Diffusion (Review)
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Transcript Diffusion (Review)
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
1
5.1- Solidification and Grain Growth
T(°C)
400
Solidification of a
phase (1st order)
Grain morphology &
spacing dependent on
cooling rate
L: Cowt%Sn
L
a
L
300
200
TE
100
a
L+a
a: Cowt%Sn
(Pb-Sn
System)
a+b
[Adapted from Fig. 9.9, Callister 6e]
0
10
20
30
Co
Co, wt%
2
(room T solubility limit)
Sn
2
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
Microstructure •Ductility
Mass
[D.R. Askeland and P. P. Phule, “The Science and
andTransport in
Formation
Engineering of Materials”,Thomson, Brooks/Cole (USA) (2003)]
Atomic Structure:
1-100 A
Properties affected:
3
•Young’s Modulus
•Thermal Growth
Microstructures and Tensile Strength of Metals Alloys
Secondary arm spacing λ 2 k 1 t
m
Relationships typically empirical
λ2
[J.W. Callister: “Introduction to Materials Science and Engineering”
6th Ed, Jon Wiley and Sons (2004)]
4
Mass Vs. Heat Transfer In alloys
Microstructure formation in alloys involves two main
mechanism:the 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
5
Length and Time Scales
Heat & mass diffusion in parent phase
Particle: heat &
mass diffusion
Diffusion length of heat
Local interface velocity
2
~αT/V Heat diffusion time ~αT/V
Diffusion length of solute
2
~
D
/VMass diffusion time ~
D
/V
d
i
f
f
u
s
i
o
n
l
e
n
g
t
h
o
f
h
e
a
tα
T 4
e
a
t
d
i
f
f
u
s
i
o
n
t
i
m
e
s
c
a
l
e
α
=
~
1
0 h
T 4
=
~
1
0
d
i
f
f
u
s
i
o
n
l
e
n
g
t
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o
f
m
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d
i
f
f
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s
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o
n
t
i
m
e
s
c
a
l
e
D
Mass transfer controls the
small-scale structures
Mass transfer is the
rate limiting step
6
5.2- Solidification Growth Mechanisms
7
Solidification into a Thermal Gradient: Columnar Dendrites
unstable solid/liquid interface
Liquid between glass slides
V
motor
cold plate(T c )
solid
liquid
L
hot plate (T h )
thermal gradient: G (T h Tc ) / L
•Aim: to understand microstructure
This process is known as
directional solidification
evolution as a function of process
parameters V and G
8
Anatomy of Solute Segregation During Solidification
C(z 0 )
+
T
Cs
z
ko
TM
Ti
Liquid diffusion profile
Cs Co
C(z=0 ) C s
-
CL
C
C C L C s (1 k o )C L
C L (z ) C o
ζ(t)= interface position
Solute concentration
In solid state
Mass Transport in Microstructure
Formation
9
Mass Transfer Kinetics of Solidification in 1D
C (z 0 )
+
Solid eventually
will reach Co
Cs
ko
Liquid diffusion
C (z= 0 ) C o
-
C L (z ) C o
ζ(t)
Diffusion in solid and Liquid
C L
t
CL
2
DL
z
Boundary conditions at moving interface
(C L C S )A d (J L J S )A dt
2
V (t)
d
dt
JL
C
DL
C L
(1-k o )C L z
z n (y)
Apply to liquid & neglect
diffusion in solid phase
Interface velocity
10
Planar Concentration Profile
Steady 1D diffusion profile:
C L
t
0
Steady state 1D concentration profile solution of: V
C L
z
2
DL
DL
+ boundary conditions
d CL
dz
2
dC L
dz
0
V(1 k o )C L (z 0)
z0
C L (z 0)
Co
ko
C L (z ) C o
Vz
(1
k
)
D
o
Steady state solution: C ssL (z) C o 1
e L
ko
, z 0
lD
2D L
solute diffusion length
V
11
“Perturbation” of Solidification Front
y
solid
T
z
liquid
T To Gz
TM
Ti
Portion Phase diagram
z
Cs Co
z ( y, t)
V (y)
C
C C L C s (1 k o )C L
