1. Materials Process Design and Control Laboratory COMPUTATIONAL TECHNIQUES FOR THE ANALYSIS AND CONTROL OF ALLOY SOLIDIFICATION PROCESSES DEEP SAMANTA Presentation for Thesis Defense (B-exam) Date: 21 December 2005 Sibley School of Mechanical and Aerospace Engineering Cornell University

2. Materials Process Design and Control Laboratory ACKNOWLEDGEMENTS SPECIAL COMMITTEE:  Prof. Nicholas Zabaras, M & A.E., Cornell University  Prof. Ruediger Dieckmann, M.S & E., Cornell University  Prof. Lance Collins, M & A.E., Cornell University FUNDING SOURCES:  National Aeronautics and Space Administration (NASA), Department of Energy (DoE), Aluminum Corporation of America (ALCOA)  Cornell Theory Center (CTC)  Sibley School of Mechanical & Aerospace Engineering Materials Process Design and Control Laboratory (MPDC)

3. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Introduction – alloy solidification processes.  Objectives of the current research.  Numerical model of alloy solidification with external magnetic fields.  FEM based computational techniques employed.  Numerical Examples.  Optimization problem for alloy solidification using magnetic fields.  Surface defect formation in aluminum alloys.  Exploring the role of mold surface topography.  Numerical Examples (parametric analysis).  Important observations and conclusions.  Suggestions for future study.

4. Materials Process Design and Control Laboratory Introduction and objectives of the current research

5. Materials Process Design and Control Laboratory INTRODUCTION Alloy Solidification Processes Used in industry to obtain near net shaped objects – casting, welding, directional and rapid solidification etc. Highly coupled process that involves several underlying phenomena – fluid flow, heat transfer, solute transfer, latent heat release and microstructure formation. Influences the underlying microstructure and properties of cast products. Most cast or solidified alloys characterized by defects. Phase change process involving more than one chemical species Appearance of solid and or crystalline phases Alloy solidification

6. Materials Process Design and Control Laboratory INTRODUCTION COMPLEXITIES IN ALLOY SOLIDIFICATION PROCESSES solid Mushy zone liquid ~10 -1 - 10 0 m (b) Microscopic scale ~ 10 -4 – 10 -5 m solid liquid (a) Macroscopic scale q os g

7. Materials Process Design and Control Laboratory Alloy solidification process Fluid flow Heat Transfer Mass Transfer Phase Change Deformation Shrinkage Microstructure evolution Non-equilibrium effects INTRODUCTION

8. Materials Process Design and Control Laboratory Close view of a freckle in a Nickel based super-alloy blade (Ref: Beckermann C., 2000) Freckles in a single crystal Nickel based superalloy blade Freckles in a cast ingot (Ref. Beckermann C.) DEFECTS DURING ALLOY SOLIDIFICATION Macrosegregation Oriented parallel to the direction of gravity in directionally (vertically) solidified cast alloys. Concentration of solute element inside freckles varies a lot from the bulk. Serve as sites for fatigue cracks and other types of failure Freckles defects Non – uniform solute concentration in bulk

9. Materials Process Design and Control Laboratory (b) (a) Macro-segregation patterns in a steel ingots (b) Centerline segregation in continuously cast steel (Ref: Beckermann C., 2000) (c) Freckle defects in directionally solidified blades (Ref: Tin and Pollock, 2004) (d) Freckle chain on the surface of a single crystal superalloy casting (Ref. Spowart and Mullens, 2003) DEFECTS DURING ALLOY SOLIDIFICATION (a) (d) (c)

10. Materials Process Design and Control Laboratory Piping Micro shrinkage Macro shrinkage Different types of shrinkage porosity (ref. Calcom, EPFL, Switzerland) Piping – occurs during early stages of solidification. Macro shrinkage - leads to internal defects. Micro shrinkage – occurs late during solidification and between solidifying dendrites. DEFECTS DURING ALLOY SOLIDIFICATION Surface defects in casting (Ref. ALCOA corp.) (a) Sub-surface liquation and crack formation on top surface of a cast (b) Non-uniform front and undesirable growth with non-uniform thickness (left) and non-uniform microstructure (right)

11. Materials Process Design and Control Laboratory OBJECTIVES OF THIS RESEARCH Macrosegregation Large scale distribution of solute Non – uniform properties on macro scale Thermosolutal convection Thermal and solutal buoyancy in the liquid and mushy zones Control of macro- segregation Macrosegregation Control or suppression of convection Development of freckles channels and other defects Density variations in terrestrial gravity conditions

