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1. User Guide Version 6 0 Volume 0 MICRESS Phenomenology eg MICRESS Phenomenology Resolution of partial differential equations is more about art than science Apocryphal quotation from Numerical Recipes in Fortran 2 2 4 except for large values of 2 Anonymous KC SA Douglas Adams prepared by G J Schmitz B B ttger J Eiken M Apel A Viardin A Carr G Laschet eg Contents Contents lu I Sii E E a E 1 I TET UG UG hes reellen 3 2 An engineering approach to microstructure modeling ececeeteeeeseeseeeeeeeeeeeeeeeseeeteesecaeeateeeeees 5 ASICS model SMS ET SEN an ns ee 11 3 1 Aspects OF multiphase H CM 11 3 2 sharp interface asympDlies ee 12 3 3 Aspects of computational efficiency 022044020n00nnnnnnnnnonnnnnnnnnnnnnonnnnnnnnnonnonnnnnnnnnenn 12 3 4 Coarsening and grain growth Dhenomeng 12 3 5 Coupling to concentration fields including Solute diffusion esessssssssssssssssnsnsneesesesese 13 Os Een ONT KEE 13 3 7 Coupling to thermodynamic databases 020r242020000n0nn0nnnnonnnnonnnnnnnnnnnnonnnnnnnnenn 14 3 8 Incorporation of nucleation PHENOMENA esses seeceseeeeeeeceseeececceseeecaseetseeeeeceatseeeeeeeaes 14 3 9 Incorporation OF elasticity plasticity anne ee 15 3 10 Self consistent coupling to macroscopic simulations eects ceceeecseeeeeeeseeeeeeeerenees 15 4 Applications to technical alloy grades AAA 16 E 16 4 1 Ve SOURCING
2. CALPHAD and Phase Field Modeling A Succesful Liaison Journal of Phase Equilibria and Diffusion 28 1 2007 101 MICRESS User Guide Volume 0 MICRESS Phenomenology 30 37 eg Chapter 7 References Ofori 2010 Pandat Pariser 2001 Pariser 2006 Phelan 2004 Plapp 2002 Provatas 1998 Provatas 1999 Qin 2005 Ramirez 2004 R sler 2003 Rudnizki 2010a Rudnizki 2010b N Ofori Opoku N Provatas A quantitative multi phase field model of polycrystalline alloy solidification Acta Matrialia 58 6 2010 2155 www computherm com G Pariser P Schaffnit I Steinbach W Bleck Simulation of the gammaz alpha transformation using the phase field method Steel Research 72 2001 9 p 354 360 G C Pariser Modeling the Austenite to Ferrite Phase Transformation for Steel Development Ph D Thesis Institute of Ferrous Metallurgy RWTH Aachen 2006 Shaker Aachen ISBN 3 8322 5014 X D Phelan Phase field modeling of 55wt Al Zn Si alloy system Proceedings Galvatech 2004 page 961 ff M Plapp A Karma Eutectic colony formation A phase field study Phys Rev E 66 6 2002 061608 N Provatas N Goldenfeld J Dantzig Efficient computation of dendritic microstructures using adaptive mesh refinement Phys Rev Lett 80 15 1998 3308 331 1 N Provatas N Goldenfeld J Dantzig Adaptive mesh refinement computation of solidification microstructu
3. Rudnizki 2010a MICRESS User Guide Volume 0 MICRESS Phenomenology 22 37 eg Chapter 5 Present developments 5 2 Determination of effective properties from simulated microstructures Models of microstructure evolution provide a maximum information depth in the form of spatially and temporally resolved microstructures To obtain information relevant for simulations on a larger length scale a reduction of these data and an extraction of effective properties is mandatory These effective properties are also most important for any alloy and process development and can be determined e g by methods of mathematical homogenization or by virtual testing The basic idea of mathematical homogenization is to calculate the effective properties of a composite structure like a multiphase microstructure of a real alloy on the basis of the knowledge of i the properties of the individual pure phases constituting the composite and ii their three dimensional topological arrangement Virtual tests are based on numerically mimicking experimental material tests like e g tensile tests with the adjusted boundary conditions from which the reproduced microstructure evolves Respective models by now have been applied to extract effective properties from microstructures simulated by the multiphase field method from 2 phase and multi phase structures in 3D Apel 2009b and to properties evolving during the gamma alpha transition in steels Another approach
4. www calphad org Carre 2010 A Carr B B ttger M Apel Phase field modelling of gas porosity formation during the solidification of aluminium International Journal of Materials Research 2010 04 Page 510 514 DICTRA Diffusion Controlled TRAnsformations software provided by Thermo Calc www thermocalc se MICRESS User Guide Volume 0 MICRESS Phenomenology 26 37 eg Chapter 7 References De Bruycker 2004 Diepers 1997 Diepers 1999 Diepers 2002a Diepers 2002b Diepers 2006 Echebarria 2004 Eiken 2006 Eiken 2007 Eiken 2009a Eiken 2009b Eiken 2010a E de Bruycker B C De Cooman M De Meyer Experimental Study and Microstructure Simulations of Zn Al Mg coatings Proceedings Galvatech 2004 page 723ff H J Diepers Diploma thesis Access RWTH Aachen 1997 H J Diepers C Beckermann I Steinbach Simulation of convection and ripening in a binary alloy mush using the phase field method Acta materialia 47 1999 13 p 3663 3678 H J Diepers Simulation des Primarabstandes gerichtet erstarrter Dendriten mit der Phasenfeldmethode PhD Thesis Access RWTH Aachen 2002 H J Diepers D Ma I Steinbach History effects during the selection of primary dendrite spacing Comparison of phase field simulations with experimental observations J Cryst Growth 237 239 2002 149 H J Diepers Steinbach Interaction of interdendritic convection and de
5. Phase field modelling of austenite formation from ultrafine ferrite carbide aggregates in Fe C International Journal of Materials Research 2010 04 Page 534 541 Beckermann 1999 C Beckermann H J Diepers I Steinbach A Karma X Tong Modeling Melt Convection in Phase Field Simulations of Solidification Journal of Computational Physics 154 1999 p 468 Benke 2008 S Benke A Multi phase field model including inelastic deformation for solid state transformations PAMM Volume 8 Issue 1 2008 pp 10407 10408 B ttger 2000 B B ttger U Grafe D Ma S G Fries Simulation of Microsegregation and Microstructural Evolution in Directionally Solidified Superalloys Mater Sci Technol 16 2000 1425 MICRESS User Guide Volume 0 MICRESS Phenomenology 25 37 eg Chapter 7 References B ttger 2006a B B ttger J Eiken I Steinbach Phase field simulation of equiaxed solidification in technical alloys Acta Materialia 54 10 2006 2697 B ttger 2006b B B ttger J Eiken M Ohno G Klaus M Fehlbier R Schmid Fetzer Steinbach and A B hrig Polaczek Controlling Microstructure in Magnesium alloys A combined thermodynamic experimental and simulation approach Advanced Engineering Materials 8 4 2006 241 B ttger 2008b B B ttger M Apel J Eiken P Schaffnit Steinbach Phase field simulation of solidification and solid state transformations in multicomponent steels Steel R
6. e g uphill diffusion of one alloy element in the gradient of another alloy element For simulation of complex alloy systems the consideration of the full diffusion matrix can be mandatory Regarding the amount of diffusion data necessary for a technical alloy system the use of available mobility databases thus is very attractive This can be achieved via online coupling to thermodynamic and mobility databases B ttger 2000 Eiken 2006 3 6 Considering fluid flow Besides diffusion fluid flow is a major transport mechanism for species and heat In general fluid flow however takes place on a larger length scale as compared to the evolution of the microstructure and thus may be considered by selecting suitable boundary conditions for a microstructure simulation But also on the scale of the microstructure itself fluid flow has significant influence on coarsening Diepers 1999 dendrite growth morphology Beckerman 1999 Steinbach 2009b and on dendrite spacing selection Diepers 2002a Diepers 2002b Diepers 2006 Zimmermann 2009 When implementing fluid flow into phase field models the well known no slip boundary condition for flow at interfaces has to be modified in order to account for the finite thickness of the diffuse interface This has been realized by introduction of an interfacial stress term varying across the thickness of the diffuse interface Beckermann 1999 MICRESS User Guide Volume 0 MICRESS Phenomenology
