%, Si = 1,1 wt.% Mg5Si6_B_DP-precipiate. Nucleates at bulk sites Al-fcc domain Trapping sites: Mg (2600 J/mol) and Si (3500 J/mol) solute atoms. FSAK model Page 48

49 Examples Variation of grain size (FSAK) Page 49

50 Examples Variation of grain size (FSAK) Page 50

51 Examples Variation of trapping enthalpy (FSAK) Page 51

52 Examples Variation of trapping enthalpy (FSAK) Page 52

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