# Evaluation of Volume Variation Partition:
the Breathing Coefficient and its Application on
Uniaxial Monosized Disc Packing Swelling *

Théo Boivin† and Olivier Gillia‡

Univ. Grenoble Alpes, CEA-Liten, Grenoble, F-38054, France

## Abstract

An analysis of the general concept of volume variation partition of a porous body is presented, introducing the *breathing coefficient*, defined as the ratio of two volume variations. Considering a total volume of a porous body, composed of solid volume and “void” volume, this ratio can be used to evaluate the distribution of a volume variation into both others. A full description of its physical interpretation is detailed, together with an uncertainty analysis that specifies precautions about its use. As an example of application, a case study of 2D monosized disc packing swelling is developed. The analytical model reveals the presence of minimisation points of the *breathing coefficient* dependent on the initial granular organisation, showing possible ways to minimise the breathing of a granular material.

## How to better evaluate the volume repartition of a breathing porous body ?

<figure><img src="image_1.png" /></figure>

Breathing coefficient

$$ \Delta V_{\text{body}} = \Delta V_{\text{solid}} + \Delta V_{\text{void}} $$

$$ \bar{\mathcal{V}}_{bs} = \frac{\Delta V_{\text{body}}}{\Delta V_{\text{solid}}} $$

Not equivalent to
density or porosity at low
breathing amplitude

Application of the coefficient on a case-study:
perfect uniaxial swelling of monosized discs

<figure><img src="image_2.png" /></figure>

*Main conclusion:* Considering a given swelling of particles, the initial packing organisation of particles can minimise the relative swelling of the whole particle bed.

<footnote>*CC BY* Distributed under a **CC-BY 4.0 licence**.</footnote>

<footnote>†Email: [theo.boivin@email.fr](mailto:theo.boivin@email.fr)</footnote>

<footnote>‡Email: [olivier.gillia@cea.fr](mailto:olivier.gillia@cea.fr)</footnote>
# 1 Introduction

The behaviour of a granular media subjected to volume variation of its particles is nowadays considered to describe the “breathing” of materials [1–3]. This led us to consider the analysis of volume variation in a more general way, not necessarily for a granular media, but for a generic “porous body”. A “porous body” is defined here as a system composed of two phases: a “solid” phase and a “void” phase (filled with gas, liquid, or void). The term “breathing” is used here to designate a succession of swelling and shrinking of the porous body. When the need is to manage the breathing behaviour of such a body, a major interest appears concerning the way the volume variation of the solid phase is converted into variation of void volume or variation of the global volume of the porous body (or similarly the way void is converted to solid and global, or the way global is converted to void and solid). The common quantities used to measure the relative volume fractions are the porosity $\epsilon = V_{\text{void}}/V_{\text{body}}$ or the solid fraction $\chi = V_{\text{solid}}/V_{\text{body}}$ (also called “density”), where $V_{\text{void}}$, $V_{\text{solid}}$, and $V_{\text{body}}$ are respectively the volumes of the void phase, the solid phase, and the porous body. They are really useful to describe efficiently a particular state at a given instant. However, how to compare two values of porosity or solid fractions at two different instants in time? If the porous body volume $V_{\text{body}}$ does not change, there is no particular issue. However, if it changes too, the comparison of both instants becomes unclear.

By restraining the idea of a porous body to the case of a packed particle bed, the question of volume partition through porosity and solid fraction is well studied. In this sense, the circle packing community has widely addressed the static state of a circle packing [4–6]. A common methodology to study these circle packing questions is to use a computational “compressor” technique that incrementally reaches a dense state [7–10]. Still, the attention is mostly paid to the final state but not the intermediary behaviour. The proposed axis in this article is to extend the notion of porosity and solid fraction to their derivative equivalents: using volume variation instead of instantaneous volumes. As a source of inspiration, Gomadam & Weidner (2005) [11] introduced the “swelling coefficient” to assess the volume variation of a porous electrode volume and also its porosity change.

The aim of this paper is to reduce the “swelling coefficient” to its simplest form in order to generalise it as the “breathing coefficient”. Section 2 introduces its definition and different guidelines for interpretation. As an illustration, Section 3 develops an analytical case study of a perfect uniaxial swelling of monosized disc packing.

