olevsky - theory of sintering
DESCRIPTION
Basics of sinteringTRANSCRIPT
-
SINTERING THEORY
BRIEF INTRODUCTION
BY
EUGENE A. OLEVSKY
SAN DIEGO STATE UNIVERSITY, CALIFORNIA, USA
2011 FAST
Spring School
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
PHYSICAL BASIS OF SINTERING
50 years to find out!
Surface tension phenomena
-
PHYSICAL BASIS OF SINTERING
Surface tension phenomena
-
Frenkel approach (1945) Pines approach (1946)
pore(vacancies)
coalescence of viscous particles
driven by surface tension
C Co 12
r
kT
V2
t
E 2s
evaporation of emptiness
SINTERING THEORY
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
Mass Transport in Sintering
From Swinkels and Ashby
-
Ashby Sintering Maps
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COMPLEX SHAPE PARTS PRODUCED VIA
POWDER METALLURGY ROUTE
flange pulley
palate expander parts foldable paper hole punch
metal fiber filter for
airbag inflators
auto transmission sprockets
camshaft sprocket
-
It was necessary to combine ideas of
MECHANICS
&
MATERIALS SCIENCE
The breakthrough happened in the end of 1980s
Theory of Sintering: Practical Implementation
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
The Main Constitutive Relationship
( ) 1
3ij ij ij L ij
We e P
W
externally applied material resistance sinteringstresses
Generalized
viscosity:
corresponds to the
constitutive properties of
particle material
Effective sintering stress:
function of porosity
Strain rate component
Bulk modulus:
Resistance to the volume change
function of porosity
Shear modulus:
Resistance to the shape change
function of porosity
Volume strain rate
Olevsky E.A. (1998), Theory of sintering: from discrete to continuum. Review, Mater. Sci. & Eng. R: Reports, 40-100
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Continuum Theory of Sintering
( ) 1[ ( ) ]
3ij ij ij
we
w
ij
Without considering sintering stress
is the ij component of the stress tensor;
0( ) 2w w
( )
( )
y
m
w
w Aw
Linear viscous (hot deformation of
amorphous materials; free sintering)
Plastic (cold pressing)
Power-law creep (hot deformation of
crystalline materials)
effective stress( )w
-
32
2 (1 )
3
(1 )
( ) 1[ ( ) ]
3ij ij ij
we
w
Bulk modulus
Shear modulus
0 Shear viscosity of the fully-dense material
2 21
1w e
Equivalent effective
strain rate
11 22 33iie volume change rate
ij Kronecker delta
2 2 2
1 2 2 3 3 1
1( ) ( ) ( )
3
Shape change rate
Continuum Theory of Sintering
-
Including sintering stress:
( ) 1[ ( ) ]
3ij ij ij l ij
we p
w
lp The effective sintering stress
Surface tension
ij external stress
For free sintering, no external stress, 0ij
2
0
3(1 )
2lp
r
0r Radius of the particle
Continuum Theory of Sintering
-
Problem of free sintering of a porous body
For linear viscous phase
( ) 10 [ ( ) ]
3ij ij ij l ij
we p
w
Projection on r direction: (a)
0( ) 2w w
0
12 [ ( ) ]
3r le p
Projection on z direction: (b)01
2 [ ( ) ]3
z le p
(a)*2+(b)0
12 [ (2 ) 3( ) ] 3
3r z le p
0 02 2 3 3 2r z l le e p p e
Continuity equation
1e
Continuum Theory of Sintering
-
20
3
00
3(1 )
2
2 (1 )2 12
3
lp re
s :Specific time of sintering
1
0 0
9exp( )
8s sdt
r
0 0 0 0 0 0
9 9 9ln
8 8 8dt
r r r
Continuum Theory of Sintering
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Pressing in rigid die and free sintering of a powder cylinder
E. Olevsky, G. Timmermans, M. Shtern, L. Froyen, and L. Delaey, The permeable element method for
modeling of deformation processes in porous and powder materials: Theoretical basis and checking by
experiments, Powd. Technol. - 93/2, 123-141 (1997)
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Gravity Influence: Grain Segregation Effect
E.A. Olevsky and R.M. German, Effect of gravity on dimensional change during sintering, II. Shape distortion,
Acta Mater., 48, 1167-1180 (2000)
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
Sintering theory was traditionally
developed either as the
application of complex diffusion
or viscous flow mechanisms to a
simple geometry or as complex
evolution of microstructure with
simple diffusion mechanisms. For
example, the bulk modulus can
be obtained from the solution of
the problem of hydrostatic
loading of the chosen
representative unit cell. The
disadvantage of this model basis
is the high degree of the
idealization of the grain-pore
structure.
