SINTERING OF POWDERS AND DENSE MATERIALSExperimental, Mechanical, Thermal approaches

and Modelling

Pr Ange Nzihou, EMACPr Roberto Santander, USACH

USACH, March 2006

ISINTERING OF POWDERS AND DENSE MATERIALS

Experimental, Mechanical, Thermal approaches

Pr Ange Nzihou

Geometry of the problem

unit problem representing thegeometry during neck formation

2 X

a

b

φ

t

L

h

F

F

bo

ao

a row of spherical particles in contact

grainboundary

particle surface 2

s

1s

neckv

bs velocity of the necks

unit tangent vectorsto the grain boundaryand particle surface at the necks

SINTERINGInitial point contact

D

late stage, neck growth long time

1.26 D

infinite time

Sintering model with the developmentof the particle bond during sintering

Microstructural scaleIn boundary, important atomic motion

MACROSCOPIC DESCRIPTION

Principal mechanisms

Evaporation –condensationSuperficial diffusion

Without shrinkage

With shrinkage

•Formation of pores•Mechanical resistance•Chemical reactions•Dimensional variationcontingent on temperature

Principal mechanisms

Volume diffusionflows (viscous and plastic)

PHYSICAL CHEMISTRY ASPECT

DRIVING ENERGIES:

Surface Energy : •Surface tension dSdW /=γ

Energy linked to the presenceof physical defects :

•Proximity of curved surfaces•In the crystalline network, there exists a concentration of C de lacuna expressed as flows( thermodynamic statistics):

)/exp(Co kTENn f≈=

Energy linked to the presence of pressure :

• If the interface is curved, the pressure of the vapor, in equilibrium with the solid, changesdepending on the curvature of the surface

SINTERING MECHANISMS IN SOLID PHASE

1-TANGENT SPHERES 2- SECANT SPHERES

1. Evaporation –condensation2. Superficial diffusion3. Volume diffusion 1. Viscous flow

2. Volume diffusion3. Intergranular diffusion4. Microcreep

1.1-TANGENT SPHERES: Evaporation - condensation

• That a transfer of atoms will be established, by the gaseous phase fromthe sphere’s surface towardthe lateral surface of the bridge.

00 21 <∆>∆ PetP

( ) tRTM

dkTP

rX *2*

3 2/10

3

⎥⎦

⎤⎢⎣

⎡ Ω= π

πγ

atomiquevolume=Ω

X

rρ

o

o'

o''

P1>0

P2<0

V S

1.2-TANGENT SPHERES : superficial diffusion

X

rρ

o

o'

o''V S

δ

ρ

s

J/2

J/2

C

•In proximity to the bridge’s surface, there exists and excess of lacunas; however, Nearby the sphere’s surface –far from the threshold–there exists a defect of lacunas.•The extra lacunas will diffuse.If a flux of lacunas is

established, there will be an equivalent flux of atomsin the opposite direction which will therefore contribute to build up the bridge.tkT

DrX ss **563

7

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

γδ

This exchange of lacunas and atoms will only be restricted to the superficial layer s

δ

1.3-TANGENT SPHERES : Volume diffusion

X

r

ρ

o

o'

o''S

R =

r2R =

1 ρ

• Based on the presence of anexcess of lacunas neighboring the bridge’s surface, and of a defect of lacunas nearby the sphere surfacesfar from the bridge.• This diffusion of lacunas (and ofatoms) is speculated to operate onthe volume and no longer on the surface

tkTD

rX v **2

52

5

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

γπ

2.1-SECANT SPHERES : Viscous flow

r

xx1

ρ

y

s

vv

1 2

o

o'

h

• The formation of a linking zone betweenspheres carried out by viscous flow of Newtonian-type material

• the displacement of atoms carried outunder the influence of a cut which isproportional to the gradient of speeds

itévisdtd cos== ηεησ l

dl=ε

tnKrx

nn

*/1/12

⎟⎠⎞⎜

⎝⎛=⎟

⎠⎞

⎜⎝⎛

ηγ

2.2-SECANT SPHERES : Volume diffusion

• Similar to those previously mentioned.

tkTD

rx v *202

5

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

γπ

• The model considered a variation of the distance between the centers of the spheres

2.3-SECANT SPHERES : Intergranular diffusion

• The experimental observations showthat in the majority of cases, it formsa neck within the linked zone, being AA’.

• Les lacunas, finding themselves in excessneighboring the concave surface of the bridge, will be able to diffuse toward this grain joint, instead of spreading to surfaces with a larger radius of curvature meaning the surfaces of the two spheres.

tkTD

rx jj *962

6

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

γδ

r

x

ρ

o

o'

A'A

j

s

x

2.4- SECANT SPHERES : Microcreep mechanism

r

x

ρ

o

o'

A'

σ

σσ

σ

• Creep indicates a flow of material occurring at a given temperature under the influence of a constant.