# 2 Introduction to the breathing coefficient

## 2.1 Definitions

The *breathing coefficient* is a proposal for a basic evaluation of breathing transmission of a porous body. It is based on the general partition of volume, divided into a variation of solid and void volumes, where “void” designates the absence of solid (filled with gas, liquid, or void). The fundamental partition is:
$$V_{\text{body}} = V_{\text{solid}} + V_{\text{void}} \qquad (1)$$

The partition of volume variations between two different states of the body directly comes:

$$\Delta V_{\text{body}} = \Delta V_{\text{solid}} + \Delta V_{\text{void}} \qquad (2)$$

The main question approached here is to evaluate the proportion of each contribution ($\Delta V_{\text{solid}}$ and $\Delta V_{\text{void}}$) to the global breathing of the body ($\Delta V_{\text{body}}$), in order to assess the balance of volume variation. To this aim, the proposed “breathing coefficient”, noted $\mathfrak{T}$ (Devanāgarī vowel pronounced [u]), is defined as:

$$\mathfrak{T} = \frac{\partial V_1}{\partial V_2} \qquad (3)$$

Where $V_1$ and $V_2$ can be $V_{\text{body}}$, $V_{\text{solid}}$ or $V_{\text{void}}$. Of course, in practice, this form is not appropriate when no analytical form is available for $V_1$. Thus, a simplified volume variations ratio can be used as a derivative equivalent:

$$\mathfrak{T} = \frac{\Delta V_1}{\Delta V_2} \qquad (4)$$

More generally, six definitions exist:

$$\begin{align}
\mathfrak{T}_{bs} &= \frac{\Delta V_{\text{body}}}{\Delta V_{\text{solid}}} , & \mathfrak{T}_{vs} &= \frac{\Delta V_{\text{void}}}{\Delta V_{\text{solid}}} \nonumber \\
\mathfrak{T}_{vb} &= \frac{\Delta V_{\text{void}}}{\Delta V_{\text{body}}} , & \mathfrak{T}_{sb} &= \frac{\Delta V_{\text{solid}}}{\Delta V_{\text{body}}} \tag{5} \\
\mathfrak{T}_{sv} &= \frac{\Delta V_{\text{solid}}}{\Delta V_{\text{void}}} , & \mathfrak{T}_{bv} &= \frac{\Delta V_{\text{body}}}{\Delta V_{\text{void}}} \nonumber
\end{align}$$

Of course, these definitions remain quite equivalent, and the choice of one among them depends on the context and the objective of the evaluation. The proposed inventory is intended to be exhaustive, in order to get a complete toolbox. For a quick conversion, Fig. 1 details most of the relationships between these different definitions. We can notice that, mathematically, the solid and void cases are interchangeable ($v$ can be $s$ and reciprocally), because they remain both terms of a unique sum.

To get a more visual representation of the volume variation partition, Fig. 2 displays a fast technique to represent the different possible breathing configurations via a vector graph. It must be underlined that only relative orientations matter: no negative orientation is represented. This representation sheds light on the existence of three different configurations of breathing which we have chosen to name: “opposite solid”, “oriented”, and “opposite void”. Furthermore, several limit cases of these configurations exist: “void breathing”, “balanced breathing”, “solid breathing”, and “internal transfer”. In fact, the “internal transfer” can be subdivided into two different cases, depending on the signs of $\Delta V_{\text{solid}}$ and $\Delta V_{\text{void}}$: “solid disappearance” ($\Delta V_{\text{solid}} < 0$ and $\Delta V_{\text{void}} > 0$) and “void disappearance” ($\Delta V_{\text{solid}} > 0$ and $\Delta V_{\text{void}} < 0$).
<figure><img src="image_3.png" /><figcaption>Figure 1: Relationships between the different definitions of the breathing coefficient.</figcaption></figure>

The breathing coefficient is theoretically a real number. Thus, it can get any value between $-\infty$ to $+\infty$. Its interpretation is mainly based on three major values: 0, 1, and $\pm\infty$. The values 0.5 and 2 are possibly interesting to introduce balanced breathing, but they are not generalisable to all breathing coefficient definitions. To help in the interpretation of this number, Table 1 associates the different limit cases to the different definitions and values.

## 2.2 Uncertainty analysis

As with any other ratio number, precaution should be taken concerning the uncertainty of the breathing coefficient. Indeed, the uncertainties due to computation and data dumping can generate noise, and the division can enhance this noise. To understand this issue, let us consider that the multiple mistakes due to computation, randomness generation, and data dumping create a range of uncertainty $\sigma \in \mathbb{R}_+$, with the same dimension as $\Delta V$. Let us also define $\gamma_1, \gamma_2 \in [0, 1]$, two variables that are impossible to predict due to uncertainty. Instead of Eq. (4), we obtain in reality:

$$ \mathfrak{F} = \frac{\Delta V_1 \pm \gamma_1 \sigma}{\Delta V_2 \pm \gamma_2 \sigma} \qquad (6) $$

If the variations of volumes are equal to $\lambda_1 \sigma$ and $\lambda_2 \sigma$ where $|\lambda_1|, |\lambda_2| \gg 1$, we have:

$$ \mathfrak{F} = \frac{\lambda_1 \sigma \pm \gamma_1 \sigma}{\lambda_2 \sigma \pm \gamma_2 \sigma} \approx \frac{\lambda_1 \sigma}{\lambda_2 \sigma} = \frac{\Delta V_1}{\Delta V_2} \qquad (7) $$
<figure><img src="image_4.png" /><figcaption>**Figure 2:** Different configurations and limit cases of volume variation partitions. These schemes must be read relatively; the reference magnitude of ΔV<sub>body</sub> is just for readability (the magnitude can be close to zero). Only the relative orientations matter (no negative orientation).</figcaption></figure>

