Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

263

Journal of Chemical Technology and Metallurgy, 49, 3, 2014, 263-274

A SUB-GRANULAR SCALE MODEL FOR SOLID STATE FREE SINTERING: RESULTS ON THE EVOLUTION OF TWO GRAINS

Jacques Léchelle1, Sylvain Martin1,2, Robert Boyer3, Kacem Saikouk3

1Commissariat à l’Energie Atomique et aux Energies Alternatives, Centre de Cadarache, DEN, DEC, SPUA, LMPC, F-13108 Saint Paul Lez Durance, France2TIMR, EA 4297, BP 20529, Université de Technologie de Compiègne, F-60205 Compiègne cedex, France3LATP, CMI, Aix-Marseille Université, Technopôle Château-Gombert, 39, rue F. Joliot Curie, F-13453 Marseille cedex 13, France

ABSTRACT

A sub-granular model development for solid state sintering of ceramics is ongoing to describe grain and pore size evolution during a free (or under gas pressure) sintering.

Local changes in principal curvature radii at grain free surfaces and grain boundaries induce an extra stress upon the outer boundaries of the grains. The latter are regarded as single crystals, of elastic constitutive law. Navier-Lamé equations resolution in the bulk of each grain gives the 3D-displacement field and hence the density of the elastic mechanical energy. Resulting Gibbs free energy variations along the grain interfaces (free surfaces and grain boundaries) induce surface and grain boundary mass transport (Fick’s first law). Since matter is almost incompressible, locally accumulated matter on grain surfaces makes the surface move (second Fick’s law). This results in an irreversible shape evolution.

A 3D-software has been designed. At each time step the Navier-Lamé equations are solved using a finite element method and Fick’s second law is treated by a finite volume-like method enabling boundary nodes to move in an irreversible way. Continuous media mechanics is taken as the origin of grains evolution although the granular nature of matter remains the core of the model. The software has been developed for the simplified case of a chemically homogeneous material. Results referring to the evolution of two initial spherical grains of the same size are checked against Coble’s works extended by Coblenz in the case of grain boundary and surface diffusion. A second simulation is compared to Coble’s model for grain boundary and volume diffusion. Discrepancies appear to be mainly attributed to the fineness of the mesh which accounts for the real curvature in the vicinity of the grain boundary.

Keywords: sintering, finite elements, 3d-elasticity, mass transport, grain-boundary diffusion, surface diffusion, finite-element method, finite-volume method, mesh evolution.

Received 26 September 2013Accepted 07 April 2014

INTRODUCTION

Classical analytical sintering models rely on the assumption that a single phenomenon takes place at a given time. For instance, densification is ongoing dur-ing the second stage of sintering while grains grow and

pore shape becomes rounder. A second example refers to the case in which intergranular and volume diffusion phenomena occur during the first and the second sinter-ing stage [1]. Furthermore, there are materials for which these two stages are even combined.

The model we propose does not rely on the as-

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

264

sumption of a unique predominant phenomenon at a given time. It is developed on a sub-granular scale to account for grain growth and pore size changes with a free shape evolution.

Another aim of the model is to rely on a parameter set that could be measured experimentally by means of designed experiments excluding the sintering process itself.

Several approaches are available on the sub-granular scale. They include the Monte-Carlo-like methods [2 - 5], in which the material is cut down into pixels in 2D, or into voxels in 3D. An energy value is then given for each voxel set configuration. Voxels can be moved ac-cording to a set of rules in order to change the grain/gas interface. A configuration is kept with a given probability if it leads to a decrease of the energy of the system. This probability reflects the ratio of the energy configura-tion and the thermal motion energy. In these models the curvature, is taken into consideration in counting the gas-type voxels within a cube of 3b voxels where b is the size of the area used to smooth the curvature. The relationship between the number of Monte-Carlo iterations and time can be determined in the case where a single diffusion phenomenon takes place at a time. It is less obvious when this is not the case. Nevertheless the competition between the different diffusion paths is not taken into account by these Monte-Carlo methods.

Bruchon [6 - 8] develops a model for a single com-ponent system under sintering with both volume and surface diffusion but excluding the grain boundaries.

Other approaches are found in the literature. For instance, the phase field method defines a function for each grain which takes the value of 1 in the bulk of the grain and of 0 outside and varies continuously from 1 down to 0 over the thickness GBδ of the grain bound-ary. They rely on the time dependent Ginzburg-Landau equations [9]. Such an approach requires the use of the partial derivatives of the overall Helmoltz free energy in respect to the order parameters. It has been recently extended by Nestler [10] to the multicomponent case for diffusion with surface energy and anisotropic diffusion coefficients as well as new phases appearance.

