computational thermodynamics: learning, doing and...

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PML STUTTGART

Computational Thermodynamics: Learning, Doing and Using

Matvei ZinkevichMatvei ZinkevichMaxMax--PlanckPlanck--Institut fInstitut füür Metallforschungr Metallforschung

Heisenbergstrasse 3, 70569 Stuttgart, GermanyHeisenbergstrasse 3, 70569 Stuttgart, Germany

Theoretical and Theoretical and Computational Computational Materials ScienceMaterials Science

11stst Summer School in Summer School in Stuttgart, June 2Stuttgart, June 2--6, 20036, 2003

From Microscopic Understandingto Functionality

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PML STUTTGART

ContentContent

• Introduction• Basic Thermodynamics• Thermodynamic Models• Thermodynamic Databases• Examples of Applications• Computer Demonstration

IMPRS IMPRS -- 11stst Summer School in Stuttgart, June 2Summer School in Stuttgart, June 2--6, 20036, 2003

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PML STUTTGARTIMPRS IMPRS -- 11stst Summer School in Stuttgart, June 2Summer School in Stuttgart, June 2--6, 20036, 2003

Phase diagramsPhase diagrams are visual representations of the state of a material as a function of temperature, pressure, and content of the constituent components

Binary phase diagramstwo dimensions ⇒

Ternary phase diagramsthree dimensions, only sections on plane ⇒

More complex systems⇒ ???

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Solution Solution ⇒⇒ the difficulty of graphically representing systems with many components is irrelevant for the CALculation of PHAse Diagrams (CALPHAD) andcan be customized for the materials problem of interest.

Computational Thermodynamics - the phase diagram is only a portion of the information that can be obtained from these calculations

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PML STUTTGARTIMPRS IMPRS -- 11stst Summer School in Stuttgart, Summer School in Stuttgart, JuneJune 22--6, 20036, 2003

DevelopmentDevelopment ofof thethe CALPHADCALPHAD MethodMethod

19th century: Gibbs - correlation between thermodynamics and phase equilibria

1908: van Laar – mathematical synthesis of a binary system1929: Hildebrand – regular solution concept1957: Meijering – thermodynamic analysis of Cu-Cr-Ni1963: Hume-Rothery – phase equilibria in Fe-base alloys1970: Kaufman and Bernstein – foundation of CALPHAD1972: Mager – least-squares method for optimization1973: 1st CALPHAD Meeting1977: Lukas – first computer software (Lukas Programs)1977: 1st volume of CALPHAD Journal1981: Agren, Hillert, Sundman – Compound Energy Formalismsince 1985: continuous development of models and software

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Fundamental equationsFundamental equations

• Internal Energy U(S,V)• Enthalpy H(S,P) = U + PV• Helmholtz Energy A(T,V) = U – TS• Gibbs Energy G(T,P) = H – TS• Chemical Potential G(T,P,ni) = Σ ni µi(T,P)• Gibbs-Duhem eq. Σ ni dµi = 0• Activity µi - µi* = RTlnai

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EquilibriumEquilibrium

Equilibrium condition for isobaric systems (P = const):

G = ΣNjGj = minimum

where Nj is the number of moles, and Gj is the Gibbs energy of phase j

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PML STUTTGARTIMPRS IMPRS -- 11stst Summer School in Stuttgart,Summer School in Stuttgart, JuneJune 22--6, 20036, 2003

Equilibrium condition in a simple binary systemEquilibrium condition in a simple binary system

Common tangent construction and calculated phase diagram for the Ni-Cu system

T = 1523 K

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Some definitionsSome definitions

• Elements – those from the periodic chart

• Species – an element or a combination of elements that forms an entity, like H2O, CO2, Fe2+

• A phase is a part of space that has homogeneous composition and structure.

• Constituents are the species that exist in a phase.

• Components is an irreducible subset of the species

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Thermodynamic modelsThermodynamic models

A model for a phase may contain different species. These species, called the constituents, contribute to the entropy of mixing.

