1. Materials Process Design and Control Laboratory COUPLED THERMOMECHANICAL, THERMAL TRANSPORT AND SEGREGATION ANALYSIS OF ALUMINUM ALLOYS SOLIDIFYING ON UNEVEN SURFACES Lijian Tan, Deep Samanta and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/zabaras /

2. Materials Process Design and Control Laboratory RESEARCH SPONSORS DEPARTMENT OF ENERGY (DOE) Industry partnerships for aluminum industry of the future - Office of Industrial Technologies ALUMINUM CORPORATION OF AMERICA (ALCOA) Ingot and Solidification Platform – Alcoa Technical Center CORNELL THEORY CENTER

3. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Brief introduction and motivation of the current study  Numerical model to study deformation of solidifying alloys  Closure criteria  Computational strategies for solving the coupled numerical system  Numerical examples: – preliminary studies of deformation of solidifying alloys – parametric investigations of solidification from molds with uneven mold topography (coupled thermal, solutal and momentum transport)  Conclusions  Future Work

4. Materials Process Design and Control Laboratory Introduction and motivation of the current study

5. Materials Process Design and Control Laboratory Surface defects in casting (Ref. ALCOA Corp.) (a) (b) (a) Sub-surface liquation and crack formation on top surface of a cast (b) Non-uniform front and undesirable growth with non-uniform shell thickness INTRODUCTION

6. 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

7. Materials Process Design and Control Laboratory Engineered mold surface (Ref. ALCOA Corp.) In industry, the mold surface is pre-machines 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

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

9. Materials Process Design and Control Laboratory SHEMATIC OF THE PROBLEM DEFINITION An Aluminum-copper alloy is solidified on an sinusoidal uneven surface. With growth of solid shell, air – gaps form between the solid shell and mold due to imperfect contact – which further leads to variation in 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 causes variation in air – gap sizes, front unevenness and stresses developing in the casting.

10. Materials Process Design and Control Laboratory SCHEMATIC OF THE HIGHLY COUPLED SYSTEM Casting domain Fluid flow Heat transfer Mold Contact pressure or air gap criterion Solute transport Inelastic deformation Heat transfer Phase change and mushy zone evolution Deformation or mold non-deformable There are 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.

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

12. Materials Process Design and Control Laboratory SALIENT FEATURES OF OUR NUMERICAL MODEL Volume averaging with a single domain and 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 A rate dependent hypo-elastic visco-plastic model is used for deformation of solid shell and mushy zone Dynamic air gap – contact pressure coupling at the mold – metal interface On the whole, a highly coupled model combining solidification and deformation in the casting is used.

13. Materials Process Design and Control Laboratory GOVERNING TRANSPORT EQUATIONS FOR SOLIDIFICATION (Ref :Shyll and Udaykumar, 1996) (Ref: C. Beckermann et al., explicit modeling of Interfacial terms) (Ref: Incropera, 1987-2000 mixture theory) Initial conditions : Isotropic permeability :

14. Materials Process Design and Control Laboratory CLOSURE RELATIONSHIPS FOR FINDING CONCENTRATION AND FRACTION Lever Rule : (Infinite back-diffusion) Scheil Rule : (Zero back-diffusion) ClCl C T ( assumed constant for all problems)

15. Materials Process Design and Control Laboratory MODELING DEFORMATION IN MUSHY ZONE Liquid or low solid fraction mush - any deformation induced by thermal expansion is permanent. (Without any strength) Solid or high solid fraction mush - plastic deformation is developed only 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.

16. Materials Process Design and Control Laboratory MODEL FOR DEFORMATION OF SOLIDIFYING ALLOY For deformation, we assume the total strain can 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

17. Materials Process Design and Control Laboratory Parameters for simulation of 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)

18. Materials Process Design and Control Laboratory THERMAL RESISTANCE AT THE METAL-MOLD INTERFACE Contact resistance: At the 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 the 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 may accelerates distortion or warping of the casting shell.

