Finite element simulation of mass transport during ...bruchon/articles/JACS_2012.pdf · possibilities of the numerical approach are shown by presenting the changes occurring by lattice

JOURNAL OF THE AMERICAN CERAMIC SOCIETY

J. Am. Ceram. Soc 0000; 00:1–23

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1111/acs

Finite element simulation of mass transport during sintering of a

granular packing. Part I. Surface and lattice diffusions

Julien Bruchon∗, Daniel Pino-Munoz, Francois Valdivieso, Sylvain Drapier

LCG CNRS UMR 5146 - Ecole des Mines de Saint-Etienne - Centre Sciences des Materiaux et des Structures - 158

cours Fauriel - 42023 Saint-Etienne - Cedex 2

SUMMARY

This paper proposes a numerical strategy to simulate the mass transport by surface and lattice diffusion

into a granular packing. This strategy is based on two cornerstones. First, the developed approach is based

on a Eulerian description of the problem: the grains are described by using a Level-Set function, and can

evolve through a fixed mesh, with respect to the physical laws. In this way, the mesh does not experience

large distortions and topological changes, such as the formation of necks or of closed porosity, are implicitly

taken into account by the Level-Set method. Second, the computation of the mechanical state into the grains

is directly performed when considering the lattice diffusion route. Hence, a mechanical problem, coupling

the grain elastic behavior to the fluid behavior of the surrounding phase, is established and solved by finite

element. The diffusion flux is then related to the gradient of the pressure field. The results obtained with this

numerical strategy are compared with success to the usual geometrical models for two spherical grains. The

possibilities of the numerical approach are shown by presenting the changes occurring by lattice diffusion

into a granular packing. Copyright c© 0000 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: sintering; numerical simulation; surface diffusion; lattice diffusion

∗Correspondence to: [email protected]

Copyright c© 0000 John Wiley & Sons, Ltd.

Prepared using nmeauth.cls [Version: 2010/05/13 v3.00]

2 J. BRUCHON ET AL.

1. INTRODUCTION

During the sintering process, the changes occurring in the microstructure of a powder compact are

due to the thermo-activation of diffusion paths [1]. In this work, three paths are considered: surface,

grain boundary and lattice diffusion (see Figure 1). The scenario of the free solid-state sintering is

then as following. First, driven by the reduction in surface energy, the matter flows over the grain free

surface toward the maximum (in absolute value) of curvature, i.e. toward the neck formed between

two grains. This corresponds to the transport by surface diffusion. Owing to the neck formation,

stress gradients appear in the material, giving rise to grain boundary and lattice diffusion.

In the middle of the twentieth century [2], simplified geometrical models have been developed

for studying the sintering kinetics, analyzing the densification, and evaluating the effects of various

processing parameters. The last three decades have witnessed the emergence of the numerical

simulation as a new paradigm for studying the evolution of microstructures during sintering. Without

being exhaustive, one can cite References [3, 4] that numerically investigate the surface diffusion

between two spherical grains of same size; References [5, 6, 7] that simulate the evolution of pores

between grains; References [8, 9, 10] that study the evolution and the rearrangement occurring in

a microstructure during sintering, by finite element, discrete element, and Monte-Carlo methods

respectively. These contributions show the substantial progresses made over the last thirty years,

both in physical modelling and numerical techniques.

However, simulating sintering at the microstructure level remains an open and still challenging

subject. Regarding the sintering literature, the following remarks can be addressed. First, most of

the papers are based on a Lagrangian description of the grains: the grain surface is discretized by

a surface mesh (a set of segments or a set of facets depending on the dimension of the considered

simulation) that deforms with the material. This type of approach makes it difficult taking into

account large distortions of the material and topological changes arising during the sintering, such

as the formation of necks or of closed porosity [11]. The second remark is that in the sintering

literature, except a few papers such as Reference [12] which are limited to a local study of the

Copyright c© 0000 John Wiley & Sons, Ltd. J. Am. Ceram. Soc (0000)

Prepared using nmeauth.cls DOI: 10.1111/acs

FINITE ELEMENT SIMULATION OF MASS TRANSPORT 3

microstructure, there is no numerical model based on a direct mechanical computation of stresses

into the grains. The mechanical equilibrium is directly enforced through restrictions on the grain

geometry (shape of grains, value of the dihedral angle, ...). Consequently, the evolution of the

microstructure, even when it is calculated using numerical methods, remains dependent on some

micromechanical models (see another example in [14]). In addition to these conventional modelling

techniques, a very different model of microstructural evolution, called phase-field model, has been

developed over the last two decades. Instead of following the interfaces in a material, the phase-field

approach describes the microstructure by a continuous variation of the properties (the phase-field

variables) within a narrow region. The equations for the evolution of these phase-field variables

are derived from general thermodynamic and kinetic principles. Following this way, complex

microstructural evolutions of powder compacts during sintering can be simulated as shown in [15].

However, phase-field models contain at least two difficulties [16]: first, the equations describing the

physical problem have to be turned in terms of phase-field variables; second, these models involve

a large number of phenomenological parameters, which have to be determined in order to obtain

quantitative results. In contrast with the phase-field approach, the work presented here is based on

an interface capturing technique and benefits from the progresses made in this domain [17] and

more generally in the high performance computing field over the last decade. Also, this approach

allows the physical model to be formulated in the usual continuum mechanics framework.