MULTI-SCALE MODELING OF SINTERING
Idealized unit-cell used for the
determination
of the effective constitutive parameters
strainvolume
stresschydrostati~
-
Normalized shear modulus Normalized bulk modulus
Kuhn & Downey 2
3(2 (1 )2 )(1 )
2
9(2 )(1 )
for Green 2
3
(11/3)2
(3 21/ 4)(1 )
8
9
(1 1/3) 2 ln 2
(3 21/4)(1)
plastic Shima & Oyane 2
9(1)
4
2
3
(1 )9
2.49 0 .5 14
flow Doraivelu et al. 2(2(1)2 1)
3(2 (1 )2)(1 )
2(2(1 )2 1)
9(2 )(1 )
Skorohod (1 )2
2
3
(1 )3
Gurson (Doraivelu et al.approximation)
2
9
1 3
1 2
2
9
1 3
1 21
2
for Ponte Castaneda (1)2 / ( m 1 )
1 2 3
27(1 )2 ( m 1 )
8
power-law Cocks (1)2 / ( m 1 )
1 2 3
(m 1)(1 )(1) 2 ( m 1 )
3
creep Duva & Crow (1)2 / ( m 1 )
1 2 3
2
3
1m
mm
2 (m 1)
(m is thecreep
exponent)
McMeeking & Sofronis 1
1
2 (m1)
2
3
1m
mm
2 (m 1)
CONSTITUTIVE PARAMETERS OF MODELS FOR POROUS
MATERIAL DENSIFICATION
-
grain growth
change pixel color
We use a digitized microstructure
pore migration
swap pixels
Monte Carlo Model was used to simulate grain growth,
vacancy diffusion and vacancy annihilation
vacancy annihilation
move pixel out
N
i j
ji qqE1
8
1
,12
1Energy
E. Olevsky, V. Tikare, and T. Garino, Multi-scale modeling of sintering A Review, J. Amer. Ceram. Soc., 89 (6),
1914-1922 (2006)
-
Mesoscale Simulation Using the Potts Model
E. Olevsky, V. Tikare, and T. Garino, Multi-scalemodeling of sintering A Review, J. Amer.
Ceram. Soc., 89 (6), 1914-1922 (2006)
E. A. Olevsky, B. Kushnarev, A. Maximenko, V.Tikare, and M. Braginsky, Modeling anisotropic
sintering in nanocrystalline ceramics, Phil.
Mag., 85, 2123-2146 (2005)
V. Tikare, M. Braginsky, E. Olevsky, and D. L.Johnson, Numerical simulation of anisotropic
shrinkage in a 2D compact of elongated
particles, J. Amer. Ceram. Soc., 88, 1, 59-65
(2005)
M. Braginsky, V. Tikare, and E. Olevsky,Numerical simulation of solid state sintering,
Int. J. Solids and Structures, 42, 621-636 (2005)
E. Olevsky, B. Kushnarev, A. Maximenko, andV. Tikare, Modeling of sintering at multiple
length scales: anisotropy phenomena, TMS
Letters, 3, 55-56 (2004)
V. Tikare, M. Braginsky, and E.A. Olevsky,Numerical simulation of solid-state sintering: I,
Sintering of three particles, J. Amer. Ceram.
Soc., 86, 49-53 (2003)
First publication:V. Tikare, E.A. Olevsky, and M.V. Braginsky,
Combined macro-meso scale modeling of
sintering, in: Recent Developments in Computer
Modeling of Powder Metallurgy Processes, ed. A.
Zavaliangos and A. Laptev, IOS Press, 85-104
(2001)
-
Results: Simulation of Microstructural Evolution during
Sintering
Time, t = 0 MCS t = 2,000 MCS t = 50,000 MCS
Digitized images can be mined for many types of data
Vacancy anihilation: jump and shift algorithms
-
Diffusion mass
transport
Vacancy anihilation
Potts Model
Meso-Scale FEM
Macro-Scale FEM
Macroscopic shape distortions
Density distribution
Macroscopic damage
Macroscopic stress-strain state
Schematics of Multi-Scale Modeling
Two possible approaches:
Direct determination of the macroscopic constitutiveparameters based on the mesoscale simulations.
