SINTERING THEORY

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

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











SUMMARY

Mass Transport in Sintering

From Swinkels and Ashby

Ashby Sintering Maps

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It was necessary to combine ideas of

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The breakthrough happened in the end of 1980s

Theory of Sintering: Practical Implementation











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


Shear modulus:

Resistance to the shape change


Volume strain rate

Olevsky E.A. (1998), Theory of sintering: from discrete to continuum. Review, Mater. Sci. & Eng. R: Reports, 40-100

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

3

2

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


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


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)0

12 [ ( ) ]

3z 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


2

0

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


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)

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)











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 2

1

2

for Ponte Castaneda (1)2 / ( m 1 )

1 23

27(1 )2 ( m 1 )

8

power-law Cocks (1)2 / ( m 1 )

1 23

(m 1)(1 )(1) 2 ( m 1 )

3

creep Duva & Crow (1)2 / ( m 1 )

1 23

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-scale

modeling 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, and

V. 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 constitutive

parameters based on the mesoscale simulations.

The macroscopic level envelopes the mesoscopic

simulators.

CONSTITUTIVE PARAMETERS

sintering stress bulk and shear moduli grain growth kinetics

DETERMINATION

Theoretical:

Mesoscale Simulation

Experimental:

Sinter-forging and free

sintering experiments

d

c)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

Str

ess

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











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” – a

distinguishing set of factors rendering different spark-

plasma vs. conventional hot pressing and sintering

results.

MODELING OF SPS

Heating rate 20°C/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 450°C-80 MPa

FAST 400°C-149 MPa

FAST 350°C- 229 MPa

HP 450°C-80 MPa

HP 400°C-275 MPa

HP 350°C -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

( ) ;x

c

cr

0 0 0xx y

y

2

2 3

3 1 1 3 3 1 1 3sin sin

2 2 2 2 2 2 2x 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).

2

3 1 1sin

2

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 that

an 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.) that

the 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 fundamental

comparative 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.


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

θ= e= ε

1-θ

3

1.3400

fd GG G

G

G is the porous material’s grain growth rate, 0

fdG

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


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


-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 effect

was studied by a number of authors including it’s 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 was

analyzed 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 sintering

D. 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

*vf

DC TJ H Q

kT 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


22 2gb Ttd gb gb e

T

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

θ= e= ε

1-θ

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


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


The intensity of thermal diffusion increases for

higher 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 frequencies

enhance sintering at the early stages of SPS and

hinder sintering at the late stages of SPS

process.

In some experimental studies the pulse frequency

was 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

Blech’s 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, MRS

Symposium 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 Boltzman’s

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




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




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=673

0K, 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)











SUMMARY

( el V ) 0

Cp

T

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


-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]











SUMMARY

Development of on-line sintering damage

criteria

Modeling of nano-powder sintering

Modeling of sintering with phase

transformations or chemical reactions

Modeling of field-assisted sintering

Development of sintering optimization

approaches

Multi-scale modeling of sintering

Further prospects

SINTERING THEORY

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Transcript of SINTERING THEORY