More precisely, the aim of this work is to develop a numerical strategy for the simulation by

finite element of the sintering of a granular packing, without restriction concerning the number of

grains, their shape and their interaction: the evolution of the microstructure depends only on the

physical description of the problem. This strategy is based on two cornerstones: first, the use of the

so-called level-set method [18, 19] to describe the microstructural morphology and its subsequent

evolution; second, the computation by stabilized mixed finite element of the pressure field into the

grains which are assumed to behave as isotropic linear elastic materials. Combining these features,

the lattice flux can then be deduced from the gradient of the pressure.



4 J. BRUCHON ET AL.

The level-set method consists in capturing a moving surface by considering it as the zero level

set of a higher dimensional function. The advantage of this technique is that topological changes

are handled naturally. Geometric properties of the surface (normal vector, curvature) can then be

’easily’ deduced from this level-set function [19, 20]. The level-set method is used in literature for

simulating a large variety of phenomena involving moving interface, in particular surface diffusion

problems [21, 22, 23, 24], and mass transport by surface diffusion [25]. The computation by finite

element of the pressure inside the elastic grains is another difficult problem to address. Indeed, in

the Eulerian approach adopted here, the computational domain contains, not only the solid grains,

but also the surrounding gaseous medium. This permits to deal with any pressure-loaded sintering

process. But in turn, solving the mechanical problem requires to perform a fluid - solid coupling,

which is still a research domain in the numerical literature.

This work is published in two parts. This paper constitutes the first part, concerned only with

the surface and lattice diffusion. The second part, dealing with the grain boundary diffusion, will

be published in a companion article. This paper is organized as follows. The physical model,

diffusion fluxes and associated interface velocities, are detailed in Section 2. Key points of numerical

techniques developed to perform sintering simulations are given in Section 3. Finally, numerical

results are discussed in Section 4.

2. PHYSICAL MODEL

Let Ω be a bounded region of IRd (typically, the unit square in the two-dimensional case d = 2, or

the unit cube if d = 3) referred to as the computational domain. Ω is filled with a set of solid grains,

denoted Ωs, embedded into a surrounding gaseous medium denoted Ωa, say the air. Solid and air

are immiscible, consequently Ωs ∩ Ωa = ∅. Furthermore, the intersection between the grains and

the boundary of Ω (denoted ∂Ω) is assumed to be empty: Ωs ∩ ∂Ω = ∅.

From a physical point of view, the set of grains is modeled as a single crystal, without taking

into account the grain orientation. Temperature T is taken uniform in the computational domain and

constant over time (isothermal approximation). The aim of the physical model is to describe the




changes occurring in the grain packing during the sintering process. These changes are induced by

thermo-activated matter diffusion routes, here through free surface and lattice. Diffusion following

route p is characterized in terms of diffusive flux,

jp = −mp∇µ (1)

where mp is the diffusion mobility for that diffusion route, and µ the chemical potential. Their

expressions depend on whether the diffusion is in surface or in volume. Once these expressions are

established, the sintering rate can be calculated, and the changes occurring in the grain packing can

be captured.

2.1. Surface diffusion

Over the grain free surface, the chemical potential is expressed as

µ = µs0 + Ωmγκ

where µs0 is a fixed reference potential. Ωm is the molar volume, γ is the surface tension coefficient

and κ the surface mean curvature. The surface diffusion mobility is ms = DsδsCa/RT , with R

the gas constant, Ds the surface diffusion coefficient, Ca the atom concentration, while δs is the

thickness of the diffusion layer. In this work, the chemical equilibrium is assumed to be reached

instantaneously: the atomic concentration Ca is assumed to remain constant throughout the process

as a first approximation, and Ca = 1/Ωm. Consequently, the surface flux is

js = −δsDsγ

RT∇s κ (2)

where operator ∇s denotes the surface gradient, defined as the tangential component of the gradient.

By assuming that there is no accumulation of matter, the displacement of the grain free surface can

then be described through a velocity vs normal to the surface and given by the following expression

(see Mullins [26] or recently [25])

vs = −Ωm(∇s ·js)n =δsDsγΩm

RT(∆s κ)n (3)



6 J. BRUCHON ET AL.

where ∇s · denotes the surface divergence operator, ∆s the surface Laplacian operator, or Laplace-

Beltrami operator, and n the unit normal vector pointing outward the grain surface.

2.2. Lattice diffusion

The approach proposed in this paper is based on the work of K. Garikipati et al. [27], which

allows the coupling between the continuum mechanics scale and a description of the diffusion

mechanism at the lattice scale. Crystal is seen as a vacancy - atom binary system, characterized

by two continuum fields, the atom concentration Ca and the concentration in vacancies Cv. When

a vacancy swaps places with an adjacent atom, the surrounding lattice is assumed to experience

a local relaxation effect. This relaxation is characterized through an isotropic strain of magnitude

(1− f)Ωa0 , where Ωa0 is the stress-free atomic volume and f is a parameter such that fΩa0 is the

effective volume of a vacancy [27]. This term is accounted for in the internal energy density e of the

binary system:

e(ε, θv, Cv, η) =1

2

(ε(u)− 1

3θvI

): C :

(ε(u)− 1

3θvI

)+ eη(η) + efCv (4)

where ε(u) = (∇u+ t ∇u)/2 is the strain tensor, u the displacement, η the entropy, eη the

entropic energy, and efCv the vacancy formation energy, with ef a constant parameter. The fourth-

order elasticity tensor is given by C = KI⊗ I+ 2G(I− 13I⊗ I), where K is the bulk modulus,

G the shear modulus, I the second-order identity tensor and I the fourth-order identity tensor.