12. Materials Process Design and Control Laboratory MEANS OF SUPPRESSING CONVECTION Control the boundary heat flux Multiple-zone controllable furnace design Rotation of the furnace Micro-gravity growth Electromagnetic fields Constant magnetic fields Time varying fields Rotating magnetic fields Combination of magnetic field and field gradients OBJECTIVES OF THIS RESEARCH

13. Materials Process Design and Control Laboratory Engineered mold surface (Ref. ALCOA Corp.) In industry, the mold surface is pre-machined to control heat extraction in directional solidification This periodic groove surface topography allows multi-directional heat flow on the metal-mold interface However, the wavelengths should be with the appropriate value to obtain anticipated benefits. Uniform front growth (left) and uniform microstructure (right) – obtained using grooved molds OBJECTIVES OF THIS RESEARCH

14. Materials Process Design and Control Laboratory  Exploring methods to reduce defects during alloy solidification processes.  Developing a computational framework for modeling alloy solidification processes.  Studying the role of convection on macrosegregation.  Employing constant or time varying magnetic fields to reduce macrosegregation based defects.  Designing appropriate mold surface topographies to reduce surface defects in alloys. Alloy solidification process Formation of various defects Material, monetary and energy losses OBJECTIVES OF THIS RESEARCH

15. Materials Process Design and Control Laboratory Numerical model of alloy solidification under the influence of magnetic fields

16. Materials Process Design and Control Laboratory PREVIOUS WORK Effect of magnetic field on transport phenomena in Bridgman crystal growth – Oreper et al. (1984) and Motakef (1990). Numerical study of convection in the horizontal Bridgman configuration under the influence of constant magnetic fields – Ben Hadid et al. (1997). Simulation of freckles during directional solidification of binary and multicomponent alloys – Poirier, Felliceli and Heinrich (1997-04). Effects of low magnetic fields on the solidification of a Pb-Sn alloy in terrestrial gravity conditions – Prescott and Incropera (1993). Effect of magnetic gradient fields on Rayleigh Benard convection in water and oxygen – Tagawa et al.(2002-04). Suppression of thermosolutal convection by exploiting the temperature/composition dependence of magnetic susceptibility – Evans (2000). Solidification of metals and alloys with negligible mushy zone under the influence of magnetic fields and gradients; Control of solidification of conducting and non – conducting materials using tailored magnetic fields – B.Ganapathysubramanian and Zabaras (2004-05)

17. Materials Process Design and Control Laboratory B Mushy zone MELT SOLID g qsqs Solidification of a metallic alloy inside a cavity in terrestrial gravity conditions– heat removal from left Strong thermosolutal convection present – drives convection during solidification Application of magnetic field on an electrically conducting moving fluid produces additional body force – Lorentz force. This force is used for damping flow during solidification of electrically conducting metals and alloys. The main aim of the current study is to investigate its effect on macro- - segregation during alloy solidification. PROBLEM DEFINITION qlql

18. Materials Process Design and Control Laboratory NUMERICAL MODEL Single domain model based on volume averaging is used. Single set of transport equations for mass, momentum, energy and species transport. Individual phase boundaries are not explicitly tracked. Complex geometrical modeling of interfaces avoided. Single grid used with a single set of boundary conditions. Solidification microstructures are not modeled here and empirical relationships used for drag force due to permeability. SALIENT FEATURES : Microscopic transport equations Volume- averaging process Macroscopic governing equations wkwk dA k (Ref: Gray et al., 1977)

19. Materials Process Design and Control Laboratory IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS Only two phases present – solid and liquid with the solid phase assumed to be stationary. The densities of both phases are assumed to be equal and constant except in the Boussinesq approximation term for thermosolutal buoyancy in the momentum equations. Interfacial resistance in the mushy zone modeled using Darcy’s law. The mushy zone permeability is assumed to vary only with the liquid volume fraction and is either isotropic or anisotropic. The solid is assumed to be stress free and pore formation is neglected. Material properties uniform (μ, k etc.) in an averaging volume dV k but can globally vary. TRANSPORT EQUATIONS FOR SOLIDIFICATION

20. Materials Process Design and Control Laboratory IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS Phenomenological cross effects – galvomagnetic, thermoelectric and thermomagnetic – are neglected. The induced magnetic field is negligible, only field is the externally applied field. Magnetic field assumed to be quasistatic. The current density is solenoidal. The external magnetic field is applied only in a single direction. The magnetic field is assumed to constant in space. Charge density is negligible MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS

21. Materials Process Design and Control Laboratory GOVERNING EQUATIONS Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Ganapathysubramanian B. and Zabaras (2004-05), Samanta and Zabaras, (2005) Magnetic damping force Thermosolutal buoyancy force term Continuity equation Momentum equation Darcy damping term Intertial and advective terms Pressure terms Viscous or diffusive terms