Here, no particular issue occurs. The uncertainty remains negligible, and the original definition is respected. In addition, if ΔV₁ is now equal to μ₁σ where |μ₁| ≪ 1, it comes:

$$ \bar{\sigma} = \frac{\mu_1 \sigma \pm \gamma_1 \sigma}{\lambda_2 \sigma \pm \gamma_2 \sigma} \approx \frac{\mu_1 \pm \gamma_1}{\lambda_2} \rightarrow 0 \qquad (8) $$
<table><caption>**Table 1:** Abacus of breathing limit cases — InTr: Internal transfer, VBr: Void breathing, BalBr: Balanced breathing, SBr: Solid breathing.</caption><thead><tr><th></th><th></th><th colspan="5">Ξ</th></tr><tr><th>Definition</th><th></th><th>-∞</th><th>0</th><th>0.5</th><th>1</th><th>2</th><th>+∞</th></tr></thead><tbody><tr><td>bs</td><td>VBr</td><td>InTr</td><td>-</td><td>SBr</td><td>BalBr</td><td>VBr</td></tr><tr><td>vs</td><td>VBr</td><td>SBr</td><td>-</td><td>BalBr</td><td>-</td><td>VBr</td></tr><tr><td>vb</td><td>InTr</td><td>SBr</td><td>BalBr</td><td>VBr</td><td>-</td><td>InTr</td></tr><tr><td>sb</td><td>InTr</td><td>VBr</td><td>BalBr</td><td>SBr</td><td>-</td><td>InTr</td></tr><tr><td>sv</td><td>SBr</td><td>VBr</td><td>-</td><td>BalBr</td><td>-</td><td>SBr</td></tr><tr><td>bv</td><td>SBr</td><td>InTr</td><td>-</td><td>VBr</td><td>BalBr</td><td>SBr</td></tr></tbody></table>

In this case, there is still no particular issue. However, reciprocally, if $\Delta V_2$ is equal to $\mu_2\sigma$ where $|\mu_2| \ll 1$, it follows:

$$ \Xi = \frac{\lambda_1 \sigma \pm \gamma_1 \sigma}{\mu_2 \sigma \pm \gamma_2 \sigma} \approx \frac{\lambda_1}{\mu_2 \pm \gamma_2} \rightarrow \pm\infty \quad (9) $$

This time, an uncertainty appears in the final result, more precisely about the sign. The value goes to $\infty$, but the sign is highly sensitive to the unpredictable value $\gamma_2$. It means that when $\Delta V_2$ is of the order of magnitude of the uncertainty, it is impossible to make any conclusion about the real sign of $\Delta V_1$ relative to $\Delta V_2$ only by analysing the *breathing coefficient*. Moreover, it is especially problematic when both volume variations $\Delta V_1$ and $\Delta V_2$ have orders of magnitude of uncertainty, i.e. are respectively equal to $\mu_1\sigma$ and $\mu_2\sigma$:

$$ \Xi = \frac{\mu_1 \sigma \pm \gamma_1 \sigma}{\mu_2 \sigma \pm \gamma_2 \sigma} = \frac{\mu_1 \pm \gamma_1}{\mu_2 \pm \gamma_2} \quad (10) $$

In this case, this is an unsolvable limit, highly sensitive to the unpredictable values $\gamma_1$ and $\gamma_2$. Now, not only the sign is concerned by uncertainty, but also the value: the whole result is non-exploitable. As a consequence, a systematic routine to check the validity of the *breathing coefficient* should be applied when using it, in order to avoid any misinterpretation of its result. To check the value validity of $\Xi$ (whatever its sign), we must have, at least:

$$ |\Delta V_1| \gg \sigma \quad \text{or} \quad |\Delta V_2| \gg \sigma \quad (11) $$

To check the validity of both value and sign, we must have:

$$ |\Delta V_2| \gg \sigma \quad (12) $$
# 3 Case study of the breathing coefficient: perfect uniaxial swelling of monosized disc packing

## 3.1 Problem description

This section presents a particular application of the breathing coefficient through the analytical modelling of a perfect uniaxial swelling of a monosized disc packing, as schemed in Fig. 3a. In terms of terminology, because the solid/void phases consideration is at the core of the breathing coefficient, the expression “disc packing” is preferred over the common expression “circle packing”. However, it designates the same study area. The problem is considered in 2D, so the breathing coefficient is based on disc area instead of volume. For this case study, the definition of breathing coefficient is the “bs” one:

$$ \mathcal{F}_{bs} = \frac{\Delta A_{\text{body}}}{\Delta A_{\text{solid}}} \qquad (13) $$

