III – MORPHOLOGY

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A. Bertei III - MORPHOLOGY

III – MORPHOLOGY

III.1 – Introduction to the morphology of random packing of spheres; III.2 – Percolation theory; III.3 – Overlapping of particles; III.4 – Apparent conductivity; III.5 – Effects of porosity; III.6 – Calculation of morphological parameters; III.7 – Morphological results; III.8 – The real morphology; III.9 – References.

III.1 – Introduction to the morphology of random packing of spheres

The structure of the central membrane and of the electrodes of the IDEAL-Cell (and, in general, of the electrodes of other kinds of SOFCs) can be assimilated to a random packing of spheres; particles seem to have spherical shape and the preparation of each layer of the cell (cathode, anode or central membrane) does not follow any particular procedures to obtain a structured packing: particles of different materials (e.g., BCY15, proton-conducting phase, and YDC15, anion-conducting phase, in the central membrane) are preventively mixed together, deposited and then sintered.

The simplest model that we can take into account to represent this composite structure is to consider all particles as rigid spheres; then we distinguish between at least 2 kinds of particles, named k and l: all k-particles have the same radius rk, such as all l-particles

have radius equal to rl (but we could consider polydisperse mixtures of particles). We

use this model in par. III.2; then, in the following paragraphs, we will consider that particles are not rigid, so we will take into account the effects of the overlapping of particles.

The purpose of this chapter is to estimate some structural properties, such as the length of TPB per unit volume and the apparent conductivity of each phase, that enter into the model equations of the central membrane (or of the electrodes) as functional parameters. But to obtain these properties we shall pass through other quantities strictly connected to the physical-mathematical description of the random structure, for example the number of contacts that each particle makes with others or the probability that a particle belongs to a connected cluster. The chapter is structured as follows:

• par. III.2 – Percolation theory: with the model of rigid spheres, here we calculate the average number of contacts among particles, the percolation thresholds and the percolation probabilities;

• par. III.3 – Overlapping of particles: we consider the effects of overlapping of spheres and define the angles of contact among different types of particles;

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• par. III.4 – Apparent conductivity: we describe methods for the estimation of the apparent conductivity of a random structure and, in particular, our simulated results;

• par. III.5 – Effects of porosity: we describe how the results of the previous theories change in the presence of pore formers;

• par. III.6 – Calculation of morphological parameters: we resume all the results of previous paragraphs to determine the structural parameters that model equations need;

• par. III.7 – Morphological results: we present some results from the theories explained in the previous paragraphs;

• par. III.8 – The real morphology: we discuss about the differences between modeled and real morphology.

Before entering in the details of the assumptions and calculations, we just report below the glossary and the nomenclature that we will use specifically in this chapter (the dimensions of the quantity described are mentioned in square brackets); see also fig. III.1-3 to visualize the concepts cited and (eq. III.1.1-4). All the relationships are referred to a binary mixture of monosized particles (k and l).

Definitions:

• connected (or percolating) cluster (A-type): cluster of particles of the same kind entirely connected from each side of the structure (fig. III.1);

• partly connected cluster (B-type): cluster of particles of the same kind connected only to the respective collector (fig. III.1);

• isolated cluster (C-type):cluster of particles of the same kind not connected to its respective collector (fig. III.1);

Glossary:

• adlv: double layer specific area (contact area between connected k and l-particles

per unit volume) [m2/m3];

• avk: total surface area of k-particles exposed to gas phase per unit volume

[m2/m3];

• dp: mean diameter of pores [m];

• Dp: mean diameter of k and l-particles [m];

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• Nk,l: average number of contacts between a k-particle and l-particles when ζl

approaches to one [l-parts/k-part];

• nkv: number of k-particles per unit volume (voids included) [k-parts/m3];

• P: ratio of radii of particles as in (eq. III.1.1) [-];

• pk: percolation probability of k-particles (i.e. probability that a k-particle belongs

to a A-cluster) [-];

• Rh: hydraulic radius [m];

• rk: radius of k-particles [m];

• Sk: surface area fraction of all k-particles as in (eq. III.1.2) [-];

• Z: overall average coordination number of all solids particles [parts/part];

• Zk: average coordination number of k-particles (i.e. number of particles in contact

for each k-particle, fig. III.2) [parts/k-part];

• Zk,l: average number of contacts between a k-particle and l-particles (i.e. average

number of l-particles in contact for each k-particle, fig. III.2) [l-parts/k-part];

• ζk: number fraction of k-particles as in (eq. III.1.4) [-].

• θkl: angle of contact between a k-particle and a l-particle (fig. III.3) [rad];

• λv

TPB: length of three phase boundary per unit volume [m/m3];

• σ: conductivity [S/m];

• φg: porosity (i.e. void degree) [-];

•

ψk

: volume fraction of k-particles relative to the total solid as in (eq. III.1.3) [-]; List of equations (relationships among primitive quantities mentioned above):

l k r r P= (eq. III.1.1)

(

)

₍

₎

k k k k 2 k 2 k 2 l v l 2 k v k 2 k v k k 1 P P 1 P P r 4 n r 4 n r 4 n S

ψ

ζ

π

− + = − + = + = (eq. III.1.2)

(

k

)

3 k 3 k 3 l 3 4 v l 3 k 3 4 v k 3 k 3 4 v k k 1 P P r n r n r n

ζ

π

ψ

− + = + = (eq. III.1.3)

(

k

)

3 k 3 k v l v k v k k 1 P P n n n

ψ

ζ

− + = + = (eq. III.1.4)

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Clarifying figures

fig. III.1 – Types of clusters in a binary random packing.

Zk,l = 5

Zk,k = 2

Zk = 7

fig. III.2 – Explanation of the meaning of Zk,l and Zk.

fig. III.3 – Angles of contact between overlapping particles. k-collector l-collector

A

B

C

l l k k l l k l rl rk θkl θlk k l

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III.2 – Percolation theory

Considering the model of rigid spheres in a binary mixture of particles of different radii, but not polydisperse, in this paragraph we will describe methods to estimate the number of contacts, the percolation thresholds and the probability of connection; then, we will extend the theory to a binary mixture of polydisperse spheres. It should be noted that the model of rigid spheres means that all the angles of contacts among particles are equal to zero (no-overlapping particles, see fig. III.3 with θkl = θlk = 0).

III.2.1 – Methods for calculation of number of contacts

There are several theories to estimate the coordination numbers in a random mixture of spheres; we will take into account three which are most used and agree better with experimental results.

Suzuki and Oshima (1983) proposed a simple model to estimate coordination numbers for a random binary mixture of spheres; in particular, they assumed that average number of contacts, such as Zk,l just to make an example, is proportional to the surface area

fraction of particles in contact with the reference particle, in this case Sl and Nk,l (i.e. the

limit average number of contacts of a k-particle with l-particle). Then, they suggested a method, based on geometrical considerations, to estimate Nk,l and Nl,k closing in this

way the system. Their results are marked with the superscript “S”.