22. Materials Process Design and Control Laboratory GOVERNING EQUATIONS where : Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Ganapathysubramanian B. and Zabaras (2004-05), Samanta and Zabaras, (2005) Electric Potential equation Solute equation Energy equation convective term diffusive term Latent heat term Convective term diffusive term Transient term Transient term

23. Materials Process Design and Control Laboratory Anisotropic permeability (obtained experimentally and from regression analysis for directional solidification of binary alloys, Heinrich et al., 1997) Isotropic permeability (empirical relation based on Kozeny – Carman relationship) d = dendrite arm spacing – important microstructural parameter. ε = Volume fraction of liquid phase. PERMEABILITY EXPRESSIONS IN ALLOY SOLIDIFICATION K x = K y = K z = fn(ε,d) K x = K y ≠ K z K x = K y = fn 1 (ε,d) K z = fn 2 (ε,d)

24. Materials Process Design and Control Laboratory CLOSURE RELATIONSHIPS INVOLVING BINARY PHASE DIAGRAM Lever Rule : (Infinite back-diffusion) Scheil Rule : (Zero back-diffusion) ClCl C T ( assumed constant for all problems) Phase diagram relationships depend on the state of the alloy – solid, liquid or mushy. These relationships are used for obtaining mass fractions and solute concentrations of liquid and solid phases. Lead to strong coupling of the thermal and solutal problems.

25. Materials Process Design and Control Laboratory FEM based numerical techniques

26. Materials Process Design and Control Laboratory COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES In the presence of strong convection, standard FEM techniques lead to oscillations in solution. Inappropriate choice of interpolation functions for pressure and velocity can lead to oscillations in pressure. Stabilized FEM techniques – 1) prevent oscillations 2) allow the use of same finite element spaces for interpolating pressure and velocity for the fluid flow problem. Stabilizing terms take into account the dominant underlying phenomena (convective, diffusive or Darcy flow regimes) Convection stabilizing term Darcy drag stabilizing term Pressure stabilizing term Convection stabilizing term Fluid flow problem Thermal and species transport problems Standard Galerkin FE formulation Stabilizing terms Stabilized FE formulation +

27. Materials Process Design and Control Laboratory COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES Multistep Predictor – Corrector method used for thermal and solute problems. Backward – Euler fully implicit method for time discretization and Newton-Raphson method for solving fluid flow problem. Thermal and solutal transport problems along with the thermodynamic update scheme solved repeatedly in a inner loop in each time step. Fluid flow and electric potential problems decoupled from this iterative loop and solved only once in each time step. For the thermal and solute sub-problems, SUPG technique used for discretization. The fluid flow sub-problem is discretized using a modified form of the SUPG-PSPG technique (Tezduyar et al.) incorporating the effects of Darcy drag force in the mushy zone ( Ref:Zabaras and Samanta: 04,05 ). Both velocity and pressure and solved simultaneously and convergence rate is improved. Combination of direct and iterative solvers used to realize the transient solution.

28. Materials Process Design and Control Laboratory All fields known at time t n Advance the time to t n+1 Solve for the concentration field (solute equation) Solve for the temperature field (energy equation) Solve for liquid concentration, liquid volume fraction (Thermodynamic relations) Inner iteration loop Segregation model (Scheil rule) SOLUTION ALGORITHM AT EACH TIME STEP Is the error in liquid concentration and liquid mass fraction less than tolerance No Yes (Ref: Heinrich, et al.) COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES n = n +1 Solve for velocity and pressure fields (momentum equation) Decoupled momentum solution only once in each time step Check if convergence satisfied Solve for the induced electric potential (3D only)

29. Materials Process Design and Control Laboratory Numerical Examples 1) Solidification of an aqueous binary alloy – effect of convection (no magnetic fields). 2) Convection damping during horizontal solidification of a Pb-Sn alloy. 3) Convection damping during directional solidification of a Pb-Sn alloy.