Other sub-granular models are developed [11]. They consider both grain boundary and surface diffusion to simulate the evolution of the shape of grains undergo-ing sintering. This evolution relies upon Brakke surface evolver [12]. The same author described from a theoreti-

cal point of view the evolution of grain aggregates as a solution of a tensor virial equation [13].

The strong form of the model we have developed [14-16], is presented in this paper. Algorithms as well as numerical methods are implemented in a C++ software called “Salammbo” linked with diffpack® library [17] with the outlook of a subsequent possible parallelisation.

COMPUTATIONAL MODEL

Assumptions of the modelThe simulation does not take into account the grain

rearrangement, i.e. grain sliding of particles in contact. It implies that once a contact has been established between two grains, this contact remains. Neither subsequent sliding, nor rolling occurs between these grains in con-tact. This phenomenon has to be considered in case of a pressure-assisted sintering, as well as in the case of the simulation of a set of more than two particles.

Grains are thought to be single-crystals even if their point defects are organized into edge or screw disloca-tions, into dislocation loops or into small tilt angle grain boundaries. During the heating stage of the sintering thermal cycle these defects are partly annealed by means of volume diffusion. The latter is slower than other solid state diffusion phenomena. Thus these defects may still be present at the end of the sintering. Three angles ( )cba

,, can be attributed to the single-crystal grains to bind their Bravais lattice orientation ( )cba

,, with the laboratory axes ( )321 ,, eee

. The sintering is supposed to be a free type one, i.e. the applied stress is weak enough so that the grain strain constitutive law can be regarded as elastic. It is worth adding that the diffusion leads to an irreversible behaviour of the system.

The gradient of the mechanical energy stored in the grains makes the matter flow in an irreversible way.

The gravity effect has not been taken into account in this simulation.

In order to illustrate the model and for current tests, the case of the contact of two grains is considered. The free surface of grain 1 is denoted by 1Γ , that of grain 2 by 2Γ , the grain boundary – by 21Γ and the triple line - by 21∆ . The volume of the grains is denoted by

iΩ (i = 1 or 2).Each grain is regarded as a continuous medium

characterized by three angles making a link to the atomic scale for direction dependent properties.

Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

265

Effects of surface curvatureThe grain outer surface curvature, whether a free sur-

face or a grain boundary, is at the origin of stress changes along grain surfaces. The curvature applies a force fd

of Laplace type upon an element dS of the surface. The resulting force is perpendicular to the surface.

1 2

1 1df n dSR R

γ

= − +

(1)

where 1R and 2R are the principal curvature radii.A typical grain size is 0.1 micrometer against c.a.

10 micrometers after sintering. It can be noticed that the origin of diffusion is very sensitive in respect to the value of the mean surface curvature defined by

+=

21

1121

RRH (2)

Numerical computation of a curvatureIn the case of a meshed surface, evaluation of the

local mean curvature H is critical for a good simulation of surface evolution. Several techniques were used and probed [16]. This problem is the cornerstone of surface evolution and is at the origin of a lot of difficulties [18, 19]. A specialized numerical library called CGAL for Computational Geometry Algorithms Library was chosen. It adjusts, at each node for a given set of first neighbours in a least square sense, a Monge form in a frame. Hereafter z axis is the outward normal and the node under consideration is at its origin.

( ) ( )

( )

( )

2 21 2

3 2 2 30 1 2 3

22 2

1,2

1 3 36

,

z x y k x k y

b x b x y b xy b y

x y x yε

= + +

+ + + +

+ +

(3)

The number of first neighbour layers to be consid-ered for a given node is a critical parameter. The curva-ture information is noisy if there are too few neighbour layers, while the curvature information is too averaged if there are too many of them. In both cases matter fluxes due to these curvature changes are not correct.

Mechanical impact of surface energyThe quasi-static equilibrium of forces is given under

the assumption of small deformations on the ground of the Navier-Lamé equation by:

3

10, 1,...,3ij

ii j

f ixσ

ρ=

∂+ = ∈

∂∑ (4)

where ijσ refers to the component ij of the stress tensor, while ρ is the density of the solid. This approach, of small deformations type, matches well our case because the aim is only to render elastic energy density changes stored into the solid at any time due to mean curvature changes along grain surfaces. The overall strain, which is rather large

0

( 15%)LL∆

≈ −

within the sintering is not due to elasticity but to a de-formation linked to irreversible matter fluxes.

Neumann boundary conditionsSome of the closure conditions of the problem are

of von Neumann type.For a grain-gas interface they reproduce a Laplace

force exerted locally upon the surface due to stress which originates from the mean curvature:

( ) 2gas SV SVn n p Hσ γ⋅ = − −

(5)

Index SV refers to the grain/gas interface (solid/vapor).

