Chapter 5: Mass-Transfer Controlled Solidification

1
Chapter 5 Mass-Transfer Controlled Solidification

• What You Will Learn
• Solidification and Grain Growth
• Solidification Growth Mechanisms
• Numerical Simulation of Dendrite Spacing in Slab
  Casting
• More Complex Numerical Models for Dendritic
  Solidification

2
5.1- Solidification and Grain Growth

• Solidification of phase (1st order)
• Grain morphology
• spacing dependent on
• cooling rate

Adapted from Fig. 9.9, Callister 6e
3
A Deeper Look at Grain Structure
Secondary arms

• Macro-Scale
• Engine Block
• 1m
• Performance criteria
• Power generated
• Efficiency
• Durability
• Cost

• Mesostructure
• grains
• 1-10 mm
• Properties affected
• High cycle fatigue
• Ductility

• Microstructure
• dendrites phases
• 50-500 um
• Properties affected
• Yield strength
• Tensile strength
• High/low cycle fatigue
• Thermal growth
• Ductility

• Nano-structure Precipitates
• 3-100 nm
• Properties affected
• Yield strength
• Tensile strength
• Low cycle fatigue
• Ductility

• Atomic Structure
• 1-100 A
• Properties affected
• Youngs Modulus
• Thermal Growth

D.R. Askeland and P. P. Phule, The Science and
and Engineering of Materials,Thomson,
Brooks/Cole (USA) (2003)
4
Microstructures and Tensile Strength of Metals
Alloys

• Secondary arm spacing
• Relationships typically empirical

J.W. Callister Introduction to Materials
Science and Engineering 6th Ed, Jon Wiley and
Sons (2004)
5
Mass Vs. Heat Transfer In alloys

• Microstructure formation in alloys involves two
  main mechanismthe release (in the case of
  exothermic reactions) and subsequent diffusion of
  heat and the rejection and diffusion of solute
• The two diffusion processes described by similar
  mathematical processes, however diffusion of
  solute occurs on much smaller length scales and
  shorter time scales that the diffusion of heat ?
  mass diffusion is the rate limiting step

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Length and Time Scales
Heat mass diffusion in parent phase
Particle heat mass diffusion
Local interface velocity
Diffusion length of heat
Heat diffusion time
Diffusion length of solute
Mass diffusion time

Mass transfer controls the small-scale
structures
Mass transfer is the rate limiting step
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5.2- Solidification Growth Mechanisms
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Solidification into a Thermal Gradient Columnar
Dendrites
unstable solid/liquid interface
Liquid between glass slides
motor
solid
liquid
cold plate
thermal gradient
This process is known as directional
solidification

• Aim to understand microstructure
• evolution as a function of process
• parameters

and
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Anatomy of Solute Segregation During
Solidification
10
Mass Transfer Kinetics of Solidification in 1D
Solid eventually will reach Co
Liquid diffusion
Boundary conditions at moving interface
Diffusion in solid and Liquid
Apply to liquid neglect diffusion in solid
phase
Interface velocity
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Planar Concentration Profile

• Steady 1D diffusion profile

• Steady state 1D concentration profile solution of

boundary conditions
solute diffusion length

• Steady state solution

12
Perturbation of Solidification Front
y
liquid
solid
z

• Temperature dissipation gtgtsolute diffusion
  Mass transport dominates
• Solute diffusion in metals negligible

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Perturbing the Steady-State Planar Solidification
Front

• Consider the initial interface profile in the
  form of a since wave

growth amplitude
instability frequency
growth rate

• Assume perturbation of interface creates
  corresponding perturbation to concentration
  profile

2D-Disturbance superimposed on the 1D
steady-state profile
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Effects of Interface Perturbation of Kinetics

• Substitute trial function
  into
  solidification model

Equation of solute diffusion
Boundary conditions

Find conditions on such
that trial function is a valid solution
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Growth Rate of Perturbation Depends on its
Wavelength
Fundamental length scales
Perturbation grows
Perturbation decays back to planar interface
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Fundamental Length Scales of Solidification
(Thermal length)
(Diffusion length)
(Capillary length)
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Growth Rate of Perturbation Depends on its
Wavelength
All wavelengths stable
Unstable range of wavelengths

• Range of such that
• Unstable growing perturbations

• Range of such that
• Planar solidification interface results

V
Stable, planar interfaces
Dendrites and cells
G
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Significance of Fastest Growing Unstable
Wavelength

• Consider being within unstable range of

• Within unstable range of V G, fastest
  wavelength ( )
• grows out first!