30. Materials Process Design and Control Laboratory SOLIDIFICATION OF AN AQUEOUS BINARY ALLOY Ra T = 1.938x10 7 Ra C = -2.514x10 7 Temperature of hot wall, T hot = 311 K Temperature of cold wall, T cold = 223K Initial temperature, T 0 = 311K Initial concentration, C 0 = 0.7 Solutal flux on all boundaries = 0 (adiabatic flux condition) v x = v y = 0.0 on all boundaries Initial and boundary conditions (a) (b) (c)(d) (a)Velocity and mass fraction (b) isotherms (c) solute concentration (d) liquid solute concentration Thermal solutal convection is very strong and large scale solute distribution occurs Effect of thermosolutal convection seen in all other fields

31. Materials Process Design and Control Laboratory DAMPING CONVECTION IN HORIZONTAL ALLOY SOLIDIFICATION Mushy zone MELT SOLID g q s = h(T – T amb ) Solidification of Pb – 10% Sn alloy studied under the influence of magnetic fields (initial temperature = 600 K) (Ra T = -2.678x10 7 Ra C = 4.941x10 8 ). This alloy is characterized by a large mushy zone and strong convection. Macrosegregation is severe and extent of segregated zone is large. A magnetic field of 5 T applied in the z direction. Lorentz force responsible for convection damping. Effect of Lorentz force on macrosegregation to be studied. L = 0.08 m H = 0.02 m

32. Materials Process Design and Control Laboratory (a) No magnetic field (b) A magnetic field of 5 T in the z direction (i) Isotherms (ii) velocity vectors and liquid mass fractions (iii) isochors of Sn (iv) liquid solute concentration (iv) (i) (ii) (iii) HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn)

33. Materials Process Design and Control Laboratory FRECKLE SUPPRESSION IN 2D DIRECTIONAL SOLIDIFICATION u x = u z = 0  T/  t = r  T/  x = 0  T/  z = G  C/  x = 0 T(x,z,0) = T 0 + Gz C(x,z,0) = C 0 Mushy zone permeability assumed to be anisotropic Formation of freckles and channels due to thermosolutal convection Lorentz force occurs once magnetic field is applied. Important parameters L x B = 0.04m x 0.007m C 0 = 10% by weight Tin (Sn) Insulated boundaries on the rest of faces g Direction of solidification Constant magnetic field of 3.5 T applied in x direction B0B0 (Ra C = 6.177x10 7 )

34. Materials Process Design and Control Laboratory (b) (i) C Sn (ii) f l (i) C Sn (ii) f l (a) (a) No magnetic field (b) Magnetic field (3.5 T) Significant damping of convection throughout the cavity Freckle formation is totally suppressed  homogeneous solute distribution (a) ΔC = C max – C min = 2.63 wt %Sn (t = 800 s) (b) ΔC = C max – C min = 1.3 wt % Sn (t = 800 s) FRECKLE SUPPRESSION IN 2D DIRECTIONAL SOLIDIFICATION

35. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD Mushy zone MELT SOLID g qsqs L H Micro-gravity based growth is purely diffusion based Objective is to achieve some sort of reduced gravity growth qlql Growth under diffusion dominated conditions leads to : A uniform solute concentration profile due to reduced convection. Reduction of defects and sites of fatigue cracking. Uniform properties in the final cast alloy. Reduction in rejection rate of cast alloy components

36. Materials Process Design and Control Laboratory Time varying magnetic fields Temporal variations in thermosolutal convection Non-linear finite dimensional optimal control problem to determine time variation Design parameter set {b} = {b 1 b 2, …,b n } Measure of convection in the entire domain and time interval considered Cost Functional: Minimization of this cost functional yields design parameter set that leads to a growth regime where convection is minimized. OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD

37. Materials Process Design and Control Laboratory Define the inverse solidification problem as an unconstrained spatio – temporal optimization problem Find a quasi – solution : B ({b} k ) such that J(B{b} k )  J(B{b})  {b}; an optimum design variable set {b} k sought Gradient of the cost functional: Sensitivity of velocity field : n sensitivity problems to be solved Gradient information Obtained from sensitivity field Direct Problem Continuum sensitivity equations Design differentiate with respect to Non – linear conjugate gradient method OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD

38. Materials Process Design and Control Laboratory DESIGN OBJECTIVES Find the optimal magnetic field B(t) in [0,t max ]determined by the set {b} such that, in the presence of coupled thermosolutal buoyancy, and electromagnetic forces in the melt, diffusion dominated growth is obtained leading to minimum macrosegregation in the cast alloy Direct Problem Single domain volume averaged equations for alloy solidification Differentiate with respect to design parameters Discretize in space and time Continuum sensitivity equations Optimization problem Sensitivity of each variable with respect to the design parameters Gradient information, step size in nonlinear CG algorithm OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD

39. Materials Process Design and Control Laboratory (t) CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION T = T i = 580 K C = C 0 = 10 % by wt. Sn Isotropic permeability (Kozeny Carman relationship) L = 0.08 m H = 0.02 m

40. Materials Process Design and Control Laboratory CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION Time varying optimal magnetic field Cost functional 5 design variables (5 CSM problems solved)

