Modelling Rapid Solidification Kinetics Quantitatively using Phase field Models Nikolas Provatas McGill University, Department of Physics Centre for the Physics of Materials Spring Meeting, Sept 26-27, 2021 

“Phase Field crystal“ (PFC) 10-10 10-8 10-6 10-4 10-2 100 Length Scale (m) 104 102 100 10-2 10-4 10-6 10-8 10-10 10-12 Time Scale (s) FEM Atomic potentials MD/DFT/MC Atomic kinetics Phase Field (PF) Heat Transfer Casting Solidification Time and Length Scales of Materials Phenomena Precipitation Grain boundaries Solute trapping Solute diffusion Dendritic solidification 

Large-scale heat transfer Alloy design Microstructure design Melt pool flow Process design Computational thermodynamics Property and performance design Phase field method Micromechanics and crystal plasticity Integration of Phase Field Simulations with Integrated computational materials engineering (ICME) Powder manufacturing Microstructure imaging, XRD, EDS Nanoindentation, fatigue and corrosion testing, … Rapid solidification tests, process monitoring, AM DoEs Partnership with VTT Technical Research Centre of Finland 

Phase Field Modelling of Solidification 

Multi-Phase & Multi-Component Solidification ➢ Chemical potentials: ➢ One order parameter for each grain/phase: ➢ Concentrations in each phase : ➢ Grand potential of each phase : ➢ Local atomic mobility: ➢ Susceptibility tensor of each phase: ➢ Local susceptibility: Greenwood et al, 2018, Comp. Mat. Sci. 142 

Parabolic Free Energy Approximation J.Heulens, B. Blanpain and N. Moelans Acta Materialia, 59 (2011) Database Calc Parabolic fitting 

Quantitative Contact with Solidification Benchmarks formulated for Dilute Binary Alloys steady state concentration sharp interface limit 

Simulating Across Multiple Scales 

W 1mm 40mm 130mm Color map represents impurity concentration (C) ~102 -103nm Dendrite Spacing » L Interface Width » w capillary length = d o dendrite radius = r Length Scale Selection in Directional Solidification [2D: M. Greenwood et al, Phys. Rev. Lett, Vol. 93, 246101 (2004) 3D: Provatas et. Al, Int. J. Mod. Phys. Vol. 19, 4525 (2005)] 

domain subdomain subdomain mini-mesh →finite difference 3D Adaptive Mesh Refinement (AMR): Architecture Oct -tree in 3D [M. Greenwood, et. al J. Com. Mat. Sci. Vol. 142, 153 (2018)] 

Scalability Across Distributed Processors 

Scalability Across Distributed Processors equivalent uniform mesh nodes= system dimensions: 128 dendrites each dendrite occupies ~32 domains, spread over 8 cores # or cores= 8 X dendrites in system 

Scalability Across Distributed Processors 8 16 32 64 512 1024 8 16 32 64 512 1024 128 twall =time of one dendrite simulated alone, in one core twall =time of full simulation domain Y-axis→ X-axis→ 

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https://www.eos.info/additive_manufacturing/for_technology_interested Application: Metal Laser Sintering of Al-Si-Mg In Collaboration with Mohsen Mohammadi and Hossein Azizi @ Marine Additive Manufacturing Centre of Excellence University of New Brunswick 

Single-layer powder base metal system (X-Z plane) at t=0 7 Result 8.0e+00 4 3 6 .0e-04 2 1 5 400 $m Powder 40 $m Base X Y X Z &' &( &) 1.9e+03 1800 1600 Field_30 800 1200 1.9e+03 1800 1400 1000 Horizontal X Y X Z P Vertical Laser direction P X Y Unsteady Heat Transfer in Semi-infinite So Solidification process of the coating layer during a ther operation is an unsteady heat transfer problem. As we d earlier, thermal spray process deposits thin layer of coat materials on surface for protection and thermal resistant as shown. The heated, molten materials will attach to th and cool down rapidly. The cooling process is importan the accumulation of residual thermal stresses in the coa solid liquid Coating with density , latent heat of fusion: hsf Substrate, k, Centerline 1600 Field_30 1200 1.9e+03 1800 1400 1000 Horizontal X Y X Z P Vertical Laser direction P X Y Unsteady Heat Transfer in Semi-infinite Solids Solidification process of the coating layer during a thermal spray operation is an unsteady heat transfer problem. As we discuss earlier, thermal spray process deposits thin layer of coating materials on surface for protection and thermal resistant purposes as shown. The heated, molten materials will attach to the substra and cool down rapidly. The cooling process is important to preve the accumulation of residual thermal stresses in the coating layer. S(t) solid liquid Coating with density , latent heat of fusion: hsf Substrate, k, Centerline Initial temperature profiles of the cross section of the melt pool (X-Z plane) 1.9e+03 1600 Field_30 1200 1.9e+03 1800 1400 1000 Horizontal X Y X Z P Vertical Laser direction P X Y Unsteady Heat Transfer in Semi-infinite Solids Solidification process of the coating layer during a thermal spray operation is an unsteady heat transfer problem. As we discuss earlier, thermal spray process deposits thin layer of coating materials on surface for protection and thermal resistant purposes, as shown. The heated, molten materials will attach to the substrate and cool down rapidly. The cooling process is important to prevent the accumulation of residual thermal stresses in the coating layer. S(t) solid liquid Coating with density , latent heat of fusion: hsf Substrate, k, Centerline Case Study in Dilute Al-Si System: FEM Modelling of Thermal Transfer Conditions 