For a grain/grain interface they reproduce the action of a grain upon the other one as well as the action of a Laplace force locally exerted upon the surface due to curvature driven stress:

( ) ( )1 1 1 2 1 1 2 GB GBn n n n Hσ σ γ⋅ = ⋅ − (6)

Index GB refers to the grain/grain interface (grain boundary), while index 1 or 2 refers to the domain (in the mathematical partial derivative sense, i.e. grain) for which σ is computed.

Algebraic conditions Other closure conditions are of an algebraic type

and enable the elimination of rigid body motion of the set of solutions of the problem.

Conservation of translational and angular momenta leads to the following algebraic conditions with the fol-lowing notations:

The gravity centre G becomes G’ after deformation,

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

266

a point M of the solid becomes M’’ after deformation so that 'MM u=

. Hence:

(7)1 2

1 2' '

0

' ' 0 '

' 0

GM d definition of G

G M d definition of G

GG motionless gravity center

ρ τ

ρ τ

Ω Ω

Ω Ω

=

=

=

∫∫∫

∫∫∫

where ρ is the density of the material that is supposed to be constant (quasi-incompressible solid). Thus:

∫∫∫ΩΩ

=21

0

τρ du (8)

The absence of two grains rotation around the grav-ity centre gives:

1 2

0GM u dρ τΩ Ω

∧ =∫∫∫

(9)

Numerical resolution of the problemThe displacement field u is computed using the

finite elements method with 2P type elements so that the second derivatives of u driving fluxes are non-zero.

The weak form for an isolated grain under von Neumann type boundary conditions can be obtained by integrating by parts, eq. (4):

( )( ) 0.21

=+∈∀ ∫∫∫ΩΩ

τρσ dwfdivfieldsadmissiblew

(10)Thus:

[ ]( )( ) [ ]

( )

1 2

1 2

:

. .

w admissible fields w f d

n w d

σ ε ρ τ

σ τ

Ω Ω

∂ Ω Ω

∀ ∈ + =

=

∫∫∫

∫∫

(11)

Elastic constitutive lawThe elastic behaviour of a cubic material (space

group 3Fm m ) leads to an elastic tensor with 3 inde-pendent coefficients C11, C12 and C44 in the frame of the axes of the crystal, i.e. of a grain (with Voïgt notations):

(12)

11 12 12

12 11 12

12 12 11

44

44

44

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

c c cc c cc c c

cc

cc

=

The cubic case is very far from the isotropic one for UO2 [20] at room temperature. The following relation would hold

( )44 11 1212

c c c= − :

11

12

44

389 396119 12159.7 64.1

c GPac GPa

c GPa

= − = − = −

(13)

The cubic case is implemented within the software although the values of the constant are chosen to represent an isotropic elastic tensor for the test cases. The elastic constitutive law is given by:

εσ :c= with (14)

12

jiij

j i

uux x

ε ∂∂

= + ∂ ∂ Each grain g has its own orientation ( )ggg ψθϕ ,,

which will impact the mechanical quasi-static equilib-rium equations. In the software these equations are used in the laboratory frame.

'c αβγδ denotes the elastic tensor components of the grain g in the frame of the laboratory while Pg is the change-of-basis matrix from the laboratory frame to that of the grain g (for change of referential Voïgt notations have been avoided).

lgkgjgigijkl PPPPcc δγβααβγδ =' (15)

where i,j,k,l are indices of rows of the change-of-basis matrix. The latter gives the composition of three rota-tions the angles of which are ( )uxg ,=ψ , around z

( ),g x uψ =

around u , and ( )aug ,=ϕ around c .After this change of frame the system of partial

derivative equations is as follows:

Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

267

=∂∂

∂

=∂∂

∂

=∂∂

∂

Ω∈∀

∑∑∑

∑∑∑

∑∑∑

= = =

= = =

= = =

3

1

3

1

3

1

2

3

3

1

3

1

3

1

2

2

3

1

3

1

3

1

2

1

0

0

0

,

i j k Mkj

iijk

i j k Mkj

iijk

i j k Mkj

iijk

g

xxu

C

xxu

C

xxu

C

M (16)

with ljkiljikijkl ccC '21'

21

+=

Weak form of the mechanical problemThe component 3,2,1∈i of the field u is devel-

oped over a basis of polynomials of second degree cN with the values c

iU of the function at the nodes of the mesh as coordinates:

( )10

1 2 31

, ,c ci i

cu U N x x x

=

= ∑ (17)

The integration over a tetrahedron K of a mesh T of the sub-domain gΩ , ( 2,1∈g denoting the grain number) gives:

3 3 3

11 1 1

3 3 3

21 1 1

3

0

1,...,10 , 0

cb cb c b

ijk ii j k c j kK K

cb cb c b

ijk ii j k c j kK K

cb cb c b

ijk ic j kK K

N N NC N U d N dSx x n

N N Nb C N U d N dSx x n

N N NC N U d N dSx x n

τ

τ

τ

= = = ∂

= = = ∂

∂

∂ ∂ ∂− =

∂ ∂ ∂ ∂ ∂ ∂

∀ ∈ − = ∂ ∂ ∂

∂ ∂ ∂−

∂ ∂ ∂

∑∑∑∑ ∫∫∫ ∫∫

∑∑∑∑ ∫∫∫ ∫∫

∑ ∫∫∫ ∫∫3 3 3

1 1 10

i j k= = =

= ∑∑∑

(18)

This system is solved in terms of elementary matrices whose components ij

rse A are computed by our software:

, , , ,1 , 1 1 1 , 1 1

12

d d n d d ne ij l m

rs rslm j j m i s rslm k k l i ss l m j s l m k

A C u N N C u N N= = = = = =

= +

∑ ∑ ∑ ∑ ∑ ∑

(19) For the right hand member:

+= ∑∑∑∫∫ ∑=== =

n

klk

mk

n

jmj

lj

d

s T

d

mlrslmi

e NuNuCF1

,1

,1 1, 2

1

(20)Following the assembly stage the form of the sys-

tem is: Au F= (21)

or, if terms involving only u values over nodes of iΩ (which do not pertain to the grain boundary Γ ) are

taken apart:

=

ΓΓΓΓΓ

Γ

FF

UU

AAAA

t11

1

111 (22)

The matrix elements A and F are computed and assembled for every grain. The non-translation (eq. 8) and non-rotation (eq. 9) conditions can be expressed by means of a matrix B with U vector of unknowns at the mesh nodes: ( )( ) ( )0=UB (23)

This relation corresponds to 6 scalar equations which are constraints imposed to the field u . It was chosen to impose them in a least square sense by addition of a vector of extra unknowns (Lagrangian parameters)

6ℜ∈L leading to the following system:

1 111 1 11

1 2 1 2

2 22 22 22

11 1 2 22

0

000

t

t t t t

t

U FA A BU FA A A B BU FA A BLB B B B

Γ

Γ ΓΓ ΓΓ Γ Γ Γ

Γ

Γ Γ

+ = +

(24)

An augmented Lagrangian method can be used (Fortin method [21]) in order to enhance the condition-ing number of the system. There r is a weighting factor. The system becomes:

( )( ) ( )( ) ( )

( )

11 1

1 2

2 22

11 11 11 1 2 11 22

1 2 11 1 2 1 2 1 2 22

22 11 22 1 2 22 22

1 11

1 2

2 22

0

0

t t

t t t

t t t t t t

t t t

n t

t

A AA A A r

A A

B B B B B B B

B B B B B B B B B B

B B B B B B B

U BU B BU B

Γ

Γ ΓΓ Γ

Γ

Γ Γ

Γ Γ Γ Γ Γ Γ Γ Γ

Γ Γ

Γ Γ Γ

+

+ + + + + +

+ +

( )1

2

nF

L FF

Γ

=

(25)

0L is obtained solving the non-augmented Lagrangian. Then 1+nL is given by:

1 0 2n n nn nL L BU with rρ ρ+ = + (26)

The non-augmented Lagrangian is used in this simulation.

Traction exerted upon an element σ( )( )21 ΩΩ∂= Kσ of the surface is given by:

1 2

1 1F p n dSR Rσ

σ

γ Σ Σ

= − − +

∫∫

(27)

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

268

Generalized chemical potentialThe generalized chemical potential can be expressed

by a stress tensor [16] as described hereafter.

0 0

0

'3

ik

T ii ik ikV VG u d

n n n

σ

µ σ σ∂ ∂∂= = −

∂ ∂ ∂ ∫ (28)

where n is the number of moles of the chemical species whose chemical potential is computed and V0 the total volume.

In terms of stress only, by means of the compliance matrix s, with Q the molar volume, it is described by:

( ) ( : : )3 2

tr sµ σ σ σQ Q= − − (29)

The Onsager relation [22, 23] states proportionality between fluxes and their origin, i.e.:

∇−=

TLJ µ

(30)

Taking into consideration the Fick first law in terms of concentration it becomes [16]:

Q=

RDL (31)

where R is the perfect gases constant. This expression of L is used in eq. 30 to give the flugs of the generalized chemical potential of elastic origin:

( ) ( : : )3 2

DJ tr sRT

σ σ σQ Q = ∇ +

(32)

Evolution of interfaces between sub domains

Flux ModellingA volume flux g

VJ is defined for every grain. It refers both to the bulk of a grain and to the half-thickness GBδ of the grain boundary, as shown in Fig. 2. Moreover, a grain boundary flux is defined g

GBJ , for each one of the grains sharing a grain-boundary. It is linked to the gradi-ent of chemical potential at the surface of its grain. It is given by concentration variation:

2g gGB

SGB GBJ D cδ= − ∇

(33)

The concentration in the boundary layer is denoted by gc in eq. 33. Although GBδ for the grain boundary is chosen to be equal to 10 Å, we use , ,V bulk V GBD D=for the volume flux flowing from the bulk towards the grain boundaries, i.e. we accept that the grain boundary thickness is zero.