• Volume subject to the strain ofcompression σ1

⎟⎟⎠

⎞⎜⎜⎝

⎛+≅

21

11γγγσ

Cases:

external load

external load

sint

erin

g

load

ext

erna

l

Sintering of a homogeneus body under uniaxial loading (tension or compression)

• Sinter forging• Plastic flow and sintering

under the action of externalforces

• overall compression• uniaxial loading• torsion

Free sintering • free sintering of linear – viscous porous material

• power – law creep (nonlinearity of the constitutiveproperties)

Evolution des solides traités (contenant des polluants)

Illustration of the sintering phenomenon:

Reduction of the specific surface area

0

20

40

60

80

100

100 300 500 700 900 1100 1300température de calcination (°C)

surf

ace

spéc

ifiqu

e (m

²/g) TCP palier de 30 min

HAP palier de 120 min

CaO palier de 120 min

Typical evolution

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

0 5 0 1 0 0 1 5 0t e m p s ( m i n )

Sp

(m²/g

)

4 0 05 0 06 0 07 0 08 0 09 0 01 0 0 01 1 0 01 2 0 0

/

0,E RT

ii i i

D ij grad D D eRT

σ−

= − =

• Diffusion in gas phase (1)

• Surface diffusion (2)

• Volume diffusion (3)

ndS kSdt

− =Kinetics:

Effect of Calcination on HAp properties

Other physical changes

Chemical changes

Amorphousto

cristalline structure

3

00 LL1

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∆

Shrinkage

Calculation of properties during the sintering

VVmm

Vm

0

0∆−∆−

==ρ

30

20

0

)100

dilatation%1(hR

)100

TG%1(m

+π

−=ρ

Density

theo1

ρρ

−=εPorosity:

600

700

800

900

1000

1100

1200

1300

1400

0 200 400 600 800 1000 1200Temperature (°C)

dens

ity (k

g/m

3 ),po

rosi

ty %

*10

-22

-18

-14

-10

-6

-2

2

wei

ght l

oss

%,

shrin

kage

%

densityporosityshrinkage %weight loss %

L0L

Force

-30

-25

-20

-15

-10

-5

0

5

0 200 400 600 800 1000 1200température (°C)

% d

ilata

tion HAP TCP

HAP sto

Density and porositycalculation from TMA,

Thermomechanical Analyzer

Modeling for the description of the sintering process

Typical representation of processes involved:

r2

2p HrTr

rr1

tT)T(c)t,T( ∆ν+⎟

⎠⎞

⎜⎝⎛

∂∂

λ∂∂

=∂∂

ρ ν: Kinetic of reaction

p

λ(T,t)α=ρ(T,t)C (T,t)

Continious measurement of the properties:

H= 2 cm

Di= 1 cm

mHAP= 2gTintText

0

50

100

150

200

250

300

1000 2000 3000 4000 5000 6000 7000 8000time (s)

Text

-Tin

t, °C

400

500

600

700

800

900

1000

1100

1200

Text

, °C

constant parametersvariable parameters

Text

Better description of the sintering phenomenon

ISINTERING OF POWDERS AND DENSE MATERIALS

MODELLING

Pr Roberto Santander

Model for Sintering and Coarsening of Rows of SphericalParticles

Literature : • Parhami F. et al; Mechanics of Materials – 31 pp 43-61 (1999)• Svoboda and Riedel; Acta Mettal. Mater – Vol 43 N 1pp 1-10 (1995)• Olevsky E; Materials Science and Engineering R23 (1998)

Goal :Model for the formation of interparticle contacts and neck growth between powder particles by grain boundary and surface diffusion.

Methodology :Model is based on a Thermodynamic Variational Principle arising from the governingequations of mass transport on the free surface and grain boundaries

Maximization of the rate of dissipation of Gibbs free energy(for a pair of particles through grain boundary and surface diffusion)

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

1. Porous medium is considered as a two phase material (phase of substance – body skleleton – and phase voids –pores)

2. The skleton is assumed to be made of individual particles havingnonlinear – viscous incompresible isotropic behavior.

3. The voids (pores) are isotropically distributed.4. The overall response is therefore isotropic5. The free energy F per unit mass of porous medium is by hypothesis,

a function of the absolute temperature T and of the specific volumev.

Second principle de la thermodynamique des milieux continus

0:...