7. pg 1 2008 Advances in Physics 59 3 2010 257 R Herzog N Warnken Steinbach B Hallstedt C Walter J M ller D Hajas E M nstermann J M Schneider R Nickel D Parkot K Bobzin E Lugscheider P Bednarz O Trunova L Singheiser Integrated Approach for the Development of Advanced Coated Gas Turbine Blades Advanced Engineering Materials 8 6 2006 535 M Hillert The role of interfacial energy during solid state phase transformations Jerekont Ann 147 1957 757 JMatPro ThermoTech http www sentesoftware co uk S S Khan N Hort J Eiken 1 Steinbach and S Schmauder Numerical Determination of Heat Distribution and Castability Simulations of as Cast Mg Al Alloys Advanced Engineering Materials 11 3 2009 162 A Karma Phase field formulation for quantitative modelling of alloy solidification Phys Rev Lett 8711 11 115701 2001 D Kauzlaric J Lienemann L Pastewka A Greiner J G Korvink Integrated process simulation of primary shaping multi scale Approaches Microsyst Technol 14 2008 1789 1796 A G Khachaturyan S Semenovskaya T Tsakalakos Elastic strain energy of inhomogeneous solids Phys Rev B 52 22 1999 15909 S G Kim W T Kim T Suzuki M Ode Phase field modelling of eutectic solidification J Cryst Growth 261 1 2004 135 T Kitashima Coupling of the phase field and CALPHAD methods for predicting multicomponent solid state phase transformations Philosophical
8. 13 37 eg Chapter 3 Basic model development 3 7 Coupling to thermodynamic databases Crucial for modelling of multicomponent and multiphase technical alloys is a proper description of the thermodynamic properties of the alloy The CALPHAD approach Calphad has proven to be very powerful for calculating phase equilibria in complex alloy systems Databases for many important classes of technical alloys are available nowadays e g Thermo Calc JMatPro FactSage Pandat putting together a vast quantity of experimental data in binary ternary and higher order alloy systems Mobility data for some alloy systems are available as well Thermo Calc A consequent continuation of this idea is the online coupling of such databases to the multiphase field model by replacing the global equilibrium consideration with local equilibrium conditions The first steps in this direction have been made by coupling e g the thermodynamic software Thermo Calc Thermo Calc to a multiphase field model Steinbach 1996 using the TQ Fortran interface Grafe 2000 a b c B ttger 2000 From these early models the quasi equilibrium approach was developed Eiken 2006 which is implemented in the software package MICRESS MICRESS and has been successfully applied to different alloy systems as described below Comprehensive reviews about coupling of thermodynamic data to phase field models detail a number of different aspects Qin 2005 Steinbach 2007a Kit
9. 2 blue contribution A pure diffusion approach however would lead to a smear out of an initially sharp interface eventually ending up with a smooth and flat curve In order to describe a stable stationary interface an additional term thus is needed fig 2 2 green contribution which stabilizes the interface Note that this contribution is negative for 0 lt lt 0 5 and positive for 0 5 lt lt 1 This term thus balances the effect of the diffusion term blue leading to a stationary stabilized interface profile Depending on the actual choice of this term different stationary interface profiles may result e g a hyperbolic tangent profile for a double well potential or a sine profile for a double obstacle potential Eventually any deviation from equilibrium fig 2 2 red contribution will lead to a movement of the stationary interface profile The deviation from equilibrium is characterized by AG Depending on the sign of AG the motion will result either in growth or shrinkage of the respective phase When equilibrium is reached AG 0 the profile characterizing the interface position will become stationary and stable Further variables in the respective equation denote the interfacial energy o the interfacial thickness n and the interfacial mobility u Figure 2 2 The phase field equation in a very simple analysis See text for explanation of the individual colors and terms of the equation graphics courtesy Nils W
10. Magazine Vol 88 No 11 11 April 2008 1615 1637 R Kobayashi Modeling and numerical simulations of dendritic crystal growth Physica D 63 1993 410 423 l Kovacevic Simulation of spheroidisation of elongated Si particle in Al Si alloys by the phase field model Mater Sci Eng A 496 1 2 2008 345 G Laschet M Apel Thermo elastic homogenization of 3 D steel microstructure simulated by phase field method Steel Res Int 81 8 2010 637 MICRESS User Guide Volume 0 MICRESS Phenomenology 29 37 eg Chapter 7 References Ma 1999 Mecozzi 2003 Mecozzi 2005 Mecozzi 2007 MICRESS 2010 Militzer 2006 Moelans 2006 Moelans 2008 Nakajima 2006 Nestler 1998 Nestler 2002 Nomoto 2009 NRC 2008 D Ma U Grafe Microsegregation in directionally solidified dendritic cellular structure of superalloy CMSX 4 Materials Science and Engineering A270 1999 p 339 342 M G Mecozzi J Sietsma S van der Zwaag M Apel P Schaffnit Steinbach Analysis of the gamma alpha transition in C Mn steels by dilatometry Laser confocal scanning microscopy and phase field modelling Proceedings MS amp T 2003 pages 353 ff M G Mecozzi J Sietsma S van der Zwaag M Apel P Schaffnit Steinbach Analysis of the gamma alpha Transformation in C Mn steel by phase field modelling Metallurgical and Materials Transactions A 36A 9 2005 2327 M G Mecozzi Phase Field
11. and experimentally verified Mecozzi 2003 Mecozzi 2005 Mecozzi 2007 and applied to model the heat affected zone during welding of low carbon steel Thiessen 2006 a b c Recent simulations of the gamma alpha transition in 3D reveal the importance of different nucleation sites not occurring in 2D simulations like quadruple points or triple lines Militzer 2006 and the effects of stresses affecting the transition Apel 2009b Work on austenitization upon heating indicates this process not being the simple reverse of the ferrite formation Savran 2009 Nucleation of austenite may start from ultrafine ferrite carbide aggregates Azizi 2010 The successful use of a recently developed NPLE non partitioning local equilibrium model was demonstrated by simulation of austenite formation from an experimental ferrite plus pearlite microstructure and comparison to experimental results Rudnizki 201 0d 4 1 3 pearlite formation Pearlite transformation is a well known eutectoid transformation where a solid parent phase decomposes into two solid phases simultaneously It is similar to eutectic solidification where the phase state of the parent phase is the liquid Both transformations can lead to a lamellar microstructure and diffusion plays a major role for the spacing selection in this structure First multiphase field investigations on pearlite formation thus addressed the diffusion in both ferrite and austenite and aimed at describing the re
12. is the determination of effective properties from simulated microstructures by mathematical homogenization Laschet 2010 MICRESS User Guide Volume 0 MICRESS Phenomenology 23 37 eg Chapter 6 Future directions 6 Future directions 6 1 Data generation by numerical methods Although already quite an amount of data required to model microstructures is available in terms of thermodynamic databases and mobility databases numerous parameters still have to be determined experimentally Respective data comprise the temperature and composition dependant properties of the individual thermodynamic phases like thermal conductivities bulk elastic moduli density etc but also the Gibb s energies for phases where this value can experimentally only hardly be determined There is a perspective that respective data may be gained from ab initio calculations and molecular dynamics simulations in the near future even for complex alloy systems Even more interesting is the calculation of interfacial properties like interfacial energy and interfacial mobility including their anisotropies and their dependence on composition and temperature Guerdane 2010 In summary phase field and multiphase field methods during the last decade have developed into powerful tools for materials engineering The present situation may be compared to the development of FEM methods which have revolutionized mechanical engineering about twenty years ago Phase Field me