The macroscopic level envelopes the mesoscopicsimulators.
-
CONSTITUTIVE PARAMETERS
sintering stress bulk and shear moduli grain growth kinetics
DETERMINATION
Theoretical:
Mesoscale Simulation
Experimental:
Sinter-forging and free
sintering experiments
-
dc)1(
3
2
26.0L )1(7.1P
0
20
40
60
80
100
120
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Porosity
Bu
lk M
od
ulu
s
Normalized Bulk Modulus (Potts) Normalized Bulk Modulus (Skorohod)
Normalized Bulk Modulus (approx)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.7 0.75 0.8 0.85 0.9 0.95
Relative Density
Sin
teri
ng S
tress
Potts Model Approximation Skorohod Model
Sintering Stress and Bulk Modulus Approximations
Based on Mesoscale Simulations
bL )1(aP
12.1
23.2)1(
3
2
E. Olevsky, V. Tikare, and T. Garino, Multi-scale modeling of sintering A Review, J. Amer. Ceram. Soc., 89 (6),
1914-1922 (2006)
-
Boundary conditions
initial state current state
each element
at each time step
Multi-Scale Virtual Reality of Powder Processing
-
Sample problem solution: sintering with inclusion
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
Overwhelming majority of publications on SPS describe
empiric trial-and-error attempts to consolidate various
powder material systems.
The conducted theoretical studies are mostly reduced
to the modeling of temperature and electric current
density distributions. In practically all of the
publications the role of electrical field is narrowed down
to the generation of Joule heat, which thereby reduces
the theoretical framework, required for the description
of shrinkage and grain growth, to the existing
constitutive models of powder consolidation.
Generic physically-based modeling concepts are
currently in strong demand to enable the understanding
and control of the thermal and field effects adistinguishing set of factors rendering different spark-
plasma vs. conventional hot pressing and sintering
results.
MODELING OF SPS
-
Heating rate 20C/min
60
65
70
75
80
85
90
95
100
0 50 100 150 200 250 300 350 400 450
Temperature (C)
Rel
ati
ve
den
sity
(%
)FAST 450C-80 MPa
FAST 400C-149 MPa
FAST 350C- 229 MPa
HP 450C-80 MPa
HP 400C-275 MPa
HP 350C -460 MPa
Courtesy S. Kandukuri & L. Froyen
Comparative study of SPS HP of hypereutectic Al-Si-Fe-X powder
-
electromigration (diffusion enhancement)
electroplasticity (electron wind,
magnetic depinning of
dislocations)
dielectric breakdown of oxide films at grain
boundaries
ponderomotive forces pinch effect surface plasmons
Field Effects in SPS
high heating rates high local non-
uniformities of
temperature distribution
(local melting and
sublimation)
macroscopic temperature gradients
thermal diffusion thermal stresses
Thermal Effects in SPS
SPS: ENHANCEMENT OF MASS TRANSPORT
-
SPS: ENHANCEMENT OF MASS TRANSPORT
E. Olevsky and L. Froyen, Constitutive modeling of spark-plasma sintering of conductive
materials, Scripta Mater., 55, 1175-1178 (2006)
E. Olevsky, S. Kandukuri, and L. Froyen, Consolidation enhancement in spark-plasma sintering:
Impact of high heating rates, J. App. Phys., 102, 114913-114924 (2007)
E. Olevsky and L. Froyen, Influence of thermal diffusion on spark-plasma sintering, J. Amer.
Ceram. Soc., 92, S122-132 (2009)
electromigration(diffusion enhancement)
electroplasticity(electron wind,
magnetic depinning of
dislocations)
dielectric breakdown of oxide films at grain
boundaries
ponderomotive forces pinch effect surface plasmons
Field Effects in SPS
high heating rates high local non-
uniformities of
temperature distribution
(local melting and
sublimation)
macroscopic temperature gradients
thermal diffusion thermal stresses
Thermal Effects in SPS
-
Micromechanical Model
E. A. Olevsky, B. Kushnarev, A.
Maximenko, V. Tikare and M.