≥⋅−+⎟⎠⎞⎜

⎝⎛ − T

TdgraqesTρ

ρεσρénergie libre spécifique F

TseF −=internalstress

internal energy heat rate

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes;

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

Clausius’ inequality – Duhem with hypothesis F=F(T,v) and0=Tdgra

ρCauchy stress tensorσ

ε0: ≥−− TsFijij

&&& ρρεσ strain rate tensorpe εεε +=

)(ijij

f εσ =

)(ij

fS ε=Plasticdistortion (irreversible)

T&but not on for a viscous material

Elastic distortion(reversible)

TFS ∂

∂−=

0)( ≥−ijijLij

P εδσ &

)TL v

FP ∂∂=

Laplace pressure or sintering stress (result of thecollective action of local capillary stresses in a porous material)

where

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

The condition Is satisfied if there exists a dissipative potential Ddefined as a homogeneous function of order m+1of the strain rate ij

ε&

0)1( ≥+=∂∂ DmD

ijij εε &&

ijijLij

DP εδσ &∂∂=− and

total

pores

VV

)()( θfDorvfD == θ=porosity=

For D, cases three

1. Linear incompressible viscous material with voids

2. Incompressible nonlinear – viscous material3. Nonlinear – viscous porous material

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

1. Linear viscous incompressible material with voids

dissipation potential 22

21 eD ζηγ +=

effective shear moduleeffective bulk module

(viscosity for example)

''

ijijεεγ &&=

ρρεε &&& −===

iitre

deviator of thestrain rate tensorshrinkage rate

002ψηζϕηη ==

2)1( θϕ −=

shear modulus of the porousbody skeleton

θθψ

3)1(32 −=

2

0)1( WD ηθ−=the potential D can be expressed as:

θψϕγ

−+= 1

22 eW

The constitutive law is:

ijLijijijPe δδψεϕησ ++= )(2 '

0&

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

2. Incompressible nonlinear – viscous material

0=θ ∞→→ ψϕ and10→e γψ →→ Wthereforee 02

•incompressible material•incompressible matric

2

0γη=D•the dissipative potential of a linear – viscous fluid is

• an extension into nonlinear –viscous behavior is obtained by

1

1+

+= m

mAD γ

material parameter depends on temperature strain rate sensitivity

• the deviatoric stress is obtained from

ij

m

ijij

AD εγεσ &&1' +=∂

∂= )( '

ijijεε &&=

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes;

3. Nonlinear – viscous porous material

• using a power law dependence, the dissipation of a porous material is

1)1(1 +−+= mWmAD θ

• matrix incompressible, nonlinear-viscous

1

1+

+= m

matrix mAD γ

γθ == Whavewe0(incompressible nonlinear viscousmaterial)

021 η== Ahavewem

(linear viscous porous material)

• constitutive law

ijLijii

m

ijPeAW δδψεϕσ ++= − )( '1 &

ifγϕτ 1−= mWA L

1-m P e W += ψAp

mWA =σ

substituting leads to the following relationship betweenthe equivalent stress σ and the equivalent strain rate W

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes;

)( Wσσ =• a generalization of the relationship, mWA =σ between equivalent stress and equivalent strain

rate arbitrary function of W

• in this general case, the constitutive relationship for a nonlinear –viscous porous material can be represented in the form (sintering under pressure):

ijLiiijPδδψεϕσσ ++= ) e (W

(W) ij

'& « basic constitutive expression for thecontinuum theory of sintering »

material resistance

externally appliedstresses

influence of capillary stresses(sintering factor)

if :

• sintering free 0 →=ij

σ

•

•

sintering without pressureby t treatemen 0 →=LP

pressureunder sintering 0 P and 0 L

→≠≠ij

σ

Phenomenological model of sintering based upon the ideas ofthermodynamics of irreversible processes

we have

P e W(W) p and W

(W) L

+== ψσγϕστ

• an important relationship between the invariants of stress – strainrate state is:

e )( ψτγϕ =−L

Pp

Model Formulation

• At hight temperature, atoms travel along the free surfaces and the interparticlecontacts to reduce the total free energy of surfaces and interfaces of the system.

ENERGY BALANCE BETWEEN SOURCES AND SINKS

∏ +=ss

RG&Gs : rate of change of the free energy of system

Rs : one-half of the rate of energy dissipation

ssbbdA J 1 2

1 dA J 1 21

ρρρρ∫∫ ⋅+⋅=

sb As

sAb

bs

JDJDRFvAAGbbsss−+= &&& γγ

areaunit per energiesboundary grain and surface b→∧ γγ

s

areasboundary grain and surface A b

→∧s

A

force applied →F

other the toparticles of row theof end one of velocity →v

kTD

D ss

Ω=

)s

δkTD

D bbb

Ω=

)δ

parametersdiffusion →∧sb

DD

surface free on the andboundary grain on the material of fluxes →∧sb

JJρρ

iesdiffusivit atom surface andboundary grain →∧sb

DD))

surfaces andboundary grain on the occursdiffusion ch within wies thickness→∧sb

δδ

volumeatomic →Ω

constant sBoltzman' →k

re temperatuabsolute →T

Model Formulation

Important unit timein length unit through diffusion by passing material of volume→J

internal theof change of rate theis system theofenergy free theof change of rate the→s