13. 009 6166 6175 U Grafe B B ttger J Tiaden S G Fries Coupling of Multicomponent Thermodynamic Databases to a Phase Field Model Application to Solidification and Solid State Transformations of Superalloys Scripta Materialia 42 12 2000 1179 1186 U Grafe B B ttger J Tiaden S G Fries Simulations of the Initial Transient during Directional Solidification of Multicomponent Alloys Using the Phase Field Method Modelling and Simulation in Materials Science and Engineering 8 6 2000 871 U Grafe Numerische Simulation der Gef gebildung bei der Erstarrung und W rmebehandlung von Elnkristallsuperlegierungen PhD thesis Access RWTH Aachen 2000 L Granasy Egry L Ratke et al Diffuse Interface model of bulk heterogeneous nucleation Scripta Metallurgica et Materialia 31 5 MICRESS User Guide Volume 0 MICRESS Phenomenology 1994 601 606 eg Chapter 7 References Guerdane 2010 Hecht 2010 Herzog 2006 Hillert 1957 JMatPro Kahn 2009 Karma 2001 Kauzlaric 2008 Khatchaturyan 1999 Kim 2004 Kitashima 2008 Kobayashi 1993 Kovacevic 2008 Laschet 2010 M Guerdane F Wendler D Danilov H Teichler B Nestler Crystal growth and melting in NiZr alloy Linking phase field modeling to molecular dynamics simulations Phys Rev B 81 22 2010 224108 U Hecht et al Advances of and by phase field modelling in condensed matter physics vol 57
14. 999 and Wang 2010 Elasticity has been incorporated into the multiphase field model Steinbach 2006 and has been applied e g to describe the pearlitic transformation in steels Steinbach 2007c Nowadays it can be also used to derive effective mechanical engineering constants e g the Young s modulus from simulated complex multiphase microstructures Laschet 2010 Effects of inelastic deformations on solid state transformations have been investigated in Benke 2008 3 10 Self consistent coupling to macroscopic simulations Consistent coupling of microstructure simulation to macroscopic temperature fields in technical processes like e g sand castings is not an easy task because latent heat plays an important role While latent heat production is directly linked to microstructure formation heat conduction is rather a phenomenon on the macroscopic length scale Thus a temperature solution which is consistent with microstructure formation can only be obtained directly if microstructure is solved simultaneously on the whole casting This obviously is impossible if complex and time consuming microstructure models like the phase field method are used Recently an iterative approach was developed to achieve self consistency between the macroscopic temperature evolution and microstructure formation which has been applied to an equiaxed AlCu casting B ttger2009a B ttger200 The approach couples a one dimensional macroscopic temperatur
15. ANGI Ol SIEB anne 16 2 1 2 gama albhaltaneiiion east ee een 17 MICRESS User Guide Volume 0 MICRESS Phenomenology 1 37 eg Contents 4 1 3 POAT tu le TD 17 4 1 4 grain growth erleben 18 BAAS MON Mess en a a ESTER aes eases 18 Ee ee 19 E ee 19 45 E Sic ee ee ae ee een 20 le 21 4 7 Intermetallic compounds 21 4 8 other alloy Systems ae een 21 5 Present development see en 22 5 1 Integrative Computational Materials Engineering 22 5 2 Determination of effective properties from simulated microstructures 23 6 Fut re SACS INOS ee ee meer 24 6 1 Data generation by numerical methods 2r24240n00nannonnnnonnnnonnnnnnnnnnnnnnnnnnnnnenn 24 7 Cl 25 MICRESS User Guide Volume 0 MICRESS Phenomenology 2 37 eg Chapter 1 Introduction 1 Introduction Any production is based on materials becoming components of a final product Materials properties thus are of great importance for productivity and reliability of processing during production as well as for application and reliability of product components A sound prediction of materials properties therefore is highly important Material properties are highly linked to their internal microstructure and properties evolution along the entire component life cycle starting from a homogeneous isotropic and stress free melt and eventually ending in failure under operational load is accompanied by microstructure formation and further evo
16. Materials Schmitz 2009b G J Schmitz and U Prahl Toward a Virtual Platform for Materials Processing JOM 61 5 2009 26 2009 Senk 2010 D Senk S Stratemeier B B ttger K G hler and Steinbach Modeling of Hot Ductility During Solidification of Steel Grades in Continuous Casting Part I Advanced Engineering Materials 12 4 2010 94 MICRESS User Guide Volume 0 MICRESS Phenomenology 32 37 eg Chapter 7 References Sommerfeld 2008 A Sommerfeld B B ttger B Tonn Graphite nucleation in cast iron melts based on solidification experiments and microstructure simulation Journal of Materials Science and Technology 24 3 2008 321 324 Steinbach 1996 Steinbach F Pezzolla B Nestler M SeeBelberg R Prieler G J Schmitz J L L Rezende A phase field concept for multiphase systems Physica D 94 1996 p 135 147 Steinbach 1999a Steinbach F Pezzolla A generalized field method for multiphase transformations using interface fields Physica D 134 1999 385 Steinbach 1999b Steinbach B Kauerauf C Beckermann J Guo Q Li Three Dimensional Modeling of Equiaxed Dendritic Growth on a Mesoscopic Scale Acta Materialia 47 3 1999 971 Steinbach 2000 Steinbach Ein Multi Phasen Feld Modell f r facettiertes Kristallwachstum PhD thesis Access RWTH Aachen 2000 Steinbach 2006 Steinbach M Apel Multi Phase Field Model for Solid State Transformation
17. Modelling of the Austenite to Ferrite Transformation in Steels PhD Thesis TU Delft 2007 MICRESS the MICRostructure Evolution Simulation Software www micress de M Militzer M G Mecozzi J Sietsma S van der Zwaag Three dimensional phase field modelling of the austenite to ferrite transformation Acta Materialia 54 15 2006 3961 3972 N Moelans B Blanpain P Wollants Phase field simulations of grain growth in two dimensional systems containing finely dispersed second phase particles Acta Materialia 54 4 2006 1175 1184 N Moelans B Blanpain P Wollants Quantitative phase field approach for simulating grain growth in anisotropic systems with arbitrary inclination and misorientation dependence Phys Rev Lett 101 2 2008 025502 K Nakajima M Apel I Steinbach The role of carbon diffusion in ferrite on the kinetics of cooperative growth of pearlite A Multi Phase Field study Acta Materialia 54 2006 3665 3672 B Nestler A A Wheeler Anisotropic multiphase field model Interfaces and junctions Phys Rev E 57 3 1998 2602 B Nestler A A Wheeler Phase field modeling of multiphase solidification Computer Physics Comm 147 1 2 2002 230 S Nomoto S Minamoto and K Nakajima Numerical Simulation for Grain Refinement of Aluminum Alloy by Multi phase field Model Coupled with CALPHAD ISIJ International 49 7 2009 1019 Steinbach B B ttger J Eiken N Warnken S G Fries
18. acts evoked by this artificial numerical interface thickness become possible in the framework of thin interface asymptotics where the limiting length scale is the diffusion length lg i e in cases where la gt n These corrections speed up simulations tremendously Further important work towards quantitative phase field modeling has been contributed by Plapp and co workers Plapp 2002 Folch 2005 Echebarria 2004 3 3 Aspects of computational efficiency The numerical solution of the coupled system of a large number of partial differential equations requires a high computational efficiency In contrast the correct resolution of details of the microstructure requires a very fine meshing Besides solving the coupled system of PDE s only at the interfaces using interface fields Steinbach 1999a especially adaptive grids providing a local remeshing at the interfaces have turned out to be beneficial Provatas 1998 Provatas 1999 Ofori 2010 In view of coupling with thermodynamic databases a multibinary extrapolation scheme reduces the computational load as compared to regular calls to databases Eiken 2006 3 4 Coarsening and grain growth phenomena Even in absence of any explicit thermodynamic driving force the phase field equations lead to a further evolution of the microstructure as the respective equations tend to minimize the total interfacial area of the system as is also noted in the Allen Cahn equation Allen 1979 which r