• Wavelength sets initial scale of solidification
  front

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Relating the Initial Unstable Wavelength to
Materials and Processing Parameters

• To determine set

• Solving Eq. on page 14 gives

This approximation valid valid when Which is
called the constitutional supercooling limit
Instability wavelength is mean of two length
scales
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Relating Dendrite Final Wavelength to Tip Radius

• Assume dendrite is an ellipsoid, described by

where

• Radius of curvature

z

• 3D cross section view

Eutectic temperature
Relates wavelength to radius of curvature
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Relating Dendrite Tip Radius to Initial Wavelength

• Experiments have shown that tip radius roughly
  the same
• as the initial interface wavelength

• Recalling the form of the initial instability
  wavelength satisfies
• from 1.4 gives

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Relating Final Dendrite Spacing to V and G

• Cell spacing depends on

K/wt
alloy
binary

of

liquidus

of

slope
M
L

g

interface

id
solid/liqu

of
energy
tension
surface
Material parameters
z


liquid

of
constant
diffusion
D
L

fusion

of
heat
latent
L
Eutectic temperature

length"
capillary
"
d
Material process parameters
o

length"
diffusion
"
l
D
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5.3- Numerical Simulation of Dendrite Spacing in
Slab Casting
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Crude Model of Thermal Gradient and
Solidification Rate
Use these in formula
Average solidification front speed
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Crude Estimate of Columnar Spacing

• Substituting time dependent values V(t) and G(t)
  into

• Inter-dendrite spacing widens as solidification
• front moves toward the centre of cast

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Numerical Algorithm of Heat Transfer
Solidification
integers i,j real8 arrays
T(0N,0N), GRAD(1N-1,1N-1) arrays
DER(1,N-1) C(i,j)f(i,j) for time1,Nmax
increment time by for i1,N-1
for j1,N-1 Apply Interior
Node explicit update of Diffusion Equation
end end for j1,N-1
Left/Right Surface boundary update (i0 N)
end for for i1,N-1
Top/bot Surface boundary update (j0,N)
end run
update 0,0 node update 0,N node update
N,0 node update N,N node Find position of
liquidus temperature Calculate gradient and V
Compute Dendrite spacing Print Temperature array
at specified times END time loop
Define variables
Initialize T
Time loop
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Updating the Interior Nodes
Point-wise explicit time marching based on
temperatures at previous time step. (Like the
mass transfer code in Ch 4)
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Loosing Heat Via the Boundary Conditions
Heat Transfer Coefficient
Left/Right wall Example
i0 i1
IN-1 iN
Apply
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Remainder of code to be developed in a project

• Require
• Start will 2D mass transfer (i.e. diffusion
• code from Chapter 4) and Change C(i,j)?T(i,j)
• Gut the previous initial and boundary
  conditions and replace with the ones defined on
  the previous page
• Heat Transfer Coefficient
• Predict
• Columnar Spacing using Formula on page 22

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5.4- More Complex Models of Dendrites
Solidification
Adaptive finite element mesh
Initial conditions
solid
liquid
G thermal gradient
V pulling speed
PVA-1.5mol ACE GV5K/s
liquid-side concentration
solid state concentration
See http//mse.mcmaster.ca/faculty/provatas for
movie download
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Predicted Spacing Between Dendritic Arms

• Inter-dendritic tip spacing set by interplay
  between fundamental length scales diffusion
  length, thermal length and surface tension
• Dimensionless Length Scales
• Dimensionless wavelength ?
• Dimensionless velocity ?
• Onset wavelength ?
• Scaling hypothesis

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Computer Generated Dendrite Spacing Chart

• Scaling based on physical
• length scales
• Scaling function independent
• of material parameters
• Application to Industrial alloys

Experiments
PF Simulations
Scaled velocity
Scaled wavelength
See also M. Greenwood, M. Haataja and N.
Provatas, PRL (2004)
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Fortran Program to Simulate Dendritic
Solidification

• Go to the Chapter 5 directory retrieve code
  ModelC_alloy
• This code is too complex to examine in this class
• Will use this as a computational tool to study
  solidification properties
• You will use this code in a project to
  determine
• Dendrite growth rate Vs. undercooling
• Centre-line concentration in dendrite Vs.
  undercooling

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Definition of Undercooling Supersaturation

• Steady-state tip growth rate
• Vs. Time ?
• Solid-state concentration?

Alter Supersaturation
Undercooling