The system is composed of a packing of monosized discs with initial radius $R_0$. The swelling is defined through a swelling factor $\xi > 1$ defined as:

$$ \xi = \frac{R}{R_0} \qquad (14) $$

Where R is the radius of uniformly swelled discs. The discs are perfectly packed with a periodic scheme by neglecting the border effect. The packing is defined through two parameters: its type and its angle $\alpha$. The type can be “square”, “30-hexagonal” or “60-hexagonal”, as schemed in Fig. 3b. The angle $\alpha$ is defined as the lowest positive angle between horizontal and the line joining centres of two discs in contact. Then, the square packing presents $\alpha = 0$ (i) and the 30-hexagonal packing presents $\alpha = \pi/6$ (ii). Looking at iii), the 60-hexagonal has, by definition, $\alpha = 0$, but it also represents the limit case where $\alpha \to \pi/3$. Of course, the initial packing does not necessarily begin at one of the particular packings depicted in Fig. 3b. Thus, we also define an angle $\alpha_0$ that describes the initial packing state when $\xi = 1$.

As introduced in Fig. 3a, the swelling of the packing is uniaxial along the vertical axis (y axis), which means that an edge-to-edge horizontal alignment of discs cannot swell laterally (along x axis). In this case, it necessarily leads to a granular reorganisation. This is for this reason that the 30-hexagonal and 60-hexagonal packings, which first seem equivalent, are in fact distinguished. The swelling consequently generates a transition between the three main packings (square, 30-hexagonal and 60-hexagonal). The square packing being non-compact, it only occurs as an initial packing. Then, the swelling will only be a succession of transitions between 30-hexagonal and 60-hexagonal packings, as they are the most compact packings when considering monosized discs [12]. To understand the different transitions, Fig. 4 displays different states during swelling of discs, starting with a square packing. Because the container constrains the packing to move vertically, the swelling of discs generates a reorganisation. One column out of two, i.e. coloured discs in b), moves up until reaching the first 30-hexagonal packing in c). At this stage, all discs are free to swell by moving up. When the swelling reaches the 60-hexagonal packing in e), new horizontal
<figure><img src="image_5.png" /><figcaption>**Figure 3:** Introduction schemes of case study — **a)** Scheme of the problem: a container with monosized packed discs. When the discs swell, the walls prevent the lateral movement of horizontal alignment going from edge to edge. The global swelling is then uniaxial. **b)** Three main packing types of case study.</figcaption></figure>

alignments appear, which requires another reorganisation. This reorganisation is similar to the square one, with the difference that one row over two moves laterally over a relative distance of one radius. It then leads to another 30-hexagonal packing, and the transition between 30-hexagonal and 60-hexagonal begins again. For this reason, the description of this swelling behaviour does not require a packing angle superior to $\pi/3$.

In a nutshell, we have $\alpha \in [0, \pi/3[$ and $\alpha_0 = \alpha(\xi = 1)$.
<figure><img src="image_6.png" /><figcaption>**Figure 4 (part 1):** Transitions between packing types — Solid line polygons correspond to the representative tiles for square packing in **a**) and **b**) and for 30-hexagonal packing in **c**) and **d**). Dashed polygon in **d**) corresponds to the representative tile of square packing after deformation due to packing transition. In **b**) and **d**), dashed lines in representative tiles reveal the subtiles (four equal parallelograms). Coloured discs designate the columns that relatively move up.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>(e) 60-hexagonal packing</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(f) Transition from 60 to 30-hexagonal packing</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(g) New 30-hexagonal packing</figcaption></figure>

**Figure 4 (part 2):** Transitions between packing types — Solid line polygons correspond to the representative tiles for 60-hexagonal packing. Dashed polygon in **e**) corresponds to the representative tile of 30-hexagonal packing after deformation due to packing transition. In **f**) and **g**), dashed lines in the representative tiles reveal the subtiles (two equal parallelograms and four equal parallelograms). The half-transparent duplicated tiles help in demonstrating the tessellation with the representative tile. Coloured discs designate the columns that relatively move up.
## 3.2 Analytical model development

The calculation of $\mathcal{E}_{bs}$ should be expressed only with variables $\xi$ and $\alpha_0$, in order to nondimension-ise the resolution (independent of the order of magnitude of $R$). Moreover, because the problem is considered periodic, the calculation of areas requires the definition of representative tiles that should respect two conditions:

* It constitutes a unique polygon sufficient to tessellate the whole disc packing, only by translation, and independently of the packing angle $\alpha$.

* The areas (tile and contained discs) during transition between packings are continuous.

The chosen representative tiles are the ones drawn in Fig. 4 with solid thick lines, a) for square, c) for 30-hexagonal and e) for 60-hexagonal. The tile for square packing in a) is a square that deforms at its middle. The tile for 30-hexagonal packing in c) is a parallelogram that keeps constant lengths (relative to disc radii) but whose angles are varying. Finally, the tile for 60-hexagonal packing in e) requires a more complex shape. Initially a parallelogram, it deforms differently depending on the row.