19. alues between 0 and 1 Metallurgists may relate this order parameter to the fraction of a specific phase e g corresponds to the fraction solid in fig 2 to be present at a specific point of space x and at a specific time t Figure 2 1 Description of a solidifying microstructure by an order parameter at a given moment t The color coding is explained in the following B 6 X t equais 1 100 solid O liquid BB 6 x t equals o 0 solid 100 liquid DES d x t between Oand1 diffuse boundary This method of describing microstructures has been extended to the description of multiple grains and multiple phases in the multiphase field method where multiple i e i different phase fields x t denote the individual phases or even all different grains In short any object which can be identified in the microstructure may have its own phase field variable in respective multiphase field models MICRESS User Guide Volume 0 MICRESS Phenomenology 5 37 RG Chapter 2 Microstructure Modeling Before entering multiphase field models it seems wise to understand or at least to get a feeling for a description of the evolution of the simple solidification situation depicted in fig 2 1 Describing the evolution of the microstructure thus means to identify the time derivative of the 4 x t i e the 4 x t A possible first step towards identification of a description of DICH is to start from a diffusion equation fig 2
20. arnken University of Birmingham MICRESS User Guide Volume 0 MICRESS Phenomenology 6 37 eg Chapter 2 Microstructure Modeling Another engineering approach to the phase field equation is based on the Gibbs Thomson equation giving a relation between interface velocity thermal and solutal undercooling and interface curvature and being well known to metallurgists since decades Diepers 1997 Beckermann 1999 A closer look at the phase field equation equation in fig 2 2 reveals a rotational symmetry as the diffusion equation fig 2 2 blue contribution does not comprise any anisotropy In order to include anisotropy into the model both the interfacial energy o and the interface mobility u are assumed to be anisotropic In 2 dimensions this can be accomplished by making these parameters dependant on the angle between the growth direction and the crystal orientation i e o o 8 and u u 0 For a simple cubic symmetry in 2D these functions could look like o ou 1 cos 48 and u uo 1 cos 40 For a hexagonal symmetry in 2 D functions like o oo 1 cos 60 and u uo 1 cos 60 would represent a first approach Please note that in case of spatially varying interfacial energies the Gibbs Thomson coefficient I has to be modified by including the second derivative of the interfacial energy T turns into T m m In order to describe anisotropy in 3D configurations a more complicated description becomes ne
21. ashima 2008 Fries 2009 and Steinbach 2009a 3 8 Incorporation of nucleation phenomena Nucleation typically takes place on a much smaller length scale than the further evolution of microstructure In spite of the fact that nucleation can in principle be described by phase field models on that small length scale Granasy 1994 Warren 2009 the description of nucleation in simulations on the scale of individual grains has to draw back on other nucleation models A variety of models have been implemented to phase field codes allowing e g to assign different nucleation probabilities in the bulk volume of the phases as compared to nucleation at interfaces and triple or higher order junctions Seed density models have been integrated to allow for different numbers of nuclei to become active dependant on the local undercooling B ttger 2006a B ttger 2009a MICRESS User Guide Volume 0 MICRESS Phenomenology 14 37 eg Chapter 3 Basic model development 3 9 Incorporation of elasticity plasticity Solid state transformations are often accompanied by elastic deformation or even plastic deformations Both phenomena can be naturally integrated into the phase field concept because both contribute to the driving force for the phase transformation although a thermodynamic interpretation of plasticity is still subject of ongoing discussions Important contributions to the field of elasticity and phase field have been made by Katchaturyan 1
22. cessary A possible approach is the use of linear combinations of spherical harmonics Eiken 2010b c This approach allowed for description of the three dimensional growth of dendrites in Mg alloys fig 4 4 The driving force AG depends on local conditions of external fields like temperature T or concentration c of the i different alloy elements but also stresses strains electric magnetic fields AG AG T j A non vanishing AG will lead to a finite change in phase fraction i e a finite AIS This change in phase fraction in turn will affect the external fields fig 2 3 Thus there is a need of solving the coupled system of partial differential equations for the phase field in multiphase field models the multiple phase fields and for all external fields affecting the phase transition MICRESS User Guide Volume 0 MICRESS Phenomenology 7 37 RC Chapter 2 Microstructure Modeling Figure 2 3 Evolving phase fractions influence e g the temperature field T by the release of latent heat or the concentration fields c due to the segregation of solute These changes in turn alter the local conditions for the driving force dG Technical alloys comprise multiple grains multiple phases and multiple components Their description in numerical models requires at least the introduction of multiple phase fields the description of multicomponent diffusion and thermodynamic and kinetic data The basic ideas of the multiphase fi
23. cture formation during solidification and Solution Heat Treatment of a Novel Single Crystal Superalloy PhD Thesis Access RWTH Aachen 2007 Warnken 2008 N Warnken A Drevermann D Ma S G Fries 1 Steinbach Development of a simulation approach to microstructure evolution during solidification and homogenization using the phase field method Superalloys 2008 Eds R C Reed et al TMS 2008 Warnken 2009 N Warnken D Ma A Drevermann R C Reed S G Fries and l Steinbach Phase field modelling of as cast microstructure evolution in nickel based superalloys Acta Materialia 2009 5862 Warren 2009 J A Warren T Pusztai L Kornyei et al Phase field approach to heterogeneous crystal nucleation in alloys Phys Rev B 79 1 2009 014204 Wheeler 1992 A A Wheeler W J Boettinger G B McFadden Phase field model for isothermal phase transitions in binary alloys Phys Rev A 45 10 1992 7424 MICRESS User Guide Volume 0 MICRESS Phenomenology 34 37 RC Chapter 7 References Zener 1947 C Zener Kinetics of the decomposition of austenite Wiley New York 1947 Zimmermann 2009 G Zimmermann L Sturz M Walterfang J Dagner Effect of melt flow on dendritic growth in AlSi7 based alloys during directional solidification Intern Journ of Cast Metals Res 22 1 4 2009 335 MICRESS User Guide Volume 0 MICRESS Phenomenology 35 37
24. e field to a multiphase field model using the homoenthalpic approximation which assumes a unique enthalpy temperature curve across the casting It has been shown that consistent coupling between the micro and macro scale was important for a correct microstructure prediction and that consistency was achieved after very few iterations MICRESS User Guide Volume 0 MICRESS Phenomenology 15 37 eg Chapter 4 Applications 4 Applications to technical alloy grades The following chapters will describe investigations and developments aiming at the description of microstructure evolution in technical alloy grades which have been performed with the help of the software MICRESS MICRESS They will address steels cast iron superalloys Al and Mg alloys solders intermetallic compounds and other alloys systems Along with the evolution of the underlying model basis the phenomena being tackled have become increasingly sophisticated for each of these materials 4 1 Steels Already in ancient times the complex interplay between diffusion precipitation dissolution and re precipitation as well as their control by well defined process scenarios has been exploited to develop sophisticated steel grades like e g the damascene steel revealing high tech structures on the micro and even nano scale It is however worth noting that approximately 70 of the present 2500 different steel grades have been developed during the last twenty years Steels
25. e phase transformations MICRESS User Guide Volume 0 MICRESS Phenomenology 3 37 eg Chapter 1 Introduction 1 0 z l L 3 l LL we 2 1273 1E NPMBGC A 0 94 a H 3 7 273 15 NPM FCC_A1 2 ET 27315NPM M7C3 0 84 5 1 273 15 NPM M23C6 Be Figure 1 1 0 64 L os b Equilibrium phase fractions of ge 2 different phases in a 25MoCr4 geb steel as a function of temperature 0 3 r calculated using Thermo Calc 0 2 d and the TCFe6 database Di L 4 oss 2 T T T A 600 800 1000 1200 1400 1600 TEMPERATURE_CELSIUS Continuing from the knowledge about equilibrium phase fractions which do not provide any information about how fast this equilibrium is reached subsequent developments aimed at describing the kinetics of diffusion controlled phase transitions One example for a software tool especially suitable for the description of multicomponent diffusion using respective databases is DICTRA The underlying approach here is based on 1 D systems like e g diffusion couples concentric cylinders or concentric spheres Under some specific assumptions phenomena like coarsening of a precipitate distribution can also be tackled Most interesting for metallurgists and materials engineers however is the microstructure and even further the properties of a material being based on its microstructure The simulation of microstructures in technical alloy systems probably has its origin in the first dendrites bein