19. 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 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

20. 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

21. 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 Larsen 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

22. Materials Process Design and Control Laboratory Numerical examples

23. Materials Process Design and Control Laboratory SOLIDIFICATION OF Al ON UNEVEN SURFACES Hypoelastic model without plastic deformation (Hector et al. 2000) Heat transfer in the mold, solid shell and melt. Heat transfer causes deformation (thermal stress). Gaps or contact pressure affect heat transfer. Solidification after air-gap nucleation not modeled.

24. Materials Process Design and Control Laboratory GAP NUCLEATION TIME: EFFECTS OF WAVELENGTH At the very early stages of aluminum solidification, contact pressure between mold and solid shell will drop at the trough due to thermal stress development. When this contact pressure drops to zero, gap nucleation is assumed to take place. This study compares very well with Hector’s semi-analytical study. It shows that gap nucleation is faster for smaller wavelength, smaller liquid pressure and better heat conductivity of the mold.. For rigid mold (with an topography amplitude=1 µm, wavelength=1-5 mm), under liquid pressure 8000 Pa, the gap nucleation time is in the order of seconds. Physical Conditions: Liquid pressure P=8000 Pa Thermal resistance at mold-shell interface R=10 -5 m 2 o C sec J -1

25. Materials Process Design and Control Laboratory GAP NUCLEATION TIME: EFFECTS OF MOLD CONDUCTIVITY Mold conductivity affects gap nucleation time The higher the conductivity, the quicker the gaps nucleate from the mold surface In this calculations, the deformation of the mold is neglected to illustrate the effects of mold conductivity. Physical conditions: Liquid pressure P=10000 Pa Mold thickness h=0.5 mm Thermal resistance at mold-shell interface R=10 -5 m 2 o C sec J -1 Wavelength=2 mm

26. Materials Process Design and Control Laboratory When the wavelength is relatively small, the evolution of the contact pressure at the trough is mainly affected by the conductivity of the mold, i.e. the deformation of the mold does not play a crucial role. GAP NUCLEATION TIME: EFFECTS OF MOLD MATERIAL ( deformable mold ) Physical Conditions: Liquid pressure P=10000 Pa Mold thickness h=0.5 mm Thermal resistance at mold-shell interface R=10 -5 m 2 o C sec J -1 Wavelength=10 mm, (20 mm, 30 mm in the next two slides)

27. Materials Process Design and Control Laboratory When the wavelength increases, the Ptr-t line is about to show a turn-around pattern when pressure reaches zero. This is defined as the `critical wavelength’ in the analytical studies of L. Hector. From this figure, we can say that the critical wavelength is slightly above 20 mm. In Hector’s analytical study, the critical wavelength is 16.60 mm, for iron mold and 14.03 mm for lead mold under the same conditions. GAP NUCLEATION TIME: EFFECTS OF MOLD MATERIAL (deformable mold)

28. Materials Process Design and Control Laboratory Notice that when the wavelength is greater than the critical value, the pressure-time curve shows a turn- around pattern before the contact pressure reaches zero. GAP NUCLEATION TIME: EFFECTS OF MOLD MATERIAL (deformable mold) This implies that a large wavelength is preferred since the contact pressure won’t decrease to zero to generate gap nucleation. But in practice, we can never get a such a smooth mold topography with amplitude 1 µm and wavelength 30 mm as in these examples. Gap nucleation occurs for most casting processes.

29. 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 change 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

30. 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.

31. Materials Process Design and Control Laboratory TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 5 mm) (a)Temperature (b)Solute concentration (c)Equivalent stress (d) Liquid mass fraction (d) (c) (b) (a) 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).

32. Materials Process Design and Control Laboratory TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 3 mm) (d) (c) (b) (a) (a)Temperature (b)Solute concentration (c)Equivalent stress (d) Liquid mass fraction For smaller 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 wavelength 3mm, the solid shell unevenness decreases faster than the case of 5mm wavelength.