Braginsky, Modelling of
anisotropic sintering in crystalline
ceramics, Philosophical Magazine,
85, (19), 2123-2146 (2005)
2
p
a
p
cr
a
2
p
c
p
ar
c
2
1 2 3x x x xb y b y b
2
1 2 3y y y yb x b x b
0
sin2
ap
xx
c cdx c
c
( ) ;xc
cr
0 0 0xx yy
22 33 1 1 3 3 1 1 3
sin sin2 2 2 2 2 2 2
x xx p p
c c
c c y c cc r c c c r c
where is the surface tension, is the dihedral angle, a and c
are the grain semi-axes; x - effective (far-field) external stress in
the x-direction (compressive x is negative). Parameter
px
c c
c
is a local stress on the grain boundary (
pc c
c
is the
stress concentration factor).
23 1 1
sin2
gb gb pxgbx
cp p
D c c
kT c r c ca a c c
gb gbgb xy
DJ
kT y
( )
2
gb
y
gbx
p p
J c
a a c c
gb
yJ is the flux of matter in the direction of the
axis y caused by the grain boundary diffusion,
gbD is the coefficient of the grain boundary
diffusion, gb is the grain boundary thickness,
k Boltzman constant; T absolute temperature.
-
Influence of High Heating Rates
Experimentally, it has been shown in a number of investigations thatan increase in heating rate considerably increases the consolidation
rate of conductive and non-conductive powders during SPS.
For example, it was shown for an alumina powder (Zhou et al.) thatthe increase of heating rate from 50 to 300C/min with the same
maximum temperature and the corresponding six time decrease of
sintering time allowed obtaining the same final density. Physically,
this was attempted to be explained as a result of the existence of
additional defects in the material directly related to high heating rates
and short time of the process. They could be initial biographic
defects resulting from processes of powder synthesis (Ivensen or
defects in grain-boundaries between particles (Dabhade et al.).
Gillia and Bouvard have conducted a series of fundamentalcomparative experiments on sintering of WC-Co powder system with
different heating cycles. They employed cycles with the same average
heating rate but with various temperature histories (by employing
sequences of steady ramps and isothermal periods). Their results
indicate the dependence of the densification rate on the average
heating rate but no dependence on the temperature history.
-
Influence of High Heating Rates
E. Olevsky, S. Kandukuri, and L. Froyen, Consolidation
enhancement in spark-plasma sintering: Impact of high
heating rates, J. App. Phys. 102, 114913-114924 (2007)
For an aluminum alloy
powder
, ,x gbx crx f G
4
22
4 2
31 1 1
8
s sD
kTG
x
= e=
1-
3
1.3400
fd GG GG
G is the porous materials grain growth rate, 0fdG
is the grain growth rate of the fully-dense material
with the grain size 0G , 0G is the initial grain size of
the porous (powder) material
Du and Cocks
4 16.67 10 3.55 10
0
fd fd TG G t
Beck et al. fdG is the current grain size of the fully-dense material; 0
fdG is the initial grain size of the fully-
dense material; t is time, s; and T is temperature, K
3
4 1.3400
1 235 /6.67 10 ln , 533
0, 533
GK sG if T K
G K G
if T K
dT
dt = const is the heating rate, K/s
-
Influence of High Heating Rates
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1000 2000 3000
Time, s
Po
ros
ity
200C/min
100C/min
50C/min
25C/min
10C/min
For aluminum powder
-
Influence of High Heating Rates
-4.E-03
-3.E-03
-2.E-03
-5.E-04
150 250 350 450 550
Temperature, C
Sh
rin
ka
ge
Ra
te, 1
/s
200C/min
100C/min
50C/min
25C/min
10C/min
-7.E-03
-5.E-03
-3.E-03
-1.E-03
150 250 350 450 550
Temperature, C
Sh
rin
ka
ge
Ra
te, 1
/s
200 C/min
100 C/min
50 C/min
For aluminum powder
Model
Experiment
-
Influence of Thermal Diffusion
J is the vacancy flux, D is the coefficient of diffusion, vC is the vacancy concentration,
vC is the vacancy concentration gradient, *Q is the heat of vacancy transport, T is the
temperature gradient.
*
v v
Q TJ D C C
kT T
-
Influence of Thermal Diffusion Ludwig-Soret effect of thermal diffusion causes concentration gradients in
initially homogeneous two-component systems subjected to a temperature
gradient.
J. Chipman, The Soret effect, Journal of the American Chemical Society, 48, 2577-2589 (1926)
For the case of atomic and vacancy diffusion in crystalline solids, this effectwas studied by a number of authors including its theoretical interpretation by
Shewmon and Schottky.
