The CALPHAD method basis and applications

Transcription

1 basis and applications John Ågren Materials Science and Engineering KTH (Royal Institute of Technology) SE Stockhol, Sweden

2 Content 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusions

3 1. Introduction - A shift in Paradig! New possibilities: Developent tie for new aterials can be decreased fro years to 3-4 years. Materials enoe Databases and odels ICME - Integrated Coputational Materials Engineering Integration of odels Materials design Method for a purpose

4 CALPHAD - the first aterials genoe because it is... the ost efficient way of integrating various pieces of inforation - different character (therocheistry, phase diagras etc) - coherent and useful for extendable far beyond the traditional therocheistry increasingly being used outside the traditional CALPHAD counity A ajor enabling technology in Materials science and engineering.

5 CALPHAD a brief history Was born 1970 with the book (Kaufan and Bernstein: Coputer calculation of phase diagras, 1970) In the 50:ies and 60:ies Kaufan and Cohen and Hillert had outlined the general principles. Early predecessors van Laar 1908 and later several others. CALPHAD was introduced as an alternative to the quantubased approach PHACOMP (1964). The annual eetings (Later called CALPHAD conferences) and the CALPHAD journal both initiated by Kaufan were instruental for the rapid expansion of the field.

6 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusions

7 2. eneral idea behind CALPHAD Properties such as Phase equilibriu - Phase diagras - Coposition of phases and copounds - Partition coefficients and equilibriu constants... Therocheical data - Heat of reactions and transforations - Heat capacities - Vapour pressures... Elastic properties - Bulk odulus - Elastic constants... Volue and its theral expansion... all ste fro a single therodynaic function of the syste of interest.

8 If that function is known... then all these properties ay be calculated using that function. also other quantities of interest ay be calculated such as - Driving forces for reactions and transforations if the syste is not at equilibriu (e.g. In phase-field siulations) - Properties of etastable states, e.g. etastable phase diagras.

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10 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusions

11 3. Therodynaic basis What is the equilibriu state of a syste aged under given external conditions? What are the driving forces for internal changes if the syste is not in equilibriu?

12 Iportant reark: Therodynaics does not only apply at equilibriu, i.e. the condition when nothing ore can change. easureents can be perfored soeties quite far outside equilibriu and be extrapolated further into the non-equilibriu regie. Thus therodynaics applies considerably outside equilibriu. But how far outside would it apply? Here we will deonstrate how therodynaics ay be applied as far as needed to solve certain probles.

13 Overview of therodynaics State variables External Internal First law: Energy, heat and work Second law: Entropy, equilibriu, driving force Multicoponent systes cheical potential

14 Surroundings External conditions: P, V, T, n k Syste Internal constitution Interactions between syste and surroundings: coposition C theral 1 echanical 1 C + 2 After fixing the external conditions the syste will approach a state of equilibriu at which there are no further changes.

15 C + 2 external variables ust be fixed in order to define a unique equilibriu state. The equilibriu state ay be represented by a point in a C + 2 diensional space; a state diagra. A state diagra containing inforation about what phase is stable is a phase diagra.

16 Exaple: 1-coponent syste, n = fix, let P and T vary => 2-diensional state diagra. Add inforation on phases => unary phase diagra T L S P

17 Exaple: Phase diagra of pure iron. Pressure in Pa and teperature in K.

18 External and internal state variables External state variables ay be controlled outside the syste by the experientalist or the processing conditions Internal state variables In a syste out of equilibriu additional variables are needed to fully characterize the syste. These variables describe the internal constitution of the syste vary until they reach their equilibriu values and the syste coes to rest

19 Intensive variables If a new syste is fored by erging two identical systes the values of all intensive variables reain constant. Ex: P, T, ρ, c k, H etc Extensive variables If a new systes is fored by erging two identical systes the values of all extensive variables will becoe twice as large. Ex: V, U, n k, Additative rule: V = V 1 + V 2

20 Two types of intensive variables: - Potentials ex: P, T - Densities or specific variables = extensive variable size ass k k ρ= ck = xk = volue volue n j n n...

21 The state of equilibriu at given pressure, teperature and coposition Equilibriu for the value of the internal variable ξ that gives the lowest ibbs energy.

22 Change in ibbs energy during a phase transforation. 2:nd law of therodynaics

23 a) Metastable equilibriu b) Unstable equilibriu (critical state). a) b)

24 Cobined 1:st and 2:nd law of therodynaics in an open syste (i.e. Exchange of atter with surroundings), i.e. ulticoponent du du = TdS PdV + µ dn Td S 1 ds Td i = is = Dd D jd j ξ j 0 j T j D = driving force for process j dξ U j = = TdS 0 = U( SV,, N, ξ ) SV,, N, ξ k j PdV k j + k k µ dn k k k k i Td (2:nd law) Process j is frozen in are natural variables for U! i j S (cobined law)

25 Fro the cobined law: Each derivative tells how uch the internal energy changes per unit aount of an extensive quantity, i.e. entropy, volue or nuber of oles of k, added to the syste: Potentials! U S = T Theral potential (teperature) U V U N U ξ j k = P = µ k = D j Volue potential (negative pressure) Cheical potential Reaction potential (negative driving force)