Finally, θv = −(1− f)Ωm(Cv − Ceqv0 ) is the strain-like term describing the relaxation effect. Since

the quantities involved in this model are expressed with respect to a number of moles, and not a

number of individual atoms, the molar volume Ωm has been substituted to Ωa0. Furthermore, this

molar volume is assumed, as a first approximation, to be constant and therefore equal to the stress-

free one. Finally, Ceqv0

is the vacancy concentration at equilibrium in a stress-free state. Note that,

contrary to [27], the number of vacancies is assumed to be constant; moreover, creep and thermal

strains are neglected. However, they might be easily added if necessary.




The Helmotz free energy is then defined as H = e− Tη. According to the developments made

in [27] and [28], the first and second thermodynamics laws lead to the following expression of the

chemical potential:

µ = µv0 +∂H

∂Cv= µv0 + (1− f)ΩmKtr(ε)

where µv0is a fixed reference potential. The previous expression has been obtained with the

assumption that Cv = Ceqv0

. Furthermore, the hydrostatic pressure is usually defined as p =

−13 tr(σ), where the Cauchy stress tensor, defined as σ = ∂H

∂ε , is reduced to its usual expression

σ = C : ε when Cv is assumed to be equal to Ceqv0

. In this case, it can be shown that p = Ktr(ε).

Consequently, the chemical potential is finally expressed as

µ = µv0 + (1− f)Ωmp (5)

Assuming that the mobility ml is equal to DlCm/kT , where Dl is the lattice diffusion coefficient,

the lattice diffusion flux is given by

jl = −Dl(1− f)

RT∇ p (6)

The displacement of the grain free surface induced by this flux is then characterized through the

following interface velocity:

vl = Ωm(jl · n)n = − (1− f)ΩmDl

RT(∇ p · n)n (7)

It has to be pointed out that assuming a constant atom concentration means that the mass

conservation is implicitly satisfied, without requiring the first Fick’s law. In particular, atom

flux (6) and corresponding interface velocity (7) are not divergence-free, and the grain volume is

consequently not guaranteed to remain unchanged during the diffusion process.



8 J. BRUCHON ET AL.

2.3. Mechanical problem

Grains are solid bodies which are assumed to have an isotropic linear elastic behavior. Momentum

conservation law is expressed in terms of both displacement and pressure fields u and p:

∇·σ = 0 ⇔ ∇·(2Gε(u)− I(1− 2G

3K)∇·u

)−∇ p = 0 in Ωs (8)

This equation has to be complemented with the relation between the hydrostatic pressure and the

displacement:

p+K∇·u = 0 in Ωs (9)

Further details can be found in the unpublished Reference †. One of the specificity of the presented

approach is the Eulerian description of the sintering step: grain kinetics is described in a fixed

mesh frame. This description implies that air into which grains are embedded is included into the

computational domain. However an accurate description of the air behavior is not required to achieve

the simulation of sintering. That is why, air is assumed to be an incompressible Newtonian fluid

with a low viscosity Gf . In this way, the surrounding fluid medium does not perturb the changes

occurring in the grain shape, but transmits stresses from the boundary of the computational domain

to the grain free surface. Momentum and mass balances are expressed through the Stokes system in

terms of fluid velocity v and hydrostatic pressure p:

∇ ·σ = 0 ⇔ ∇·(2Gf ε(v)

)−∇ p = 0 in Ωa (10)

∇·v = 0 in Ωa (11)

where ε(v) = (∇v + t ∇v)/2 is the strain rate tensor. Equation (11) expresses the fluid

incompressibility. The stress vector σ · n is enforced as boundary condition on ∂Ω. Finally, the

†D. Pino-Munoz, J. Bruchon, F. Valdivieso and S. Drapier, A finite element based level-set method for fluid - elastic

solid interaction with surface tension, submitted for publication in Int. J. Numer. Meth. Eng.




grain - air coupling is achieved by considering the Laplace’s law which rules the discontinuity of

the normal stress over the grain free surface:

[(σ · n) · n] = γκ (12)

where [.] denotes the jump operator across the grain free surface, γ is the surface tension parameter

and κ the mean curvature.

3. NUMERICAL STRATEGY

Above equations are discretized and solved by using a finite element framework. The computational

domain is discretized using a mesh made up of triangles in 2D or tetrahedrons in 3D. Velocities,

pressure and displacements are approximated with continuous and piecewise linear functions. The

numerical method used to simulate the surface diffusion mechanism is detailed in [25], while the

lattice diffusion case is detailed in Reference†. Here, only the level-set method, which is a key point

of the presented numerical strategy, is enlightened. This method consists in defining, as an input

data of the simulation, a real-valued function α(x, t = 0), chosen as the signed distance function to

the grain free surface Γ:

α(x, t = 0) =

dist(x,Γ) if x ∈ Ωa

−dist(x,Γ) if x ∈ Ωs

0 if x ∈ Γ = Ωs ∩ Ωa

where dist(x,Γ) is the distance from any point x to the grain free surface Γ. Once the function

α(x, t) is known at a time t ≥ 0, the grain free surface is represented by the isosurface α(x, t) =

0; moreover, first and second derivatives of α with respect to the position x, provide both normal

vector n and curvature κ [25]. The diffusion flux and the corresponding interface velocities can

then be computed by using a finite element method. The grain surface can then be “transported”

from time t to time t+∆t with the diffusion velocity vs (Equation (3) for surface diffusion) or vl



10 J. BRUCHON ET AL.

(Equation (7) for lattice diffusion) by computing the values of α(x, t+∆t). This is achieved by

solving, on the time interval [t, t+∆t], the transport equation

∂α

∂t+ vp · ∇α = 0 (13)

with p = s or p = l, or vp = vs + vl.