Chen et al. (2009) used an approach very close to Suzuki and Oshima (1983), in particular only the method to estimate Nk,l and Nl,k changes. In this model of

percolation, marked with “C”, the overall average coordination number of all solids particles Z is assigned (note that in the previous model Z is calculated) and the balance of the total number of contacts between l-particles and k-particles, that shall be equal to the total number of contacts between k and l-particles, is used to estimate Nk,l and Nl,k.

Bouvard and Lange (1991) used a different approach: they suggested that the average number of contacts in excess of the minimum required for stability is proportional to the surface area of particles, in this way it is possible to calculate Zk and Zl. Then, they

assumed that the fraction of contacts of k-particles with l-particles Zkl/Zk is proportional

to the average number of contacts of l-particles Zl within the medium (for l-particles it is

the same, just change indexes). This method, marked with “B”, does not need correlation for Nk,l and Nl,k because it does not use them; the overall average

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Results from each theory are compared with results obtained by computer simulations, marked with *. Computer simulations (Suzuki and Oshima, 1983) follow this algorithm to generate a random packing of spheres: k or l-spheres are selected by uniform random number and dropped one by one into a rectangular container, with a square base of side 7.5 to 15 times the diameter of coarse particles, from a random point in the plane above the container. Each particle rests only when it achieves 3 points of contact (with floor, walls or other particles). When a falling particle meets a packed particle it rolls over the surface without inertia or friction until a stable position is achieved. The total number of packed particle is around 6’000.

After this brief introduction, let us enter in details of the models.

By definition, the average coordination number of k-particles Zk must be the sum of

the average number of contacts that a k-particle makes with other k-particles (Zk,k) and

l-particles (Zk,l): l , k k , k k Z Z Z = + (eq. III.2.1)

and the same for l-particles.

To respect the total balance of contacts between k and l-particles, which affirms that the total number of contacts between k-particles and l-particles must be equal to the total number of contacts between l-particles and k-particles, there is a relationship that links together the quantities Zk,l and Zl,k as:

(

k

)

l,k

l, k

kZ 1

ζ

Z

ζ

= − (eq. III.2.2)

We can also define the overall average coordination number Z as:

(

k

)

l

k

kZ 1 Z

Z =

ζ

+ −

ζ

(eq. III.2.3)

For a random packing of monosized spheres of the same type, Z is called Nc, usually

assumed equal to 6 (Suzuki et al., 1981).

According to Bouvard and Lange (1991), considering that to reach the static equilibrium of the particles there shall be at least 3 points of contact for each sphere, they assume that the average number of contacts in excess of the minimum required for stability is proportional to the surface area of particles:

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A. Bertei III - MORPHOLOGY 2 B k B l P 1 3 Z 3 Z = − − (eq. III.2.4)

By combining (eq. III.2.3) and (eq. III.2.4) they obtain:

(

)

(

k

)

2 k 2 B B k 1 P P 3 Z 3 Z

ζ

+ − − + = (eq. III.2.5)

(

)

(

k

)

2 k B B l 1 P 3 Z 3 Z

ζ

+ − − + = (eq. III.2.6)

To obtain the average number of contacts between different particles, they assume that the fraction of contacts of generic j-particles with i-particles, Zj,iB/ZjB, is proportional to

the average number of contacts of an i-particle within the medium:

B B i i B j B i , j Z Z Z Z =

ζ

(eq. III.2.7)

By applying (eq. III.2.7) to the case of k and l-particles, we obtain:

B B l l B k B l , k Z Z Z Z =

ζ

(eq. III.2.8) B B k k B k B k , k Z Z Z Z =

ζ

(eq. III.2.9)

and the same for l-particles.

In this model it should be noted that the parameter ZB must be assigned; Bouvard and Lange (1991) suggest setting ZB equal to 6. Another important feature is that this model respects the total balance of contacts in (eq. III.2.2).

In Suzuki and Oshima method (1983) Zk,l is proportional to surface area fraction Sl and

Nk,l as: S l , k l S l , k S N Z = (eq. III.2.10) S k , k k S k , k S N Z = (eq. III.2.11)

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and the same for l-particles. In these relationships Nk,lS represents the number of

l-particles in contact with a k-particle when ζl approaches 1, so Nk,lS is the limit of Zk,lS

when in the packing there are only a few k-particles. Thus, it should be clear that Nk,kS

and Nl,lS are equal to Nc, set equal to 6.

They propose that Nk,lS and Nl,kS can be estimated as:

(

)

(

)

(

)

(

)

0.5 c S l , k 2 P P P 1 1 P N 3 2 5 . 0 N + − + + − = (eq. III.2.12)

(

)

(

)

(

)

(

₁ ₁

)

0.5 1 1 c S k , l 2 P P P 1 1 P N 3 2 5 . 0 N + − + + − = − − − − (eq. III.2.13)

When Nk,lS (or Nl,kS) becomes lesser than 2, it must be kept equal to 2.

It should be noted that (eq. III.2.12-13) come from this consideration: as in fig. III.4, consider a k-particle in contact with a l-particle, we call As the surface area of the

hypothetical sphere of radius rk + rl and Ac the fraction of As occupied by a contact

particle l; it is assumed that Nk,l is proportional to the ratio As/Ac1.

fig. III.4 – Representation of the hypothetical sphere and the areas As and Ac.

With (eq. III.2.1) and (eq. III.2.3), ZkS, ZlS and ZS can be calculated. Note that in this

theory ZS is calculated and not assigned as in the previous theory; it is clear that it should reasonably be near to 6 (Nc) but not necessarily equal, it depends on the

1_{By geometrical considerations:}

(

)

2 l k s 4 r r A =

π

+

(

)

[

(

)

]

        + + − + = l k 2 / 3 l k k l k k c r r r 2 r r r 2 r r 2 A

π

. As Ac l k

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composition (ζk) and the ratio of radii (P). The drawback of the Suzuki and Oshima

method is that it does not respect the balance of contacts in (eq. III.2.2) for P ≠ 1.

Chen et al. (2009) propose a new method to estimate the coordination numbers: they agree with Suzuki and Oshima (1983) assuming (eq. III.2.10-11) but suggest another form for Nk,l and Nl,k. They resolve a system with (eq. III.2.1-3) and (eq. III.2.10-11)

that yields:

(

2

)

C C l , k 0.51 P Z N = + (eq. III.2.14)

(

2

)

C C k , l 0.51 P Z N = + − (eq. III.2.15) so: C l , k l C l , k S N Z = (eq. III.2.16) C k , k k C k , k S N Z = (eq. III.2.17)

and the same for l-particles; note that they assume Nk,kC = Nl,lC = ZC where ZC is an

assigned parameter set equal to 6.

