Modeling of the bcc ordering using the sublattice formalism

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1 Modeling of the bcc ordering using the sublattice formalism N. Dupina, B. Sundmanb Calcul Thermodynamique, 3 rue de l'avenir Orcet, France, nathdupin@wanadoo.fr Swedish Institute for Metals Research, Dorttning Kristinas v. 48, Stockholm, Sweden a b

2 Outline Introduction Crystallography Thermodynamic description Binary cases Ternary cases Introducing vacancies Conclusion

3 Introduction Ti Fe Fe Ni Al Fe

4 Introduction Modeling ordering (BWG, CEF, CVM, MC) Why CEF? rapidity flexibility Problems with CEF? too flexible (many parameters) not enough if not enough sublattice topology

5 Introduction

6 Introduction Goal to show the ability of CEF to describe / and L21 with a single model to present some prototype calculations. to understand the effect of the parameters available. to decrease the number of parameters optimised. to increase the inputs from ab initio

7 Crystallography D03, L21 B32 disordered bcc lattice AB ordering A3B or BC ordering

8 Crystallography D03, L21 B32 The minimum cluster needed to model these different orderings is an irregular tetrahedron.

Crystallography By tetrahedron, By cell, By atom, 4/6 8 4 2/4 6 3 4 / 24 atoms 2 atoms 6 tetrah. 12 tetrah.

9 Crystallography By tetrahedron, By cell, By atom, 4/ / / 24 atoms 2 atoms 6 tetrah. 12 tetrah. E / tetrah. : 1/6 ( eαδ + eαγ + eβδ + eβγ ) + 1/4 ( eαβ + eδγ ) E / atom : ( eαδ + eαγ + eβδ + eβγ ) + 3/2 ( eαβ + eδγ )

10 Thermodynamic description (A,B)1/4 (A,B)1/4 (A,B)1/4 (A,B)1/4 α s: β γ δ α β γ δ α β γ δ α γ β δ y i =y i =y i =y i y i =y i y i =y i B32 y i =y i y i =y i α β γ δ D03 or L21 y i =y i y i y i

11 Thermodynamic description G= ijkl α yi 1 RT 4 s ij β yj γ yk δ y l G ijkl s y i ln s i s s s y i y j L ij s yi y si y sj y rk y rl Lijsr: kl sr ijkl

12 Thermodynamic description crystallography Gijkl = Gijlk = Gjikl = Gjilk = Gklij = Glkij = Gklji = Glkji but Gijkl Gikjl ( B32)

13 Thermodynamic description crystallography α β γ δ L ij =Lij =Lij =L ij =Lij αγ αγ αδ αδ βγ βγ βδ βδ L ij : kl =L kl:ij =Lij:kl =L kl:ij =Lij:kl =Lkl:ij =L ij:kl =L kl:ij αβ αβ γδ γδ L ij: kl =L kl:ij =Lij:kl =L kl:ij

14 Binary cases 1. Constant bond energies 2. Variation of the bond energies 3. Interaction parameters 4. Reciprocal interaction parameters 5. Discussion

15 Binary cases 1. Constant bond energies, classical BWG uab energy of a 1st neighbour bond vab energy of a 2nd neighbour bond G() = 4 uab G(B32) = 2 uab + 3 vab GA3B(D03) = 2 uab vab GAB3(D03) = 2 uab vab

16 uab = 250R vab = 0 4u/R =δ =γ γ=δ =β α= α 2u 4u x xa u) 8 ( B β 2nd order / transition

17 uab = 125R vab = + 167R ( 4u + 3v)/R 4u AB 2uAB+1.5vAB + fields 4v/R

18 uab = 400R vab = 200R ( 4u + 3v)/R DO3 β α= 4u γ γ=δ 2u+1.5v 2u+1.5v δ 3v/R DO3 DO3 stabilised

19 uab = 0 vab = 333R 3v/R B32 DO3 B32 δ α= β= γ α= 2u+1.5v 2u+3v δ γ β 2nd order /B32 transition

