Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Lijian Tan and Nicholas Zabaras Level set method for simulating multi-phase multi-component dendritic solidification Level set method for simulating multi-phase multi-component dendritic solidification

Outline Materials Process Design and Control Laboratory Brief introduction Level set method & Mathematical model for multi-phase multi- component solidification systems Numerical examples Conclusions and future work

Background Materials Process Design and Control Laboratory 10  m

Phase field method Materials Process Design and Control Laboratory General kinematics equation Approximating the free energy using as Major difficulty: Parameter identification History: First developed by J. Langer (1978) as a computational technique to solve Stefan problems for pure materials Ideas: (1) enthalpy method (2) Cahn-Allen equation Phase field variable: (1) no direct physical meaning (2) can describe the real world when Easy to implement (coding), major success in the last two decades

Front tracking method Materials Process Design and Control Laboratory Major difficulty: Difficult for 3D and multiphase Ideas: (1) Uses markers to represent interface (2) Markers are moved using velocity computed from Stefan equation Sharp interface model Uses directly thermodynamic data Computationally difficult to implement

Level set method Materials Process Design and Control Laboratory History: Devised by Sethian and Osher (1988) as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions. Advantage: Interfacial geometric quantities can be easily calculated using signed distance.

Multiphase solidification system Materials Process Design and Control Laboratory We use a signed distance function for each phase. Multi-phase system: one liquid phase + one or more than one solid phases. Relation between the signed distances: (1)Exactly one signed distance would be negative (2)The smallest positive signed distance has same absolute value of the negative signed distance

Multiphase solidification system Materials Process Design and Control Laboratory We use a signed distance function for each phase. Multi-phase system: one liquid phase + one or more than one solid phases. Relation between the signed distances: (1)Exactly one signed distance would be negative (2)The smallest positive signed distance has same absolute value of the negative signed distance

Multiphase solidification system Materials Process Design and Control Laboratory We use a signed distance function for each phase. Multi-phase system: one liquid phase + one or more than one solid phases. Relation between the signed distances: (1)Exactly one signed distance would be negative (2)The smallest positive signed distance has same absolute value of the negative signed distance

Level set equation Materials Process Design and Control Laboratory Level set equation: Stabilized Galerkin form: Semi-descretized form:

Reinitialize Materials Process Design and Control Laboratory Iterative method: Fast matching method:

Reinitialize Materials Process Design and Control Laboratory At interface

Computational techniques Materials Process Design and Control Laboratory Narrow band: Adaptive meshing: 1 The level set equation is solved on a narrow band. 2 Re-initialization, heat transfer and solute transport is performed in the whole domain using adaptive meshing based on the distance from the interface.

Diffused interface Materials Process Design and Control Laboratory For numerical convenience, we assume phase change occurs in a diffused zone of width 2w that is symmetric around phase boundary.  Diffused interface feature (Convenience of whole domain method) This diffused interface allows us to use whole domain method conveniently for heat transfer and fluid flow as shown in these two figures. Consequently, a phase fraction can be defined as

Heat transfer & fluid flow Materials Process Design and Control Laboratory Heat transfer and fluid flow can be modeled using volume averaging For heat transfer: Temperature on the interface is not applied as an essential boundary condition to guarantee energy conservation of the numerical scheme (The Gibbs-Thomson relation is weakly forced by adjusting the growth velocity of phases) For fluid flow: The diffused interface is treated as a porous medium using a Kozeny-Carman approximation (This is only to avoid applying the no slip condition.)

Numerical scheme for fluid flow Materials Process Design and Control Laboratory Stabilized equal-order velocity-pressure formulation for fluid flow Derived from SUPG/PSPG formulation Additional stabilizing term for Darcy drag force incorporated Galerkin formulation for the fluid flow problem

Numerical scheme for fluid flow Materials Process Design and Control Laboratory Stabilized formulation for the fluid flow problem Advection stabilizing term Darcy drag stabilizing term Pressure stabilizing term Diffusion stabilizing term

Stabilizing parameters for fluid flow Materials Process Design and Control Laboratory advective viscous Darcy Stabilizing terms Stabilizing parameters continuity Convective and pressure stabilizing terms modified form of SUPG/PSPG terms Darcy stabilizing term obtained by least squares, necessary for convergence Viscous term with second derivatives neglected A fifth continuity stabilizing term added to the stabilized formulation pressure

Solute redistribution Materials Process Design and Control Laboratory Solute is diffused from places with high chemical potential to places with low chemical potential. Particularly, solute rejection is because chemical potential is higher in solid phase than in liquid phase. For a multi-phase multi-component system, it is only necessary to determine Solute transport in a system with n component can then be modeled as: Solute transport should also be compatible with

