PHASE FIELD MODELING OF ISOTHERMAL SLAG CRYSTALLIZATION › wp-content › uploads › 2010 › 06...
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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
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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
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Outline
3/10/20113
� 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
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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
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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
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Microstructure evolution in slags
3/10/20116
� Process slag in contact with furnace refractories� Freeze lining formed by solidification of process
slag� Tapping slag at high temperature and cooling
down
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Microstructure evolution in slags
3/10/20117
� Process slag in contact with furnace refractories� Freeze lining formed by solidification of process
slag� Tapping slag at high temperature and cooling
down
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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
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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 …)
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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
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Outline
3/10/201111
� 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
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Concepts of phase field modeling
3/10/201112
� Diffuse interfaces between different phases can treat arbitrary interface shapes
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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 −=
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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
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Outline
3/10/201115
� 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
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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)
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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 =
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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
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Numerical determination of dµ/dx
3/10/201119
Too coarse grid of finite
differences
Exceeding numerical
accuracy of ChemApp?
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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
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Diffusion potentials in liquid
3/10/201121
liquidSiO
liquidCaO
liquidCaO 2
~ µµµ −= liquidOAl
liquidOAl
liquidOAl 323232
~ µµµ −=
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Second derivative of Gibbs energy
3/10/201122
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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
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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)
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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
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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!
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Up-hill diffusion in CaO-Al2O3-SiO2
3/10/201127
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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
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Diffusion of Fe2O3 and FeO (1500°C)
3/10/201129
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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
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Outline
3/10/201131
� 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
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Isothermal mineral crystallization
3/10/201132
CaO-Al2O3-SiO2@ 1350°CCaO. SiO2 crystallizesTernary diffusion interactions
CaO. SiO2
Liquid
Intial conditionDeveloped diffusion profile
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
Stoichiometric phase at boundary of composition domain requires careful selection of
its parabolic approximation.
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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
CaO Al2O3mole fraction
L
L
2CaO.Al2O3.SiO2
Liquid CaO-Al2O3-SiO2@ 1400°CLiquid and 2CaO.Al2O3.SiO2Different tie-line selection
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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
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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
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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