26. e plus pearlite microstructure during annealing of cold rolled dual phase steel submitted Metallurgical and Materials Transactions A Savran 2009 V I Savran Austenite formation in C Mn steel PhD Thesis TU Delft 2009 Schaffnit 2007 P Schaffnit M Apel Steinbach Simulation of ideal grain growth using the multi phase field model Materials Science Forum 558 559 2007 1177 Schaffnit 2009 P Schaffnit C Stallybrass S Bez A Schneider J Konrad A Liessem Quantitative phase field simulation of the austenite grain growth between 900 C and 1400 C of a micro alloyed line pipe steel 9th International Seminar Numerical Analysis of Weldability Graz Seggau 2009 pre print Schaffnit 2010 P Schaffnit C Stallybrass J Konrad A Kulgemeyer H Meuser Dual scale phase field simulation of grain growth upon reheating of a microalloyed line pipe steel International Journal of Materials Research 2010 04 Page 549 554 Schmitz 1998 G J Schmitz B Nestler Simulation of phase transitions in multiphase systems peritectic solidification of RE Ba2Cu307 x superconductors Materials Science and Engineering B53 1998 p 23 27 Schmitz 2009a G J Schmitz B Zhou B B ttger J Villain and S Klima Phase field modeling and experimental observations of microstructures in solidifying Sn Ag Cu solders presented at the COST531 meeting in Bochum 2009 to be submitted to Journal of Electronic
27. eld approach Steinbach 1996 are e Definition of one phase field for each phase and for each grain of a phase e Pairwise interaction for each pair of phases grains like in standard phase field e Possibility of implementation of specific phase boundary grain boundary properties Further conceptual aspects comprise coupling to concentration fields Tiaden 1999a use of thermodynamic databases resp mobility databases multiphase interactions and physics of triple junctions MICRESS User Guide Volume 0 MICRESS Phenomenology 8 37 Chapter 2 Microstructure Modeling temperature liquid y austenite ferrite concentration one phase field for each phase 9 1 Figure 2 4 ds by Schematic example of the multiphase field approach for the peritectic solidification of steel One order parameter is assigned to each of the phases liquid L and The multiphase field approach thus allows for the description of multiphase equilibria at triple junctions like eutectics or peritectics which occur in general when solidifying technical alloy systems The concept is also applicable to solid state transformations like eutectoid transformations MICRESS User Guide Volume 0 MICRESS Phenomenology 9 37 eg Chapter 2 Microstructure Modeling The evolution of an initial state comprising multiple phases into a final state denoted by prime comprising the same phases can be realized as a rotation mat
28. epresents one of the roots of the phase field method Available work on coarsening phenomena comprises Ostwald ripening of solids in a liquid alloy with or without fluid flow Diepers 1999 or grain growth in solids Fan 1997 Moelans 2006 Moelans 2008 with or without stresses in pure substances and in alloys Respective investigations have also addressed ideal grain growth Schaffnit 2007 and later have been extended to phase field models comprising pinning of grain boundaries by impurities a recent overview is Apel 2009a MICRESS User Guide Volume 0 MICRESS Phenomenology 12 37 eg Chapter 3 Basic model development 3 5 Coupling to concentration fields including solute diffusion Coupling to concentration fields was first discussed for isothermal phase transitions in binary alloys by Wheeler 1992 One of the challenges to allow for the description of technical alloys was the correct treatment of solute redistribution and the calculation of the driving forces across diffuse interfaces revealing a numerically finite thickness This has been first realized for a multiphase binary system Tiaden 1998 Kim 2004 Aspects of combined heat and solute diffusion during solidification of a binary alloy have been treated by Ramirez 2004 While diffusion in binary alloys can still be handled by a single diffusion equation the simulation of diffusion in complex alloy system is much more challenging in view of effects like cross diffusion
29. erent thermodynamic phases has been considered in this simulation B ttger 2009c Recent work on Al Alloys comprises effects of flow on dendritic growth Zimmermann 2009 simulations on grain refinement Nomoto 2009 rheo casting of Al alloys B nck 2010 and porosity formation during solidification of A356 Carre 2010 4 5 Mg Alloys Mg based alloys are gaining increasing technical importance due to the high demand for weight reduction especially in transportation industry A specific feature of magnesium solidification is the hexagonal anisotropy of the hcp lattice Equiaxed solidification of the magnesium alloy AZ31 has been simulated using a two dimensional hexagonal anisotropy and a seed density model for the description of nucleation of the primary dendrites B ttger 2006b Major objectives of further studies were the influence of alloy composition and process parameters on the grain size Eiken 2007 Eiken 2010b c Phase field simulations of solidification of Mg alloys in three dimensions have been applied in order to investigate the role of the Mg specific hexagonal dendrite morphology in the process of competitive grain growth and the resulting selection mechanisms Eiken 2009a Eiken 2010a c Further work e g addresses the castability of technical Mg alloy grades Kahn 2009 Figure 4 4 3D simulation of texture evolution in Mg 6 Al Only few grains prevail after a short distance of directional solidification T
30. es Intern 79 8 2008 608 B ttger 2009a B B ttger J Eiken M Apel Phase field simulation of microstructure formation in technical castings A self consistent homoenthalpic approach to the micro macro problem J Comput Phys 2009 6784 6795 B ttger 2009b B B ttger M Apel J Eiken P Schaffnit G J Schmitz and l Steinbach Phase Field Simulation of equiaxed solidification A homoenthalpic approach to the micro macro problem Modelling of Casting Welding and Advanced Solidification Processes MCWASP XII Vancouver 2009 Pages 119 127 B ttger 2009c B B ttger A Carr G J Schmitz J Eiken and M Apel Simulation of the Microstructure Formation in Technical Aluminum Alloys using the Multi Phase Field Method Transactions of the Indian Institute of Metals 62 4 5 2009 299 B ttger 2010 B B ttger S Stratemeier E Subasic K G hler I Steinbach and D Senk Modeling of Hot Ductility During Solidification of Steel Grades in Continuous Casting Part II Advanced Engineering Materials 12 4 2010 101 B nck 2010 M B nck N Warnken A B hrig Polaczek Microstructure evolution of rheo cast A356 aluminium alloy in consideration of different cooling conditions by means of the cooling channel process Journal of Materials Processing Technology 210 4 2010 624 Calphad N Saunders A Miodownik CALPHAD calculation of phase diagrams a comprehensive guide Elsevier 1998 and
31. f graphite lamellae Respective simulations could be confirmed by experiments MICRESS User Guide Volume 0 MICRESS Phenomenology 18 37 eg Chapter 4 Applications Figure 4 2 Solidification simulation in cast iron The formation and the growth of tiny MnS particles in the liquid influence the subsequent formation of graphite Sommerfeld 2008 4 3 Superalloys Nickel based superalloys find widespread use in high temperature applications e g in turbines for aero engines gas or steam turbines for power generation R sler 2003 Herzog 2006 Many of the respective components like turbine blades and or vanes are produced using methods of investment casting and subsequent directional solidification Solidification can then cause melt related defects in these components The morphological evolution of the dendritic structure and the subsequent solid state transformations upon cooling and homogenization heat treatments thus are important for applications Multiphase field models coupled to thermodynamic databases can account for the full compositional complexity of technically relevant superalloys Warnken 2007 Microsegregation the phase fractions in the as cast and directionally solidified Ma 1999 Grafe 2000c B ttger 2000 microstructures formation of eutectic islands Warnken 2005 the solidification rate dependent dimensions of the mushy zone and the sequence of phase formation can be correctly predicted For phase transf