Evolution of grain/gas interfacesThe mass conservation equation (the second Fick’s

law) accepts that flux through the interface in the inter-face frame is nil. Let n and t

be the outer normal vector and a tangential vector at the surface which is smoothed at a given point M [16].

( )( )( )int . . .V S S S VS S

V ndS J n div J J t t dSδ= Q + +∫∫ ∫∫

(34)

Fig. 1. Axes of the laboratory and of the crystal. Fig. 2. Flux used in the model.

Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

269

where Q is the molar volume, while Sδ is the thickness of the surface layer.

The operator Sdiv brings into play partial deriva-tives with respect to a surface the equation of which is unknown (except maybe at t=0) and which is not param-eterized. The relation between the surface metrics and that of 3D-space cannot be taken into account. Surface and grain boundary fluxes themselves involve a second surface operator; S∇ :

∇−=

TDJ SSS

µ (35)

These particular operators are used in plate mechan-ics [24].

Evolution of grain boundariesThe mass conservation at the grain boundaries is

described by [16]:

( ) ( )( )( )int 1

1 2 1 2 1 21

.

. .

S

V V S GB GB GB V VS

V n dS

J J n div J J J J t t dSδ

=

= Q + + + + −

∫∫

∫∫

(36)where Q is the molar volume and δ is the thickness of the surface layer.

Flux expressionInterface velocity equations show that the volume

flux VJ and the divergence of the surface flux have to be known. Thus it is not necessary to know how to express surface flux [16]:

3

1

6 6

1 1

1, 2,3 ,3

2

V ik

i k

jV iij i j

i j k k

Dk jRT x

D sRT x x

σ

σ σσ σ

=

= =

∂∀ ∈ = − +

∂

∂ ∂+ + ∂ ∂

∑

∑∑

(37)

where s is the compliance matrix (the inverse of that of elasticity).

Interface velocity computationThe flux of the surface divergence is given by the

circulation of the vector, which gives the interface ve-locity. One of Duduchava theorem [25] facilitates the computation of the circulation of the surface gradient as far as this gradient can be simply calculated as the usual scalar product of the vector and the unit vector in the derivation direction. At the mesh nodes it is given, as the vector product of the outer normal vector of the smoothed surface given by CGAL library and the tangent

vector of the edge of a surface triangle. The normal vec-tor is interpolated between two nodes of a triangle edge as vector circulation has to be computed along the edge.

Mass conservation considerationsBoth eqs. (34) and (36) are integral ones giving the

evolution of a surface element. With the discretization linked to the mesh, a mean velocity can be computed over a triangle σ at the surface. But there is no obvious way to compute it at nodes. Difficulties appear keeping the number and the connectivity of the surface nodes while remeshing the deformed solid at the end of the time step.

Another solution consists in using a level-set method. Other approaches can be found in the literature. Recently, Bruchon et al. [7] and Pino Muñoz et al. [8] developed a 3D model of sintering using a level set method. It seems to be the most flexible approach, but nevertheless it needs further specific development. This is not done in our software.

Classical Lagrangian description of the particles is used in this study. The interfaces are discretized accord-ing to the mesh. The motion is computed separately for every triangle that belongs to a surface. According to eq. 36, the flux balance over a triangle gives the accumula-tion of matter during a time step.

The area of the triangle is approximated to remain constant with motion during a time step. If q.n is the displacement of the triangle, then:

S

tVq

∆=

int (38)

This approximation is valid only in case when the normal vectors at the nodes of the triangle are equal to

Fig. 3. Velocity of the nodes at the surface of the grains.

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

270

the normal vector of the triangle. In other words, the curvature radius should be much greater than the size of the mesh. This assumption is not valid near the triple points where the curvature radius is very small. Thus, the variation of the triangle area has to be taken into account and the motion of nodes has to be computed along their own normal vector.

We note the normal vector at the node i of the triangle by in , while the angle between n and in by

iθ as shown in Fig. 3. The new position of the nodes is calculated in cor-

respondence with:

( )'

cosi i ii

qc c nθ

= +

(39)

Then, the new area of the triangle, 'S can be com-puted from the new position of the nodes, ic' , and the corrected value of h:

2'

'int

SS

tVq

+

∆= (40)

After the calculation of the triangle motion, q’, the motion of each node is computed as the average of its displacements over the triangles of which it is a vertex. For example, if node i belongs to k triangles of surface, its new position at the end of the time step will be com-puted as follows:

( )1

'1cos

knew li i i

l il

qc c nk θ=

= + ∑

(41)

where ilθ is the angle between the vector normal to the surface of the triangle k and the vector normal to node i , while lh' is the displacement of the surface of triangle l calculated with (40).