Firstly, these tiles respect the condition of tessellation only with translation. This is quite trivial for both first tiles, and f) and g) help in demonstrating the tessellation with the third one. Secondly, the advantage of this choice is that, whatever the value of $\alpha$, each of these representative tiles include the area of four discs. Whatever the angle $\alpha$, we then get:

$$A_{\text{solid}} = 4\pi R^2 \qquad (15)$$

To obtain the areas of the tiles, some geometrical analytics is required by observing the interme-diary schemes, in a), d) and f). The first transition tile (from square to 30-hexagonal packings) is decomposed into four parallelograms, whose symmetric areas are directly deductible from the angle $\alpha$ (we have $\beta = \pi/2 - \alpha$). The reasoning is equivalent for the second transition tile (from 30-hexagonal to 60-hexagonal packings), also decomposed into four parallelograms, where $\beta = 2\alpha$. The third transition tile (from 60-hexagonal to 30-hexagonal) is decomposed into four equilateral triangles and two parallelograms with varying angle $\beta = 2(\pi/3 - \alpha)$. Thanks to this analysis, we obtain the respective expressions of $A_{\text{body}}$ equal to the areas of transition tiles:

$$A_{\text{body}} = \begin{cases} 16R^2 \cos(\alpha) & , \text{if } \alpha \in [0, \frac{\pi}{6}] \quad (\text{square at } \alpha = 0) \\ 4R^2 \left[ \sqrt{3} + 2 \sin \left( \frac{2\pi}{3} - 2\alpha \right) \right] & , \text{if } \alpha \in [0, \frac{\pi}{6}] \quad (\text{60-hexagonal at } \alpha = 0) \\ 32R^2 \cos(\alpha) \sin(\alpha) & , \text{if } \alpha \in \left[ \frac{\pi}{6}, \frac{\pi}{3} \right] \end{cases} \qquad (16)$$

Here again, the condition of continuity is respected. Indeed, all limits equal $8R^2\sqrt{3}$, except for limit at 0 for the square expression, this limit being not a value of transition but only an initial stage never reached again during swelling.
The last step consists in defining the relationship between the disc radii change and the angle: $\alpha(\xi, \alpha_0)$. To do so, the swelling factor $\xi$ is subdivided as the product of three sub-swelling factors:

$$ \xi = \xi_{\pi/3} \xi_n \xi_r = \frac{R_{\pi/3}}{R_0} \times \frac{R_n}{R_{\pi/3}} \times \frac{R}{R_n} \qquad (17) $$

Where $R_{\pi/3}$ is the radius when $\pi/3$ is reached for the first time and $R_n$ the radius after $n-1$ full transitions (0 to $\pi/3$). In other words, $\xi_{\pi/3}$ is the swelling required to reach $\pi/3$ for the first time, $\xi_n$ the swelling factor required to make $n-1$ full transitions (0 to $\pi/3$) and $\xi_r$ the remaining swelling, insufficient to reach $\pi/3$. Called the “transition index”, $n$ is defined as the number of times the packing reaches the 60-hexagonal packing ($\alpha \to \pi/3$). We can notice that, starting from a horizontal alignment of discs ($\alpha = 0$, square or 60-hexagonal), the distance between both centres does not change (for each line, relatively, one disc out of two moves up, but not laterally). The only difference between square and 60-hexagonal packings is that, in the second case, one line out of two laterally moves during the transition to the 30-hexagonal packing. Then, the initial horizontal distance between centres $d_h$ remains constant until $\alpha$ reaches $\pi/3$:

$$ d_h = 2\xi_{\pi/3}R_0 \cos\left(\frac{\pi}{3}\right) = 2R_0 \cos(\alpha_0) \implies \xi_{\pi/3} = 2\cos(\alpha_0) \qquad (18) $$

Where, with $\alpha_0 = 0$, it comes $\xi_{\pi/3} = 2$. By consequence, the transition of $\alpha$ from 0 to $\pi/3$ (what-ever it starts from a square or a 60-hexagonal packing) requires the disc radius to be doubled. This conclusion implies that $\xi_n$ is of the form:

$$ \xi_n = 2^{n(\xi, \alpha_0)-1} \qquad (19) $$

Where $n$ depends on $\xi$ and can be determined with the Algorithm 1. The remaining swelling factor is directly defined by the final packing angle $\alpha$:

$$ \xi_r = \frac{1}{\cos(\alpha)} = \frac{\xi}{\xi_{\pi/3}\xi_n} \qquad (20) $$

It comes the final expression of packing angle $\alpha$:

$$ \alpha = \arccos\left(\frac{2^{n(\xi, \alpha_0)} \cos(\alpha_0)}{\xi}\right) \qquad (21) $$