Chen et al. method substantially imposes the respect of the total balance of contacts to calculate Nk,lC and Nl,kC and uses the hypothesis that Zk,lS is proportional to Sl and Nk,lC

as Suzuki and Oshima (1983). It is interesting to note that ZkC and ZlC, calculated by

using (eq. III.2.1), coincide with Bouvard and Lange results, even if Zk,lC ≠ Zk,lB and

Zk,kC ≠ Zk,kB.

Fig. III.5 shows Nk,l as predicted from Suzuki and Oshima theory (blue line) and Chen

et al. (red line); simulated results (dots) in the range P = [0.25,4], as reported in Suzuki and Oshima (1983), are also shown. Note that P can vary from a minimum to a maximum value with Pmin = 2/ 3−1 to Pmax =

(

)

1 1 3 /

2 − − , outside of this range there could be segregation (i.e. smaller particles pass through the voids left by bigger particles)2.

Note that Nk,lC ≥ Nk,lS except in the range [1,1.327]. There is a good agreement of Nk,lS

with simulated results for P ≥ 1 while, for P < 1, Nk,lS < Nk,l*. The agreement of Chen et

al. with simulated results for P > 1 is not good while it is quite good for P < 1.

2

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A. Bertei III - MORPHOLOGY 0 5 10 15 20 25 30 35 40 0 1 2 3 4 P N k ,l

Suzuki and Oshima Chen et al. Simulated

fig. III.5 – Estimation of Nk,l by using Suzuki and Oshima or Chen et al. theory and

comparison with simulated results in the range P = [0.25,4].

fig. III.6 – Comparison of coordination numbers predicted by several theories and results from computer simulations: a) P = 3; b) P = 0.33.

Coordination number Zk as obtained by different theories (blue: Suzuki and Oshima;

red: Chen et al.; light blue: Bouvard and Lange) is compared with results from computer simulations (dot, as reported by Suzuki and Oshima, 1983) in fig. III.6, in a) for P = 3 and in b) for P = 0.33. Note that Chen et al. results coincide with Bouvard and Lange results as said before. For P = 3 (and in general for P ≥ 1) Suzuki and Oshima results agree with Zk* in all the range of compositions; at the same time, ZkC does not agree

0 5 10 15 20 25 30 0 0.2 0.4 _ζ 0.6 0.8 1 l Zk

Suzuki and Oshima Chen et al.

Bouvard and Lange Simulated P = 3 a) 0 1 2 3 4 5 6 7 0 0.2 0.4 _ζ 0.6 0.8 1 l Zk b) P = 0.33

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well with Zk* especially when ζl approaches 1. For P = 0.33, Suzuki and Oshima

method predicts ZkS < Zk* while ZkC ≈ Zk*.

By analyzing fig. III.5-6, we can draw the following considerations:

1. for P ≥ 1 Suzuki and Oshima method yields results for k-particles that agree very well with simulated results: in these cases predicted ZkS is substantially equal to

Zk* determined by computer simulations (fig. III.6a);

2. in particular, for P ≥ 1 when ζl ≈ 1 Zk ≈ Zk,l, so the good agreement of ZkS with

Zk* shows that the relationships to calculate Zk,lS and Nk,lS are good (fig. III.6a

and fig. III.5);

3. when ζl ≈ 0 for P ≥ 1, Zk ≈ Zk,k and so the good agreement of ZkS with Zk* shows

that the relationship to calculate Zk,kS is good (fig. III.6a);

4. when ζl ≈ 0, for P ≥ 1 Zl ≈ Zl,k and so the bad agreement of ZlS with Zl* shows

that the relationships to calculate Zl,kS and Nl,kS are not so good; this consideration

is evident in fig. III.6b and fig. III.5 for P < 1, change k with l and vice versa3; 5. according to points 2 and 4, for P ≥ 1 we can estimate Zl,k by using the total

balance of contacts instead of Suzuki and Oshima form, point 3 leads to think that we can estimate Zl,l in the same way as we estimate Zk,kS. For P < 1 we could

repeat the same things but changing k with l and vice versa;

6. the method of estimation of Nk,lS seems to be suited for P ≥ 1 and not for P < 1

(fig. III.4 and fig. III.5);

7. Chen et al. method yields ZkC > Zk* especially when ζl approaches 1 for P ≥ 1

(fig. III.6a); in the same way, Nk,lC > Nk,l* for P > 1 (fig. III.5).

Our suggestion (marked with the superscript “N”) is this: checked that Suzuki and Oshima method yields results that agree very well with simulated results for P ≥ 1 for k-particles but not for l-k-particles and it does not respect the balance of contacts, we propose to modify the form for Nl,k in the way to respect (eq. III.2.2). So, assuming the

form given by Suzuki and Oshima:

(

)

(

)

(

)

(

)

0.5 c N l , k 2 P P P 1 1 P N 3 2 5 . 0 N + − + + − = (eq. III.2.18)

we calculate Nl,kN by using (eq. III.2.2); it yields:

3

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(

)

(

)

(

)

(

)

[

0.5

]

2 c N k , l P 2 P P P 1 1 P N 3 2 5 . 0 N + − + + − = (eq. III.2.19)

This is valid when P ≥ 1; when P ≤ 1 l-particles are bigger than k-particles, so we must calculate Nk,lN by using (eq. III.2.2) assuming Nl,kN = Nl,kS. Finally, it results as:

(

)

(

)

(

)

(

)

[

]

(

)

(

)

(

)

(

)

[

]

       + − + + − = + − + + − = ≥ 2 5 . 0 c N k , l 5 . 0 c N l , k P 2 P P P 1 1 P N 3 2 5 . 0 N 2 P P P 1 1 P N 3 2 5 . 0 N 1 P (eq. III.2.20)

(

)

(

)

(

)

(

)

[

]

(

)

(

)

(

)

(

)

[

]

       + − + + − = + − + + − = ≤ − − − − − − − − − 2 5 . 0 1 1 1 1 c N l , k 5 . 0 1 1 1 1 c N k , l P 2 P P P 1 1 P N 3 2 5 . 0 N 2 P P P 1 1 P N 3 2 5 . 0 N 1 P (eq. III.2.21) and, obviously: N l , k l N l , k S N Z = (eq. III.2.22) N k , k k N k , k S N Z = (eq. III.2.23)

and the same for l-particles. As in Suzuki and Oshima theory, Nk,kN = Nl,lN = Nc = 6.

Thus, for P ≥ 1 our new method reproduces the good results of Suzuki and Oshima for k-particles (because the relationships are the same) and for l-particles it predicts results that agree better with Zl* because there is the physical link due to the total balance of

contacts (that we take into account); for P ≤ 1 we should repeat the same considerations changing k with l and vice versa. Moreover, our model does not need to assign Z, which can be calculated by using (eq. III.2.3).