32. fe 2000a Grafe 2000b Grafe 2000c Granasy 1994 J Eiken Phase field simulation of microstructure formation in technical magnesium alloys International Journal of Materials Research 2010 04 Page 503 509 J Eiken A Phase Field Model for technical Alloy Solidification PhD thesis Access RWTH Aachen 2010 H Emmerich Advances of and by phase field modelling in condensed matter physics Advances in Physics 57 2008 1 FactSage www gtt technologies de D Fan L Q Chen Computer Simulation of grain growth using a continuum field model Acta Mater 45 2 1997 611 622 P Fayek U Prahl W Bleck Phase field simulations of the austenite grain growth of a microalloyed line pipe steel including particle pinning during the reheating process submitted Steel Res Int 2010 R Folch M Plapp Quantitative phase field modeling of two phase growth Phys Rev E 72 1 2005 011602 S G Fries B B ttger J Eiken 1 Steinbach Upgrading CALPHAD to microstructure simulation the phase field method Int J Mat Res 100 2009 2 S Fukumoto and S Nomoto Microstructure Simulation for Solidification of Stainless Steel by Multi Phase Field Model J Japan Inst Metals Vol 73 No 7 2009 pp 502 508 in Japanese P K Galenko S Reutzel D M Herlach S G Fries Steinbach M Apel Dendritic solidification in undercooled Ni Zr Al melts Experiments and modeling Acta Materialia 57 2
33. g simulated using the phase field method Kobayashi 1993 and the subsequent extension of the phase field method to multiple phase fields Steinbach 1996 allowing early simulations of eutectic and peritectic systems This multiphase field model later has been coupled to thermodynamic and mobility databases thus providing the basis for all the examples on simulations of technical alloy grades being depicted in this paper For reviews of these developments the reader is referred to Steinbach 2007a Kitashima 2008 Fries 2009 and Steinbach 2009a MICRESS User Guide Volume 0 MICRESS Phenomenology 4 37 RC Chapter 2 Microstructure Modeling 2 An engineering approach to microstructure modeling The phase field method can be rigorously derived from thermodynamic principles and theories of phase transitions and a lot of dedicated literature is available covering these fundamental and mathematical aspects for a review see e g Emmerich 2008 Hecht 2010 In this paper we will give a phenomenological approach for a rather intuitive interpretation of the phase field concept and equations The first step towards the simulation of the dynamics of microstructure evolution is the basic description of a static microstructure fig 2 1 A simple approach is to use a so called order parameter d for simulations of microstructure evolution in a simple solid liquid system d itself is a function of space x and time t i e o x t and may take v
34. g to macroscopic simulations 3 1 Aspects of multiphase equilibria One of the major questions arising shortly after publication of the first model was whether the multibinary description of a triple junction was sufficient to describe the correct equilibrium situation being macroscopically defined by Young s law or whether additional terms have to be introduced into the formulations The need for such additional terms is identified in a number of publications Nestler 1998 Nestler 2002 From an engineering perspective such terms may also be related to catalyst reactions where the mere presence of a third phase influences the transition between two other phases MICRESS User Guide Volume 0 MICRESS Phenomenology 11 37 eg Chapter 3 Basic model development 3 2 Sharp interface asymptotics Solution of the system of partial differential equations PDEs on a reasonable grid size leads to an unphysical interface thickness of some numerical grid cells in contrast to the physical thickness of some monolayers Nevertheless the paradigm behind phase field modelling is that the phase field method is able to reproduce the sharp interface solutions e g the Gibbs Thomson equation in a solidification problem From the numerical point of view this asymptotic is not feasible because this would require a numerical interface thickness on the scale of the physical interface thickness Karma 2001 however could show how corrections for artef
35. he simulation has been started from 50 initial nuclei being randomly oriented Eiken 201 0b c MICRESS User Guide Volume 0 MICRESS Phenomenology 20 37 eg Chapter 4 Applications 4 6 Solders Failure of electronic components often occurs at solder joints and particularly at microstructural features like e g phase boundaries with intermetallics Especially new solder alloys on the basis of ternary and higher alloyed solder systems like Sn Ag Cu are gaining importance and cannot be easily described by analytical approaches Increasing integration density moreover leads to additional constituent elements originating from either boards components or from their surface finish These constituents have also to be considered for microstructure evolution In addition to recent thermodynamic and kinetic modeling describing the range of possible stable phases the phase field approach allows describing their spatial distribution A respective microstructure Schmitz 2009a may serve as a basis for future lifetime and reliability prediction of the respective solder joint Special interest may originate from modeling electric current distributions in the microstructure and their influence on inhomogeneous heating during operation of the joint or on electromigration of components affecting microstructure evolution 4 7 Intermetallic compounds The properties of modern TiAl based intermetallic alloys critically depend on the solidified microst
36. lution Scope of this handbook is to review the current state of the art with respect to simulation of microstructure evolution based on the phase field approach in technical alloy grades Starting from a short overview about computational thermodynamics and kinetics and respective databases for technical alloys an engineering approach to phase field and multiphase field models will be depicted in order to allow for a basic explanation of these methods in general being developed by physicists and mathematicians for materials scientists and metallurgists Binary and ternary phase diagrams being available in printed form in books or publications have provided the basis for the development of materials ever since Increasing availability of computers has allowed for the continuous development of computational thermodynamics and respective databases in the last decades Such software tools and databases are nowadays available for complex alloy systems comprising a number of alloy elements e g Thermo Calc Pandat FactSage JMatPro Their databases are established using a well defined assessment scheme Calphad They allow determining phase diagrams calculating the sequence of phase transitions the amount of phase fractions being stable at a given temperature and other thermodynamic properties fig 1 Even more important for describing the evolution of a microstructure is that such models also allow the calculation of the driving forces for th
37. mmon and open standard for information exchange between different simulation tools allowing daisy chaining a number of different process simulation tools along the production chain and also coupling the different scales being relevant for microstructure evolution and accordingly for the resulting properties Schmitz 2009b Results and boundary conditions from the macroscopic process simulations can be used as boundary conditions to model microscopic effects like abnormal grain growth during carburizing The resulting microstructures for each step provide the basis for the determination of effective properties which can be either used to optimize alloy concepts or can be coupled back as local values to the process simulations in order to improve their accuracy and predictive capabilities with respect to the final properties of the component vonmMe heati rolli cooli forgin carburizing E S s rie E s component sub u mikro boundary averaged conditions properties Figure 5 1 Sketch of a virtual platform for materials processing The information exchange between different simulation tools on the component scale as well as the information exchange between different length scales is based on an open standardized data format This strategy allows tracking the influence e g of microsegregation during solidification on subsequent precipitate formation and eventually on grain stability during carburizing Schmitz 2009b
