0305 Olevsky - Theory of Sintering

7/27/2019 0305 Olevsky - Theory of Sintering

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SINTERING THEORY

BRIEF INTRODUCTION

BY

EUGENE A. OLEVSKY

SAN DIEGO STATE UNIVERSITY, CALIFORNIA, USA

2011 FAST

Spring School


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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 Sintering5. Extrapolation of Sintering Concepts Towards

Constitutive Modeling of SPS

6. Sample SPS Problem Solutions

7. Further prospects of sintering modeling

SUMMARY


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SUMMARY


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PHYSICAL BASIS OF SINTERING

50 years to find out!

Surface tension phenomena


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PHYSICAL BASIS OF SINTERING

Surface tension phenomena


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Frenkel approach (1945) Pines approach (1946)

pore(vacancies)

coalescence of viscous particles

driven by surface tension

C Co 12

r

kT

V2t

E 2s

evaporation of emptiness

SINTERING THEORY


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SUMMARY


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Mass Transport in Sintering

From Swinkels and Ashby


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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


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It was necessary to combine ideas of

MECHANICS

&

MATERIALS SCIENCE

The breakthrough happened in the end of 1980s

Theory of Sintering: Practical Implementation


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SUMMARY


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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


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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


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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



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Including sintering stress:

( ) 1[ ( ) ]

3ij ij ij l ij

we p

w

lp The effective sintering stress

Surface tension

ij external stressFor free sintering, no external stress, 0ij

2

0

3

(1 )2lp r

0r Radius of the particle



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Problem of free sinter ing of a porous body

For linear viscous phase

( ) 10 [ ( ) ]3

ij ij ij l ijw e pw

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 l e e p p e

Continuity equation

1e



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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 9ln8 8 8

dtr r r


P i i i id di d f i t i f d li d


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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 byexperiments, 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)


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SUMMARY


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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 withsimple diffusion mechanisms. For

example, the bulk modulus can

be obtained from the solution of

the problem of hydrostatic

loading of the chosenrepresentative 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~


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Normalized shear modulus Normalized bulk modulus

Kuhn & Downey 2

3(2 (1 )2)(1 )

2

9(2 )(1 )

for Green 2

3

(1 1/3 )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 .514

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

1 m

mm

2 (m 1)

(m is thecreep

exponent)

McMeeking & Sofronis 1

1

2 (m1)

2

3

1 m

mm

2 (m 1)

CONSTITUTIVE PARAMETERS OF MODELS FOR POROUS

MATERIAL DENSIFICATION


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grain growthchange pixel color

We use a digitized microstructure

pore migrationswap pixels

Monte Carlo Model was used to simulate grain growth,

vacancy diffusion and vacancy annihilation

vacancy annihilationmove pixel out

N

i j

ji qqE1

8

1

,12

1Energy

E. Olevsky, V. Tikare, and T. Garino, Multi-scale modeling of sinteringA Review,J. Amer. Ceram. Soc., 89 (6),1914-1922 (2006)


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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 elongatedparticles, 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 multiplelength 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)


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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


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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.


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CONSTITUTIVE PARAMETERS

sintering stress bulk and shear moduli grain growth kinetics

DETERMINATION

Theoretical:Mesoscale Simulation

Experimental:Sinter-forging and free

sintering experiments

Sintering Stress and Bulk Modulus Approximations


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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

lkModulus

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

Sintering

Stres

s

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 sinteringA Review,J. Amer. Ceram. Soc., 89 (6),1914-1922 (2006)

Multi-Scale Virtual Reality of Powder Processing


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Boundary conditions

initial state current state

each element

at each time step

Multi-Scale Virtual Reality of Powder Processing


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Sample problem solution: sintering with inclusion


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Historical Development2. Classical Models of Sintering: Viscous and







SUMMARY

MODELING OF SPS


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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 downto 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 sinteringresults.

MODELING OF SPS


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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)

Relative

density(%)

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


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electromigration

(diffusion enhancement)

electroplasticity

(electron wind,

magnetic depinning ofdislocations)

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 andsublimation)

macroscopic

temperature gradients

thermal diffusion

thermal stresses

Thermal Effects in SPS

SPS: ENHANCEMENT OF MASS TRANSPORT


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SPS: ENHANCEMENT OF MASS TRANSPORT

E. Olevsky and L. Froyen, Constitutive modeling of spark-plasma sintering of conductive

materials, Scrip ta Mater., 55, 1175-1178 (2006)

E. Olevsky, S. Kandukuri, and L. Froyen, Consolidation enhancement in spark-plasma sintering:

Impact of high heating rates, J. Ap p. 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

Mi h i l M d l


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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

sin

2

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 c a 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 ycaused by the grain boundary diffusion,

gbD is the coefficient of the grain boundary

diffusion,gb

is the grain boundary thickness,

k Boltzman constant; T absolutetemperature.