Solute redistribution Materials Process Design and Control Laboratory (1) Define the chemical potential equal to the concentration in the coexistence liquid phase. (2) Within each phase, chemical potential is only related with concentration Assumptions:

Interface kinematics Materials Process Design and Control Laboratory Equilibrium temperature: Given chemical potential (or concentration) of all components, we can get the equilibrium temperature from phase diagram. Gibbs-Thomson relation (Incorporate surface tension and kinetic effects)

Interface kinetics Materials Process Design and Control Laboratory Interface velocity can then be derived from energy conservation at diffused interface assuming the interface temperature approaches equilibrium temperature exponentially with a form similar to Newton’s Cooling law. Loops (augmentations) may be necessary to make interface temperature equal to equilibrium temperature. Given V Interface position Phase fractions

Nucleation Materials Process Design and Control Laboratory The nucleation rate is proportional to the number of critical clusters with an adsorption rate and has the form of The number of clusters with n atoms in equilibrium is A new phase is generated through nucleation process. A schematic of nucleation in eutectic growth is shown in the bottom left figure. As the interface of alpha phase becomes unstable, solute B becomes richer and richer in these valleys. When the solute concentration of component B beyond a certain point, a beta phase will be nucleated in the valley and keep growing. For simplicity, we currently only considered this type of nucleation in our numerical examples.

Numerical examples Materials Process Design and Control Laboratory Pure material Binary alloy Eutectic growth Ternary alloy (Single phase multi-component alloy) 3D examples with fluid flow

Numerical examples (pure material case 1) Materials Process Design and Control Laboratory Initial crystal shape Domain size Initial temperature Boundary conditions adiabatic With a grid of 64by64, we get Results using finer mesh are compared with other researcher’s results in the next slide.

Numerical examples (pure material case 1) Materials Process Design and Control Laboratory Our method Osher et. (1997) Triggavason et. (1996) Heinrich et. (2003)

Numerical examples (pure material case 1) Materials Process Design and Control Laboratory

Numerical examples (pure material case 2) Materials Process Design and Control Laboratory

Numerical examples (binary alloy case) Materials Process Design and Control Laboratory Initial crystal shape Domain size Initial temperature Boundary conditions not heat/solute flux Initial concentration

Numerical examples (binary alloy case) Materials Process Design and Control Laboratory Results are presented only within domain [0,2]by[0,2]. Micro-segregation can be observed in crystal.

Numerical examples (Pb-Sb binary alloy) Materials Process Design and Control Laboratory Pb-Sb alloy dendritic growth

Numerical examples (Eutectic growth) Materials Process Design and Control Laboratory

Numerical examples (Ternary alloy Ni-5.8%Al-15.2%Ta) Materials Process Design and Control Laboratory Important parameters Insulated boundaries on the rest of faces 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

Numerical examples (Ternary alloy Ni-5.8%Al-15.2%Ta) Materials Process Design and Control Laboratory (a)Interface position (b) Al concentration (c) Ta concentration Pattern of concentration for Al and Ta are the similar due to the assumption of equal diffusion coefficient in liquid for both component and similar partition coefficient. ( a ) ( b ) ( c )

Two dimension crystal growth with convection Materials Process Design and Control Laboratory With fluid flow, the crystal tips will tilt in the upstream direction. Whole domain method with convective and pressure stabilizing terms (SUPG/PSPG) The diffused interface is treated as a porous medium using a Kozeny- Carman approximation. (Using stabilize term DSPG )

Three dimension crystal growth with convection Materials Process Design and Control Laboratory As in the 2D case, the crystal tips will tilt in the upstream direction. Low undercooling High undercooling

Conclusions and future work Materials Process Design and Control Laboratory A level set method is implemented using the finite element technique for multi- phase evolution. Fast marching is implemented for re-initialization. Techniques such as narrow band computation and adaptive meshing is implemented with the aid of signed distance. A volume averaging model with diffused interface is used for heat transfer, fluid flow and solute transport. Because of the energy-conserving features of the diffused interface model, our model compares well with sharp interface models, and shows improvement over the phase field method with a much coarser mesh. The numerical algorithm is tested for 2d and 3d solidification of pure materials with convection, binary alloy solidification, and eutectic growth. We are currently extending this framework to various practical alloy systems and plan for developing multiscale solidification design algorithms for explicit control of the microstructure and mechanical properties.

Contact information Materials Process Design and Control Laboratory