P. Shewmon, Thermal diffusion of vacancies in zinc, Journal of Chemical Physics, 29, (5), 1032-1036 (1958)
G. Schottky, A theory of thermal diffusion based on lattice dynamics of a linear chain, Physica Status Solidi, 8, (1),
357 (1965)
For the electric-current assisted sintering, the effect of thermal diffusion wasanalyzed by Kornyushin and co-workers. Later, for rapid densification, the role
of temperature gradients was studied by Searcy and by Young and McPherson.
Y. V. Kornyushin, Influence of external magnetic and electric-fields on sintering, structure and properties, Journal of
Materials Science, 15, (3), 799-801 (1980)
A. W. Searcy, Theory for sintering in temperature-gradients - role of long-range mass-transport, Journal of the
American Ceramic Society, 70, (3), C61-C62 (1987)
R. M. Young and R. McPherson, Temperature-gradient-driven diffusion in rapid-rate sintering, Journal of the
American Ceramic Society, 72, (6), 1080 (1989)
Johnson argued against thermal diffusion significance in microwave sinteringD. L. Johnson, Microwave-heating of grain-boundaries in ceramics, Journal of the American Ceramic Society, 74, (4),
849-850 (1991)
We demonstrate a possible significance of thermal diffusion for SPSE. Olevsky and L. Froyen, Influence of thermal diffusion on spark-plasma sintering, J. Amer. Ceram. Soc. 92, S122-
132 (2009)
-
Influence of Thermal DiffusionJ is the vacancy flux, D is the coefficient of diffusion, vC is the vacancy concentration,
vC is the vacancy concentration gradient, *Q is the heat of vacancy transport, T is the
temperature gradient.
*
v v
Q TJ D C C
kT T
2
v fC HC T
kT
*v fDC T
J H QkT T
*
m fQ H H
Schottky:
Young &
McPherson:
Wirtz:
Kornyushin:
mH is the enthalpy of vacancy migration;
fH is the enthalpy of vacancy formation
vm
DC TJ H
kT T
;
v m f TT
C H HJ D T
k T T
did not include the term vC ! Otherwise:
T is the thermal diffusion ratio ( T is
the spatial average of temperature)
v mT
C H
k T We re-define:
TdivJ D TT
The driving force for
the vacancy migration:
T
TT q
dt
C
Heat transfer equation:
T is the thermal conductivity; C is heat capacity; t is time; and q is the
heat production per unit volume of the material and per unit time, which in the case of SPS can be represented as
2
eq E , where e is the specific
electric conductivity, and E is the electric field intensity 2T
e
T
TdivJ D E
T t
C
-
Influence of Thermal Diffusion
22 2gb Ttd gb gb eT
TJ divJ G D E G
T t
C2T e
T
TdivJ D E
T t
C
2
2 2
2
gbgb gb Ttd td
gbx e
Tp p
DJ T GE
T tG r G r
C
_ ,gbx gbx
curvature driven th diffusion driven
x crx f G
x
= e=
1-
3
10 1.3401.5 10 /G
G m sG
E. Olevsky and L. Froyen, Influence of thermal diffusion on spark-plasma sintering, J. Amer. Ceram. Soc. 92, S122-132 (2009)
T is the thermal conductivity; C is heat capacity; t is time; and q is the
heat production per unit volume of the material and per unit time, which in the case of SPS can be represented as
2
eq E , where e is the specific
electric conductivity, and E is the electric field intensity
is porosity; G is the average grain size
-
Influence of Thermal Diffusion
25
125
225
325
425
525
625
0 200 400 600 800 1000
Time, s
Te
mp
era
ture
, C
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Po
ros
ity
Temperature
Porosity - Model
Porosity - Experiment
25
207
389
571
753
936
1118
1300
0 70 141 211 281 352 422
Time, s
Te
mp
era
ture
, C
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Po
ros
ity
Temperature
Porosity - Model
Porosity - Experiment
Porosity kinetics during SPS of aluminum
powder. Comparison of the developed model
taking into account the impact of thermal
diffusion with experimental data of Xie et al.,
Effect of interface behavior between particles on
properties of pure al powder compacts by spark
plasma sintering, Materials Transactions, 42, (9),
1846-1849 (2001)
Porosity kinetics during SPS of alumina powder.