Finally, an important difficulty raised by this approach has to be underlined. The velocity

associated with the lattice diffusion (Equation (7)) involves the evaluation of the pressure gradient

over the grain surface. However, by virtue of the Laplace’s law (Equation (12)), the normal stress

is discontinuous across this interface, and consequently the pressure gradient is not defined at this

place. That means that in the grain, just under the grain surface, the pressure field is equal to the

grain pressure (driven by the curvature through the Laplace’s law), while just over the surface,

the pressure field is equal to the pressure of the surrounding medium. Consequently, its gradient,

computed in a finite element sense, is not correct at the interface, because it takes into account the

values of both grain pressure and surrounding medium pressure. Furthermore in a finite element

framework, this problem does not occur only in the mesh elements crossed by the interface, but in a

few layers of elements around the interface. This difficulty is overcome by a numerical trick using

the metric properties of the level-set (or distance) function α. Figure 2 shows the isovalues of the

function α in the vicinity of the interface, that is the lines described by α = β for several values

of β, |β| 1. These lines can be considered, at least locally, as linear transformations (geometric

expansions or contractions) of the surface α = 0 (the black solid line in Figure 2). Consequently,

the curvatures of these lines are approximatively equal to the curvature of the surface α = 0 when

β is very small. The numerical trick consists in solving the mechanical system (8)-(9) using a shifted

distance function, say α′ = α− λ, with λ a positive constant parameter. In this way, the surface is

shifted and becomes the dashed line in Figure 2. The pressure is then discontinuous over this virtual

surface α′ = 0, and remains continuous and differentiable over the true surface α = 0, where

the pressure gradient, the flux (Equation (6)) and the velocity (Equation (7)) can now be numerically

computed.




From a practical point of view, the value of λ is chosen by a numerical test, as the smallest distance

from the grain surface α = 0 at which the pressure and the gradient of pressure are found to be

appropriate (the criterion is the sign of the normal component of the pressure gradient) when the

Laplace’s law is enforced over this surface. This choice provides a value of λ equal to twice or thrice

the mesh size. The use of a mesh adaptation technique (see next Section) guarantees that this value

is small when compared to the grain size.

4. NUMERICAL RESULTS AND DISCUSSION

Previous developments have been implemented in the finite element library CimLib. This highly

parallel C++ library, is developed at Centre for Material Forming (Mines ParisTech, CNRS UMR

7635) by T. Coupez and co-workers [29]. Furthermore, the simulations presented in this paper have

been carried out by using a mesh adaptation strategy which consists in refining the mesh in the

vicinity of the grain surface with a method described in [30]. This method consists in adapting the

mesh size with respect to the second derivatives of a primal variable such as the level-set function

or the pressure. This method results in more accurate simulations while keeping a “reasonable”

number of mesh elements.

The numerical developments exposed in this work allow the surface diffusion to be combined with

the lattice diffusion into a single simulation. However, in this paper, these diffusion mechanisms are

presented separately, in order first to validate our approach and, second, to investigate the effect of

each diffusion route.

4.1. Surface diffusion simulation

The first case investigated is the neck growth by surface diffusion between two spherical grains

of same radii. The initial grain surfaces (described by the set α(x, 0) = 0) are chosen to be

nearly tangential as shown in Figure 3(a). The initial curvature is then ill-defined at the triple

point. However, due to the diffusion, as shown in Figure 3 and analysed in [25], the contact surface




becomes quickly smooth, the curvature is then well-defined everywhere and the neck can grow. A

well-known geometrical model of the literature [2] states

(x(t)

r

)n

=AδsDsγΩm

RTr4t = At∗

where r is the grain radius, x is the neck radius, A a constant and n is a parameter depending on

the diffusion route. Figure 4 presents, in logarithmic scales, the growth of the dimensionless neck

radius x/r versus the dimensionless time t∗ = δsDsγΩm

RTr4 t, obtained by finite element simulation for

different grain radii, ranging from 0.1 to 2.5. The best curve fitting these data, obtained by a least-

square approximation of the numerical results, is x/r = 1.3t∗1/7 and is referred to as “Simulation,

1/7” in Figure 4. The value n = 7 for the surface diffusion route corresponds to the value provided

by the analytical model developed by Kuczynski in 1949. However, it has to be underlined that

this value corresponds to a kind of mean value which takes into account the different stages of

the sintering. The same remark is addressed in [3]. To illustrate that, a curve corresponding to a

1/6-power-law, and referred to as “1/6 law”, has been plotted in Figure 4. It can be shown that this

1/6-law provides a better approximation of the first stage of the sintering (0.025 ≤ x/r ≤ 0.053,

obtained with r = 2.5). Furthermore, it can be observed in this figure a kind of undercutting effect

in the early stage of each simulation. This effect, first described in [31], can be explained in the

present case by the fact that, when the simulation starts, the neck between two grains is defined with

an accuracy which depends on the mesh size. When the neck size becomes “reasonable” compared

to the mesh size, this effect vanishes and does not affect any more the subsequent neck growth.

