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Nagy El-Kaddah
MTE 449 Powder Metallurgy
Chapter 4
Solidification and Microstructure ofAtomized Powders
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Analysis of Solidification Processes
To produce the desired microstructure of the powder in
atomization processes, one need to control process
variables that influence the rate of solidification of the
atomized droplets
The first step toward this goal is to identify key
solidification parameters through the analysis of heat
transfer processes and nucleation and growth kinetics inthe droplet
Due to complexity of solidification phenomena, the
analysis should involve every available experimentallydeveloped relationships for describing microstructure
features of the material system
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Formulation of Heat Transfer Problem for a Droplet
For an atomized metal droplet in tens of microns size range, due to high
thermal conductivity, the rate of heat flow from the droplet to the surround iscontrolled principally by convective resistance with practically no temperature
gradient in the droplet.
By assuming that the droplet is space wise isothermal and temperature varying
only with time, from overall energy balance, the temperature history of the
droplet is given by
)()();()(
)(
)(
2244
,
,
**
ooradoradorad
oconvconv
ssp
sl
sl
p
llp
pradconvp
TTTThTThTTq
TThq
TTc
TTT
TT
Lc
TTc
CqqA
td
TdCV
+===
=
>>
+
=+=
where V and A are volume and surface area of the droplet, Cp is specific heat, L
is latent heat of fusion, To is room temperature, hconv and hradare convective and
radiation heat transfer coefficients, is Stefan-Boltzmann constant and is
emissivity of material
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Calculation of Cooling Rate and Solidification Time
Cooling Rate
Calculation of the cooling rate to predict the grain structure of solidified the
particle is based on the cooling rate of the droplet at the liquidus temperature
(TM), which may be written as
)()(
6
,
oMradconv
VlpTThhDc
k
td
Td
+=
where DV is equivalent volume diameter and k is shape factor. The convective
heat transfer coefficient, hconv, may be estimated from the following correlation
g
ggp
g
Vgg
g
Vconv
K
cDV
K
DhNu
Nu
,
3/1805.0
PrRe
PrRe0266.0
====
The temperature history of droplet during cooling from its initial temperature toits freezing temperature is given by
where KV is thermal conductivity of the surrounding gas
+=
Vlp
radconv
oi
o
Dc
thhk
TT
TT
,
)(6exp
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Calculation of Cooling Rate and Solidification Time (cont.)
Solidification time
Solidification time is the time taken to cool the droplet to the liquidus
temperature and to freeze the droplet.
The cooling time can be evaluated from temperature history equation when
the droplet reached TM
Since the latent heat is much higher than the specific heat, the freezing time
can be directly estimated by equating total heat losses to total heat released
during solidification
+=
oM
oi
radconv
Vlp
CTT
TT
hhk
Dct ln
)(6
,
)()(6 oMradconv
Vf
TThhk
DLt
+=
From these two equation the solidification time is
+
+
=
)(
ln
)(6
,
oMoM
oilp
radconv
VS
TT
L
TT
TTc
hhk
Dt
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Prediction of Grain Structure
For all solidification processes except rotating disk, the structure of solidified
powders is dendritic, and dependence of the secondary arm spacing, whichrelates to the grain size, on the cooling rate (class 8) is given by
where C and n are constants specific to the alloy.
The value of n is between 0.5 and 1
n
dt
dTC
=
From the above analysis, the process variables affecting cooling rate, and
hence the grain structure are
Size of the droplet
Droplet velocity
Temperature of the melt
Gas used in the atomization chamber
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Typical Cooling Rates and Microstructure Characteristics
of Atomized Steel Powders
Process D50, m , m dT/dt, C/s h, J/m2.s.CGas atomization 75 2 2. 104 1. 103
Centrifugal atomization 150 3 5. 103
5. 103
Water atomization 1000 7 4. 102 3. 103
Melt explosion 650 7 4. 102 2. 103
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Nucleation and Growth Kinetics
Solidification starts with nucleation of the solid in the melt and followed by
growth of formed nuclei.Nucleation
It is a non-equilibrium process, and transformation of the liquid to solid takes
place in under cooled liquid below its melting points.
The deviation of the bulk free energy of the liquid from its equilibrium value at
the melting point (Gv =Go T) causes the atoms to form solid clusters in themelt (homogeneous nucleation) or to deposit on the surface of dispersed solid
phases such as nonmetallic inclusions in the melt (heterogeneous nucleation)Homogeneous Nucleation Theory
Clustering of the atoms is a probabilistic process involving clustering few atoms
to form a solid phase and the growth of the cluster (diffusion).
This process involves an increase of the energy of the cluster due to the
formation a new surface (surface energy)
As a result not all formed clusters survive, only the ones (nuclei) which do not
increase the energy of the system
N l i d G h Ki i ( )
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Nucleation and Growth Kinetics (cont.)
The rate nucleation is determined from
1. Extent of deviation from equilibrium state, i.e. (Gv)2. Rate of growth of initially formed cluster to its critical size
Undercooling has opposite effects on these two kinetic parameters.
Higher undercooling favors formation of clusters through Gv, and hindersgrowth by slowing mobility of atoms
=
Tk
E
TTHk
TII
f
Mo exp
3
16exp
22
23
043
4 23 =+ RGR v
where Io is nucleation rate constant andE is activationenergy for atom motion
Maximum nucleation rate occur at moderate
undercooling High undercooling suppress nucleation and
favors solidification of amorphous metals
nucleation
rate, I
Homogeneous Nucleation Theory (cont.)
The condition for formation of a stable cluster
vGR
=
3 The critical size nucleus
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Nucleation and Growth Kinetics
Heterogeneous Nucleation
The heterogeneous solid phases in melt act as nuclei if they overcome interfacialbarrier to form a grow on them
Based on this concept a number of models were proposed to predict nucleation
rate in terms to
Number of available heterogeneous particles in the melt (Ns)
Undercooling of the melt and empirical growth rate constants ().
Hunts model
Growth Kinetics
The growth of formed nuclei is described empirically in terms of undercooling
2TdtdR =
=2
21 exp
T
NI s
The nucleation and growth theories are foundation of all solidification models
for predicting grain and microstructure of solidified materials.
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Time-Temperature-Transformation (TTT) diagram
The TTT diagram provides a
practical way to predict solidified
phases and their amounts for any
material during cooling
It maps the times for nuclei toform and to grow in crystalline
phase as function of temperature.
The shortest nucleation time is
intermediate undercooling wherenucleation rate is maximum.
The form phases are determined
from superimposing the cooling
rate from heat transfer analysis
path i normal cooling, crystalline solid
path ii rapid solidification, amorphous solid
path iii partial transformation, mixed microstructure; some crystals and some amorphouspath iv atomization with intermediate quench - gives all amorphous
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Effect of Particle Size on Amorphous/Crystalline Ratio
Smaller particles have higher cooling rates than larger ones
They are more likely to experience high undercooling which favors
solidification of amorphous structure
The percentage of amorphous phase increases with
Decreasing particle size Increasing heat transfer coefficient by using gases with higher thermal
conductivity such as H2 and He.
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