41. Materials Process Design and Control Laboratory CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION (a) (b) (c) (i) No magnetic field Comparison of results at time t = t max = 120 s (a) Isotherms (b) Solute concentration (c) Liquid mass fractions and velocity vectors - Convection is almost fully damped throughout the solidification process. - Significant reduction in macrosegregation - Use of a time varying optimal magnetic field results in a near diffusion based growth - Near homogeneous solute concentration profile obtained. (ii) Optimal magnetic field

42. Materials Process Design and Control Laboratory CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION Time (in sec) Maximum velocity magnitude (mm/s) – No magnetic field Maximum velocity magnitude (mm/s) – Optimal magnetic field 40.0 120.0 74.9 96.9 0.310 0.170 Time (in sec) ΔC = C max – C min (wt % Sn) - No magnetic field ΔC = C max - C min (wt % Sn) - Optimal magnetic field 40.0 120.0 10.04 17.57 0.85 1.52 Time (in sec) Elemental Peclet number (Pe) – No magnetic field Elemental Peclet number (Pe) – Optimal magnetic field 40.0 120.0 6.810 8.811 0.028 0.012 Elemental length, h = 1 x 10 -3 m Convection damping Macro- -segregation suppression Diffusion dominated growth regime

43. Materials Process Design and Control Laboratory z x y  T/  z = G  T/  t = r g Direction of solidification Mushy zone permeability assumed to be anisotropic Formation of freckles and channels due to thermal – solutal convection Lorentz force – primary damping force once magnetic field is applied. Important parameters L x B x H = 0.01m x 0.01m x 0.02m C 0 = 10% by weight Tin v x = v y = v z = 0 on all surfaces A magnetic field applied in x Insulated boundaries on the rest of faces FRECKLE SUPPRESSION IN 3D DIRECTIONAL SOLIDIFICATION (Ra C = 7.721x10 6 )

44. Materials Process Design and Control Laboratory - Formation of freckles in the absence of magnetic field (t = 800 s). - Thermosolutal convection is strong and leads to large scale solute distribution (at t = 400 s, ΔC = C max – C min = 5.97 wt. % Sn at t = 800 s, ΔC = C max – C min = 7.4 wt. % Sn) (a) Concentration of Sn (t = 800 s) (b) Liquid Mass fraction (t = 800 s) FRECKLE SUPPRESSION IN 3D DIRECTIONAL SOLIDIFICATION

45. Materials Process Design and Control Laboratory FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD Time varying optimal magnetic field Cost functional 4 design variables (4 CSM problems solved)

46. Materials Process Design and Control Laboratory FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD - Complete suppression of freckles in the presence of optimal time varying magnetic field. - Thermosolutal convection causing convection is totally suppressed. - Homogeneous solute distribution in the solidifying alloy (at t = 400 s, ΔC = C max – C min = 0.4 wt. % Sn at t = 800 s, ΔC = C max – C min = 1.65 wt. % Sn) (a)Concentration of Sn (t = 800 s) (b) Liquid mass fraction (t = 800 s)

47. Materials Process Design and Control Laboratory Surface defect formation in Aluminum alloys

48. Materials Process Design and Control Laboratory Aluminum industry relies on direct chill casting for aluminum ingots. Aluminum ingots are often characterized by defects in surface due to non-uniform heat extraction, improper contact at metal/mold interface, inverse segregation, air-gap formation and meniscus freezing etc. These surface defects are often removed by post casting process: such as scalping/milling. Post-processing leads to substantial increase of cost, waste of material and energy. The purpose of this work is to reduce scalp-depth in castings. Detailed understanding of the highly coupled phenomenon in the early stages of solidification is required. INTRODUCTION

49. Materials Process Design and Control Laboratory INTRODUCTION Surface defects in casting (Ref. ALCOA corp.) (a) (c) (b) (a)Sub-surface liquation and crack formation on top surface of a cast (b) Non-uniform front and undesirable growth with non-uniform thickness (left) and non-uniform microstructure (right) (c) Ripple formation

50. Materials Process Design and Control Laboratory Engineered mold surface (Ref. ALCOA Corp.) In industry, the mold surface is pre-machined to control heat extraction in directional solidification. This periodic groove surface topography allows multi-directional heat flow on the metal-mold interface. However, the wavelengths should be with the appropriate value to obtain anticipated benefits. INTRODUCTION Uniform front growth (left) and uniform microstructure (right) – obtained using grooved molds