26 Equilibriu conditions coe fro second law: For a closed syste with a constant energy du under constant volue: du = TdS PdV + µ kdnk TdiS k The condition ds = dis > 0 iplies that S ust increase towards a axiu which is reached at equilibriu. = 0 S S ξ ξ

27 A ore convenient equilibriu condition Equilibriu conditions fixed T and P we obtain fro the cobined law: du = TdS PdV D dξ d( U TS + PV ) = SdT + VdP D dξ At constant T and P : d( U TS + PV ) = D dξ = U TS + PV ibbs energy ξ TP = D = 0 at equilibriu. At given T and P the ibbs energy is iniized at equilibriu. Most useful!

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29 Characteristic state functions with their natural variables ),, ( ),, ( ),, ( ),, ( ),, ( ξ ξ ξ ξ ξ V U S P S H P T V T F V S U Either one of these functions fully characterize a syste.

30 If ( T, P, x, 1 x2... xn) is known, what is µ? j It is µ j = straight + x j forward to show that n x x i i= 1 i

31 Exaple: Binary syste The tangent construction μb = + (1 xb) x B μ = x A B x B x B

32 Soe properties derived fro ibbs energy of a phase: = ( TPx,,, x... ξ, ξ...) V = / P S = / T ( / )/ / c = T / T... P α = P T P µ D 2 2 k j xk j = + = ξ α α j x = N ( TPx,,, x... ξ, ξ...) x α α α α α j

33 Driving force under fixed P, T and coposition Cobined law yields: d = D d ξ j j Consider now a process where the ξ variables change fro an initial state to a final state. The driving forces ay vary in a coplicated way during the integration the integral final final start d = = Djd start ξ The integrated driving force. j

34 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusive rearks

35 4. Therodynaic odels If we know ibbs energy as a function = ( P, T, x1, x2,... ξ1, ξ2,...) for the individual phases in a syste: We ay calculate a lot of properties of practical interest. Calculate equilibriu state and phase diagra by iniizing. Calculate driving forces to use in kinetic siulations.

36 Modeling of solutions ibbs energy per ole for a solution phase is norally divided in: = + ideal + E + ph reference surface = configurational contribution (rando ixing) = ideal = xi i, RT xi ln xi, ph C i= 1 i= 1 excess ter e.g. ordering (agnetic/cheical) C E = physical contribution e.g. regular solution

37 (1 x ) + x B A B B B A ideal + E + ph A = μ A B = μ B x B

38 Regular-solution type of odels E = where i> j j x i x j L ij L ij = L ij + k ( x i x Redlich-Kister polynoial j ) k k L ij = L ij + ( x i x j ) 1 1 L ij + ( x i x j ) 2 2 L ij +... k = 0 : Regular solution k = 1: Sub-regular solution k = 2 : Sub- sub- regular solution etc

39 1 L AB 0 L AB Contributions to E : 3 2 AB L AB L

40 Phases with sublattices ( A A...) ( B B...) 1 2 a1 1 2 a2...(...) For exaple: Oxides, e.g. ( Ni, Mn)( O, N) Interstitials, e.g. ( Fe, Mn, Si...)( C, Va) + 1, K + 1 )( Cl a t 1... Carbo-nitrides, e.g. ( Cr, Fe, Mo) Salts, e.g. ( Na, F 1 ) 23 ( C, N) 6

41 The site fraction y = N / N N t t t j k j j = = nuber of forula units / N y t j is the fraction of lattice sites on sublattice t that are occupied by coponent k. One can usually not calculate the cheical potential of a coponent because the derivative N cannot be taken without violating the / k constraints.

42 ) ( ) ( ) (...), (...), (...), ( C Cr I I I I I y y y C C B B A A a a a t k t k t k t t I I a a a t eg... copound: a hypothetical is where... In general, = + = µ

43 The copound energy foralis For exaple:how to represent the ibbs energy of ( Cr, Fe) 23( C, N) 6? per ole of forula unit. Rando ixture on eachsublattice S ix / R = 23 ( y ln y + y ln y ) + 6( y ln y + y ln y ) Cr Cr Fe Fe What about the reference and excess energies? Note:in the reference ixing entropy and excess energy vanish. C C N N

44 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N Fe N Fe C Fe C Fe N Cr N Cr C Cr C Cr ref y y y y y y y y N Fe C Fe N Cr C Cr = 4 copounds: The reference energy surface

45 [ ] )... 6ln 23ln N Fe C Cr C Fe N Cr FeC CrN C Cr FeC CrN N Fe C Cr C Cr t k t k t k C Cr C Cr y y RT y y y y y y + = = + + = + + µ µ The cheical potentials

46 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples 6. Extension of CALPHAD beyond therocheistry 7. Conclusive rearks

47 5. Soe exaples d < 4% Average deviation Fro: Saunders & Miedownik: Calphad -a coprehensive review

