1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5

1. A multi-component theory ofsolid mixtures with higher gradientsand its application to binary alloys byA. Brandmair,1) T. Böhme,1),2) W. Dreyer,3)W.H. Müller1)STAMM 2008 Symposium on Trendsin Applications of Mathematics to MechanicsLevico Italy, September 24, 2008 3) Weierstraß-Institut für Angewandte Analysis und Stochastik Mohrenstr. 39 D-10117 Berlin 2) ThyssenKrupp Steel AG Werkstoffkompetenzzentrum Physikalische Technik Kaiser-Wilhelm-Straße 100 D-47166 Duisburg 1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5 D-10587 Berlin German Federal Environmental Foundation

2. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

3. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

4. Microstructural changes in solids I SMT SnPb solder joints: formation of interface cracks spinodal decompositionmicrostructural coarsening Ball Grid Arrays and solder ball before and after 4000 temperature cycles aging at RT after (a) 2h, (b) 17d and (c) 63d (a) after solidification, (b) 3h and (c) 300 h at 125°C MELF miniature resistor and solder joint before and after 3000 temperature cycles Will and, if so, how will the microstructural change influence the material properties ?

5. Introduction: Microstructural changes in solids II decomposition + coarsening in the bulk Leadfree solder, e.g., AgCu28: Ag-Cu: aging at 1000 Kelvin cracks along the phase boundaries Ag-rich Cu-rich Will and, if so, how will the microstructural change influence the material properties ? after solidification after 2h after 20h after 40h Cu with lead-free bumps SnPb solder balls Leadfree solder, e.g., SnAg3.8, SnAg3.8Cu0.7: Formation of Inter-Metallic Compounds IMCs (Cu6Sn5, Ag3Sn)at the interface and in the bulk thor.inemi.org/webdownload/newsroom/Articles/Lead-Free_Watch_Series/Oct05.pdf

6. Introduction: Microstructural changes in solids III • Triggered by different diffusion coefficients Cu atoms move to pile up IMCs. • Kirkendall voids appear, e.g., between Cu substrate and thin Cu3Sn layer due to migration of Cu atoms from Cu3Sn to Cu6Sn5, which is much faster than Sn-diffusion from Cu6Sn5 towards Cu3Sn. • Unbalanced Cu-Sn interdiffusion generates atomic vacancies at lattice sites which coalesce to voids. • Model: vacancy diffusion Will and, if so, how will the microstructural change influence the material properties ?

7. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

8. Experiments I: Setup & realization • material►eutectic Ag-Cu • temperature► 970 K • aging time► 0 - 40 h • etching (for Ag)► solution of NH3-H2O2

9. after solidification after 2h aging after 20h aging after 40h aging Experiments II: Micrographs ► instantaneous decomposition ► coarsening (Ostwald ripening) ► light: α-phase (Ag-rich) , dark: β-phase (Cu-rich)

10. Experiments III: Image analysis • determination of the β-areasand the total number of β-precipitates N by means of Digital Image Analysis (DHSTM) • Attention: 2D analysis of a 3D problem (observation of one cross-section) • Solution: statistical averaging • sufficient large areas of intersection • analysis of various micrographs at each coarsening stage (individual photo) (individual stage) (spherical phases) (oblate spheroids) (cf., Underwood, 1970)

11. Experiments IV: Coarsening rates • faster coarsening rate for oblate spheroids with • and • t 1/3- dependence well-known from LSW-theories 1/3

12. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

13. motion displacements, velocity, deformation gradient strains and stresses Nomenclature I

14. constitutive equations for sought in a situation wheresolids separated byphase boundaries “move” toward equilibrium which, again, is characterized by a boundary→ higher gradienttheories required ! Nomenclature II • Primary variables: • Variablesdetermined by partial mass balances (w/o chemical reactions), andbalance of momentum and internal energy: • total mass balance (no external forces)

15. Constitutive relations for the diffusion flux (w/o thermo-diffusion) w/o gradients with density gradients(w/o strain gradients) ( : mobility) chemical potential Helmholtzfree energydensity functional derivative:

16. Constitutive relations for the stress tensor without gradients with density gradients(no strain gradients) Castigliano’s 2nd theorem specific Helmholtz free energy pressure

17. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

18. Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

19. R. Clausius P. M. M. Duhem Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

20. B.D. Coleman W. Noll A.E. Green P.M. Nagdhi I. Müller I-S. Liu Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

21. Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles mixtures: Duhem: radiation: ideal gas:

22. Clausius: Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

23. Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

24. Entropy principle I: Historic remarks Entropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a)Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires “additional” constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

25. 2nd law Entropy principle II: Formulation Two constitutive quantities: entropy density and entropy flux .Constitutive relation of the entropy density: Local balance for entropy density: Entropy production positive definite & of the form fluxes x driving forces: (Absolute) temperature defined as follows: Stress tensor decomposed into an elasticand adissipative part:

26. Entropy principle III: The evaluation procedure H.W. Alt, I. Pawlow: On the entropy principle of phase transition models with a conserved order parameter. Advances in Mathematical Science and Applications, 6(1), pp. 291–376, 1996: Balances interpreted as “evolution equations” for variables: Balances viewed as a system ofalgebraic equations, i.e., choose right hand sidesarbitrarily& calculate left hand sides. Right hand sides of the balances + chain rule  list of arbitrary terms: Construct constitutive relations such that 2nd law is identically satisfied.

27. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principleClassical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

28. Mixtures w/ohigher gradients I: Choice of variables Simple -component mixture without reactions / viscous effects: Different representations of entropy density: with a constant number of variables, e.g.: Different representations useful under different circumstances, e.g.:

29. Mixtures w/o higher gradients II: Entropy production Local entropy balance: Chain rule for Kinematic relation chain rule , right hand side of balances (note: balance of momentum not required for exploitation, since not in ) kinematic relation

30. Mixtures w/o higher gradients III: Entropy production Substitution and rearrangement: a) separate terms into flux / force: b) separate terms linear in (cf., arbitrary terms) Definition entropy-flux(first parenthesis): Residual inequality: terms linear in drop out Galileian invariance P Q

31. experimentallyinconvenient quantity Mixtures w/ohigher gradients IV: Heat & diffusion flux Q-bit Helmholtz free energy density (chemical potential) Legendre Transformation Constraint No coupling, quadratic form (Fourier’s law of heat conduction) ( : mobility)

32. Mixtures w/ohigher gradients V : Selected results P-bit Pressure Legendre transform applied to 2nd Piola-Kirchhoff stress tensor Legendre transform applied to Gibbs-Duhem relation

33. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradientsTheory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

34. Mixtures withhigher gradients I: Functional representations functional representation of the entropy density Note: - Choice of Higher Gradients depends on the problem - Present choice: Convenient for diffusion problems & for definition of the chemical potential

35. Mixtures withhigher grad. II: Specialization to binary alloy Now:Binary alloy A-B: Relating difference of chem. pot. to derivative of Helmholtz free energy density: with Helmholtz free energy density diffusion flux for a binary alloy: (isothermal case)

36. Mixtures withhigher grad. III: Specialization to binary alloy Transformation to the reference configuration / re-definition of mobility Problem with free energy density(more general case): Decomposition & Taylor expansion HGC can be identified & calculated by microscopic atomistic theories (e.g., EAM) it follows (Böhme et al., 2007): (periodic arrangement of the lattice) Approach for elastic contributions ? ? phase diagram leading term

37. Mixtures withhigher grad. IV: Specialization to binary alloy Partial mass balance & mass concentration : Material parameters:

38. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

39. Application: Spinodal decomposition in AgCu I: Simulations in 1D (no external stress) Study concentration development along a line

40. Application: Spinodal decomposition in AgCu II: Simulations in 1D (no external stress) Fortran 95, FFTPack, explicit Euler scheme

41. Application: Spinodal decomposition in Ag-Cu II: Simulations in 1D (tensile stress: 103 MPa) stresses accelerate coarsening Fortran 95, FFTPack, explicit Euler scheme

42. Ag-rich phase Cu-rich phase Application: Spinodal Decomposition in Ag-Cu III: 2D (no external stress) initial 3000 time loops 6000 loops 60 000 loops

43. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

44. Experiment and simulation: Coarsening rates By image analysis of experiments and computer generated microstructural evolution 3D simulations required ?

45. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

46. Average elastic properties by homogenization I Homogenization performed by S.V. Sheshenin, M. Savenkova (FE-program “Elast”) Boundary value problem (plane strain analysis) for a given micrograph (= RVE): Loading sequences applied, e.g.: etc.

47. Average elastic properties by homogenization II AgCu28 simulated temporal development of microstucture 1 2 3 4 5 6 Conclusion  material is cubic, just like its constituents Ag and Cu  changing microstructure leadsto no change in elastic coefficients

48. Average elastic properties by homogenization III SnPb37 effective elastic moduli of simulated microstucture Attention  lead (Pb) is cubic = 3 elastic constants, tin (Sn) is tetragonal = 6 elastic constants  2D excerpt: Conclusions composite material is less “tetragonal” due to the slight difference between C1111 and C2222 for Sn and the presence of the cubic Pb  laminate theory gives similar results:

49. Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

50. Conclusions and Outlook Phase field / higher gradient models to be used for microstructural changes in multi- component alloys should not simply be postulated but rather based on balance laws for physical quantities. Material theory should and can be used to derive the corresponding field equations. Material parameters in these relations should not simply be “guessed,” rather they should be obtained from experiments that are independent of the to-be-described phenomenon and, eventually, also be obtained from atomic methods (e.g., embedded atom methchnique). The spinodal decomposition observed in some solder/welding materials as well as the subsequent process of coarsening can be modeled quantitatively using such a strategy. Homogenized elastic properties for experimentally observed as well as predicted micrographs showing microstructural change have been obtained. Non-linear homogenized material properties were not obtained yet.