Fig. III.7 shows that our new method estimates Nk,lN in good agreement with Nk,l* in

the range of ratio of radii [0.25,0.4]. Fig. III.8 shows the comparison between our estimation of the coordination number ZkN with simulated results (Suzuki and Oshima,

1983) for P = 3 and P = 0.33: the agreement is good. Note that it is possible to compare fig. III.7 with fig. III.5 and fig. III.8 with fig. III.6.

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A. Bertei III - MORPHOLOGY 0 5 10 15 20 25 30 35 40 0 1 2 3 4 P N k ,l This study Simulated

fig. III.7 – Comparison between Nk,lN calculated by using our theory with simulated

results in the range P = [0.25,4].

fig. III.8 – Comparison of coordination numbers predicted by our theory and results from computer simulations: a) P = 3; b) P = 0.33.

Zk,l as calculated from the four theories (same colours) for P = 3 and P = 0.33 are

compared in fig. III.9. Note that for P > 1 is Zk,lC > Zk,lS = Zk,lN while Zk,lB reaches Zk,lC

at ζl = 1; for P < 1 it is the same but Zk,lN > Zk,lS as expected because Nk,lN > Nk,lS

(compare fig. III.5 with fig. III.7).

Zk,k are reported in fig. III.10: blue line for Zk,kS = Zk,kC = Zk,kN and light blue for Zk,kB.

Note the strange shape made by Zk,kB for P > 1: it also explains why Zk,lB is so little near

0 5 10 15 20 25 30 0 0.2 0.4 _ζ 0.6 0.8 1 l Z k This study Simulated 0 1 2 3 4 5 6 7 0 0.2 0.4 _ζ 0.6 0.8 1 l Z k P = 3 P = 0.33 a) b)

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A. Bertei III - MORPHOLOGY 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 _ζ 0.6 0.8 1 l Zk ,l P = 0.33 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 ζl Z k ,l

Bouvard and Lange This study P = 3 1 10 0 0.2 0.4 ζl 0.6 0.8 1 Z k ,k P = 0.33 0.1 1 10 0 0.2 0.4 ζ_l 0.6 0.8 1 Z k ,k

Suzuki and Oshima; Chen et al.; This study

Bouvard and Lange P = 3

ζl = 0.8 in fig. III.9 (remember that Zk,lB + Zk,kB = ZkB and that ZkB = ZkC: if Zk,kB > Zk,kC

then Zk,lB < Zk,lC, all calculated at the same P and ζl).

fig. III.9 – Comparison among Zk,l estimated by different theories.

fig. III.10 – Comparison among Zk,k estimated by different theories.

Thus, the conclusions that we can draw are:

1. according to fig. III.7, our new method seems to predict Nk,l in good agreement

with simulated results, better than Nk,lC and Nk,lS where Nk,lC > Nk,lN in general;

2. Chen et al. method predicts coordination numbers bigger (or equal) than other theories, maybe because it yields high Nk,lC;

3. Bouvard and Lange results show different features compared with other theories, especially in the estimation of Zk,k;

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A. Bertei III - MORPHOLOGY 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ζl Z

Bouvard and Lange This study

Simulated P = 3

4. our new model seems to reach good results in the estimation of each coordination number.

The explanation of these conclusions could be found in fig. III.11 where Z calculated from the theories and simulation results (extracted from Suzuki and Oshima, 1983) are plotted for P = 3. While ZC and ZB are always equal to 6, because those values are assigned, ZS and ZN calculated are lesser than 6; results from simulations agree with this fact. This may explain why Chen et al. method (and Bouvard and Lange theory too) predicts coordination numbers bigger than other theories. This is another signal that confirms how our new method could describe better than others the morphological properties of a random packing of spheres. Then, in the following section, results concerning percolation thresholds will still confirm the validity of our model.

It is important to say that all the methods described yield the same results (i.e. Zk, Zk,l,

Zk,k, Z) when P = 1.

fig. III.11 – Estimation of Z and comparison with simulated results.

III.2.2 – Percolation thresholds and probability of connection

It is straightforward to imagine that when the number fraction of a phase, for example l-particles, goes from 1 towards 0, in the packing there will be a smaller number of percolating clusters (type A) and there will be a value of ζl (marked with the subscript

“th”) at which that kind of clusters disappears: we call ζl-th percolation threshold for

l-particles. In the same way, we can find a percolation threshold ζk-th for k-particles and,

obviously, we can traduce the numerical fractions into volume fractions (i.e. ψl-th and

ψk-th). Thus, below the percolation threshold for l-particles there is not any A-type

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A. Bertei III - MORPHOLOGY 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1 10 P ζ l-th This study

Bouvard and Lange This study

Bouvard and Lange Simul. and exper.

current, above the threshold there will be a flow of charges while below it the flow will be 0.

Considering l-particles (but the same considerations could be done for k-particles), the value of ζl-th depends only on P. Another interesting consideration is that there should

be a link between the presence of A-type clusters of l-particles and Zl,l whatever P we

choose: a percolating cluster will exist only if there are several contacts between l-particles, so when Zl,l is above a certain critical value. Thus, we can predict percolation

thresholds (i.e. ζl-th) by using the percolation theories described before just assuming a

Zl,l-th.

Kuo and Gupta (1995) showed that the threshold for l-particles is reached when Zl,l

equals 1.764 and not 2 as suggested by Bouvard and Lange (1991); the agreement between the predicted value of ζl-th (and ψl-th) and the simulated or experimental results

is better if we use Zl,l-th = 1.764 instead of 2.

The predicted values of ζl-th from our theory4 (in green) or Bouvard and Lange theory

(light blue) are shown in fig. III.12 and compared with simulated and experimental results (dots) extracted from Kuo and Gupta (1995). Solid lines represent percolation thresholds calculated by using Zl,l-th = 1.764 while in dotted lines by using Zl,l-th = 2.

Note that ζl-thN are calculated by using (eq. III.2.23) for l-particles and ζl-thB by using

(eq. III.2.9) where Zl,l-th is assigned. Fig. III.13 shows the same results in terms of ψl-th.5

fig. III.12 – Comparison between ζl-th predicted by different theories and simulated or

experimental results.

4_{It is important to note that Z}

l,lN = Zl,lS = Zl,lC because all the theories use the same hypothesis. 5

We represent percolation threshold for l-particles instead of k-particles to compare directly fig. III.12-13 with figures reported in Kuo and Gupta (1995).

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A. Bertei III - MORPHOLOGY 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1 10 P ψ l-th This study Bouvard and Lange This study Bouvard and Lange Simul. and exper.

fig. III.13 – Comparison between ψl-th predicted by different theories and simulated or

experimental results.