Finally, the displacement of the nodes is calculated as follows:

• First of all, eq. 38 is computed to get the initial value for q;

• Then, eqs. 39 and 40 are computed iteratively until an adequate conservation of mass is reached;

• Finally, the new position of the vertexes is calculated using (41)

Stability of the meshing stepOnce the position of the surface nodes is updated,

the mesh generator GMSH [26] is executed to update

the 3D-grid. The number of surface nodes and elements is kept

constant over the simulation. Hence, the meshing step can become critical when the surface triangles tend to be very irregular. This happens usually near the triple points where the largest deformations occur. In order to overcome this issue, a surface mesh control algorithm is called prior to the 3D-mesh generator execution.

A single point of failure occurs when a triangle is flat. Hence, the principle of the mesh control algorithm is to keep the angles of the surface elements (P2 triangles) at a reasonable value.

First of all, the nodes of the surfaces are classified into rows which correspond to the distance from their position to the triple line. Then, the triangles of the surfaces are scanned. If an angle of the tested triangle is below the limit, the opposite vertex which belongs to the highest row is moved. In order to ensure mass and shape conservation, the displacement of the node which is initially computed in the triangle plane, is projected onto the plane defined by the normal vector at the vertex.

RESULTS AND DISCUSSION

Simulations were run for two identical in size par-ticles in contact. The initial mesh was generated using NETGEN [27] for two truncated spheres. This corre-sponds to a neck radius r/R of about 0.1, being the best compromise between mesh size, stability and accuracy of the particles evolution. The grain boundary is initially defined by the symmetry plane of the two particles.

Fig. 4. Initial (a) and intermediate stage (b) of sintering for two spherical particles.

a)

b)

Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

271

Surface and grain boundary diffusionThe modelling of sintering for two identical in size

spherical particles considering coupled surface and grain boundary diffusion was studied by several authors [28 - 31]. Hence, simple analytical results were found to give a good description of the evolution and can be used for the validation of our model [31].

The neck radius r, the particle radius R and the center to center approach distance h are the main parameters. Dimensionless time, τ

t , is used, where

4

GB S

R kTD

τδ γ

=Q

Bouvard and McMeeking [31] showed that the evolution of the neck radius is well fitted by ( )1 654t τ for a ratio between grain boundary and surface diffusion equal to 2GB

S

DD

δδ

= [32].

Fig. 5 shows that our simulations underestimate the evolution of the neck radius during the first stage. This is a consequence of the approximate shape of the neck, which does not allow an accurate representation of the curvature radius along the neck. Thus, the grain bound-ary diffusion is negligible while it should contribute to the increase of the neck size. Once the neck radius reaches 0.3, the shape becomes more realistic and the neck starts to grow faster.

Rendering a realistic representation of the curvature at the neck is very challenging. It is necessary to reach a compromise between the mesh size, which directly impacts the calculation time and the preservation of the shape of particles. Moreover, the curvature radius is very sensitive to the mesh-fineness. At the very first stage of sintering, the surface near the neck undergoes a

curvature change which can only be obtained by means of a fine mesh. The local mesh size would have to be adapted to the local curvature value. As simulation goes on, surface nodes are moved leading to bad surface ele-ments (irregular 2P triangles) which cannot be used in case of finite element methods application. The control algorithm of the current surface remeshing implemented in our software modifies surface nodes position keep-ing their connectivity and delays the appearance of this phenomenon. But it leads to underestimation of the stress gradient along the grain boundary at the very first instants of simulation. It should be noted that this is an issue when the grain boundaries are taken into account. In this case, the curvature radius along the triple line has to be computed separately for each particle while the curvature for the nodes at the neck corresponds to its limit value for the surface of the grain under consid-eration. When the particles in contact are considered as a continuous medium [7, 8], the curvature at the neck surface is more easily computed.

Hence, the major deformation observed at the very first stages, is connected with the increase of the neck radius.

The deformation h can be evaluated from a geo-metrical point of view. According to Coble’s assump-tions [33], the evolution can be described considering truncated particles. Using this assumption, an analytical solution for the evolution of the neck radius is described by [34]:

Rh

Rr 2

= (42)

It is worth adding that the deformation in our model is computed on the ground of the motion of the gravity centre of the two particles. Hence, the comparison with Coble’s assumption is limited.

The evolution of the neck radius with respect to the centre to centre approach distance is shown in Fig. 6. It is evident that the neck is bigger than that predicted by the geometric model because of the low grain boundary diffusion.