51. Materials Process Design and Control Laboratory Numerical model for deformation of solidifying alloys

52. Materials Process Design and Control Laboratory PREVIOUS WORK Zabaras and Richmond (1990,91) – hypoelastic rate-dependent small deformation model to study the deformation of solidifying body. Rappaz (1999), Mo (2004) – deformation in mushy zone with a volume averaging model: Continuum model for deformation of mushy zone in a solidifying alloy and development of a hot tearing criterion. Mo et al. (1995-98) – Surface segregation and air gap formation in DC cast Aluminum alloys. Hector and Yigit (2000) – semi – analytical studies of air gap nucleation during solidification of pure metals using a hypoelastic perturbation theory. Hector and Barber (1994,95) – Effect of strain rate relaxation on the stability of solid front growth morphology during solidification of pure metals. Chen et al. (1991 – 93), Heinrich et al. (1993,97) – Inverse segregation caused by shrinkage driven flows or combined shrinkage and buoyancy driven flows during alloy solidification. A thermo-mechanical study of the effects of mold topography on the solidification of Al alloys - Tan and Zabaras (2005)

53. Materials Process Design and Control Laboratory PROBLEM DEFINITION Solidification of Aluminum-copper alloys on sinusoidal mold surfaces. With growth of solid shell, air – gaps form between the solid shell and mold due to imperfect contact – which leads to variation in thermal boundary conditions. The solid shell undergoes plastic deformation and development of thermal and plastic strain occurs in the mushy zone also. Inverse segregation caused by shrinkage driven flow affects variation in air – gap sizes, front unevenness and stresses developing in the casting.

54. Materials Process Design and Control Laboratory Fluid flow Heat transfer Casting domain Heat transfer Mold Contact pressure/ air gap criterion Solute transport Inelastic deformation Phase change and mushy zone evolution Deformable or non-deformable mold SCHEMATIC OF THE HIGHLY COUPLED SYSTEM There is heat transfer and deformation in both mold and casting region interacting with the contact pressure or air gap size between mold and casting. The solidification, solute transport, fluid flow will also play important roles.

55. Materials Process Design and Control Laboratory GOVERNING TRANSPORT EQUATIONS FOR SOLIDIFICATION Initial conditions : Isotropic permeability : Continuity equation Momentum equation Energy equation Solute equation

56. Materials Process Design and Control Laboratory MODEL FOR DEFORMATION OF SOLIDIFYING ALLOY For deformation, we assume the total strain to be decomposed into three parts: elastic strain, thermal strain and plastic strain. Elastic strain rate is related with stress rate through an hypo-elastic constitutive law Plastic strain evolution satisfy this creep law with its parameters determined from experiments (Strangeland et al. (2004)). The thermal strain evolution is determined from temperature decrease and shrinkage. Strain measure : Elastic strain Thermal strain Plastic strain

57. Materials Process Design and Control Laboratory MODELING DEFORMATION IN MUSHY ZONE Liquid or low solid fraction mush - zero thermal and plastic strains. (Without any strength) Solid or high solid fraction mush - thermal and plastic strains start developing gradually. The parameter w is defined as: Low solid fractions usually accompanied by melt feeding and no deformation due to weak or non – existent dendrites  leads to zero thermal strain. With increase in solid fraction, there is an increase in strength and bonding ability of dendrites  to non – zero thermal strain. The presence of a critical solid volume fraction is observed in experiment and varies for different alloys.

58. Materials Process Design and Control Laboratory IMPORTANT PARAMETERS FOR DEFORMATION IN MUSHY ZONE Volumetric thermal expansion coefficient Volumetric shrinkage coefficient Strain-rate scaling factor Stress scaling factor Activation energy Creep law exponent Mushy zone softening parameter Creep law for plastic deformation Ref. Strangeland et al. (2004) Critical solid fraction for different copper concentrations in aluminum-copper alloy Ref: Mo et al.(2004) Al – Cu – 0.3 pct Mg

59. Materials Process Design and Control Laboratory THERMAL RESISTANCE AT THE METAL-MOLD INTERFACE Contact resistance: At very early stages, the solid shell is in contact with the mold and the thermal resistance between the shell and the mold is determined by contact conditions Example: Aluminum-Ceramic Contact Before gap nucleation, the thermal resistance is determined by pressure After gap nucleation, the thermal resistance is determined by the size of the gap Heat transfer retarded due to gap formation Uneven contact condition generates an uneven thermal stress development and accelerates distortion or warping of the casting shell.