G&

sum theisenergy internal system, In the rate. work external theminusenergy energiesboundary grain and surface of

∏→ vand A , A , J ,J of s variationcompatible respect to with valueminimum stationary a has bssb

settingby achieved ison minimizati Ritz -Rayleigh The

0=+=∏ ssRG δδδ &

δδδδδδ

a&

b&L&h&

t&x&

δΠ must be zero for variations degrees of freedom of theVariational functional Π

Model Formulation ……………… compatibility condition

∑ ∫∫ +⋅+=b 21

)ˆˆ(all neck

neckA

snsdLssvdAvA

s

κ& ∑ ⋅=b

ˆall bneckb

dLsvA&

∑ ∫=b

all A

bbb

vdAFv σ

sum of the principal curvatures of the particle surface

rate of motion of the particle surfacein the outward normal direction

motion of the locus of point of connectionbetween the grain boundary and free surface

integral of the strees over each grain boundary

0=⋅∇+s

sn

Jvρ

0=⋅∇+bbb

Jvρ

0ˆˆˆ2211

=⋅+⋅+⋅bbss

JsJsJsρρρ

Lon 21

σκγκγ −==ss

volume conservation on As

volume conservation on grain boundary

flux continuity at the necks requires

continuity of chemical potential

Equations of conservation

sAon κγ

ssssDJ ∇−=

bAon σ

bbbDJ ∇=

Model Formulation

The resulting expressions are coupled linear equations :

[ ] fk ˆˆ ˆ =δ&

rates of change of the six degrees of freedom

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

+

+=

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

+

x 2- t 2-

F x 2-

F b 2-

Fa 2-

h 2-

2-

x

t

h

L

b

a

bs

s

s

s

s

s

6

3 000 4

2 4

2

0 8

4 8

4 8

4 00

0 8

4 8

4 8

4 00

0 8

4 8

4 8

4 00

4

2 000) 2 2 2(2

b 0

4

2 0000 t)2Lag 2a x (22

γπγπ

γπ

γπ

γπ

γπ

γπ

πππ

πππ

πππ

πππ

ππ

ππ L

sDtx

sDthb

sDtLa

bDx

bDx

bDx

bDx

bDx

bDx

bDx

bDx

bDx

sDthbthbgbx

sDx

sDtLa

sxDa

&

&

&

&

&

&

the set of variables used above are not independent, by imposing volume conservationand geometrical constraints, 222222 x- b h and x- a ==L

Model Formulation

[ ] fk =δ& rates of change of the three degrees of freedom

( )( )

( ) ⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−+−−+

−+−−

−+−−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +

Fxtxx

thx

Lx

hb

La

Fxbh

hx

hxb

bx

x

FxaL

Lx

Lxa

ax

x

a

bLa

aLaa

b 2 2 x 2

2

2221h b 4

22221L a 4

x

b

a

D1

D x 3t 2

xt D 21

D x 4th x t bD 2

1D x 4

tLx t

D 21

D x 4th x t b D

g b 2D 21

Dx t h D 2

h b D 21

D x 4tLx t D 2

h b Dg a 2

D 21

Dx t L

s

222

2s

222

2s

bs

22

bsbs

bss

b4

bs

22

b

bsbs

a4

bs

22

πγγπ

γπ

γ

&

&

&

with

( ) ( )⎥⎦⎤

⎢⎣⎡ −−= 3232o

2 31-h 3

1-L V 1 hbLaxt π( ) a

Lax

xaLnga −⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

211

( ) bh

bx

xbLngb −⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

211

Numerical Procedures

• initial conditions 0 x and bb o === oaa

Case 1: very small value of x, instability numerical asymptotic approach

Case 2: elimination of the circular disc

a

b

2 X

L

h

Case 3:elimination of the spherical surfaces

2 X

t

• Runge Kutta

Case 1: very small values for the magnitude of x

• in the limit where x /a , x /b and t /x are much smalle than 1 and higher orderterms in are neglected[ ] fk =δ&

[ ] t xF 4)2( )g (gD g g D 4

32basbab

πγγ b

ba

bs Dxtggx −−++=&

( ) ( )⎥⎦⎤

⎢⎣⎡ +++−= b

1a1

4b 32V 1 433o

2xaxt π

b23

b2

sg b x 6

F Dt b t

) 2(π

γγ +−−=b

bsg

Db&a

23b

2s

g a x 6F Dt

a t ) 2(

πγγ +−−=

a

bsg

Da&

Of with 1/ga and 1/gb negligible is obtained :

generalization of Coble’s are used until :

01.0=axTe

1111

)2(4 D 96 2b6

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛ +

−⎟⎠⎞⎜

⎝⎛ +

−=

oooo

bs

ba

F

ba

xπ

γγ

elapsed timeTe →

Results

numerical matlabpaper

Results