1.1e+00 0.5 t =5.0 t =8.0 X X 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 B B C A A A A B Increasing nucleation density × 10$ %s 1.1e+00 0.5 t =5.0 t =8.0 X X 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 B B C A A A A B Increasing nucleation density × 10$ %s t = 0.5 ms 1.1e+00 0.5 t =5.0 t =8.0 X Y X Z 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 B B C A A A A B Increasing nucleation density × 10$ %s t = 0.8 ms Microstructure Evolution in Horizontal Build Sample 

Microstructure of the fully solidified layer: Horizontal VS. Vertical Build Directions 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 t =5.0 t =8.0 X Y X Z B B (c) 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 B B C A A A A B Increasing nucleation density × 10$ %s 1.1e+00 0.5 t =5 t =8 X Y X Z 9 2e-03 0.02 Field_4 0.1 0.2 1.1e+00 0.5 0.05 B B C A A A A B Increasing nucleation density × 10 

Modelling Non-Equilibrium Interfacial Kinetics in Rapid Solidification 

➢Example: consider model C ➢Can make contact with the sharp interface model even when the interface and 21 Sharp Interface Limit of Basic “Model C” phi-4 potential Tilts the wells / M p T T u L c − = l ≪1 

Separation of Length Scales (solid) f field - f = -1 Phase-field model solutions U field Sharp-Interface phases interface Outer region Outer region Inner region Classic approach: phase-field (order parameter) and temperature field approach sharp-interface profiles when →0 f =1 (liquid) -d o k -bV n Sharp-Interface U=(T-Tm )/Tm Outer region 

Scaling Relations ➢ In the limit of: and Second order in perturbation theory Capillary length Interface kinetic coefficient Matching inner and outer regions 

Quantitative Phase Field Modeling of Solute Trapping and Continuous Growth Kinetics for Rapid Solidification • Most phase field models emulate sharp interface limit where k(V) = ke • Solute trapping should affect 1. Solute partitioning k(V) > ke 2. Kinetic undercooling (generalized Gibbs-Thomson condition) • Using matched asymptotic boundary layer analysis , we incorporate solute trapping into the phase field model with the following: 1. Controllable k(V) which is [relatively] independent of the chosen interface width W 2. Follows kinetic undercooling according to Continuous Growth Model (CGM) kinetics with either full or zero solute drag 

Continuous Growth Model Description (CGM) • Popular sharp interface model for solute trapping • Kinetic undercooling depends on k(V) as follows: = 1 gives complete solute drag, and = 0 gives zero solute drag Goal: Want phase field model to follow CGM undercooling (1) and to approximate a k(V) curve Approach: Use sharp interface asymptotics by modifying the so-called ”anti- trapping” current M. J. Aziz and W. J.Boettinger, Acta Metall. Mater. 42 (1994) 

Setting Up Phase Field Model for SIM Analysis 

Recovering the CGM k(V) in the Sharp Interface Limit Anti-trapping form→ Modify anti-trapping coefficient → →full drag →no drag Modified CGM-SIM coefficients A controls solute trapping → ( k(V) made independent of W by keeping A*W constant) binary alloy 

Model Convergence Kinetic undercooling Tatu Pinomaa and Nikolas Provatas, Acta Materialia, Vol. 168, 167 (2019) 

In-situ imaging of rapid solidification Dynamic transmission electron microscopy (DTEM) Copper concentration (EDXS) STEM image quality map Inverse polar figure map McKeown, Joseph T., et al. "Time-resolved in situ measurements during rapid alloy solidification: Experimental insight for additive manufacturing." JOM 68.3 (2016): 985-999. • Explicit (in-situ) imaging of rapid solidification • Controlled conditions (~ 2D, no flow, simple heat transfer) • Crucial for calibrating and validating solidification models Application: Rapid Solidification of Aluminum Alloys Collaboration with Tatu Pinomaa and Ansi Laukkanen (VTT) & Jörg M.K. Wiezorek and Joseph T. McKeown (Laurence Livermore National Labs) 

• No latent heat release • T-independent material properties • Heat transfer 2D Modelling Thermal Transfer Conditions 

21/07/2022 VTT – beyond the obvious 31 Phase field modeling DTEM