The figures show that our model can predict percolation thresholds better than Bouvard and Lange theory when we assume Zl,l-th = 1.764; it should demonstrate that

the relationship (eq. III.2.23) works well.

Another interesting property is pl, the probability that a l-particle belongs to a

percolating cluster; it is strictly connected with Zl,l as demonstrated by Bouvard and

Lange (1991): pl depends only on Zl,l, in other words if 2 packing present the same Zl,l,

even if they are characterized by different composition and ratio of radii, the probability pl(1) will be equal to pl(2). Then, it should be clear that pl approaches 0 at the percolation

threshold for l-particles.

Bouvard and Lange (1991) produced some results by computer simulations (the algorithm is the same as describe in sec. III.2.1 for Suzuki and Oshima, 1983) and suggested an empirical correlation to fit them:

4 . 0 5 . 2 l , l B l 2 Z 4 1 p _             − − = (eq. III.2.24)

It is easy to verify that (eq. III.2.24) predicts percolation thresholds when Zl,l = 2. To

take into account that percolation thresholds are identified for Zl,l-th = 1.764 instead of 2

as explained before, several suggestions were given as listed below:

4 . 0 5 . 2 l , l C l 2 Z 764 . 3 1 p _             − −

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A. Bertei III - MORPHOLOGY 4 . 0 5 . 2 l, l Z l 472 . 2 Z 236 . 4 1 p _             ₋ −

= by Zhu and Kee (2008) (eq. III.2.26)

These last correlations predict pl = 0 at Zl,l-th = 1.764 but do not agree with simulated

results as reported by Bouvard and Lange (see fig. III.14). Our suggestion is to take a form of pl like:

δ γ

β

α

              − − = l,l l Z 1 p (eq. III.2.27)

with 4 adjustable parameters (any parameter does not have any physical meaning). Then, if we impose pl = 0 at Zl,l-th = 1.764 it yields:

764 . 1 − =α β (eq. III.2.28)

The second imposition that we should do could be pl = 1 at a certain value of Zl,l, call

this value Zl,l-con (where the subscript “con” means that all clusters are connected), in

this way we could find α (i.e. α = Zl,l-con); we do not know the value of Zl,l-con and we

can not predict it by physical-mathematical considerations. However, our aim is to produce a correlation that follows the shape of simulated results, so we can handle 3 adjustable parameters (α, γ, δ) and we do not mind about the exact value of Zl,l-con. So,

we found that the expression:

              − − = 7 . 3 l , l N l 472 . 2 Z 236 . 4 1 p (eq. III.2.29)

agrees well with simulated results.

Fig. III.14 shows the comparison between pl predicted with several correlations (light

blue: Bouvard and Lange; red: Chen et al.; blue: Zhu and Kee; green: our suggestion) and simulated results (dots) as reported by Bouvard and Lange (1991). While plC and plZ

predict percolation threshold at Zl,l = 1.764 but do not agree with simulated results, our

correlation agrees both with Zl,l = 1.764 and with simulated results. It is obvious that

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A. Bertei III - MORPHOLOGY 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 Zl,l 3 4 5 p l

Bouvard and Lange Chen et al.

Zhu and Kee This study Simulated

fig. III.14 – Comparison between pl predicted by correlations and simulated results.

III.2.3 – Extension to polydisperse mixtures

An advantage of our percolation model (but also of Suzuki and Oshima and Chen et al. theories) is that it can be applied to multicomponent mixtures; in particular, we take into account the situation of a binary mixture of particles, k and l, made with polydisperse powders (i.e. k and l-particles have a distribution of radii).

Consider a packing that includes m particle sizes for k-particles (i.e. different radii rk1,

rk2…rkm) and n particle sizes for l-particles. So, for a generic hi-particle (k or l),

characterized by a radius rhi, we define:

hj hi hj , hi r r

P = check that Pmin < Phi,hj < Pmax to avoid segregation (eq. III.2.30)

∑

= = = = = = + = + = + = _n 1 j lj lj m 1 j kj kj hi hi n 1 j 2 lj lj m 1 j 2 kj kj 2 hi hi n 1 j 2 lj v lj m 1 j 2 kj v kj 2 hi v hi hi r r r r r r r 4 n r 4 n r 4 n S

ψ

ζ

π

(eq. III.2.31)

∑

= = = = + = + = _n 1 j 3 lj lj m 1 j 3 kj kj 3 hi hi n 1 j 3 lj 3 4 v lj m 1 j 3 kj 3 4 v kj 3 hi 3 4 v hi hi r r r r n r n r n

ζ

π

ψ

(eq. III.2.32)

∑

= = = = + = + = _n 1 j 3 lj lj m 1 j 3 kj kj 3 hi hi n 1 j v lj m 1 j v kj v hi hi r r r n n n

ψ

ζ

(eq. III.2.33)

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Using (eq. III.2.20-21):

(

)

(

)

(

)

(

)

[

]

(

)

(

)

(

)

(

)

[

]

        ≤ + − + + − ≥ + − + + − = − − − − − 1 P if P 2 P P P 1 1 P N 3 2 5 . 0 1 P if 2 P P P 1 1 P N 3 2 5 . 0 N hj , hi 2 hj , hi 5 . 0 1 hj , hi 1 hj , hi 1 hj , hi 1 hj , hi c hj , hi 5 . 0 hj , hi hj , hi hj , hi hj , hi c N hj , hi (eq. III.2.34) and then: N hj , hi hj N hj , hi S N Z = (eq. III.2.35)

So, we can also calculate the coordination numbers for particle hi:

∑

= = m 1 j N kj , hi N k , hi Z Z (eq. III.2.36)

∑

= = n 1 j N lj , hi N l , hi Z Z (eq. III.2.37) N l , hi N k , hi n 1 j N lj , hi m 1 j N kj , hi n m 1 j N hj , hi N hi Z Z Z Z Z Z =

∑

=

∑

+

∑

= + = = + = (eq. III.2.38)

and the overall average coordination number:

∑

+ = =m n 1 i N hi hi N Z Z ζ (eq. III.2.39)

The overall average coordination numbers for all k-particles and all l-particles are proposed to be:

∑

= = = _m 1 i ki m 1 i N k , ki ki N k , k Z Z ζ ζ (eq. III.2.40)

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∑

= = = _n 1 i li n 1 i N l , li li N l, l Z Z ζ ζ (eq. III.2.41)

These relationships take into account the total number of contacts k-k (or l-l) in the mixture divided by the total number of k-particles (or l-particles).

The probability of connection, pkN or plN, can be calculated by substitution of Zk,kN or

Zl,lN into (eq. III.2.29).