The ideal entropy of mixing:

S = Rln(?(Ni)!/(SNi)!)

or

Sm = –RSxiln(xi)

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Pure elements, stoichiometric phases and end-members of solutions:

G = a + bT + cTlnT + SdnTn

Note that the enthalpy, entropy, heat capacity, etc. can be calculated from G:

H = a – cT – S(n-1)dnTn

S = –b – c – clnT – SndnTn-1

Cp = –c – Sn(n-1)dnTn-1

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The Gibbs energy per mole for a solution phase is normally divided into four parts:

Gm = Gmrsf – TSm

cfg + EGm + Gmphys

• Gmrsf is the reference surface for Gibbs energy

• Smcfg is the configurational entropy

• EGm is the excess Gibbs energy

• Gmphys is a physical contribution (e.g., magnetic)

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Ideal solution (gas, liquid, dilute solid solutions):

Smcfg = –RSxiln(xi) EGm = 0

Regular solution:

Smcfg = –RSxiln(xi) EGm = Hmix ≠ 0

Subregular solution:

Smcfg = –RSxiln(xi) EGm = f(T,xi) ≠ 0

EGm = Si Sj>i xi xj Lij Lij = binary interaction parameter

Lij = S??Lij(xi – xj)? (Redlich-Kister)

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Topological features of phase diagrams calculated using regular solution theory (Mager-matrix)

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Thermodynamic modelsThermodynamic modelsLattice stability:

Gmrsf(component i in phase j) – Gm

rsf(SER)

• based on metastable extrapolations

• requires standartization (SGTE)

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Compound Energy Formalism (CEF) or sublattice model:The Gibbs energy expression for (A,B,...)a(C,D,...)c

• Gmrsf = Si Sj y'iy''j °Gij

• Smcfg = –R(aSi y'i ln(y'i) +cSjy''jln(y''j))

• °Gij is the Gibbs energy of formation of the end-member compound ia jc• y'i and y''j are site fractions• a and c are the site ratios• The excess and physical contributions are as for a (sub)regular solution on each sublattice.

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CEF can be used describe interstitial solutions, carbides, oxides, intermetallic phases etc.In each particular case, models should be based on the crystallographic information.

A1 type or Cu type orα-brass, cF4

A2 type orW type orβ-brass, cI2

B2 type or CsCl type orβ-brass, cP2

A3 type orMg type orε, hP2

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Excess Gibbs energy in CEF for (A,B)a(D,E)cEGm = y'Ay'B(y''DLA,B:D + y''ELA,B:E) + y''Dy''E(y'ALA:D,E+ y''BLB:D,E) + y'Ay'By''Dy''ELA,B:D,E

Each L can be a Redlich-Kister series

The surface of reference for the Gibbs energy of a phase (A,B)a(D,E)c , according to CEF, plotted above the composition square

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Associated solution model makes use of fictitious constituents, for example FeS in liquid Fe-S, to describe short range order around the FeS composition.

• A parameter °GFeS is added.

• Interaction parameters between Fe-FeS and FeS-S are added to those between Fe-S ⇒ similarity to a ternary system.

• A gas phase is similar to an associated solution (without excess parameters) but in this case the constituents are real.

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The ionic liquid model is the modified sublattice model, where the constituents are cations (Mq+), vacancies (Vaq–), anions (Xp–), and neutral species (Bo).

• Assumes separate „sublattices“ for Mq+ and Vaq–, Xp–, Bo, e.g., (Cu+)P(S2–, Va–, So)Q, (Ca2+, Al3+)P(O2–, AlO1.5

o)Q or (Ca2+)P(O2–, SiO44–, Va2–, SiO2

o)Q

• The number of „sites“ (P, Q) varies with composition to maintain electroneutrality.

• It is possible to handle the whole range of compositions from pure metal to pure non-metal.

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The role of magnetismThe role of magnetism

Smag(max) = Sdis(T=∞) – Sord(T=0)

Hmag(max) = Hdis(T=∞) – Hord(T=0)

Smag(T), Hmag(T) = f(T, TC, µ)

Gmag(T) = Gord(T) – Gdis(T)

Model for Gmag(T):

Gmag(T) = RTln(µ + 1) g(τ) τ = T/TC

g(T) = 1 – K1(τ–1 + K2(τ3/6 + τ9/135 + τ15/600) τ < 1

g(T) = – K3(τ–5/10 + τ–15/315 + τ–25/1500) τ > 1

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Flowchart of the CALPHAD methodFlowchart of the CALPHAD method

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Connection with firstConnection with first--principle calculationsprinciple calculations

• From first principles one may calculate the energy at 0 K for different configurations of atoms on specific lattices.

• The cluster energies can then be used in a Cluster Variation Method calculation of the phase diagram for example.