20 uab = + 333R vab = 333R B32 δ γ α= δ γ β= 2u+3v B32 α= 2u+1.5v 3v/R β +B32 fields

21 Binary cases 1. Constant bond energies, classical BWG. describes many different cases. only symmetrical descriptions. only two parameters available describing phase diagram and thermodynamics. not adapted to described most of the real systems

22 Binary cases Variation of the bond energies G() = 4 uab + α G(B32) = 2 uab + 3 vab + αb32 GA3B(D03) = 2 uab vab + αa3b GAB3(D03) = 2 uab vab + αab3 examples with uab = 250R, vab = 0

23 αab3 = 250R DO3 2u+1.5v +α DO3 stabilised AB3 but also and, mostly on B side

24 αab3 = + 250R destabilisation of, and phases, mostly on B side +

25 αb32 = 250R stabilised

26 αb32 = + 250R destabilised

27 α = 250R stabilised slightly stabilised +, +

28 α = + 250R destabilised slightly destabilised DO3 stabilised

29 Binary cases 1. Variation of the bond energies. of one compound affects its stability but also the one of other orders and of the. allows asymmetrical descriptions. increases freedom to describe real systems. can be difficult to estimate in metastable area. can be estimated thanks to first principle calculations for each stoichiometric compound

30 Binary cases Interaction parameters L AB =L0 s s y A y B L1 examples with uab = 250R, vab = 0

31 L0 = 250R ( 4u + 3v)/R +2L0/R 2u+1.5v 4u +L0 stabilised less ordered

32 L0 = + 200R ( 4u + 3v)/R +2L0/R L0 destabilised + +

33 L1 = 250R asymmetry and affected +

34 Binary cases 1. Interaction parameters. may affect all the phases on the whole range of composition but in larger extent. allow asymmetrical descriptions. increase freedom to describe real systems. need experiments in order to be assessed reasonably

35 Binary cases Reciprocal parameters shown to improve the description of the topology for fcc alloys simulating the SRO, two different ones in the bcc alloys αγ L AB: AB =LR1 αβ L AB: AB =LR2 examples with uab = 250R, vab = 0

36 LR1 = 250R ¼ LR1 stabilised Tmax closer to CVM

37 LR1 = 250R ¼ LR1 stabilised Tmax closer to CVM

38 LR2 = 250R ⅛ LR2 stabilised Tmax closer to CVM

39 LR2 = 250R ⅛LR2 stabilised

40 Binary cases 1. Reciprocal parameters. allow Tmax closer to CVM calculations. affect mostly the phase. simulate SRO

41 Binary cases 1. Discussion The different parameters available should allow a close description of any real systems. Similar high temperature / phase diagram can be obtained with many different descriptions, low temperature equilibria can difficult to study or hiden by other phases; the knowledge on the thermodynamic behaviours by experiments or theoretical calculations is the basis of a reliable description.

42 Ternary cases Extrapolation of binary bond energies Gi2j2() = 4 uij Gi2j2(B32) = 2 uij + 3 vij Gi3j(D03) = 2 uij vij Gij3(D03) = 2 uij vij Giijk(L21) = 2 uij + 2 uik vjk Gikij(F4-3m) = uij + uik + ujk vij vik

43 uab = 250R uab = 250R ubc = 125R vbc = + 167R uab = 250R ubc = 125R vbc = + 167R uac = 125R vac = + 167R uab = 125R vab = + 167R uab = 250R ubc = 250R uab = 125R vab = + 167R ubc = 125R vbc = + 167R uac = 125R uac = 125R vac = + 167R vac = + 167R