There are not till now reliable simulated or experimental results concerning polydisperse or multicomponent mixtures of particles, so we can not validate our model. It should be noted that (eq. III.2.31-41) are the same obtained by Chen et al. (2009), except the definition of Nhi,hj. In particular, while (eq. III.2.34-39) could be reliable and

represent with a good approximation the properties of a random packing of polydisperse spheres, we can not take (eq. III.2.40-41) and their use into (eq. III.2.29) with reasonable certainty because it is not sure if the correlation (eq. III.2.29) made for binary mixtures could also be applied to polydisperse powders.

III.3 – Overlapping of particles

Till now we have considered that spheres do not overlap as they are rigid; but when we study the morphology of the central membrane and of electrodes in general we must take into account that particles are compressed and sintered. Then, assuming for instance that before sintering particles are spherical, after that operation we should expect that their shape shows overlapping regions.

Consider a binary, not polydisperse, randomly packed mixture of particles, k and l (i.e. the same situation described in sec. III.2.1); the model that we suggest, as already mentioned by Janardhanan et al. (2008), is based on the following assumptions:

1. particles are spheres before sintering;

2. the sintering process creates overlapping regions among particles characterized by angles of contact θij;

3. after sintering particles keep quite the same position, same radii rk and rl and the

spherical shape except in the overlapping region;

4. presence of overlapping made by 3 or more particles is excluded. Fig. III.15 shows the effects of these assumptions.

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a) b) c)

fig. III.15 – Graphic meaning of the assumptions: a) before sintering; b) sintering; c) after sintering.

It is interesting to note that point 3 leads to the disappearance of a fraction of volume of particles: if particle i overlaps on particle j, the fraction of volume of i that is in common with j disappears and the same for particle j. In a real situation we could think that in the overlapping region particles are compressed, so in the neighbourhood of the contact the solid density of each particle is higher than in the bulk. In this way, we exclude the increasing of the mean radii of particles and the presence of boundary effects at the contacts (i.e. we assume that the contact region is a perfect circle).

We can calculate the fraction of volume lost in each kind of contact; we use the symbol Viij to represent the volume of a generic i-particle lost in a contact i-j

(characterized by an angle θij). So we obtain in particular (see fig. III.3):

(

)

kl k

(

kl

)

kl k kl 2 kl 2 kl kl kl

k h 3a h , h r 1 cos and a r sin 6 V =π + = − θ = θ (eq. III.3.1)

(

)

kk k

(

kk

)

kk k kk 2 kk 2 kk kk kk

k h 3a h , h r 1 cos and a r sin 6 V =π + = − θ = θ (eq. III.3.2)

(

)

lk l

(

lk

)

lk l lk 2 lk 2 lk lk lk

l h 3a h , h r 1 cos and a r sin 6 V =π + = − θ = θ (eq. III.3.3)

(

)

ll l

(

ll

)

ll l ll 2 ll 2 ll ll ll

l h 3a h , h r 1 cos and a r sin 6

V =π + = − θ = θ (eq. III.3.4)

Note that there is the equality:

lk kl a

a = (eq. III.3.5)

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Assuming that the percolation properties of the mixture can be estimated by equations described in sec. III.2.1 (and in particular by using our new method) although particles are not rigid spheres, the total volume lost in the contacts for each particle, k or l, is:

kk k N k , k kl k N l, k lost k Z V Z V V = + (eq. III.3.6) ll l N l , l lk l N k , l lost l Z V Z V V = + (eq. III.3.7)

Thus, the effective volumes of the particles, Vkeff and Vleff, will be:

lost k 3 k 3 4 eff k r V V =

π

− (eq. III.3.8) lost l 3 l 3 4 eff l r V V =

π

− (eq. III.3.9)

These relationships are useful to calculate the total number of particles per unit volume (voids included) nv:

(

)

eff l k eff k k g v V 1 V 1 n ζ ζ φ − + − = (eq. III.3.10)

It should be easy to understand that if θij > 0, nv will be higher compared with nv that

we could calculate without considering the overlapping effects (i.e. considering that the effective volumes are equal to the volumes of the spheres).

The values of θij depend first of all on the sintering conditions and on the mechanical

properties of the materials; nevertheless, it is not easy to correlate sintering conditions to angles of contacts and, on the other hand, it could be quite difficult to measure θij. A

typical value assumed for the biggest between θkl and θlk is 15° (Costamagna et al.,

1998; Chen et al., 2009; Nam and Jeon, 2006), in the same way we can assume that also angles θkk and θll are equal to 15°: at the end all these values are handled as adjustable

parameters, so other values instead of 15° can be used.

It is interesting to note that for reasonable angles (i.e. θkk = θll = max(θkl,θlk) ≤ 15°) the

errors that we will make on nv if we do not consider the overlapping effects (i.e. using

3 i 3

4

π

_r _{instead of V}

ieff) will be less than 0.52%, so we can neglect overlapping effects in

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III.4 – Apparent conductivity

The conductivity of a material, σ (the resistivity ρ is the reverse), takes into account the resistance to the flow of current, so it is the key parameter in the Ohm law:

V

i=−σ∇ (eq. III.4.1)

where i is the density of current and V the potential. When a material is a composite made of conducting and non-conducting particles, we can solve the problem of the total conduction by using 2 different approaches:

1. we can apply Ohm law only to the domain made by conducting particles using the conductivity of that material;

2. we can apply Ohm law to the whole structure using an apparent conductivity σapp. The first procedure needs the knowledge of the exact morphology of the structure to apply the law only where there is the conducting phase, the second one needs a correlation that links σapp to the average morphological properties of the composite. In modelling it is easier to use the second approach (see par. II.7).

In the following sections we will focus on the correlations concerning σapp.

III.4.1 – Definition of apparent conductivity and methods for its estimation

Consider a composite structure of k-particles and l-particles, imagine that only phase k can conduct current (we refer to a bulk conduction and not to a surface conduction); σk

is the conductivity of the pure dense material. Take a solid dense parallelepiped of phase k with length L and cross section equal to S; if we apply a difference of potential

∆V between the 2 opposite sides of the solid, the total current I that will flow will be:

V L S

I =σk ∆ (eq. III.4.2)

Now take a container with the same geometric dimensions of the parallelepiped described before and fill it with a mixture of k and l-particles (see fig. III.16); if we apply the same difference of potential ∆V the current will be Icom with 0 ≤ Icom < I. We

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fig. III.16 – Definition of apparent conductivity.