• The cluster energies can also be used for Monte-Carlo simulations of the phase diagram.

• The energies from a first principle calculation can also be used directly in a sublattice model if the configurations correspond to the compounds.

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Experimental dataExperimental data

1. Crystallographic data

2. Phase Diagram data

3. Thermochemical data

4. Physical property data

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Crystallographic dataCrystallographic data

• Structure of the phases in the system.

• Sublattices for different constituents.

• Types of defects for non-stoichiometric compounds.

L21D03B2A2

L12L10B1A1

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Phase diagram dataPhase diagram data

• Thermal analysis: start/end temperatures of transformations

• Microscope: phase identification, determining phase amounts

• X-ray: phase identification, lattice parameters

• Microprobe: phase identification, phase compositions (tie-lines)

• X-ray and neutron diffraction: site occupancies

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Thermochemical dataThermochemical data

• Calorimetric data: enthalpy of formation, ~ of transformation, ~ of mixing

• EMF, Knudsen cell data: chemical potentials, activities

• Partial pressure: activities

• DSC: heat content, heat capacity, enthalpy of transformation

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Model selectionModel selection

• Special behaviour of data, like strong ”V”shaped enthalpy of mixing (sharp raise of activity).

Ø solid phase = long range order

Ø liquid phase = short range order

• Same phase may occur in several places in the system (or related phases like ordered superstructures).

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Thermochemical properties for FeThermochemical properties for Fe--SSThe phase diagramand some thermodynamicproperties: activity, enthalpy andentropy at 2000 K

Modeling of theliquid phase

metallic systems:sub(regular) solution model

metal-nonmetalsystems: associated or ionic liquid model.

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Numerical method for assessmentNumerical method for assessment

• The basic fitting is to vary the modelparameters to minimize the difference between the experimental value (qi

exp) and the same quantity calculated (qi

calc) from the model.

• Err = Si ((qiexp – qi

calc) wi/s i)2

wi is a weight assigned to the experimentalvalue

s i is an experimental error (standard deviation)

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Higher component systemsHigher component systemsEGm = Si Sj>i xi xj Lij + xi xj xk Lijk

Lijk = vi 0Lijk + vj

1Lijk + vk 2Lijk

vi = xi + (1 – xi – xj – xk)/3

In the ternary system vi = xi

In higher order systems Si vi =1 that guarantes the symmetry

Lijk is a ternary interaction parameter

Higher order interaction parameters can be usually omitted (a few exceptions in quaternary systems)

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Examples of applicationsExamples of applications

• Ni-Al based intermetallic alloys

• Micro-alloyed steels

• Oxide inclusions in steels

• Solders

• Simulation of microstructure

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NiNi--Al based intermetallic alloysAl based intermetallic alloys

a) Ni-25Al-18Fe; b) -15Fe; c) -13Fe

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MicroMicro--alloyed steelsalloyed steels (HSLA)(HSLA)

Projected miscibility gap for the case of (Ti,Nb)(C,N)

Calculated percentage of phases in microalloyed steels showing precipitation of TiN- and NbC-based carbides

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Oxide inclusions in steelsOxide inclusions in steels

Calculated isothermal sections CaO-FeO-SiO2 (a) and CaO-MgO-SiO2 (b)

Calculated phase fractions vs temperature for two different compositions

Calculated liquidus surface for five-component oxide system with fixed contents of Al2O3 and MgO

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Solders (AgSolders (Ag--BiBi--Sn)Sn)a) Liquidus surface; b) solidus surface

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Simulation of microstructureSimulation of microstructure

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Solidification of AlSolidification of Al--4.8at%Ag4.8at%Ag--16.0%Cu16.0%Cu

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77--Component Component ddendritic endritic ssuperalloy IN706uperalloy IN7062D-Simulation using a moving frame:

Mole fraction Ti

0.01

40.

016

0.01

80.

020

0.02

20.

024

0.01

20.

010

8·10

-5

Ni Fe Cr Ti Nb Al Cat% bal 37.7 17.1 1.83 1.8 0.55 0.05

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Solidification of IN706 using the Solidification of IN706 using the uunit nit ccell ell mmodelodel

x(Nb)

(dT/dt= -0.5 K/s, λ= 200 µm)

0.02

0.04

0.06

0.08

0.10

0.12

1

3 4

2

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Microstructure after Microstructure after ssolidificationolidification

Al C Cr

Fe Nb Ti

ηMC

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