The next study is dedicated to the neck growth by surface diffusion between two spherical grains

of different radii, say r1 and r2. References [32] and [33] state that the neck growth obtained in this

case is equal to the one obtained for two grains of same equivalent radius r defined by

r =2r1r2r1 + r2

A simulation involving two grains of different sizes (for example r1 = 0.1 and r2 = 0.2 in

Figure 5) proves the relevancy of this definition: the neck radius obtained by simulation with two




grains of equivalent radius (here r = 0.133) is shown in good agreement with the one computed for

two grains of radii 0.1 and 0.2 in Figure 5.

4.2. Lattice diffusion simulation

In the following simulations, the computational domain is a square or a cube with a side length

equal to 1µ m. Table I summarizes the values of the different parameters used in the lattice diffusion

simulations. Two remarks can be done. First, the value of the viscosity of the surrounding medium is

very high, here Gf = 1000 Pa.s. However, since the role of the surrounding fluid is only to transmit

the stresses from the computational domain boundary to the grain surfaces, an accurate description

of the dynamics of this medium is not of interest here. Consequently, the key parameter is not Gf

but the ratio Gf/(∆tG) which has to be small enough to guarantee that the surrounding medium

does not perturb the motion of the grain surfaces. The present ratio, lower that 2.10−5, satisfies

this condition. The second remark is that the parameter (1− f)ΩmDl/RT (through the diffusion

coefficient Dl) allows the time scale to be set. Here, the time is expressed in seconds. However, this

choice is arbitrary.

Two spherical grains of same size The growth of neck, by lattice diffusion between two spherical

grains of same radii is first analysed. The geometrical model developed in the literature [2] states

that

(xr

)n

=BDlγΩm

RT r4t = Bt∗

where B is a constant, t∗ = (DlγΩm)/(RT r4) the dimensionless time, and the value of the

parameter n stands between 4 et 5. Figure 6 shows the growth of the dimensionless neck radius x/r

versus t∗, computed by finite element for a grain radius ranging from 0.1 to 0.4. The best curve fitting

these data, obtained by a least-square approximation of the numerical results, is x/r = 0.36t∗1/5.6

and is referred to as “n = 5.6”. Once again this value, which is larger than the upper bound predicted

by the theory, represents a kind of mean value that takes into account the different stages of the

sintering. However, when these simulations are examined individually, the coefficient n is shown to




depend on the grain size and to vary slightly into each simulation. More precisely, the simulations

provide a coefficient n that decreases when the grain size increases: n is equal to 4.85 when r = 0.1,

to 4.23 when r = 0.2, to 4.14 when r = 0.3 and to 3.88 when r = 0.4. Understanding in details this

dependency requires deeper investigations which have not been carried out by the authors yet.

Two spherical grains of different size Figure 7 presents the evolution, by lattice diffusion, of two

spherical grains of different size. This simulation is two-dimensional, and plane-strain assumption

is considered. When the simulation starts (Figure 7(a)), the larger grain has a radius of 0.2µm, the

smaller one, a radius of 0.075 µm, while the initial neck radius is equal to 0.0065 µm. The finite

element mesh is refined in a band around the grain surface, as shown in Figures 7(a) and 7(c). In

this way, the description of the interfaces is improved, while the mesh size remains “reasonable”.

In the presented case, the mesh is made up of approximately 42,000 triangles and 21,000 nodes.

The solid line shown in these figures is the grain surface, i.e. the zero level set of the function α

which separates the solid phase from the fluid one. The pressure field is found to be constant in the

fluid domain, equal to the external pressure imposed over the computational domain boundaries.

As explained in Section 3, the mechanical problem (8)-(9) is solved by considering a “virtual”

grain surface shifted by a distance λ toward the surrounding fluid medium (here, λ = 0.009µm).

This explains why the grain pressure is “extended” beyond the grain surface in Figures 7(a)-7(c)

(the fluid domain has been removed from Figure 7(b)). Using this numerical trick, pressure is

not discontinuous any more at the interface, and its gradient can consequently be evaluated at the

interface in a usual finite element way. Note that, even if the value of the distance λ is related to the

mesh size and is therefore computed from a purely numerical point of view, it can also be considered

with a physical sound as the width of a boundary layer at the fluid - solid interface.

The pressure field computed on the initial grain configuration is represented in Figure 7(a), while

pressure field obtained after 1000 diffusion steps is shown in Figure 7(c). Figure 7(b) is a zoom-

in over the small grain after 500 diffusion steps. In this figure, the vectors of the interface velocity

(Equation (7)), induced by the lattice diffusion flux (Equation (6)), are superimposed on the pressure




field. This velocity is, as expected, oriented toward the minimum of pressure, opposite to the

pressure gradient direction. Finally, as mentioned in Section 2.2 and since the atom concentration is

taken constant, the lattice diffusion flux, and consequently the surface velocity, are not divergence-

free. The surface (2D-case) of the grains could therefore change from the initial state (Figure 7(a))

to the final one (Figure 7(c)). However, the grain surface is equal to 0.146 µm2 in the initial

configuration, and to 0.143 µm2 in the final state. Thus, the volume change is less than 2%, which

is not significant, especially when considering the possible numerical errors accumulated all along

the simulation.