38. n Physically Based Modelling of Material Response to Welding PhD Thesis TU Delft 2006 Thiessen 2007 R G Thiessen J Sietsma T A Palmer J W Elmer M Richardson Phase field modelling and synchrotron validation of phase transformations in martensitic dual phase steel Acta Materialia 55 2007 601 614 Tiaden 1998 J Tiaden B Nestler H J Diepers Steinbach The Multiphase Field Model with an Integrated Concept for Modelling Solute Diffusion Physica D 1998 115 p 73 86 Tiaden 1999 J Tiaden Phase field simulations of the peritectic solidification of Fe C Journal of Crystal Growth 198 199 1999 pp 1275 1280 Tolui 2010 M Toloui M Militzer Phase field simulation of austenite grain growth in the HAZ of microalloyed linepipe steel International Journal of Materials Research 2010 04 Page 542 548 Wang 2010 Y Z Wang J Li Phase field modeling of defects and deformation Acta Materialia 58 4 2010 1212 1235 Wang 2009 J S Wang P D Lee Quantitative Simulation of Fe rich Intermetallics in Al Si Cu Fe Alloys during Solidification Proceedings of 138th TMS Annual Meeting and Exhibition San Francisco Feb 2009 VOL 1 Materials Processing and properties Warnken 2005 N Warnken D Ma M Mathes I Steinbach Investigation of eutectic island formation in SX superalloys Mater Sci and Eng A 413 12 2005 267 271 Warnken 2007 N Warnken Simulation of microstru
39. ndritic primary spacing Phase field simulation and analytical modeling Solidification and Gravity IV Materials Science Forum 508 2006 145 B Echebarria R Folch A Karma M Plapp Quantitative phase field model of alloy solidification Phys Rev E 70 6 2004 061604 J Eiken B B ttger Steinbach Multiphase Field approach for multicomponent alloys with extrapolation scheme for numerical application Phys Rev E 73 066122 2006 J Eiken B B ttger Steinbach Simulation of Microstructure Evolution during solidification of Magnesium Based Alloys Trans Indian Inst Met 60 2 3 2007 179 184 J Eiken Dendritic growth texture evolution in Mg based alloys investigated by phase field simulation International Journal of Cast Metals Research 22 1 4 2009 86 89 J Eiken M Apel V T Witusiewicz J Zollinger and U Hecht Interplay between a Ti nucleation and growth during peritectic solidification investigated by phase field simulations J Phys Condens Matter 21 2009 464104 J Eiken Phase Field Simulations of Dendritic Orientation Selection in Mg Alloys with Hexagonal Anisotropy Materials Science Forum 649 2010 pp 199 204 MICRESS User Guide Volume 0 MICRESS Phenomenology 27 37 eg Chapter 7 References Eiken 201 0b Eiken 201 0c Emmerich 2008 FactSage Fan 1997 Fayek 2010 Folch 2005 Fries 2009 Fukumoto 2009 Galenko 2009 Gra
40. ormations occurring during solidification effects of back diffusion have been identified as being important Extensions of the method which include homogenization of the as cast microsegregation have been demonstrated Warnken 2008 Warnken 2009 Recent studies have addressed the long term behavior gt 100 000 h of precipitates in technical superalloy grades B ttger 2010 unpublished 4 4 Al alloys In the field of aluminum alloys there is a high interest in microstructure simulation originating from automotive industry being caused by demands for lightweight alloys with optimized mechanical properties Consequently several approaches for the simulation of microstructure formation in technical aluminum alloys have been used by now incorporating thermodynamic data on different levels Kovacevic 2008 Qin 2005 Wang 2009 The multiphase field model Eiken 2006 with direct coupling to thermodynamic data bases has been used for the calculation of microsegregation in the hypoeutectic alloy AA6061 the widely used A356 casting alloy and eventually the slightly hypereutectic piston alloy KS1295 comprising up to 14 thermodynamic phases B ttger 2009c MICRESS User Guide Volume 0 MICRESS Phenomenology 19 37 eg Chapter 4 Applications Figure 4 3 Simulation of the solidification of a commercial Al alloy grade comprising seven alloy elements Some of these alloy elements tend to form intermetallic phases and a total of 14 diff
41. provide a variety of different phenomena occurring both during solidification and during subsequent solid state transformations the microstructure resulting from the preceding process step in most cases being of major importance for its further evolution during the subsequent steps Probably for this reason steels have been the first technological materials being investigated by multiphase field methods 4 1 1 solidification of steels First activities aimed at modeling the peritectic solidification in a binary Fe C system Tiaden 1999 Recent work describes modeling of the solidification of technical steel grades B ttger 2008b Fukumoto 2009 and also addresses aspects like hot ductility during solidification of steel grades in continuous casting processes B ttger 2010 Senk2010 The phenomena considered in such simulations comprise e g the formation of MnS precipitates effects of cross diffusion leading to inverse segregation of specific elements like P the formation of segregation bands as consequence of discontinuous solidification conditions and many others MICRESS User Guide Volume 0 MICRESS Phenomenology 16 37 eg Chapter 4 Applications 4 1 2 gamma alpha transition Next step for the microstructure evolution in technical steel grades are solid state transformations especially the gamma alpha transition This phenomenon has first been modelled in 2D in 2001 Pariser 2001 Pariser 2006 and be further extended
42. res using dynamic data structures Journal of Computational Physics 148 1 1999 265 290 R S Qin E R Wallach R C Thomson A phase field model for the solidification of multicomponent and multiphase alloys J Cryst Growth 279 1 2 2005 163 J C Ramirez C Beckermann A Karma and H J Diepers Phase field modeling of binary alloy solidification with coupled heat and solute diffusion Phys Rev E 69 2004 051607 J R sler M G tting D Del Genovese B B ttger R Kopp M Wolske F Schubert H J Penkalla T Seliga A Thoma A Scholz C Berger Wrought Ni Base Superalloys for Steam Turbine Applications beyond 700 C Advanced Engineering Materials 5 7 2003 469 483 J Rudnizki B Zeislmair U Prahl W Bleck Prediction of abnormal grain growth during high temperature treatment Comp Mater Sci 49 2 2010 209 J Rudnizki B Zeislmair U Prahl W Bleck Thermodynamical simulation of carbon profiles and precipitation evolution during high temperature case hardening Steel Res Int 81 6 2010 472 MICRESS User Guide Volume 0 MICRESS Phenomenology 31 37 eg Chapter 7 References Rudnizki 201 0c J Rudnizki S Konovalov U Prahl W Bleck Integrative microstructure simulation during high temperature case hardening accepted Steel Res Int Rudnizki 201 0d J Rudnizki B B ttger U Prahl W Bleck Phase field modelling of austenite formation from a ferrit
43. rix operating onto the initial state while keeping its norm i e the sum of all phase fractions constant at a value of 1 A I e wld p ga de de go dr pi Ga Py dr This matrix may be split into an identical matrix with 1 on the diagonal and an antisymmetric matrix comprising transitions between the individual phase pairs CA 1 0 ola 0 ds ghia d ad 0 db l d 0 1 0 al 0 li 6 0 a 0 dy g d 0 0 ijl D d d p 0 Such an interpretation has been one of the ingredients for the early development of the multiphase field method the transition matrix elements then being related to the binary transitions known from the classical phase field equation e g for the solid liquid system Please note that these matrix elements have to be considered as operators and not as a pure multiplication and further simplifications have been made here to demonstrate the basic idea A detailed derivation is given in Steinbach 1996 For the solid liquid system the order parameter has been identified as the fraction of the solid phase x x and for the binary solid liquid system the fraction of the liquid phase thus is given as x 1 x Rewriting the equation in figure 2 2 in terms of d X and d x then yields sl sl d Ls o v d bh D AG a d The interface mobilities interface energies interface width and the driving force have been complemented by indices el indicating their rela