V L S

I_com =σ_kapp ∆ (eq. III.4.3)6

Thus, the apparent conductivity allows us to calculate the current that flows in the composite by using the geometric dimensions of the whole structure. It should be clear that σkapp depends on the morphological properties of the structure such as the

composition, the radii of particles, etc.; then, σkapp will approach zero at the percolation

threshold. There are several correlations to estimate σkapp, some examples are listed

below: •

(

)

(

)

µ µ

ζ

γ

σ

th k th k k k app k 1 ₋ − − − ⋅ = (eq. III.4.4)

as suggested by Costamagna et al. (1998), where µ is an exponent taken as 2 and γ is an adjustable parameter (set equal to 0.5) to take into account the effects of the necks (i.e. angles of contact) between particles;

•

[

(

)

]

m k k g k app k p 1 φ ψ σ σ ₌ ₋ (eq. III.4.5)

as in Nam and Jeon (2006) (but also in Zhu and Kee, 2008) where the Bruggeman factor m is equal to 1.5; Chen et al. (2004) use the same correlation with m = 1;

• Chen et al. (2009) separate the apparent conductivity into intra-particle conductivity and inter-particle contribution added in series; the first term is equal to (eq. III.4.5);

σ σ

I Icom

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A. Bertei III - MORPHOLOGY 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ψk σ k a p p /σ k

Nam and Jeon (correlation) Costamagna et al. (correlation) Choi et al. (experimental)

φg = 0.35

P = 1.431

• Chen et al. (2001) separate the apparent conductivity in 2 contributions too; in this paper the intra-granular resistivity is function of a tortuous factor τ that takes into account the effective length of the paths across the particles.

It is clear that there are several correlations but not persuasive. We think that in a correlation should appear at least the angle of contact between particles of the same type

θii (e.g. if that angle approaches 0 the apparent conductivity will become zero). In fig.

III.17 we compare the results of Choi et al. (2009) with Costamagna et al. (red) and Nam and Jeon (blue) correlations for a binary mixture of monosized (i.e. not polydisperse) particles with rk = 0.365µm and rl = 0.255µm (i.e. the ratio P is equal to

1.431) and porosity of 35%; Choi et al. results derive from computer simulations, they are validated with experimental results and show a very good agreement (so they can be considered as experimental results). Results from the 2 correlations are calculated by using (eq. III.4.4-5) in which parameters ζk-th and pk are calculated by using our

percolation theory.

fig. III.17 – Comparison among Choi et al. results with Costamagna et al. and Nam and Jeon correlations for apparent conductivity.

It is clear that there is not agreement, the errors are often of the order of 100%: these 2 common correlations are not satisfactory.

If we want to build a morphological model of the central membrane or, in general, of a random packing of particles, all the parameters should enter in the correlations and θij

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In the following sections we will describe the importance of the angle of contact in the apparent conductivity and our approach to the estimation of σapp.

III.4.2 – The importance of the angle of contact

The importance of the angle of contact between particles of the same type can be shown by a simple example: consider N spheres of radius r0 with their centers lined up

on a straight line and the same angle of contact θ among them (the 2 spheres at the opposite sides are cut to create the same angle); assume that the spheres are made of a material with conductivity σ and impose a difference of potential ∆V at the opposite sides of the chain of particles. It is obvious that the current I that flows will depend principally by the angle θ (but also by the total length of the path L).

We have simulated the situation by using COMSOL Multiphysics as solver of the Ohm law setting the following parameters: r0 = 1m, σ = 1S/m, ∆V = 1V, N = 30; fig.

III.18 shows the results of simulation for θ = 15°: the potential varies from 0 (blue) to 1V (red) while the current is shown as red lines. At several angles of contact we read the total current I that passed across the particles. Note that results are independent on particle size: chains of particles with different size (r0) but equal angle of contact will

produce the same current density.

fig. III.18 – Current across overlapping particles: simulation with COMSOL. L

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A. Bertei III - MORPHOLOGY 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 5 10 15 20 25 30 θ [°] I/ I a p p ,n o rm 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 30 θ [°] I/ I a p p ,c y l

We compare I with the total current that would pass if the section of passage was Anorm

(i.e. Iapp,norm) or Acyl (i.e. Iapp,cyl), where Anorm is the area of the circle of radius r0, Acyl the

area of the circle of radius r0*sin(θ) (i.e. the section of passage created by the contact).

In the first case, Iapp,norm represents the current that would pass if the path was a cylinder

of section Anorm and length L, in the second one Iapp,cyl is the current that would flow in a

cylinder of area Acyl and length L. The results of simulations are shown in fig. III.19 as

ratios I/Iapp,norm (that we could call σapp/σ in this situation) and I/Iapp,cyl as a function of θ.

fig. III.19 – Simulated results of current across particles at several angles of contact.

Fig. III.19a shows that the angle of contact between particles has a central role in the conduction, I increases when θ increases (see I/Iapp,norm); in other words, σapp becomes

zero as θ approaches 0.

Another interesting consideration comes out analyzing I/Iapp,cyl in fig. III.19b: when θ

is big (i.e. θ ≈ 30°) there is a “little” difference between the real morphology of the path and the ideal cylinder of area Acyl, but when θ is small (i.e. θ < 15°) the current does not

flow only within the cylinder, so the flux is bigger than Iapp,cyl and there is a high loss of

potential in the necks. The last consideration leads to think that the tortuous factor τ, defined as the ratio of the effective length of the path and the ideal straight length covered, plays a different role if the angle of contact is big or small. For small θ in a random packing the ohmic resistances are concentrated at the necks among particles, so

τ is important because it determines the number of contacts per unit of length. For big angles of contact, the current flows within the ideal path with the same section of the area of contact: in this case the loss of potential is principally due to the total length of

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the paths. These are only limit cases and represent a simplification of the reality because also parallel conducting paths are present in a random packing.7

Another confirmation to these conclusions comes out from a 2D model of conduction where we can separate the effect of the neck to the effect of the mean section of the path; in a chain of spheres while θ increases, Acyl increases but also the mean section of

the path (that we can roughly estimate as the average between Acyl and Anorm) increases.

Thus, it is impossible to separate the effects of the necks and the effects due to the change of the mean section of passage. The simplified 2D model8 that we used is represented in fig. III.20a.

b)

a) c)

fig. III.20 – a) Geometry of the 2D model; b) Limit cases; c) Example of solution.

The 2D model has 4 parameters (i.e. a, h, L, N), in particular a and h are geometric parameters to describe the necks while L, the length of the cells, represents the characteristic dimension of the bulk (note that the bases are equal to 1, so in this way we separate the dimensions of the necks to the mean section of passage). We impose a

7_{Thus, for big angles of contact, if the current flowed only within the ideal path of section}

(

)

2

0sin

r

θ

π

,

the apparent conductivity would be σ σ θ_τ

2 app ∝ _sin

; for small angles of contact, the global resistance is due mainly to losses at the contacts, so the expected relationship could be σapp ∝2r0_τ _.