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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 ratesand 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.


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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

1.3400fd GG G

G

G is the porous materials grain growth rate, 0fdG

is the grain growth rate of the fully-dense materialwith 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-densematerial; 0

fdG is the initial grain size of the fully-

dense material; tis time, s; and Tis 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 = constis the heating rate, K/s



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0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 1000 2000 3000

Time, s

Porosity

200C/min

100C/min

50C/min

25C/min

10C/min

For aluminum powder



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-4.E-03

-3.E-03

-2.E-03

-5.E-04

150 250 350 450 550

Temperature, C

ShrinkageRate,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

ShrinkageR

ate,1/s

200 C/min

100 C/min

50 C/min

For aluminum powder

Model

Experiment


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Influence of Thermal Diffusion

J is the vacancy flux, D is the coefficient ofdiffusion, vC is the vacancy concentration,

vC is the vacancy concentration gradient,*Q is the heat of vacancy transport, T is thetemperature gradient.

*

v v

Q TJ D C C

kT T



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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 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 theAmerican 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)



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ue ce o e us oJ is the vacancy flux, D is the coefficient ofdiffusion,

vC is the vacancy concentration,

vC is the vacancy concentration gradient,*

Q is the heat of vacancy transport, T is thetemperature gradient.

*

v v

Q TJ D C C

kT T

2v 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 T T

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 thematerial and per unit time, which in thecase of SPS can be represented as

2

eq E , where e is the specificelectric conductivity, and E is theelectric field intensity2T

e

T

TdivJ D E

T t

C



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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 thematerial and per unit time, which in thecase of SPS can be represented as

2

eq E , where e is the specificelectric conductivity, and E is theelectric field intensity

is porosity; G is the average grain size



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25

125

225

325

425

525

625

0 200 400 600 800 1000

Time, s

Temperatu

re,C

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Porosity

Temperature

Porosity - Model

Porosity - Experime nt

25

207

389

571

753

936

1118

1300

0 70 141 211 281 352 422

Time, s

Temperatu

re,

C

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Porosity

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



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The intensity of thermal diffusion increases for

higher pulse frequencies.

The thermal diffusion promotes components

(atoms and vacancies) separation. At early stagesof 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 diffusionconditions, 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


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Transfer During SPS (model including electromigration):

EC C J E

Nernst-Einstein equation

grain-boundary diffusion power-law creep

driving sources

externally applied load

sintering stress

electromigration

*gb gb

E q

D

C Z ekT

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 xy q

D UJ Z e

kT l y

U and l are the electric potential and the characteristic length along theelectric 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. Mul l ins,

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. Gun go r and D. Marou das, Int. J. Fractur e,

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

Symposi um Pr oceed in g s, 6p . (2000):

electromigration stress in 0.5 -thick Al film under 300V/m field should reach the level of 1.5GPa

C tit ti M d l f S k Pl Si t i


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5

2

1

3*

2 2

2 2

3 1 1 3 31 1

2 22

m

gb gb q p xx gbx cr x x

pp

D Z e G r UA

GkT l G r G G G r

G is the grain size; pr is the pore radius; A and m are power-law creep frequencyfactor and power-law creep exponent, respectively; gbD is the coefficient of the

grain boundary diffusion, gb is the grain boundary thickness, kis the Boltzmans

constant, T is the absolute temperature; is the atomic volume, *Z is thevalence 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

C t ib ti f diff t f t t h i k d SPS


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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

ShrinkageR

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

ShrinkageR

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

ShrinkageR

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

Porosity

external load

surface tension

electromigration

Contribution of different factors to shrinkage rate of aluminum powder under SPS

417U Vl m

, T=6730K, x =28.3MPa

Shrinkage kinetics during SPS of aluminum powder:


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The average particle size is 55m. The applied field is accepted to be of

500V

m(Joule heat generation balancebased 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


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SUMMARY


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(elV) 0

CpT

t

(kT T) el V2

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

TEMPERATURE DISTRIBUTION DURING SPS


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prismatic die

temperature temperature gradient

temperature temperature gradient

cylindrical die

TEMPERATURE DISTRIBUTION DURING SPS

SPS SCALABILITY (SIZE DEPENDENCE)


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( )

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



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-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.015 0.030 0.045 0.060

(Porosity(Cent

er)

Porosity(Surface))/SampleRadius

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


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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

(GrainSize(Cente

r)

GrainSize

(Surface))/Sam

pleRadius

Die Radius [m]

SUMMARY


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SUMMARY

F rther prospects


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Development of on-line sintering damagecriteria

Modeling of nano-powder sintering

Modeling of sintering with phase

transformations or chemical reactionsModeling of field-assisted sintering

Development of sintering optimization

approaches

Multi-scale modeling of sintering

Further prospects

0305 Olevsky - Theory of Sintering

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