Volume diffusionThe volume diffusion generally has a low impact

on the behavior of a homogeneous material because it is much slower than the surface or the grain boundary diffusion. But it starts to predominate in case of different materials and hence its contribution has to be under-

Fig. 5. Comparison of the evolution of the neck radius with analytical solution [32].

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

272

stood. This has been studied by Bruchon et al. [7] and Pino Muñoz et al. [8], but they considered a continuous medium without any grain boundaries.

The neck radius evolution was found in our simu-lations to fit well with the kinetic law given by Coble [33] for volume diffusion considering grain boundaries:

nV

m

PV

m

FDr tR R kT

BDh tR R kT

γ

γ

Q =

Q =

(43)

with F=32,m=3, n=4, P=2 and B=2.The evolution of the neck radius is visualized in Fig.

7, whereas the centre to centre approach (h) - in Fig. 8. If the neck radius evolution suggests that the main phe-nomenon is volume diffusion (from the grain boundary to the neck), the shrinkage graph does not really fit. It is worth noting that the h represents the particles shrinkage, which corresponds to the amount of displacement of the centre of the truncated spheres in case of two spherical particles [33]. It is not really possible to match the model, as long as the particles do not remain spherical. Besides, h corresponds, in our model to the relative displacement of the centre of gravity of both particles. This explains why the shrinkage is underestimated in our case.

CONCLUSIONS

A mechanical model of sintering is presented. It allows the simulation of coupled grain boundary, volume and surface diffusion. The shape of the grains is unconstrained during the simulation and is only linked to the Fick’s second law, i.e. to mass conserva-tion. The results representing the relative neck radius versus time are consistent with the analytical solutions described in the literature (Coble’s works) and the numerical sintering models like those in Bouvard’s papers. Nevertheless, the evolution of the neck of the grains is very sensitive to the local mesh size and the initial shape of the grain boundary. As the simulation proceeds the surface mesh node position changes keep-ing the initial surface mesh connectivity. This leads to bad (irregular) surface elements. A curved surface remeshing control algorithm is developed to delay the appearance of this problem. Grains surface has to be remeshed on the ground of updated node location and local curvature. The surface remeshing at the end of a time step needs to be improved to allow the simulation of large deformations occurring during the intermediate and final stages of sintering.

Fig. 6. Evolution of the neck radius with respect to the centre to centre approach distance. Comparison with the model of Coble [33].

Fig. 7. Evolution of the neck radius. Comparison with the model of Coble [33].

Fig. 8. Evolution of the neck radius with respect to centre to centre approach distance. Comparison with the model of Coble [33].

Jacques Léchelle, Sylvain Martin, Robert Boyer, Kacem Saikouk

273

REFERENCES

1. D.L. Johnson, A general model of the intermediate stage of sintering, Journal of the American Ceramic Society, 53, 1970, 574-577.

2. M.B. Veena Tikare, E.A. Olevsky, Numerical simu-lation of solid state sintering: I, Sintering of Three Particles, Journal of the American Ceramic Society, 86, 2003, 49-53.

3. M. Braginsky, V. Tikare, E. Olevsky, Numerical simulation of solid state sintering, Micromechanics of Materials, 42, 2005, 621-636.

4. S. Bordère, Original Monte Carlo Methodology De-voted to the Study of Sintering Processes, Journal of the American Ceramic Society, 85, 2002, 1845-1852.

5. J.H.S. Bordère, D. Gendron, D. Bernard, Monte Carlo Prediction of Non-Newtonian Viscous Sintering: Experimental Validation for the Two-Glass-Cylinder System, Journal of the American Ceramic Society, 88, 2005, 2071-2078.

6. J. Bruchon, D. Pino-Muñoz, F. Valdivieso, S. Drapier, Finite Element Simulation of Mass Transport During Sintering of a Granular Packing. Part I. Surface and Lattice Diffusions, Journal of the American Ceramic Society, 95, 2012, 2398-2405.

7. J. Bruchon, D. P. Muñoz, F. Valdivieso, S. Drapier, G. Pacquaut, 3D simulation of the matter transport by surface diffusion within a level-set context, European Journal of Computational Mechanics/Revue Europée-nne de Mécanique Numérique, 19, 2010, 281-292.

8. D. Pino Muñoz, J. Bruchon, S. Drapier, F. Valdivieso, A finite element-based level set method for fluid - elastic solid interaction with surface tension, Interna-tional Journal for Numerical Methods in Engineering, 93, 2013, 919-941.

9. L.-Q. Chen, W. Yang, Computer simulation of the domain dynamics of a quenched system with a large number of non-conserved order parameters: the graingrowth kinetics, Physical Review B, 8, 1994, 15752-15756.