Diffusion into a granular packing The last case investigated in this paper is the simulation of mass

transport by lattice diffusion in the granular packing shown in Figure 8. This packing, represented

in its initial state in Figure 8(a), is made up of 82 grains (plane-strain assumption), with an initial

porosity of 25%. The grains are distributed following three different sizes: 30% of the grains

have a radius equal to 0.04 µm; 30%, a radius equal to 0.05 µm; and 40%, a radius of 0.06

µm. The computational domain is a square with a size that is slightly larger than 1, in order to

avoid the intersections between its boundaries and the grains. It is discretized with a mesh refined

in the neighborhood of the grain surface (Figures 8(a) and 8(c)), and made up of approximately

276,000 nodes and 550,000 triangles. In this simulation, the grain surface is shifted of a distance

λ = 0.003µm when computing the mechanical problem. The pressure field is shown to be well

computed (negative in the vicinity of the grain neck, positive elsewhere and depending on the grain

size). The vectors of the diffusion velocity are well oriented, pointing in the direction opposite

to the pressure gradient, as shown in Figure 8(b). This figure, which corresponds to a focus on

the microstructure simulated after 50 diffusion increments, shows different pore shapes and the

associated mechanical state (pressure field). Progressively, as shown in Figure 8(c) (450 diffusion

increments), the pores adopt a spherical shape, due to the mass transport by lattice diffusion. The

change in the grain surface, from the initial state to the final one described in Figure 8(c), is 0.7

%, which is negligible. Finally, the CPU time of this computation which involves 450 time steps, is




of eight hours by using a parallel computing strategy on 16 cores (Intel Xeon 2.2 GHz processor).

This case shows the ability of the approach developed in this paper to simulate the evolution of a

granular packing under lattice diffusion, without any restriction regarding the shape of the grains or

their interactions. Note that similar cases involving the surface diffusion are treated in reference [25].

5. CONCLUSIONS AND OUTLOOK

In this paper, a numerical strategy has been proposed to simulate the mass transport by surface and

lattice diffusion into a granular packing. The first originality of the developed approach is to be based

on a Eulerian description of the problem: the grains are described by using a Level-Set function, and

can evolve through a fixed mesh with respect to the physical laws. In this way, the mesh does not

experience large distortions, and topological changes, such as the formation of necks or of closed

porosity, are implicitly taken into account by the Level-Set method. A mesh adaptation strategy

allows the mesh to be refined in the vicinity of the grain surface. The details of the microstructure

can then be captured, while the number of mesh elements remains reasonable when using parallel

computing, in terms of computation time cost as well as memory cost. The second originality of the

approach is to propose a direct computation of the mechanical state into the grains, when considering

the lattice diffusion route. Hence, a mechanical problem coupling the grain elastic behavior to the

fluid behavior of the surrounding phase, is established and solved by finite element. The diffusion

flux is then related to the gradient of the pressure field. The results obtained with this numerical

strategy have been successfully compared with the geometrical models, when considering two

spherical grains. The capabilities of the numerical approach have been demonstrated, by presenting

the changes occurring, by lattice diffusion route, into a packing made up of 82 grains (with plane

strain assumption).

The first outlook to this work is to describe the grain boundary diffusion. Obviously, a new

interface between two grains has to be represented, modifying the pressure field established into

each grain. The grain boundary diffusion flux is then related to the normal stress gradient. These

developments are in progress, and will be presented in a forthcoming paper. Then, the surface, lattice




and grain boundary diffusion routes will be activated separately, or combined into a whole diffusive

flux, in order to evaluate their effect on a microstructure. Furthermore, so far, the mechanical

displacement, which is computed by the finite element analysis, has not been considered when

moving the grain surfaces. However, this displacement has to be taken into account to ensure

implicitly the mechanical equilibrium. Finally, the simulations of mass transport by lattice diffusion

shown in this paper are two-dimensional simulations, carried out under a plane strain assumption.

The global objective of our work is to develop a numerical tool to simulate the sintering of fully

three-dimensional granular packings, which represents a gap in term of computation complexity

and cost, but will open realistic perspectives regarding digital material design.

References

1. M.F. Ashby, A first report on sintering diagrams, Acta Metall. Mater. 22(3) (1974), pp. 275–289.

2. M.N. Rahaman, Ceramic Processing and Sintering, Marcel Dekker, Inc., New York (1995).

3. R. M. German and J. F. Lathrop, Simulation of spherical powder sintering by surface diffusion, J. Mater. Sci. 13(5)

(1978), pp. 921–929.

4. D. Bouvard and R.M. McMeeking, Deformation of interparticle necks by Diffusion-Controlled Creep, J. Am. Ceram.

Soc. 79(3) (1996), pp. 665–672.

5. J. Svoboda and H. Riedel, New solutions describing the formation of interparticle necks in solid-state sintering, Acta

Metall. Mater. 43(1) (1995), pp. 1–10.

6. J. Svoboda and H. Riedel, Quasi-equilibrium sintering for coupled grain-boundary and surface diffusion, Acta Metall.

Mater. 43(2) (1995), pp. 499–506.

7. J. Pan, A. C. F. Cocks and S. Kucherenko, Finite element formulation of coupled grain-boundary and surface

diffusion with grain-boundary migration, Proc. R. Soc. Lond. A 453(1965) (1997), pp. 2161–2184.