44 uab = 250R ubc = + 100R uab = 250R ubc = 250R uab = 250R ubc = + 100R ubc = + 100R uab = 125R vab = + 167R ubc = 125R vbc = + 167R uab = 250R uac = + 100R uac = + 100R uab = 125R vab = + 167R ubc = 125R vbc = + 167R uac = + 100R uac = + 100R

vbc = + 167R uac = + 333R vac = 333R ubc = 125R vbc = + 167R L21 L21+ +B3 2 uab = 125R vab =

45 uab = + 333R vab = 333R uab = 250R vbc = 333R B32 uab = 250R B32 L21 L21 L21 ubc = 250R vac = 333R uab = 125R vab = + 167R ubc = 125R vbc = + 167R uab = 125R vab = + 167R ubc = 125R vbc = + 167R uac = + 333R vac = 333R ubc = 125R vbc = + 167R L21 L21+ +B3 2 uab = 125R vab = + 167R L21 L21+ +B3 uac = R vac = 333R L0AC = 200R GAC(L21) = 2 uab + 2 ubc vac 250 R

46 Ternary cases Many different phase equilibria obtained just extrapolating very simple binary systems. Ternary descriptions have to be based on careful binary modeling. Ternary parameters are available to describe more precisely real ternary systems. First principle calculations can also help to estimate the energies of ternary compounds, in particular when metastable.

47 Introducing vacancies Vacancies ( ) are known as point defects in (A,B, )1/4(A,B, )1/4(A,B, )1/4(A,B, )1/4 For, it is equivalent to (A,B, ) where are "thermal vacancies" Simplest model for "thermal vacancies": (A, )

48 Introducing vacancies CEF, for one mole of site (A, ) G ya HA = ya(ga HA) + y G + Gid + yay LA with Gid = RT(yAlnyA+y lny ) GA HA Gibbs energy of one mole totally full of A refered to its enthalpy in the reference state G Gibbs energy of one mole totally empty, should be identical whatever A, =0 or >>0? LA Interaction parameter of A and

49 G = G ya GA = y G + Gid + ya y LA G /RT Gm /RT G yp For one mole of site: a single minimum for ya 1 two minima ya 1 and y 1 yp For one mole of A minima for ya 1 divergency for y 1

50 Introducing vacancies For theoritical reasons, G = 0 favoured and used bellow And what expression for LA? yp LA = brt unreasonnable LA = a problems at high T for lower a roughly y at Tfus LA 100Tfus LA = a+brt b= 3 T /1000 K can allow a better fit if experiments

51 Introducing vacancies (A, ) with LA is mathematically equivalent to (A, )½ (A, )½ with GA: = G :A = ½ LA or to (A, )¼ (A, )¼ (A, )¼ (A, )¼ with ua =⅛ LA LA LB triple defects (A,B)(B, )

52 uab = 250R ua = + 150R ub = + 500R =δ =γ β α= α =β γ=δ On A side, : (A) (A) (B, ) (B, ) On B side, : (A,B) (A,B) (B) (B) stabilised asymmetrically

53 ua = + 150R α= β α=β=γ=δ ub = + 500R γ=δ uab = 250R and stabilised by entropy effect

54 Introducing vacancies. The introduction of vacancies allows to easily describe triple defects.. The introduction of vacancies may stabilise asymmetrically the phase.

55 Thermodynamic description The formalism G = G( x) + G(ord,y) - G(ord,y=x) can be used equivalently. The relations imposed by the crystallography remain. Once fixed, some freedom is removed to describe the different ordered phases. The parameters of G(ord,y) are not defined uniquely; there is an infinity of values giving the same overall description with G(ord,y=x) between G(,x) and 0.

56 Conclusion Conclusion It is important to know the crystallography of a phase and its defects in order to model it satisfactorily. It is important to know the effect of the different parameters of the thermodynamic model used. The experimental knowledge of a system allows to get into the real world. The use of first principle results helps to avoid unrealistic metastable extrapolation.

57 Acknowlegments to the organisers of TofA 2004 for their invitation, to Alan Dinsdale for interaction on the modeling of vacancies, to Suzana Fries for simulating discussions on ordering.

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