44. ructure Commonly a rather coarse grain structure is obtained if a Ti forms via the peritectic reaction liquid B Ti gt a Ti A multiphase field model has been applied to qualitatively simulate the interaction between nucleation and growth of the peritectic a Ti in TIAI alloys with Al content varying between 43 and 47 at Al With increasing aluminum content the fraction of the pro peritectic B Ti phase being present at the peritectic temperature decreases A higher Al content additionally increases the grain refining effect due to growth restriction Eiken 2009b 4 8 other alloy systems Besides applying multiphase field models to structural material like those detailed above there are also applications to functional materials like superconductors Schmitz 1998 solar silicon Apel 2002a b Steinbach 2007b Steinbach 2000 coatings De Bruycker 2004 Al Zn Si Phelan 2004 rapidly solidified Ni Zr Al Galenko 2009 Cu Al alloys Kauzlaric 2008 and Nb Si alloys Amancherla 2007 MICRESS User Guide Volume 0 MICRESS Phenomenology 21 37 BC Chapter 5 Present developments 5 Present developments 5 1 Integrative Computational Materials Engineering Recently the high importance of an Integrative Computational Materials Engineering ICME for the future economic development and competitiveness has been strongly emphasized NRC2008 One of the major prerequisites for an efficient ICME is the definition of a co
45. s can be derived from the Gibbs Thomson relation they implicitly tend to minimize curvature and thus allow for the description of ripening and grain growth Subsequent to models for ideal grain growth Schaffnit 2007 effects of particle pinning on the mobility of the grain boundaries have been included Apel 2009a fig 4 1 Respective models now allow for the description of abnormal grain growth Rudnizki 2010a e g during case hardening Rudnizki 2010b c or for the description of grain growth in microalloyed line pipe steels Schaffnit 2009 Schaffnit 2010 Tolui 2010 Fayek 2010 korn 000 1600 1200 800 400 1 fx RI KR Figure 4 1 3D grain growth simulation for different time steps starting from 2000 individual grains Color coded are the individual grains left On the right representation of the triple lines of intersecting grain boundaries 4 2 Cast iron Few simulations in the area of cast iron have by now addressed aspects of nucleation conditions for graphite in dependence on the segregation profile of different alloy elements In a simulation study Sommerfeld 2008 nucleation of graphite on MnS particles which form during solidification has been identified as a possible scenario for formation of lamellar graphite in gray iron Based on this scenario especially the titanium composition turned out to be a decisive factor Too high levels of titanium lead to suppression of nucleation and poor development o
46. sulting spacing growth rate Respective results Nakajima 2006 already predicted a larger growth rate as compared to classical theoretical models Zener 1947 Hillert 1957 but still could not close the discrepancies with experimental observations Further investigations revealed that the transformation strain inhibits the cooperative growth mode of cementite and ferrite and provokes the salient growth of cementite needles ahead of the ferrite front The predicted growth velocities are in the right order of magnitude as compared to the experiment and thus close the gap between predictions by classical models being based on diffusion only and experimental observations Steinbach 2007c While all above simulations locally resolve the distinct thermodynamic phases of the pearlite i e ferrite and cementite present model developments aim at describing pearlite as an effective phase without resolving the individual ferrite cementite lamella Thiessen 2007 For this purpose a combination of thermodynamic descriptions taken from databases and of linearized phase diagrams for the pearlite pseudo phase has recently been implemented into a multi phase field code MICRESS MICRESS User Guide Volume 0 MICRESS Phenomenology 17 37 RC Chapter 4 Applications 4 1 4 grain growth Phase field models do not always require an explicit thermodynamic driving force to drive the evolution of a microstructure Because the respective equation
47. thods are actually on the track to become the FEM for metallurgists and materials scientists MICRESS User Guide Volume 0 MICRESS Phenomenology 24 37 eg Chapter 7 References 7 References Allen 1979 S Allen and J W Cahn A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening Acta Metall 27 1979 1084 Amancherla 2007 S Amancherla S Kar B Bewlay Y Ying A Chang Thermodynamic and Microstructural Modelling of Nb Si Based Alloys Journal of Phase Equilibria and Diffusion 28 1 2007 2 Apel 2002a M Apel B B ttger H J Diepers I Steinbach 2D and 3D phase field simulations of lamellar and fibrous eutectic growth Journal of Crystal Growth 237 239 2002 p 154 158 Apel 2002b M Apel D Franke Steinbach Simulation of the crystallisation of silicon ribbons on substrate Crystalline Silicon for Solar Cells Solar Energy Materials and Solar Cells 72 2002 p 201 208 Apel 2009a M Apel B B ttger J Rudnizki P Schaffnit and Steinbach Grain Growth Simulations including particle pinning using the Multi Phasefield Concept ISIJ International 49 7 2009 1024 Apel 2009b M Apel S Benke 1 Steinbach Virtual Dilatometer Curves and effective Young s modulus of a 3D multiphase structure calulated by the phase field method Computational Materials Science 45 2009 589 Azizi 2010 H Azizi Alizamini M Militzer
48. tion to this particular transition This formulation allows assigning different properties like e g interfacial energies for each of the individual phase interactions in multi phase systems For the ternary situation depicted in fig 2 4 the evolution of the liquid phase fraction would e g be described as a sum over the interactions with the ferrite and the y austenite d Ha H d fe gs p rae 2 vale P d Of i p jas 2 2 a a Na D higher order junction terms J MICRESS User Guide Volume 0 MICRESS Phenomenology 10 37 eg Chapter 3 Basic model development 3 Basic model development Starting from the initial idea of describing microstructure evolution in multiphase systems Steinbach 1996 a number of further developments was necessary to make the model applicable and useful for technical alloy systems The respective major topics are shortly outlined in the following and the reader is referred to respective articles for further reading In detail amongst others the following topics have continuously been addressed since 1996 e aspects of multiphase equilibria e sharp interface asymptotics e aspects of computational efficiency e coarsening phenomena e coupling to concentration fields including solute diffusion e consideration of fluid flow e coupling to thermodynamic databases e incorporation of nucleation phenomena e incorporation of elasticity plasticity e self consistent couplin
49. with Elastic Strain Physica D 217 2006 153 Steinbach 2007a Steinbach B B ttger J Eiken N Warnken S G Fries CALPHAD and Phase Field Modeling A Succesful Liaison Journal of Phase Equilibria and Diffusion 28 1 2007 101 Steinbach 2007b Steinbach M Apel Phase field simulation of rapid crystallization of silicon on substrate Materials Science amp Engineering A 449 451 2007 95 Steinbach 2007c Steinbach M Apel The influence of lattice strain on pearlite formation in Fe C Acta Materialia 55 2007 4817 Steinbach 2009a Steinbach Phase field models in Materials Science Topical Review Modelling Simul Mater Sci Eng 17 2009 073001 Steinbach 2009b Steinbach Pattern formation in constrained dendritic growth with solutal buoyancy Acta Materialia 57 2009 2640 2645 Thermo Calc Thermo Calc Software www thermocalc se Thiessen 2006a R G Thiessen LM Richardson J Sietsma Physically based modelling of phase transformations during welding of low carbon steel Materials Science and Engineering A 427 2006 223 Thiessen 2006b R G Thiessen 1 M Richardson A Physically Based Model for Microstructure Development in a Macroscopic Heat Affected Zone Grain Growth and Recrystallization Metallurgical and Materials Transactions B 37B 2006 655 MICRESS User Guide Volume 0 MICRESS Phenomenology 33 37 eg Chapter 7 References Thiessen 2006c R G Thiesse

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