ς(y,t)
t
Temperature dissipation >>solute diffusion
Solute diffusion in metals negligible
CL
Mass transport dominates
Ds 0
12
Perturbing the Steady-State Planar Solidification Front
Consider the initial interface profile in the form of a since wave: ς(y, t) εsin(qy)e
=growth amplitude
q
=instability frequency
(q ) =growth rate
ω(q)t
2
Assume perturbation of interface creates corresponding perturbation to concentration profile:
C L (y, z, t) C L (z) εC L (y, z)e
ss
=
p
ω(q)t
+
2D-Disturbance superimposed
on the 1D steady-state profile
13
Effects of Interface Perturbation of Kinetics
p
Substitute trial function C L (y, z, t) C ss
(z) εC L (y, z)e
L
Equation of solute diffusion
C L
t
V
C L
z
ω(q)t
into solidification model
Boundary conditions
D L C L ( z ) nˆ V nˆ (1 k o ) C L ( z )
D L C L
2
+
CL (z )
Co
ko
TM
M LLp
G
M
L
CL (z ) Co
Find conditions on ω versus q such that trial function is a valid solution
Mass Transport in Microstructure
Formation
14
Growth Rate of Perturbation Depends on its Wavelength
w (q)= -2k o + 2k o -1+ 1+ (ql D )
2
lD
2
1-d
l
q
o D
2l
T
Fundamental length scales
l T lD d o
(q )
Perturbation grows
q
ς(y, t) εsin(qy)e
Perturbation decays
back to planar interface
15
ω(q)t
Fundamental Length Scales of Solidification
lT
lD
M L ΔC
(Thermal length)
G
2D
(Diffusion length)
V
γ
TM
do
L M ΔC
L
p
(Capillary length)
16
Growth Rate of Perturbation Depends on its Wavelength
(q )
Unstable range
of wavelengths
All wavelengths
stable
q
q
Range ofV, G
(q )
such that 0
Range of V, G such that 0
Unstable growing perturbations
V
Planar solidification interface results
Dendrites
and cells
Stable, planar interfaces
G
17
Significance of Fastest Growing Unstable Wavelength
Consider being within unstable range of V & G
Within unstable range of V & G, fastest wavelength ( q c )
grows out first!
Wavelength sets initial scale of solidification front
(q )
qc
q
Mass Transport in Microstructure
Formation
c
2
qc
18
Relating the Initial Unstable Wavelength to Materials
and Processing Parameters
•To determine q c set
d ω(q)
0
dq
1
2π
γT M
λc
(1 k o )
V
qc
M
C
o
L
2D L
ko
•Solving Eq. on page 14 gives
2
2D L γT M
V M ΔC
L
1
o
2
1
c l D d o 2
This approximation valid valid when
G
V
DL
Instability wavelength is
mean of two length scales
V
DL
M LC o
(1 k o )
ko
M L ΔC M L G c
Which is called the “constitutional
supercooling limit”
Mass Transport in Microstructure
Formation
19
Relating Dendrite Final Wavelength to Tip Radius
f
b
Assume dendrite is an ellipsoid, described by
Radius of curvature R
r
z
R
a
b
1
( r 0, z a )
where
r
2
b
2
z
2
a
2
z ( r )
1 ( z )
2
3
2
2
b
Ra
a
T TL TE G a
3D cross section view
Eutectic temperature
m 2b
Relates wavelength
to radius of curvature
f
TR
Mass Transport in Microstructure
Formation
G
20
1
Relating Dendrite Tip Radius to Initial Wavelength
Experiments have shown that tip radius roughly the same
m
b
as the initial interface wavelength
R c
R
r
z
a
Recalling the form of the initial instability wavelength satisfies
from 1.4 gives
T TL - TE G a
1
1
R λ c l D d o 2
Mass Transport in Microstructure
Formation
2D L γ 2
V ΔT
21
Relating Final Dendrite Spacing to V and G
•Cell spacing depends on
V and G
c
b
r
R
Material parameters
z
ML slope of liquidus of binary alloy [K/%wt]
surface tension energy of solid/liquid interface
DL diffusion constant of liquid
L latent heat of fusion
T
T
T
L
E
Material + process