Comparison of the developed model taking into
account the impact of thermal diffusion with
experimental data of Shen et al., Spark plasma
sintering of alumina, J. Amer. Ceram. Soc., 85, (8),
1921 (2002)
3
2
11
2 223
4 24
0
2
2
2
3 32 129 2 23
1 4 1 9 1 2 exp 1
3 2
2 1
m
m m
xx
gb gb ref
gbx
cr
gb gb v m
e
T
G G
D G
QkTGA G
RT
D C H TE
t Gk T
C
curvature-driven grain boundary diffusion thermal diffusion power-law creep
-
Influence of Thermal Diffusion
The intensity of thermal diffusion increases forhigher pulse frequencies.
The thermal diffusion promotes components(atoms and vacancies) separation. At early stages
of sintering, this should lead to the growth of
inter-particle necks, which corresponds to the
enhancement of sintering. At the final stages of
sintering, however, the pores may serve as
vacancy sinks under thermal diffusion
conditions, which impedes sintering.
It is possible that the increased pulse frequenciesenhance sintering at the early stages of SPS and
hinder sintering at the late stages of SPS
process.
In some experimental studies the pulse frequencywas found to have a limited impact on SPS
results - its contributions at early and late stages
of SPS could offset each other.
TJ D TT
E. Olevsky and L. Froyen, Influence of thermal diffusion on spark-plasma sintering, J. Amer. Ceram. Soc. 92, S122-132 (2009)
-
Major Components of Densification-Contributing Mass
Transfer During SPS (model including electromigration):
EC C J E
Nernst-Einstein equation
grain-boundary diffusion power-law creep
driving sources
externally applied loadsintering stress
electromigration
*gb gb
E q
DC Z e
kT
Blechs formula
gb gbD
CkT
where is the atomic volume, *Z is the valence of a migrating ion, and qe is
the electron charge (the product * qZ e is called the effective charge).
*1gb gbgb x
y q
D UJ Z e
kT l y
U and l are the electric potential and the characteristic length along the
electric field.
( )
2
gb
y
gbx
p
J c
ca a
*
2 2
3 1 1
2
gb gb q pxgbx
pp
D Z e G rU
kT l G r G GG r
is the surface tension, x - effective (far-field) external stress in the x-direction
G a c is the grain size, p p pr a c is the pore radius.
M. Scherge, C.L. Bauer, and W.W. Mullins,Acta Met. Mater., 43 (9), 3525-3538 (1995):
electromigration stress of 23MPa along grain
boundaries under an electric field of 500 V/m (in a 1-
thick film) and up to GPa range stresses for grain
structures with closed surface junctions
M.R. Gungor and D. Maroudas, Int. J. Fracture,109 (1), 47-68 (2001): electromigration stress of
140MPa in a 1 -thick film under the field of about 425
V/m
Q.F. Duan and Y.L. Shen, J. Appl. Phys. 87 (8),4039-4041 (2000): electromigration stress of
450MPa along fast-diffusion length of 15 under 650
V/m
Z. Suo, Q. Ma, and W.K. Meyer, MRSSymposium Proceedings, 6p. (2000):
electromigration stress in 0.5 -thick Al film under 300
V/m field should reach the level of 1.5GPa
-
5
2
13
*2 2
2 2
3 1 1 3 31 1
2 22
m
gb gb q pxx gbx crx x
pp
D Z e G rUA
GkT l G r G GG r
G is the grain size; pr is the pore radius; A and m are power-law creep frequency
factor and power-law creep exponent, respectively; gbD is the coefficient of the
grain boundary diffusion, gb is the grain boundary thickness, k is the Boltzmans
constant, T is the absolute temperature; is the atomic volume, *Z is the
valence of a migrating ion, and qe is the electron charge (the product *
qZ e is
called the effective charge); U and l are the electric potential and the
characteristic length along the electric field; is the surface tension; x - effective (far-field) external stress in the x-direction; is porosity.
