PHASE FIELD MODELING OF ISOTHERMAL SLAG CRYSTALLIZATION › wp-content › uploads › 2010 › 06...

PHASE FIELD MODELING OF ISOTHERMAL

SLAG CRYSTALLIZATION

J. Heulens, N. Moelans, B. Blanpain

Department of Metallurgy and Materials EngineeringKatholieke Universiteit Leuven

Kasteelpark Arenberg 44 – box 2450B-3001 Heverlee (BELGIUM)

12th Annual GTT-Technologies Workshop, June 16-18, 2010

Outline

3/10/20112

� Introduction� Research framework and organization

� Concepts of phase field modeling� Slag specific properties in the model

� Thermodynamic data� Stoichiometric phases� Facet interfaces� Redox equilibria

� Case studies (simulation results)� Conclusions

Outline

3/10/20113





Goal and framework of this research

3/10/20114

� Problem statement� Microstructure evolution of metallurgical slags

controls many aspects of pyrometallurgical processes� e.g. freeze lining, refractory wear, tapping and cooling

� Goals� Develop a multiphase and multicomponent model

to simulate microstructure evolution in slags� Perform necessary high temperature experiments

� … to compare with the model� … to supply data for the model

Microstructure evolution in slags

3/10/20115

� Process slag in contact with furnace refractories� Freeze lining formed by solidification of process

slag� Tapping slag at high temperature and cooling

down

Refractory

Metal

Slag


3/10/20116



down


3/10/20117



down

Organization of the research

3/10/20118

Phase field modeling of isothermal slag crystallization

Phase field modeling of isothermal slag crystallization

SIMULATIONS

... interpret results

• Phases• Interfaces• Phase field

model• Numerics

... develop model

HIGHT T EXPERIMENTS

... to compare with the model

HPC

CSLM

... to supply data to the model

Modeled physical phenomena

3/10/20119

Gibbs energies of all phases in the system

Solidifying CAS slag (L,CS,C2AS)

Surface energies of all interfaces

Interface anisotropy (in kinetics and/or energy)

Diffusion coefficients for all components in

all phases

Chemical potentials of all components in all

phases

BULK PHYSICO-CHEMICAL PROPERTIES

INTERFACIAL PROPERTIES

Flow in the liquid phase (convection, forced …)

Investigated slag systems

� 4-component system� Ca-Al-Si-O

� 3-component if oxides are assumed to diffuse� CaO� Al2O3

� SiO2

� pO2 independent

� 3-component system� Fe-Si-O

� Transition to FeO and Fe2O3 to study redox� FeO (Fe2+)� Fe2O3 (Fe3+)� SiO2

� pO2 ~ FeO/Fe2O33/10/201110

CaO-Al2O3-SiO2 FeO-Fe2O3-SiO2

Outline

3/10/201111





Concepts of phase field modeling

3/10/201112

� Diffuse interfaces between different phases can treat arbitrary interface shapes

Microstructural evolution

3/10/201113

� System of p phases and c components� Phase fields evolve by energy minimization:

� Interfacial energy (Double well and gradient term)

� Chemical energy (Gibbs energies of phases)

� Diffusion equation for every component:

+∇+−=

∂∂

∫ ∑=

dVfmfLt V

p

rchemr

i

i

1

2

0 2ηκ

δηδη

∇⋅∇=

∂∂

∑ ∑=

−

=

p

i

c

r

ir

irki

k Mt

x

1

1

1

~µφ iSiO

ir

ir 2

~ µµµ −=

Thermodynamic equilibrium

3/10/201114

� Local (volumetric) chemical energy is calculated as:

� Equality of diffusion potentials of c-1 components in p phases:

� Mass balance to link phase concentrations to xk:

� Set of equations is solved for phase concentrations xki

Eiken J, Bottger B, Steinbach I, Physical Review E 73(6), 2006

Kim SG, Kim WT, Suzuki T, Physical Review E 60(6), 1999

∑=

=p

i

iichem ff

1

φ

ic

ik

i11

~...~...~−==== µµµ pi ..1=

∑=

=p

i

ikik xx

1

φ 1..1 −= ck

Outline

3/10/201115





Thermodynamic data of slags

3/10/201116

� The phase field model uses:� Gibbs energies of all phases (solid and liquid)

� fi(xk,T) depends on local composition and temperature� 2nd order Taylor expansion for local approximation

� Diffusion potentials of all components� µµµµk(xk,T) depends on local composition and

temperature� Second derivative of Gibbs energies

� This data can be calculated using ChemApp� Converting c chemical potentials into c-1 diffusion

potentials� Carefully assess and use reference states for

µµµµk(xk,T)

2nd order Taylor approximation

3/10/201117

� Gibbs energy of phase i is approximated by:

� A, B and C are retrieved by ChemApp and stored in an array to load in the phase field code

� Diffusion potentials are linear in concentrations, which greatly reduces computational effort

( ) ( )( ) ( )∑ ∑−

=

−

≠

+

−+−−+−=

1

1

1

ˆˆˆˆ2

ˆc

k

ic

kl

ik

ik

ik

il

il

ik

ik

ikl

ik

ik

ikki CxxBxxxxAxx

Af

il

ik

iikl xx

fA

∂∂∂=

2iki

k

iik x

fB µ~=

∂∂= ii fC =

ChemApp calls (liquid phase)

3/10/201118

� Gibbs energy � call tqgetr(‘GM’, 0, 0, F, noerr)

� Diffusion potential of CaO� G_SiO2_0 and G_CaO_0 (standard states in liquid at certain T)

� call tqgetr('MU', i_liquid, i_SiO2, mu_SiO2, noerr)

mu_SiO2 = G_SiO2_0 + mu_SiO2

� call tqgetr('MU', i_liquid, i_CaO, mu_CaO, noerr)

mu_CaO = ((G_CaO_0 + mu_CaO) - mu_SiO2)/Vm

� Second derivative of Gibbs energy� call tqsetc('IA ', i_liquid, i_CaO, x_CaO±±±±dr, numcon, noerr)

� dmu_dCaO = (mu2-mu1)/(2.d0*dr)/Vm

Numerical determination of dµ/dx

3/10/201119

Too coarse grid of finite

differences

Exceeding numerical

accuracy of ChemApp?

Gibbs energy of liquid

� All phases suspended

� Liquid phase is entered

� Stored in an array with ∆ = 0.001 spacing

� Locally approximated with 2nd order Taylor expansion� First derivatives� Second derivatives

3/10/201120

Diffusion potentials in liquid

3/10/201121

liquidSiO

liquidCaO

liquidCaO 2

~ µµµ −= liquidOAl

liquidOAl

liquidOAl 323232

~ µµµ −=

Second derivative of Gibbs energy

3/10/201122

Stoichiometric phases in slags

3/10/201123

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.20.30.40.50.60.70.80.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SiO2

CaO Al2O3mole fraction

L

L

Liquid is ternary solution

Wollastonite (CaO.SiO2)

Gehlenite (2CaO.Al2O3.SiO2)

Anorthite (CaO.Al2O3.2SiO2)

CaO-Al2O3-SiO2 @ 1400°C

Stoichiometric phases (binary version)

3/10/201124

~xB2

Hu SY, Murray J, Weiland H, Liu ZK, Chen LQ, CALPHAD 31(2), 2007

FACTSage

Second derivative must be large enough too ensure minimal solubility

but not too different from solution phase (numerics)

Diffusion matrix of 42CaO-10Al2O3-48SiO2

3/10/201125

� Kinetic coefficients in diffusion equations can be determined by literature data and ChemApp

� Diffusion equation for component k:

� M matrix relates to interdiffusion matrix D:

∇⋅∇=

∂∂

∑ ∑=

−

=

p

i

k

r

ir

irki

k Mt

x

1

1

1

~µφ

1

2

2

1

2

2

1

1

1

2221

1211

2221

1211~~

~~ −

∂∂

∂∂

∂∂

∂∂

⋅

=

xx

xxDD

DD

MM

MMµµ

µµ

MODEL

LITERATURE

CHEMAPP

Diffusion matrix of 42CaO-10Al2O3-48SiO2

3/10/201126

� Interdiffusion matrix D at 1350°CSugawara et al., Met. Trans. 8B, 1977, 605

� Thermodynamic factors matrix G at 1350°CChemApp

� Resulting M matrix for the model can now be calculated

1110977.5948.3

464.2734.8

323232

32 −

−−

−− ⋅

−−

=

OAlOAlOAlCaO

OAlCaOCaOCaO

DD

DD

10

2

2

1

2

2

1

1

1

10307.4420.1

420.1375.1~~

~~

⋅

=

∂∂

∂∂

∂∂

∂∂

xx

xxµµ

µµ

Uphill diffusion!

Up-hill diffusion in CaO-Al2O3-SiO2

3/10/201127

Redox equilibria of multivalent cations

3/10/201128

� Oxygen dependent system� FeO-Fe2O3-SiO2

� Open systems with at least one boundary in contact with O2 atmosphere (with fixed pO2)

� Assumptions made regarding to redox reactions� Redox equilibria are locally always in equilibrium

� Fe3+ / Fe2+ is a direct measure for local oxygen potential� O2 can only diffuse into the slag by changing the local

multivalent composition

� Special boundary condition to preserve Fe but not O

Diffusion of Fe2O3 and FeO (1500°C)

3/10/201129

Faceted growth of minerals

3/10/201130

� Conditions for a faceted interface:� Interface controlled growth (diffusion is faster)� Interface mobility is heavily dependent on

orientation� Cusps in the relation to orientation

� Tanh type function has physical meaning (Burton-Carbrera-Frank)

Uehara T, Sekerka RF, Journal of Crystal Growth 254, 2003

Outline

3/10/201131





Isothermal mineral crystallization

3/10/201132

CaO-Al2O3-SiO2@ 1350°CCaO. SiO2 crystallizesTernary diffusion interactions

CaO. SiO2

Liquid

Intial conditionDeveloped diffusion profile

0.1

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0.5

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0.10.20.30.40.50.60.70.80.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SiO2


Stoichiometric phase at boundary of composition domain requires careful selection of

its parabolic approximation.

Ternary diffusion paths in slags

3/10/201133

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.20.30.40.50.60.70.80.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SiO2


L

L

2CaO.Al2O3.SiO2

Liquid CaO-Al2O3-SiO2@ 1400°CLiquid and 2CaO.Al2O3.SiO2Different tie-line selection

FeO-Fe2O3-SiO2 system at 1400°C

3/10/201134

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.20.30.40.50.60.70.80.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SiO2

FeO Fe2O3mole fraction

LIQUID

FeO.Fe2O3

Open boundary in FeO-Fe2O3-SiO2

3/10/201135

� Initial slag is in equilibrium with pO2 = 2.4E-5� Upper boundary has pO2 = 1.5E-3

3.75 µm

5 ms

Conclusions

3/10/201136

� Phase field model linked with FACT database� Gibbs energies and diffusion potentials are

retrieved� Stoichiometric phases � parabolic Gibbs

energies� Faceted growth with cusp function in mobility

� Case studies of simulations� Mineral growth and diffusion path calculations

� Stoichiometric phase in contact with liquid

� Mineral growing with varying oxygen potential� Two competing diffusion fields

� Isothermal mineral crystallization in liquid is possible

PHASE FIELD MODELING OF ISOTHERMAL SLAG CRYSTALLIZATION › wp-content › uploads › 2010 › 06...

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