33. Materials Process Design and Control Laboratory VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS Air-gap sizes increase with time Increasing melt superheat leads to some suppression of air gaps Initially, stresses higher for lower superheat At later times, the difference is small λ = 5 mm, C Cu = 5 wt.%, mold material = Cu Increasing melt superheat leads to some suppression of air gaps and a smaller stress at beginning stages. At later times, the difference of equivalent stresses is however small.

34. Materials Process Design and Control Laboratory EFFECT OF WAVELENGTH ON AIR-GAP SIZES AND MAX EQUIVALENT STRESS 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

35. 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.

36. Materials Process Design and Control Laboratory VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS Equivalent stress far lower for Cu molds than Fe or Pb molds Air gap sizes higher for Cu molds than Fe or Pb molds ΔT melt = 0 o C, λ = 5 mm, C Cu = 5 wt.% Gap nucleation and stress development are prominent for a mold of higher thermal conductivity like Cu. For Fe or Pb molds, heat removal is inhibited due to their lower thermal conductivity. This in turn inhibits air-gap formation and development of stresses..

37. 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 By comparing the result with modeling inverse segregation and without modeling inverse segregation, we can find that inverse segregation actually plays an important role in air- gap evolution.

38. 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 the wavelength greater than 5mm, both unevenness and stress increase-> implies wavelength less than 5 mm is optimum Equivalent stress at dendrite roots The highest stress observed for 1.8% copper alloy suggest 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

39. Materials Process Design and Control Laboratory EFFECTS OF SURFACE ROUGHNESS AND MOLD COATINGS Effect of uneven surface topography and non – uniform contact on microstructure evolution. Incorporating the effects of surface tension and surface coatings to study solidification on microscale. Studying the effects of surface roughness on solidification on microscale. Optimal design of a mold surface topography to minimize surface defects.

40. Materials Process Design and Control Laboratory PRELIMINARY STUDY OF EFFECTS OF SURFACE TENSION Materials Process Design and Control Laboratory In the macro-scale, the liquid pressure exerted by the droplet can overcome surface tension and causes the molten Aluminum droplet to contact the bottom of the cavity.

41. Materials Process Design and Control Laboratory EFFECT OF SURFACE TENSION Materials Process Design and Control Laboratory However, in the microscale, a change of surface tension could drastically change the solidification speed at very early stages of solidification. This suggests taking account of surface tension in our future study is very important.

42. Materials Process Design and Control Laboratory CURRENT AND FUTURE RESEARCH Microstructure evolution Lap marks, ripples, cold shuts Surface parameters and mold topography in transport processes Varying stresses in solid Air gap formation (non uniform contact and shell remelting) Meniscus instability Metal/mold interaction Shell growth kinetics uneven growth distortion Macrosegregation Inverse segregation Interfacial heat transfer

43. Materials Process Design and Control Laboratory CONCLUSIONS 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. Effect of melt pressure on solidification beyond gap nucleation was found to be negligible. Effects of surface topography more pronounced for a mold with higher thermal conductivity Computation results suggests that aluminum copper alloy with 1.8% copper is most susceptible for 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.

44. Materials Process Design and Control Laboratory RELEVANT PUBLICATIONS D. Samanta and N. Zabaras, “A numerical study of macrosegregation in Aluminum alloys solidifying on uneven surfaces”, in press in International Journal of Heat and Mass Transfer. L. Tan and N. Zabaras, “A thermomechanical study of the effects of mold topography on the solidification of Aluminum alloys”, in press in Materials Science and Engineering: A. D. Samanta and N. Zabaras, “A coupled thermomechanical, thermal transport and segregation analysis of the solidification of Aluminum alloys on molds of uneven topographies ”, submitted for publication in the Materials Science and Engineering: A. CONTACT INFORMATION http://mpdc.mae.cornell.edu/