parameters
1
Eutectic temperature
do " capillary length"
lD " diffusion length"
1
3T 2 2 D L TM 4
c
G V L T
22
5.3- Numerical Simulation of Dendrite Spacing in Slab Casting
T liquid
T mold
z
Liquid
mold
th e rm a l g ra d ie n t G (t)
Nucleation of solid seed
crystals near mold wall
Tliquid T melt
Tmold Tmelt
m
Mold
wall
z
at
Dendrites
early time
Thermal length scale
Columnar front
later times
a thermal diffusion constant
Mass Transport in Microstructure
Formation
23
Crude Model of Thermal Gradient and Solidification Rate
T liquid
T mold
z
Liquid
mold
th e rm a l g ra d ie n t G (t)
Tliquid T melt
G
T liquid T mold
at
Tmold Tmelt
z
V (t )
at
Use these in c formula
a
t
Average solidification front speed
Thermal length scale
a thermal diffusion constant
Mass Transport in Microstructure
Formation
24
Crude Estimate of Columnar Spacing
1
1
3 T 2 2 D LTM
Substituting time dependent values V(t) and G(t) into: c
G V T
1
4
1
2
3
T
2 D LTM
c
T a
T
at
t
4
Inter-dendrite spacing widens as solidification
front moves toward the centre of cast
Mass Transport in Microstructure
Formation
3
c const t 8
c (t )
25
Numerical Algorithm of Heat Transfer & Solidification
integers ::i,j
Define
variables
Initialize T
update 0,0 node
real*8
::
arrays
:: T(0:N,0:N), GRAD(1:N-1,1:N-1)
update N,0 node
arrays
::DER(1,N-1)
update N,N node
t
update 0,N node
C(i,j)=f(i,j)
Find position of liquidus temperature
Calculate gradient and V
for time=1,Nmax
t
increment time by
for i=1,N-1
Compute Dendrite spacing
Print Temperature array at specified times
END time loop
for j=1,N-1
Apply Interior Node explicit update of Diffusion Equation
Time loop
end
end
for j=1,N-1
Left/Right Surface boundary update (i=0 & N)
end for
for i=1,N-1
Top/bot Surface boundary update (j=0,N)
end run
26
Updating the Interior Nodes
T
n 1
T n i 1, j 2T n i, j T n i 1, j
2
x
n
i, j T i, j ta
T n i 1, j 2T n i, j T n i 1, j
2
y
Point-wise “explicit” time marching based
on temperatures at previous time step.
(Like the mass transfer code in Ch 4)
A pply to (i=1,...,N -1, j=1,...,N -1) nodes
27
Loosing Heat Via the Boundary Conditions
Heat Transfer Coefficient
Left/Right wall Example:
T
k
.
Apply:
T0 T1
wall
.
i=0
T m ould
x
h T m elt T m ould
.
i=1
.
.
I=N-1
.
i=N
T m ould
T w all
xh T1 Tmould
k
T N T N 1
T w all
xh T N 1 Tmould
k
28
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
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
GV=5K/s
liquid-side concentration
solid state concentration
30
See http://mse.mcmaster.ca/faculty/provatas for movie download
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
lD
lD
clT
1 1
vlT *
lD lD
(,
d
l,
l)
c
lT
d
o
2 D
v
m c
G
T
L
m
c
oD
T
clT
Scaling hypothesis:
lD
lT
fc
lD
31
Computer Generated Dendrite Spacing Chart
Scaling based on physical
length scales
LT therm
al length
lD
LD diffusionlength
C lT
c (lT,lD,do)
Experiments
PF Simulations
Scaling function independent
of material parameters
Application to Industrial alloys
1 1
lT *
lD lD
Scaled
wavelength
Scaled velocity
See also: [M. Greenwood, M. Haataja and N. Provatas, PRL (2004)]
32
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
33
Definition of Undercooling & Supersaturation
T
Steady-state tip growth rate
Vs. Time ?
Alter Supersaturation:
(T )
CL Co
Solid-state concentration?
CL Cs
TM
(T1 ) 0
T1
Undercooling:
T
(T 3 ) 1
T3
Cs
Co
CL
C
Mass Transport in Microstructure
Formation
34