Diffusion (Review)

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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
h
o
f
m
a
s
sD
m
a
s
s
d
i
f
f
u
s
i
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)
z0
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
 3T  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 LTM
Substituting time dependent values V(t) and G(t) into: c  
 
 G   V T
1
4


1

2

 
3

T
  2 D LTM
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
vlT   * 
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
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