This leads to the final form of the *breathing coefficient* for this case study. Because of the ratio, it can be simplified with nondimensionalised areas (independency from the radius value), by dividing by $4R_0^2$:

$$ \mathfrak{F}_{bs} = \frac{\widetilde{\Delta A}_{\text{body}}}{\widetilde{\Delta A}_{\text{solid}}} \qquad (22) $$

Where:

$$ \left\{ \begin{array}{l} \widetilde{\Delta A}_{\text{body}} = \tilde{A}_{\text{body}} - \tilde{A}_{\text{body},0} \\ \widetilde{\Delta A}_{\text{solid}} = \pi(\xi^2 - 1) \end{array} \right. \qquad (23) $$
<caption>**Algorithm 1: n COMPUTATION**</caption>

**Input:** $\xi$, $\alpha_0$
1 $n = 0$
2 $\xi' = \frac{\xi}{\cos(\alpha_0)}$ # Reset angle to zero
3 **while** $\xi' > 2$ **do**
4 $\lfloor n \leftarrow n + 1$, $\xi' \leftarrow \frac{\xi'}{2}$
**Output:** $n$

The expression of $\tilde{\Delta}A_{\text{body}}$ is based on Eq. (16), depending on the packing angle $\alpha$, through the Algorithm 2. Finally, an interesting common parameter is the solid fraction, directly defined based on dimensionless areas:

$$ \chi = \frac{\tilde{A}_{\text{solid}}}{\tilde{A}_{\text{body}}} \qquad (24) $$

## 3.3 Results discussion

As a particular case of the model, Fig. 5 displays the evolution of the breathing coefficient $\mathcal{T}_{bs}$, the solid fraction $\chi$ and the packing angle $\alpha$ as a function of the swelling factor $\xi$. It is more especially restrained to the case with a null initial packing angle $\alpha_0$, for both square and 60-hexagonal initial packings. An overview of the graph indicates an order of magnitude of $\mathcal{T}_{bs}$ between 1 and 1.5, which is close to a “solid breathing” (see Table 1 for interpretation). In other words, most of the swelling of the body is induced by the disc swelling, and the generation of void area is relatively low. However, a singular behaviour appears at the beginning of the 60-hexagonal initial packing case (dashed line in Fig. 5), where $\mathcal{T}_{bs}$ tends towards infinity for low values of swelling factor $\xi$. In this case, the swelling is mostly due to void area variation. This is coherent with the fact that the 60-hexagonal packing is the most compact packing for monosized discs, so at this state, the slightest variation of disc area necessarily leads to a high variation of void. On the opposite, the square initial packing displays a low breathing coefficient at a low amount of swelling, presenting a limit for $\xi \to 1$ equal to $2/\pi \approx 0.634$. At this stage, the swelling is at the transition between “internal transfer” and “solid breathing” (see Table 1 for interpretation): the area variation of discs is compensated by an opposite area variation of void. These particular behaviours changes when the swelling is enough to reach the 30-hexagonal packing ($\alpha = \pi/6$). For both square and 60-hexagonal packings, the breathing coefficient suddenly increases with swelling factor $\xi$, which is coherent, once again considering the compact characteristic of the 30-hexagonal packing. A first transition finishes when the swelling factor $\xi$ reaches 2. From this point, both square and 60-hexagonal initial packings tend to a similar behaviour, revealing breathing coefficient local minimisation points for every $\xi$ value of $2^n$ (at each 60-hexagonal packing) and $2^{n+1}\sqrt{3}/3$ (at each 30-hexagonal packing). In
<caption>**Algorithm 2: $\tilde{\Delta}A_{\text{BODY}}$ COMPUTATION**</caption>

**Input:** $\xi$, $\alpha_0$
1 &nbsp;&nbsp;$\alpha$ = Eq. (21)
2 &nbsp;&nbsp;**if** $\alpha < \frac{\pi}{6}$ **then**
3 &nbsp;&nbsp;&nbsp;&nbsp;**if** Square initial packing and $n = 0$ **then**
4 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body}} = 4\xi^2 \cos(\alpha)$
5 &nbsp;&nbsp;&nbsp;&nbsp;**else**
6 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body}} = \xi^2 [\sqrt{3} + 2 \sin(\frac{2\pi}{3} - 2\alpha)]$
7 &nbsp;&nbsp;**else**
8 &nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body}} = 8\xi^2 \cos(\alpha) \sin(\alpha)$
9 &nbsp;&nbsp;**if** $\alpha_0 < \frac{\pi}{6}$ **then**
10 &nbsp;&nbsp;&nbsp;&nbsp;**if** Square initial packing **then**
11 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body},0} = 4 \cos(\alpha_0)$
12 &nbsp;&nbsp;**if** 60-Hexagonal initial packing **then**
13 &nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body},0} = \sqrt{3} + 2 \sin(\frac{2\pi}{3} - 2\alpha_0)$
14 &nbsp;&nbsp;**else**
15 &nbsp;&nbsp;&nbsp;&nbsp;$\tilde{A}_{\text{body}} = 8 \cos(\alpha_0) \sin(\alpha_0)$
16 &nbsp;&nbsp; $\tilde{\Delta}A_{\text{body}} = \tilde{A}_{\text{body}} - \tilde{A}_{\text{body},0}$
**Output:** $\tilde{\Delta}A_{\text{body}}$