60. Materials Process Design and Control Laboratory MOLD – METAL BOUNDARY CONDITIONS The actual air – gap sizes or contact pressure are determined from the contact sub problem. This modeling of heat transfer mechanism due to imperfect contact is very crucial for studying the non-uniform growth at early stages of solidification. Consequently, heat flux at the mold – metal interface is a function of air gap size or contact pressure: = Air-gap size at the interface = Contact pressure at the interface

61. Materials Process Design and Control Laboratory The thermal problem is solved in a region consisting of both mold and casting to account for non-linear (contact pressure/air gap dependent) boundary conditions at the mold – metal interface. Deformation problem is solved in both casting and mold (if mold deformable) or only the casting (if mold rigid, for most of our numerical studies). Solute and momentum transport equations is only solved in casting with multistep predictor – Corrector method for solute problems, and Newton-Raphson method for solving heat transfer, fluid flow and deformation problems. Backward – Euler fully implicit method is utilized for time discretization to make the numerical scheme unconditionally stable. The contact sub-problem is solved using augmentations (using the scheme introduced by Laursen in 2002). All the matrix computations for individual problems are performed using the parallel iterative Krylov solvers based on the PETSc library. COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

62. Materials Process Design and Control Laboratory All fields known at time t n Advance the time to t n+1 Solve for the concentration field (solute equation) Solve for the temperature field (energy equation) Solve for liquid concentration, mass fraction and density (Thermodynamic relations) Inner iteration loop Segregation model (Scheil rule) Is the error in liquid concentration and liquid mass fraction less than tolerance No Solve for velocity and pressure fields (momentum equation) Yes (Ref: Heinrich, et al.) Decoupled momentum solver SOLUTION ALGORITHM AT EACH TIME STEP n = n +1 Solve for displacement and stresses in the casting (Deformation problem) Contact pressure or air gap obtained from Contact sub-problem Check if convergence satisfied Convergence criteria based on gap sizes or contact pressure in iterations

63. Materials Process Design and Control Laboratory Numerical examples

64. Materials Process Design and Control Laboratory SOLIDIFICATION OF Al-Cu ALLOY ON UNEVEN SURFACES Deformation problem Heat Transfer (Mold is rigid and non- deformable) Solidification problem We carried out a parametric analysis by changing these four parameters 1) Wavelength of surfaces (λ) 2) Solute concentration (C Cu ) 3) Melt superheat (ΔT melt ) 4) Mold material (Cu, Fe and Pb) Both the domain sizes are on the mm scale Combined thermal, solutal and momentum transport in casting. Assume the mold is rigid. Imperfect contact and air gap formation at metal – mold interface

65. Materials Process Design and Control Laboratory SOLIDIFICATION COUPLED WITH DEFORMATION AND AIR-GAP FORMATION Because of plastic deformation, the gap formed initially will gradually decrease. As shown in the movies, a 1mm wavelength mold would lead to more uniform growth and less fluid flow. Important parameters 1) Mold material - Cu 2) C Cu = 8 wt.% 3) ΔT melt = 0 o C Air gap is magnified 200 times. Preferential formation of solid occurs at the crests and air gap formation occurs at the trough, which in turn causes re-melting.

66. Materials Process Design and Control Laboratory TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 3 mm) (a)Temperature (b)Solute concentration (c)Equivalent stress (d) Liquid mass fraction Important parameters 1) Mold material - Cu 2) C Cu = 5 wt.% 3) ΔT melt = 0 o C We take into account solute transport and the densities of solid and liquid phases are assumed to be different. Inverse segregation, caused by shrinkage driven flow, occurs at the casting bottom.This is observed in (b). (d) (c) (b) (a)

67. Materials Process Design and Control Laboratory TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 5 mm) For other wavelengths, similar result is observed: (1) preferential formation of solid occurs at the crests (2) remelting at the trough due to the formation of air gap. For λ = 3mm, the solid shell unevenness decreases faster than for λ = 5mm. (d) (c) (b) (a) (a)Temperature (b)Solute concentration (c)Equivalent stress (d) Liquid mass fraction

68. Materials Process Design and Control Laboratory Max. equivalent stress σ eq variation with λ σ eq first increases and then decreases Initially, σ eq is higher for greater λ Later (t=100 ms), stress is lowest for 5 mm wavelength. Air-gap size variation with wavelength λ Initially, air-gap sizes nearly same for different λ At later times, air-gap sizes increase with increasing λ ΔT melt = 0 o C, C Cu = 5 wt.%, mold material = Cu VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS

69. Materials Process Design and Control Laboratory VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS σ eq first increases and then decreases Variation of σ eq with Cu concentration is negligible after initial times Air-gap sizes increase with time Increasing Cu concentration leads to increase in air-gap sizes ΔT melt = 0 o C, λ = 5 mm, mold material = Cu Increase of solute concentration leads to increase in air-gap sizes, but its effect on stresses are small.