E. Olevsky and L. Froyen, Constitutive modeling of spark-plasma sintering of conductive materials, Scripta Mater. 55, 1175-1178 (2006)
shrinkage due to grain-boundary diffusion
shrinkage due to dislocation creep
Constitutive Model of Spark-Plasma Sintering
-
Densification map for aluminum powder,
T=673K, =28.3MPa
Contribution of different factors to shrinkage under SPS
E. Olevsky and L. Froyen, Constitutive modeling of spark-
plasma sintering of conductive materials, Scripta
Mater. 55, 1175-1178 (2006)
1.E-10
1.E-07
1.E-04
1.E-01
1.E+02
1.E+05
1.E+08
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Porosity
Sh
rin
kag
e R
ate
, 1/s
shrinkage rate due to electromigration (electric current)
shrinkage rate due to sintering stress (surface tension)
shrinkage rate due to power-law creep (punch load)
1.E-10
1.E-07
1.E-04
1.E-01
1.E+02
1.E+05
1.E+08
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Porosity
Sh
rin
kag
e R
ate
, 1/s
shrinkage rate due to electromigration (electric current)
shrinkage rate due to sintering stress (surface tension)
shrinkage rate due to power-law creep (punch load)
1.E-10
1.E-07
1.E-04
1.E-01
1.E+02
1.E+05
1.E+08
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Porosity
Sh
rin
kag
e R
ate
, 1/s
shrinkage rate due to electromigration (electric current)
shrinkage rate due to sintering stress (surface tension)
shrinkage rate due to power-law creep (punch load)
Grain Size: 1Grain Size: 40Grain Size: 100nm
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04
Grain Size, m
Po
rosit
y
external load
surface tension
electromigration
Contribution of different factors to shrinkage rate of aluminum powder under SPS
417U V
l m , T=6730K, x =28.3MPa
-
The average particle size is 55m. The applied field is accepted to be of
500V
m (Joule heat generation balance based estimation), the pressure is
constant and equal to 23.5 MPa.
Shrinkage kinetics during SPS of aluminum powder:
comparison with experiments
Pressure 10 MPa
Field 250 V/m
10 MPa
250 V/m
E. Olevsky and L. Froyen, Constitutive modeling of spark-plasma sintering of conductive materials, Scripta Mater. 55, 1175-1178 (2006)
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
( el V ) 0
CpT
t (kT T) el V
2
ij (W )
Wij
.
1
3
e
.
ij
PLij
.
1 e
.
Conductive DC
Heat Transfer by Conduction
Stress-Strain Analysis
Densification
Coupled electro-thermo-mechanical FEM calculations
-
prismatic die
temperature temperature gradient
temperature temperature gradient
cylindrical die
TEMPERATURE DISTRIBUTION DURING SPS
-
SPS SCALABILITY (SIZE DEPENDENCE)
Size 1 Size 2 Size 3 Size 4
SampleHeight [mm] 4 8 12 16Radius [mm] 7.5 15 22.5 30
DieHeight [mm] 30 60 90 120Radius [mm] 15 30 45 60
PunchHeight [mm] 20 40 60 80
RamHeight [mm] 40 80 120 160Radius [mm] 40 80 120 160
Alumina Disk-Shape Specimens (Same Aspect Ratio):
experimental verification
(size 2):
temperature evolution porosity evolution
-
SPS SCALABILITY (SIZE DEPENDENCE)
-
SPS SCALABILITY (SIZE DEPENDENCE)
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.015 0.030 0.045 0.060
(Po
ros
ity (C
en
ter)
P
oro
sit
y (S
urf
ac
e))
/ S
am
ple
Ra
diu
s
Die Radius [m]
Porosity Gradient
0.219
0.106
0.216
0.187
0.195
0.153
0.175
0.140
-
SPS SCALABILITY (SIZE DEPENDENCE): GRAIN GROWTH
SPS Setup Geometry
Grain Size Evolution at Sample Center Grain Size Evolution at Sample Surface
Plane used for
displaying results
Die
Ram
Punch
Ram
Grain Size Gradient
0.0E+00
5.0E-09
1.0E-08
1.5E-08
2.0E-08
2.5E-08
3.0E-08
3.5E-08
0.015 0.030 0.045 0.060
(Gra
in S
ize (
Cen
ter)
G
rain
Siz
e
(Su
rface))
/ S
am
ple
Rad
ius
Die Radius [m]
-
1. Science of Sintering: Fundamentals and
Historical Development
2. Classical Models of Sintering: Viscous and
Diffusion Mechanisms of Mass Transport
3. Continuum Modeling of Powder Consolidation
4. Multi-Scale Modeling of Sintering
5. Extrapolation of Sintering Concepts Towards
Constitutive Modeling of SPS
6. Sample SPS Problem Solutions
7. Further prospects of sintering modeling
SUMMARY
-
Development of on-line sintering damagecriteria
Modeling of nano-powder sinteringModeling of sintering with phase
transformations or chemical reactions
Modeling of field-assisted sinteringDevelopment of sintering optimization
approaches
Multi-scale modeling of sintering
Further prospects