8. Y.U. Wang, Computer modelling and simulation of solid-state sintering: A phase field approach, Acta Mater. 54(4)

(2006), pp. 953–961.

9. B. Henrich, A. Wonisch, T. Kraft, M. Mosele and H. Riedel, Simulations of the influence of rearrangement during

sintering, Acta Mater. 55(2) (2007), pp. 756–762.

10. M. Braginsky, V. Tikare and E. Olevsky, Numerical simulation of solid state sintering, Int. J. Solids Struct. 42(2)

(2005), pp. 621–636.

11. J. Pan, Modelling sintering at different length scales, Int. Mater. Rev. 48(2) (2003), pp. 69–85.




12. K. Shinagawa and K. Darcovich, Coupled Finite Element Analysis of Grain Boundary Diffusion and Particle

Deformation in Sintering, J.G.Heinrich and C. Aneziris, Proc. 10th ECerS Conf., Goller Verlag, Baden-Baden, 2007,

pp. 141–147, ISBN: 3-87264-022-4.

13. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, A PDE-Based Fast Local Level Set Method, J. Comput.

Phys. 155(2) (1999), pp. 410–438.

14. J. Pan, H.N. Ch’ng and A.C.F. Cocks, Sintering kinetics of large pores, Mech. Mater. 37(6) (2005), pp. 705–721.

15. Y. U. Wang, Computer modeling and simulation of solid-state sintering: A phase field approach, Acta Mater., 54(4)

(2006), pp. 953–961.

16. N. Moelans, B. Blanpain, P. Wollants, An introduction to phase-field modeling of microstructure evolution, Calphad,

32(2) (2008), pp. 268–294.

17. L. Ville, L. Silva, T. Coupez, Convected Level Set method for the numerical simulation of Fluid Buckling, Int. J.

Numer. Meth. Fl., 66(3) (2011), pp. 224–244.

18. J.A. Sethian, Level Sets Methods and Fast Marching Methods, Cambridge Monograph on Applied and

Computational Mathematics, vol. 3. (1999).

19. S. Osher and F. Fedkiw, Level Set Methods: An Overview and Some Recent Results, J. Comput. Phys. 169(2) (2001),

pp. 463–502.

20. J. Bruchon, H. Digonnet and T. Coupez, Using a signed distance function for the simulation of metal forming

processes: Formulation of the contact condition and mesh adaptation. From a Lagrangian approach to an Eulerian

approach, Int. J. Numer. Meth. Eng. 78(8) (2009), pp. 980–1008.

21. D. Adalsteinsson and J. A. Sethian, A Level Set Approach to a Unified Model for Etching, Deposition, and

Lithography - III: Redeposition, Reemission, Surface Diffusion, and Complex Simulations, J. Comput. Phys. 138(1)

(1997), pp. 193–223.

22. Z. Li, H. Zhao and H. Gaoz A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a

Fixed Cartesian Grid, J. Comput. Phys. 152(1) (1999), pp. 281–304.

23. D. L. Chopp and J. A. Sethian Motion by Intrinsic Laplacian of Curvature, Interfaces and Free Boundaries 1(1)

(1999), pp. 107–123.

24. P. Smereka, Semi-Implicit Level Set Method for Curvature and Surface Diffusion Motion, J. Sci. Comput. 19(1-3)

(2003), pp. 439–456.

25. J. Bruchon, S. Drapier and F. Valdivieso, 3D finite element simulation of the matter flow by surface diffusion using

a level-set method Int. J. Numer. Meth. Eng. 86(7) (2011), pp. 845–861.

26. W. W. Mullins, Theory of Thermal Grooving, J. Appl. Phys., 28(3) (1957), pp. 333–339.

27. K. Garikipati, L. Bassman and M. Deal, Lattice-based Micromechanical Continuum Formulation for Stress-driven

Mass Transport in Polycrystalline Solids, J. Mech. Phy. Sol. 49(6) (2001), pp. 1209–1237.

28. E. Fried and M.E. Gurtin, A unified treatment of evolving interfaces accounting for small deformations and atomic

transport: grain-boundaries, phase transitions, epitaxy, Adv. Appl. Mech. 40 (2004), pp. 1–177.




29. Y. Mesri, H. Digonnet and T. Coupez, Advanced parallel computing in material forming with CIMLib, European

Journal of Computational Mechanics 18(7-8) (2009), pp. 669–694.

30. Y. Mesri, W. Zerguine, H. Digonnet, L. Silva and Thierry Coupez, Dynamic Parallel Adaption for Three Dimensional

Unstructured Meshes: Application to Interface Tracking, Proceedings, 17th International Meshing Roundtable,

Springer-Verlag, pp.195–212, October 12-15 2008.

31. F. A. Nichols, W. W. Mullins, Morphological Changes of a Surface of Revolution due to Capillarity Induced Surface

Diffusion, J. Appl. Phys., 36(6) (1965), pp. 1826–1835.

32. J. Pan, Le H. Kucherenko and J.A. Yeomans, A model for sintering of spherical particles of different sizes by solid

state diffusion, Acta Mater. 46(13) (1998), pp. 4671–4690.

33. C.L. Martin, D. Bouvard and S. Shima, Study of particle rearrangement during powder compaction by discrete

element method, J. Mech. Phys. Solids. 51(4) (2003), pp. 667–693.