Note that we are neglecting the presence of parallel conducting paths and talking about limit cases. cell (i) A B C limit case (1) limit case (2) cell (N) cell (i) cell (2) cell (1) V = 0V V = 1V a h L 1

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A. Bertei III - MORPHOLOGY 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 0.2 0.4

a

0.6 0.8 1

∆

V

n

/

∆

V

to t

[%

]

sim lim (1) lim (2) 0 5 10 15 20 25 30 35 40 0 0.2 0.4

_L

0.6 0.8 1

∆

V

n

/

∆

V

to t

[%

]

sim lim (1) lim (2) difference of potential equal to 1V and consider σ = 1S/m, then solve the problem (by using COMSOL Multiphysics as solver of the Ohm law).

For a generic cell i (in the middle of the structure to exclude boundary effects) we call:

B C n V V V = − ∆ (eq. III.4.6) A C tot V V V = − ∆ (eq. III.4.7)

∆Vn is relative to the loss of potential due to the neck, ∆Vtot is the total loss of potential

in the cell. There are 2 limit cases as represented in fig. III.20b: in case (1) the current flows in the whole cell, in the second current flows across a channel of the same opening of the neck. By applying Ohm law it yields:

L a h a h V V ) 1 ( tot n + =       ∆ ∆ (eq. III.4.8) L h h V V ) 2 ( tot n + =       ∆ ∆ (eq. III.4.9)

Note that in case (1) there is a strong contribution of the necks on the loss of potential while in case (2) it is not like that. The results of simulations are reported in fig. III.21.

fig. III.21 – Results from 2D simulations compared with ∆Vn/∆Vtot as calculated in

limit cases with N = 51. L = 1

h = 0.001

a = 0.01 h = 0.001

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Note that is:

) 1 ( tot n tot n ) 2 ( tot n V V V V V V       ≤ ≤       ∆ ∆ ∆ ∆ ∆ ∆ (eq. III.4.10)

When L is constant, if a increases the ratio ∆Vn/∆Vtot decreases. For each a, especially

when it approaches 0, ∆Vn/∆Vtot is far from limit (2), so it confirms that for small

sections of contact the current does not flow within a channel of the same section of the contact (see for instance fig. III.20c); at the same time, ∆Vn/∆Vtot is much different from

limit (1), so it hints that there are dead regions where current does not flow (i.e. near the 4 angles of the cells). Limit (2) is reached only when L approaches 0 but it is obvious that this situation has not got a physical meaning, a sphere is likely represented by the 2D model when L = 1.

Thus, the conclusion that we can draw about the results of the 2D model for L = 1 is that for small or reasonable openings (i.e a) the loss of potential is far from limit (1) as limit (2), so we can not link the apparent resistivity either to the effective length of paths or to the number of contacts, the real behaviour is a combination of the two aspects. Then, there are other effects to consider, first of all the presence of parallel paths. All these considerations lead us to build a specific model of 3D random packing of overlapping spheres to estimate apparent conductivity by using specific simulations.

III.4.3 – Generation of 3D random structures

It is not easy to create a mathematical model of a random packing of spheres that has got a physical meaning. For example, when we consider particles with the same radius, the simulated structure must show some properties such as a solid degree (i.e 1 – φg) of

0.64 (Bouvard and Lange, 1991) and an average coordination number equal to 6 (Suzuki et al., 1981). Moreover, we want to consider the effects of the angles of contact, so particles must partly overlap.

The simplest way to generate a random distribution of spheres is to assign a random coordinate (x, y, z) within a cube for N times, where N is the number of particles that we want to fill in the container; although this model creates a random structure that respects the mean properties described before (by using an adequate number N and radius r compared with the dimensions of the container), the structure presents wide empty regions and areas where particles are too overlapped.

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The model described in sec. III.2.1, in which particles are dropped one by one from a random position into a container and they fall and roll until they find a stable position, has been used by several authors (Suzuki and Oshima, 1983; Bouvard and Lange, 1991). The structure that we obtain has got a physical meaning but it does not take into account the overlapping of spheres; Kenney et al. (2009) improved this model to allow to particles to partly overlap9; anyway, the computational weight is very high.

We propose a model that starts from a structured packing, in particular a BCC structure (i.e. Body Centered Cubic, see fig. III.22a). This configuration presents 8 contacts per particle and a solid degree of 0.680 without any overlapping (i.e. θ = 0); but if we consider to grow the radii of particles from r0 (i.e. the initial radius where there

is not overlapping) to r =r0 /cosθ (where θ is the angle of contact) and remove the

12.5% of particles, the structure will present 6 contacts per particle and a solid degree that will vary from 0.595 to 0.704 when θ goes from 0 to 20° (and for θ = 15° the solid degree is 0.656, very close to 0.64). Thus, the structure has got the geometric properties of a random packing but particles are centered in not random positions. So, we change the position of the center of each particle assuming a random coordinate within the cell that contains the particle (fig. III.22b): it creates a random structure with the mean properties calculated as the ordered configuration described above; then, if we extract a section of the structure, it is very similar to an image of a cross section of random packing simulated by Suzuki and Oshima (1983) but with some particles overlapped, see fig. III.23a.

a) b)

fig. III.22 – a) BCC configuration; b) Changing of the position of the center within the cell.

9_{In particular, dropping and rolling particles do not stop when the distance between their centers and the}

centers of packed particles is equal to the summation of the respective radii but when that distance is equal to a specified value lesser that the summation of radii; it creates overlapping regions.

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Therefore, we create a 3D cubic grid (of length equal to 1) and fill randomly only the BCC positions, from 0 to 87.5% (i.e. if we want to simulate a packing with a number fraction ζ of conducting particles, the program fills only ζ*87.5% of positions)10; in

each occupied position a sphere of radius r (i.e. we can choose the angle of contact θ) is created and its center is put in a random point inside the cell. The maximum number of particles put in the grid (i.e. when ζ = 1) is about 10’000 so that the simulation is significant (Bouvard and Lange, 1991). The algorithm that creates the structure is implemented with MATLAB; the computing time does not go past 60s. The drawback of this model is that it can not simulate composites with different particles radii.

The structure obtained is implemented into COMSOL or ANSYS: we cut the boundaries of the ideal cube that surrounds the packing so that the apparent dimensions are length and section equal respectively to 1m and 1m2, set as conductivity of the material σ = 1S/m and impose a difference of potential equal to 1V at the opposite sides of the cube. Then, the software meshes the geometry and solves the Ohm law in the whole domain: the value of the total current that flows through the structure is the apparent conductivity for this situation (i.e. for ζ and θ imposed before). In this way, we can obtain results as σapp(ζ,θ). Note that we are combining intra-particles (i.e. the bulk property of the material) and inter-particles resistivities (i.e. the additional resistivity due to the passage of charge from a particle to another at their boundary) just considering σ defined as the conductivity of the sintered dense material.

a) b)

fig. III.23 – a) Cross section and b) 3D view of a simulated structure with a single component (radius: 2.56*10-2m; mean angle of contact: 15°; porosity: 34.4%).

10_{The program allows filling more than 1 type of particles, e.g. 3 types of materials, but for our purposes,}