fact, these local minimisation points can be quite interesting for any process that uses granular breathing, where the estimation of volume variation can be important for container design. For example, by taking the 60-hexagonal initial packing case (dashed line in Fig. 5), for a swelling factor comprised between 2 and 4, the minimum value of $\mathfrak{T}_{bs}$ is around 1.103 (at $\xi = 4$) and the maximum is around 1.298 (at $\xi = 2.802$). The respective equivalent values by taking the definition “vs” of *breathing coefficient* are 0.103 and 0.298. In other words, at $\xi = 4$, the swelling of discs is higher than at $\xi = 2.802$, but a lower proportion of void area is generated (10.3% against 29.8%). Consequently, an optimisation step seems necessary for any process where, for example, the density of the granular material is important. More precisely, looking at the solid fraction curve $\chi$, the respective values at $\xi = 4$ and $\xi = 2.802$ are around 0.907 and 0.785. The case at $\xi = 4$ is the hexagonal configuration proved to be the densest for a monosized disc packing, with $\chi_{\text{max}} = \pi/\sqrt{12}$, [12].
<figure><img src="image_10.png" /><figcaption>**Figure 5:** Breathing coefficient $\mathcal{F}_{bs}$, solid fraction $\chi$ and packing angle $\alpha$ as a function of swelling factor $\xi$ and initial packing (square in solid line, 60-hexagonal in dashed line), with $\alpha_0 = 0$. Vertical dotted lines correspond to 30-hexagonal packing ($\alpha = \pi/6$).</figcaption></figure>

The initial angle $\alpha_0$ also has an important influence on the distribution of local minimisation points of *breathing coefficient*. In this sense, Fig. 6 displays the *breathing coefficient* $\mathcal{F}_{bs}$ and the packing angle $\alpha$ as a function of the swelling factor $\xi$, and this for several values of initial angle $\alpha_0$. Both square and 60-hexagonal initial packings are displayed for the *breathing coefficient*. Expectedly, the difference between square and 60-hexagonal initial packings only occurs for $\alpha_0 < \pi/6$. Indeed, above $\pi/6$, even in the square initial packing case, the packing is already hexagonal. As observed above, the difference is more pronounced for a swelling factor $\xi$ comprised between 1 and 2, where the square initial packing generates lower values of *breathing coefficient*, especially for $\alpha < \pi/6$. Generally speaking, for both square and 60-hexagonal initial packings, the increase of $\alpha_0$ implies a phase shift to the left, as drawn by the arrow on the $\alpha$ curve. Consequently, the formation of 30-hexagonal or 60-hexagonal packings does not occur at the same amount of swelling. More precisely, these points are more generally met at every $\xi$ value of $2^n \cos(\alpha_0)$ (at
<figure><img src="image_11.png" /><figcaption>**Figure 6:** Breathing coefficient $\mathcal{E}_{bs}$ and packing angle $\alpha$ as a function of swelling factor $\xi$ and initial angle $\alpha_0$. For breathing coefficient, top curve is 60-hexagonal initial packing, bottom curve is square initial packing. The horizontal arrow draws the phase shift due to variation of initial angle $\alpha_0$.</figcaption></figure>

each 60-hexagonal packing) and $2^{n+1} \cos(\alpha_0)\sqrt{3}/3$ (at each 30-hexagonal packing). This formulation includes the particular case addressed above, for $\alpha_0 = 0$. This observation is interesting for a practical application where the maximum swelling factor is imposed. In this case, a judicious choice of initial granular organisation (through $\alpha_0$) would imply a minimal *breathing coefficient* at the highest amount of swelling.

As we can observe, the model is not linear with swelling factor $\xi$, but logarithmic in base 2 (new $\alpha$ transition every $2^n$). The local minimisation points of the *breathing coefficient* of these transitions revealed to be at hexagonal packings ($\alpha = 0$ and $\alpha = \pi/6$). More precisely, the expressions of the *breathing coefficient* at these points are, respectively:
$$ 
\left\{
\begin{aligned}
\mathcal{T}_{bs}(\alpha = 0) &= \frac{2^{2n+1}\sqrt{3}\cos^2(\alpha_0) - \tilde{A}_{body,0}}{\pi(2^{2n}\cos^2(\alpha_0) - 1)} \\
\mathcal{T}_{bs}(\alpha = \frac{\pi}{6}) &= \frac{2^{2n+3}\sqrt{3}\cos^2(\alpha_0) - 3\tilde{A}_{body,0}}{\pi(2^{2n+2}\cos^2(\alpha_0) - 3)}
\end{aligned}
\right.
\qquad (25)
$$