LIST OF TABLES

I Parameters used in simulations of lattice diffusion . . . . . . . . . . . . . . . . . . 21




Parameters

Mechanical Problem G = 156 GPa, K = 390 GPa

Equations (8), (9), (10) (Poisson’s coefficient ν = 0.25), Gf = 1000 Pa.s

Equation (12) γ = 0.9 N.m−1

Diffusion Equation (7) (1− f)ΩmDl/RT = 0.013 m4.s−1.N−1

Time step 3.10−4 s ≤ ∆t ≤ 10−3 s

Table I. Parameters used in simulations of lattice diffusion




LIST OF FIGURES

1 Different diffusion routes: surface, grain boundary and lattice (or volume) diffusion 23

2 Isovalues of the distance function α in the vicinity of the interface. Solid line:

α = 0 (interface), dashed line: α = 0.009, or α′ = 0 with α′ = α− λ and

λ = 0.009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Neck growth by surface diffusion between two spherical grains of equal size [25] . 23

4 Growth by surface diffusion of the dimensionless neck radius x/r over

dimensionless time t∗ (logarithmic scale) for different values of r [25] . . . . . . . 24

5 Growth by surface diffusion of the dimensionless neck radius x/r over

dimensionless time t∗ (logarithmic scale) for two spherical grains of different radii

(r1 = 0.1 and r2 = 0.2) and for two spherical grains of same equivalent radius

(r = 0.133) [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Growth by lattice diffusion of the dimensionless neck radius x/r over dimensionless

time t∗ = (logarithmic scale) for different values of r . . . . . . . . . . . . . . . . 24

7 Two-dimensional simulation, under plane-strain assumption, of lattice diffusion in

two cylindrical grains of different sizes: (a) and (c) mesh refined around the interface

and pressure isovalues (MPa), respectively on the initial configuration and after

1000 computation increments, i.e. at time t = 1 s; (b) pressure isovalues (MPa) and

induced diffusion velocity after 500 computation increments, i.e. at time t = 0.5 s . 25

8 Two-dimensional simulation, under plane-strain assumption, of lattice diffusion

in a granular packing made up of 82 grains of different sizes: (a) and (c) mesh

refined around the interface and pressure isovalues (MPa), respectively on the

initial configuration and after 450 computation increments, i.e. at time t = 0.02 s;

(b) pressure isovalues (MPa) and induced diffusion velocity after 50 computation

increments, i.e. at time t = 0.18s . . . . . . . . . . . . . . . . . . . . . . . . . . . 26




Figure 1. Different diffusion routes: surface, grain boundary and lattice (or volume) diffusion

Figure 2. Isovalues of the distance function α in the vicinity of the interface. Solid line: α = 0 (interface),

dashed line: α = 0.009, or α′ = 0 with α′ = α− λ and λ = 0.009

(a) t0 (b) t1 (c) t2

Figure 3. Neck growth by surface diffusion between two spherical grains of equal size [25]




0.01

0.1

1

1e-10 1e-08 1e-06 0.0001D

imen

sion

less

rad

ius

x/r

Dimensionless time t’

r = 0.1r = 0.2r = 0.3r = 0.5r = 2.5

Simulation, 1/71/6 law

Figure 4. Growth by surface diffusion of the dimensionless neck radius x/r over dimensionless time t∗

(logarithmic scale) for different values of r [25]

0.19

0.3

0.45

3.1959e-06 1e-05 3e-05 0.0001 0.00032

Dim

ensi

onle

ss r

adiu

s x/

r

Dimensionless time t*

Two different radiiOne equivalent radius

Figure 5. Growth by surface diffusion of the dimensionless neck radius x/r over dimensionless time t∗

(logarithmic scale) for two spherical grains of different radii (r1 = 0.1 and r2 = 0.2) and for two spherical

grains of same equivalent radius (r = 0.133) [25].

0.01

0.1

1

1e-06 1e-05 0.0001 0.001 0.01 0.1

Dim

ensi

onle

ss r

adiu

s x/

r

Dimensionless time t*

r=0.1r=0.15r=0.2r=0.3r=0.4

n = 5.6

Figure 6. Growth by lattice diffusion of the dimensionless neck radius x/r over dimensionless time t∗ =

(logarithmic scale) for different values of r




(a) Time t = 0 s (b) Time t = 0.5 s, focus on small grain

(c) Time t = 1 s

Figure 7. Two-dimensional simulation, under plane-strain assumption, of lattice diffusion in two cylindrical

grains of different sizes: (a) and (c) mesh refined around the interface and pressure isovalues (MPa),

respectively on the initial configuration and after 1000 computation increments, i.e. at time t = 1 s; (b)

pressure isovalues (MPa) and induced diffusion velocity after 500 computation increments, i.e. at time

t = 0.5 s




(a) Time t = 0 s (b) Time t = 0.02s, focus on the microstructure (truncated

pressure scale)

(c) Time t = 0.18 s

Figure 8. Two-dimensional simulation, under plane-strain assumption, of lattice diffusion in a granular

packing made up of 82 grains of different sizes: (a) and (c) mesh refined around the interface and pressure

isovalues (MPa), respectively on the initial configuration and after 450 computation increments, i.e. at time

t = 0.02 s; (b) pressure isovalues (MPa) and induced diffusion velocity after 50 computation increments,

i.e. at time t = 0.18s



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