48 Noinal coposition (SAF 2507): Fe 25% Cr 7% Ni 4% Mo 0.27% N 0.02% C Predict the teperature when siga-phase becoes stable within soe coposition variation: FeBase Cr 23 27% Ni 6 8% Mo 3 5% N % C %

49 Oxidation of Ni-base alloy Ni-2 ass% Al at 1200 C (TCFE7) :PO2,NPM(SPINEL) Mole fraction oxides NPM(*) Al 2 O 3 4 NiAl 2 O NiO 2 3:PO2,NPM(FCC_A1) 1:PO2,NPM(HALITE) 4:PO2,NPM(CORUNDUM_M2O3) P O2 FUNCTION PO2 NiAl 2 O 4 +NiO

50 The data challenge (ex. Ni-base alloys) Al B C Co Cr Fe Hf Mo N Nb Ni Pd Pt Re Si Ta Ti V W B x C x x Co x x x Cr x x x x Fe x x x x x Hf x x x x x x Mo x x x x x x x N x x x x x x Nb x x x x x x x x x Ni x x x x x x x x x x Pd x x x x x x x x x x Pt x x x x x x x x x x Re x x x x x x x x x x x x Si x x x x x x x x x x x x x x Ta x x x x x x x x x x x x x x x Ti x x x x x x x x x x x x x x x x V x x x x x x x x x x x x x x x x x W x x x x x x x x x x x x x x x x x x Zr x x x x x x x x x x x x x x x x x x x 20 eleents: 190 binary systes 184 of the 190 binary systes assessed for full range coposition All Ni containing ternaries plus other ternary systes also assessed to full range of coposition (184 in total) 292 interetallic and solution phases

51 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusive rearks

52 6. Extension of CALPHAD beyond therocheistry Diffusion and phase transforations: driving forces fro CALPHAD therodynaics Other properties - Diffusion coefficients - Interfacial obilities - Interfacial energies - Stacking fault energy - Elastic constants (anisotropic behaviour) - Optical - Electronic - Magnetic - Strength, ductility etc...

53 Fro atos to icrostructure: The ai is to predict icrostructure evolution and aterials properties. Interfacial energy & Volue & Elastic constants Therodynaics: ibbs energy CALPHAD Phase Field Method Langer-Schwartz f(ρ) First Principles Calculation ρ Diffusion: Mobility

54 enoic database for diffusion Multicoponent systes: any diffusion coefficients! Various type of coupling effects ay ake it ore coplicated than Fick s law. A CALPHAD-type of approach was suggested for inforation on diffusion kinetics (Andersson-Ågren 1992) - Allowed systeatic representatation of the kinetic behaviour of ulticoponent alloy systes. DICTRA was developed in the 1990s for nuerical solution of ulticoponent diffusion probles in siple geoetries.

55 Diffusion in Ni-base alloys: Capbell et al Siple substitutional odel

56 rowth of oxides controlled by diffusion Metal α Oxide β O xme+yo->me x O y Atosphere with O 2 Me xme+yo->me x O y - Oxygen diffusion in the oxide layer gives inward growth - Metal diffusion in the oxide layer gives outward growth - Internal oxidation needs oxygen diffusion into the etal, i.e. Oxygen diffusion through the oxide scale and the etal

57 Defect based odels of diffusion in oxides Vacancy echanis operative on different sublattices. The defect structure, the vcancy content, calculated fro CALPHAD databases, and the obility paraeters are stored in obility databases. eneralization the Wagner odel! Account for type A grain-boundary diffusion.

58 Experiental data on Fe tracer diffusion in spinel - Optiization of Fe obilities (Hallströ et al. 2011)

59 Siulation of oxidation of iron using the hoogenization odel in DICTRA Maxiu volue fraction porosity Distance fro surface [µ] Jonsson et al Mat Sci Foru (2008)1005

60 Mechanical properties Solution hardening (use the copound energy foralis ): Forula unit: ( M1M 2...)( C, N, Va) b Yield stress σ y σ = SSH y ij = y ' i ij y '' j σ k yij + σ ( ' ' ) n '' ' y y y A y ( '' '' + y y ) i j SSH y k ijk i k In classical odels n = = 2 / 3 and the A paraeters represent a cobination of isatch in lattice paraeter and elastic constants. Here they are taken as adjustable paraeters! i k A ik

61 1. Introduction A shift in Paradig! 2. eneral idea behind CALPHAD 3. Therodynaic basis 4. Therodynaic odels 5. Soe Exaples: 6. Extension of CALPHAD beyond therocheistry 7. Conclusive rearks

62 7. Conclusive rearks Models in CALPHAD can have any level of sophistication and account for: - Crystallographic structure - Lattice vibrations - Cheical order and disorder - Other phenoena such as agnetic order disorder... Different phases in the sae aterial can be represented by different odels and require different type of paraeters. Quantities which are experientally unknown, uncertain or show a large scatter ay be calcaluated by ab-initio ethods. CALPHAD is the ost efficient ethod to organise our experiental and theoretical knowledge on therodynaics and phase equilibria.