Microstructural Rapid Solidification Model

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Microstructural Rapid Solidification Model

A. R. S. M.

04. 15. 2019

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Introduction – Solidification Microstructures

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Various Microstructures in Solidification

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Introduction

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As‐cast(built)

After heat treatment Conventional Casting Additive Manufacturing

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Introduction

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Unique microstructure due to Rapid Solidification Chemical homogeneity due to Rapid Solidification

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Macroscopic view for Normal Solidification

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• Field : Temperature(T), Concentration(C)

• Thermodynamic non‐equilibrium process (Kinetics: V)

• Microstructure evolution due to gradients of T and C

>> Solving T and C diffusion equations (Fick’s law)

>> Moving boundary(Solid/Liquid Interface) condition must be included

Nature in Solidification

initial transient field

steady state field

𝑇 𝐺 · 𝑉

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Local equilibrium at the Solid/Liquid interface

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• Equilibrium partition coefficient

At equilibrium,

For small concentration

• Liquidus slope = m

k = C_s/C_l

slope = m

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Morphological instability of S/L interface

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• Simple unstable condition

• But it ignores the effect of the surface tension

• Only solutal gradient in liquid is considered

• Instability due to temperature gradient should also be considered unstable

stable

Solute rejection in case k < 1

mG_c G

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Interface stability analysis

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Perturbation analysis

where Thus,

λ_i : critical perturbation which means the minimum wavelength to be unstable

→

Gibbs‐Thomson effect considered

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Interface stability analysis

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Marginal stability

To obtain time dependence ε and λ

where

Some assumptions are applied for simplicity

when

→ The criterions are the same

Instability due to T distribution considered

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Morphological categories

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Const. V Const. G

𝑇 𝐺 · 𝑉

Solidification map

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Dendrite Tip Radius (DTR)

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< LMK model >

• only valid for low growth rate

• The shape of the dendrite tip is not derived

2R

Morphology of Dendrite

LMK model

Simple model to estimate radius

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Dendrite Tip Radius (DTR)

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• Combining LMK criterion and Ivantsov’s solution

• Since some coefficient(C_tl, P_c, ξ_c) contain radius r in the final equation, it is solved iteratively using Newton‐Raphson method

• Excellent agreement with the experimental data in various range of velocity

KGT model

To determine radius at not only low velocity but also high velocity

where

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cf) There is very simple model as for the industrial use

Primary Dendrite Arm Spacing (PDAS)

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Kurz and Fisher / Trivedi model

• Most‐accepted models in PDAS calculation

• Assuming the shape of dendrite as an ellipse with a hexagonal arrangement

• Trivedi model predicts a minimum in the solute Peclet number as a function of V

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Morphological Transition

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Cellular to columnar Dendrite Transition (CDT)

Columnar dendrite to Equiaxed Transition (CET)

Fully equiaxed condition

• It is obvious in experimental data that there are maximum PDAS at CDT region (effect of CR)

• Increasing the solute promotes columnar dendrite over cellular growth (effect of C₀)

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Secondary Dendrite Arm Spacing (SDAS)

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Kattamis and Fleming model

• Growth SDAS is analogous to the Ostwald ripening of precipitates

• The driving force for the ripening process is the difference in chemical potential of crystals with differing interfacial energies due to differing curvatures

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Solute redistribution

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Mass balance in DS Rapid diffusion in liquid

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Microsegregation (MS)

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Rapid diffusion in liquid

Two extreme cases

Equilibrium cooling (α’ = 0.5)

Scheil cooling (α’ = 0)

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Secondary Phase Fraction (SPF)

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Roosz model and Voller’s approach

To calculate the secondary phase fraction and solute profile across the SDAS

1. Total dendrite tip undercooling : 2. Microsegregation calculation 3. eutectic undercooling

4. Total eutectic phase(secondary phase) fraction calculation

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Secondary Phase Fraction (SPF)

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SDAS also increases with time >> Coarsening of SDAS

where

Coarsening of SDAS

Mass balance

Solute back diffusion

Roosz model and Voller’s approach

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Secondary Phase Fraction (SPF)

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Discretizing by Euler forward treatment

• Finite Difference Control Volume method

• The calculation steps are terminated until eutectic temperature is reached

• Secondary phase(eutectic) fraction can be calculated as 1 – X_s(t)/X₀(t)

Roosz model and Voller’s approach

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Microstructural Solidification Model Summary

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DTR PDAS

SDAS

CDT/CET

SPF MS

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Departure from Local Equilibrium

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Local thermodynamic equilibrium at S/L interface

k = C_s/C_l

slope = m

Non‐equilibrium condition at S/L interface

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T0 condition

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Thermodynamic criterion

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T0 condition

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At equilibrium

At non‐equilibrium Then,

From the relationship

it follows that,

Approximation gives

Baker and Cahn model

Thermodynamic criterion

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T0 condition

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Baker and Cahn model

Consider kinetics of Atom rearrangement Adding equations The relationship bet’n

delta G and V >>

Combining eqs.

for Planar interface for DT

Set R and V=∞, k_v=1 < T0‐line > only for dilute solution Thermodynamic criterion

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Non-equilibrium Partition Coefficient

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Aziz model

• Solute Trapping model

• Relationship between k and V

The flux proportional to chemical potential gradient

where

and then,

The source term of solute rejection at the interface

Therefore

Again for dilute solution, or

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Absolute Stability

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Absolute stability

Upper limit of stability

(cf) marginal stability = Lower limit of Stability

Diffusion field near to absolute stability

laser remelting experiments

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