70. Materials Process Design and Control Laboratory EFFECT OF INVERSE SEGREGATION – AIR GAP SIZES Differences in air-gap sizes for different solute concentrations are more pronounced in the presence of inverse segregation. (a) With inverse segregation(b) Without inverse segregation inverse segregation actually plays an important role in air-gap evolution.

71. Materials Process Design and Control Laboratory Value of front unevenness and maximum equivalent stress for various wavelengths one cannot simultaneously reduce both stress and front unevenness when wavelength greater than 5mm, both unevenness and stress increase  wavelength less than 5 mm is optimum Equivalent stress at dendrite roots The highest stress observed for 1.8% copper alloy suggests that aluminum copper alloy with 1.8% copper is most susceptible to hot tearing Phenomenon is also observed experi- mentally Rappaz(99), Strangehold(04) VARIATION OF EQUIVALENT STRESSES AND FRONT UNEVENNESS Time t = 100 ms

72. Materials Process Design and Control Laboratory CONCLUSIONS AND OBSERVATIONS Magnetic fields are successfully used to damp convection during solidification of metallic alloys in terrestrial gravity conditions. Near homogeneous solute element distributions obtained. Suppression of freckle defects during directional solidification of alloys achieved. An optimization problem solved to determine time varying magnetic fields that damp convection and minimize macrosegregation in solidifying alloys. Optimal time varying magnetic fields take into account variations in thermosolutal convection – superior to constant magnetic fields. Reduction in current and power requirements possible.

73. Materials Process Design and Control Laboratory Early stage solidification of Al-Cu alloys significantly affected by non – uniform boundary conditions at the metal mold interface. Variation in surface topography leads to variation in transport phenomena, air-gap sizes and equivalent stresses in the solidifying alloy. Air-gap nucleation and growth significantly affects heat transfer between metal and mold. Distribution of solute primarily caused by shrinkage driven flows and leads to inverse segregation at the casting bottom. Presence of inverse segregation leads to an increase in gap sizes and front unevenness. Effects of surface topography more pronounced for a mold with higher thermal conductivity Computation results suggest that an Al-Cu alloy with 1.8% Cu is the most susceptible to hot tearing defects. An optimum mold wavelength should be less than 5mm. Overall aim is to develop techniques to reduce surface defects in Al alloys by modifying mold surface topography. CONCLUSIONS AND OBSERVATIONS

74. Materials Process Design and Control Laboratory Suggestions for future research

75. Materials Process Design and Control Laboratory MULTILENGTH SCALE SOLIDIFICATION MODELING Ability to resolve morphology of microstructural entities like dendrites. Effect of various instabilities on the actual growth front morphology can be studied. Key to understanding effects of micro – scale phenomenon on macro scale and vice – versa. Can avoid very fine grids for macro – scale simulations. Lay the foundations of multi – length scale robust design. Importance of multi – length scale modeling Large scale casting metres Small scale phenomena Dendritic growth Macro defects Degrade quality of casting Macroscopic transport phenomenon Evolution of microstructure Multi – length scale solidification model

76. Materials Process Design and Control Laboratory compute average of physical quantities on fine scale Macro scale grid compute equivalent parameters (permeability) MULTILENGTH SCALE SOLIDIFICATION MODELING Macroscopic governing equations based on volume averaging to compute fields like velocity, temperature and solute concentration Boundary conditions for micro problem Microstructure evolution model On each element micro scale grid 1) Interface temperature condition 2) Thermal Flux jump at the interface 3) Concentration flux jump at the interface 4) Thermal, solutal and momentum problems in the liquid phase 5) Thermal problem in the solid phase. 6) Curvature effects 7) Tracking the interface position Averaging techniques Upscaling methods based on homogenization Allows variation of macro variables on both scales Multi- length scale direct problem

77. Materials Process Design and Control Laboratory Multi-length scale design problem Design macro variables (magnetic field or boundary heat flux to get a desired microstructure MULTILENGTH SCALE SOLIDIFICATION MODELING Multi-length scale direct problem Volume – averaged continuum macro model Microstructure evolution model Probabilistic nucleation model Averaging/ upscaling techniques CSM based optimization method Cast components with desired properties and microstructure

78. Materials Process Design and Control Laboratory MICROSTRUCTURE EVOLUTION DURING EARLY STAGE SOLIDIFICATION Microstructure evolution Surface parameters and mold topography in transport processes Interfacial heat transfer Varying stresses in solid Inverse segregation Air gap formation (non uniform contact and shell remelting) Metal/mold interaction Shell growth kinetics uneven growth distortion Combined effect of several phenomena on microstructure evolution (during early stages of solidification)

79. Materials Process Design and Control Laboratory THANK YOU FOR YOUR ATTENTION