10. B. Nestler, in: A. Miranville (Ed.), Phase-field models for crystal growth and alloy solidification, in Mathematical Methods and Models, volume Volume 53, Nova Science Publishers, Inc., 2005, pp. 197-227.

11. F. Wakai, K. A. Brakke, Mechanics of sintering for coupled grain boundary and surface diffusion, Acta

materialia, 59, 2011, 5379-5387. 12. K.A. Brakke, The Surface Evolver, Experimental

Mathematics, 1, 1992, 141-165. 13. F. Wakai, K.A. Brakke, Tensor virial equation of

evolving surfaces in sintering of aggregates of particles by diffusion, Acta materialia, 61, 2013, 4103-4112.

14. J. Lechelle, M. Trotabas, A mechanistic approach for the modelling of MOX fuel pellet sintering, in: “Advanced methods of process/quality control in nuclear reactor fuel manufacture”, proceedings of a technical committee held in Lingen, Germany, 18-22 October 1999, v. IAEA-TECDOC-1166, IAEA, 1999, pp. 65-77.

15. J. Léchelle, R. Boyer, M. Trotabas, A mechanistic approach of the sintering of nuclear fuel ceramics, Materials Chemistry and Physics, 67, 2001, 120-132.

16. M. Ajdour, Développement d’un code de calcul pour la simulation du frittage en phase solide, Ph.D. thesis - science et génie des matériaux - n°432 SGM, Ecole Nationale Supérieure des Mines de Saint-Etienne, 2006.

17. H.P. Langtangen, Computational Partial Differential Equations: Numerical Methods and Diffpack Pro-gramming, Springer, 2003.

18. T. Surazhsky, E. Magid, O. Soldea, G. Elber, E. Riv-lin, A Comparison of Gaussian and Mean Curva-tures Estimation Methods on Triangular Meshes, in: Proceedings of IEEE International Conference on Robotics and Automation (ICRA2003), Taipei, Taiwan, pp. 1021-1026.

19. D. Cohen-Steiner, J.-M. Morvan, Restricted Delau-nay triangulations and normal cycle, in: Proceedings of the nineteenth annual symposium on Computa-tional geometry, SCG ’03, ACM, New York, NY, USA, 2003, p. 312-321.

20. S. Prof. Phillpot, The Development of Models to Optimize Selection of Nuclear Fuels through Atomic-Level Simulation, DOE report ID 14649, 2009.

21. M. Fortin, Méthodes de lagrangien augmenté: ap-plications à la résolution numérique de problèmes aux limites, number 9 in Méthodes mathématiques de l’informatique, Dunod, Paris, 1982.

22. L. Onsager, Reciprocal Relations in Irreversible Processes. I., Physical Review, 37, 1931a, 405-426.

Journal of Chemical Technology and Metallurgy, 49, 3, 2014

274

23. L. Onsager, Reciprocal Relations in Irreversible Processes. II., Physical Review, 38, 1931b, 2265-2279.

24. J. Garrigues, Modèles de comportement élastique, http://cel.archives-ouvertes.fr/cel-00827790, 2013.

25. L. R. Duduchava, D. Mitrea, M. Mitrea, Differential operators and boundary value problems on hyper-surfaces, Mathematische Nachrichten 279 (2006) 9961023.

26. C. Geuzaine, J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, 79, 2009, 1309-1331.

27. J. Schöberl, J. Gerstmayr, R. Gaisbauer, NETGEN - automatic 3d tetrahedral mesh generator., http://www.hpfem.jku.at/netgen/, 2003.

28. J. Pan, H. Le, S. Kucherenko, J. Yeomans, A model for the sintering of spherical particles of different sizes by solid state diffusion, Acta Materialia, 46, 1998, 4671-4690.

29. J. Svoboda, H. Riedel, New solutions describing

the formation of interparticle necks in solid-state sintering, Acta Metallurgica et Materialia, 43, 1995, 499-506.

30. W. Zhang, I. Gladwell, Sintering of two particles by surface and grain boundary diffusion - a three-dimensional model and a numerical study, Compu-tational Materials Science, 12, 1998, 84-104.

31. D. Bouvard, R.M. McMeeking, Deformation of Interparticle Necks by Diffusion-Controlled Creep, Journal of the American Ceramic Society, 79, 1996, 666-672.

32. W.S. Coblenz, J.M. Dynys, R.M. Cannon, R.L. Coble, Initial stage solid state sintering models. A critical analysis and assessment, In Proceedings of the Fifth International Conference on sintering and related phenomena, 1980, 141.

33. R.L. Coble, Initial Sintering of Alumina and Hema-tite, Journal of the American Ceramic Society, 41, 1958, 55-62.

34. F. Parhami, R. McMeeking, A network model for initial stage sintering, Mechanics of Materials, 27, 1998, 111-124.