From these expressions, a particular limit value emerges at high transition index n:

$$ \lim_{n \to \infty} \mathcal{T}_{bs}(\alpha = 0) = \lim_{n \to \infty} \mathcal{T}_{bs}\left(\alpha = \frac{\pi}{6}\right) = \frac{\sqrt{12}}{\pi} \approx 1.102657791 \qquad (26) $$

Considering the high amount of swelling of monosized discs supposed to reorganise in hexagonal packings, this value can be seen as the theoretical local minimisation value of the *breathing coefficient*. It must be noticed that significantly lower values of the *breathing coefficient* can be obtained but only at low values of swelling (n < 4). However, for higher swelling (n ≥ 4), the minimisation to $\sqrt{12}/\pi$ becomes particularly valid. In this optic, Fig. 7 shows the *breathing coefficient* as a function of packing angle $\alpha$ for both square and 60-hexagonal initial packings, with a null initial packing angle $\alpha_0$, and this for different transition indexes n. A difference appears between square and 60-hexagonal initial packings, where in the square case the limit to the minimisation value $\sqrt{12}/\pi$ is made with inferior values, whereas in the 60-hexagonal case, it is made with superior values. In Fig. 7, the curves for n > 4 (not displayed here) are quite merged with the curve at n = 4, which confirms the transition index limit stated above. After n = 4, the distinction between square and 60-hexagonal initial packings becomes negligible.

In fact, this limit value of $\sqrt{12}/\pi$ is not a surprise at all, because it corresponds to the reverse of the maximum theoretical solid fraction:

$$ \lim_{n \to \infty} \mathcal{T}_{bs}(\alpha = 0) = \frac{1}{\chi_{\max}} \qquad (27) $$

This observation reveals an almost equivalence between porosity, solid fraction, and *breathing coefficient*. When looking at Eq. (23), it is obvious that:

$$ \lim_{\xi \to \infty} \mathcal{T}_{bs} = \frac{1}{\chi} \qquad (28) $$

And also, by taking the “vs” definition, the equivalence is made with porosity:

$$ \lim_{\xi \to \infty} \mathcal{T}_{vs} = \frac{1}{\varepsilon} \qquad (29) $$

In other words, when the initial areas $A_0$ (or, more generally speaking, initial volumes $V_0$) become negligible compared to their respective variations, the *breathing coefficient* becomes useless, bringing nothing additional compared to the traditional porosity and solid fraction. A first impression of equivalency between porosity or solid fraction compared to *breathing coefficient* may come
<figure><img src="image_12.png" /><figcaption>**Figure 7:** Breathing coefficient $\mathfrak{T}_{bs}$ as a function of packing angle $\alpha$ and initial packing. The horizontal dashed lines represent the limit minimisation value of $\sqrt{12}/\pi$. The initial packing angle $\alpha_0$ is null.</figcaption></figure>

to mind. In fact, this is only true when $\Delta V_1 \gg V_{1,0}$ and $\Delta V_2 \gg V_{2,0}$, where $V_{1,0}$ and $V_{2,0}$ designate the initial volumes. In practice, it is quite expected that the volume variations, if they are not at the same order of magnitude, will be much lower than the absolute volumes (for example, thinking about a thermal dilation of a particle bed or a gas storage within a hybrid porous material). In this case, the way the volume variations between body, solid, and void are distributed remains hard to apprehend. It is for this reason that the *breathing coefficient* invites thinking in terms of volume variations instead of absolute volumes.

# 4 Conclusions

The aim of this paper was to introduce an analytical description of the *breathing coefficient*, noted $\mathfrak{T}$ and defined as $\Delta V_1/\Delta V_2$, where $\Delta V_1$ and $\Delta V_2$ are two volume variations. It is mostly useful for a porous body whose total volume changes due to a volume variation of solid and another of “void” (filled with gas, liquid, or void). Based on a basic volume variation partition, this parameter presents six different definitions, whose choice depends on the applied problem. The interest of this number is to evaluate the conversion of a volume variation into both others. Together with
the description of an interpretation guide, an uncertainty analysis of this parameter was made. Considering a given uncertainty $\sigma$, we suggest a systematic routine in order to check the validity of the *breathing coefficient*, with $|\Delta V_1| \gg \sigma$ or $|\Delta V_2| \gg \sigma$ to validate the value, and $|\Delta V_2| \gg \sigma$ to validate both value and sign.

As an example of application, a case study of perfect uniaxial swelling of monosized disc packing was carried out, using a 2D analytical model. In this case study, the discs swelling generates a global swelling of the discs bed, where periodic packing types are met. Focusing on the definition $\Xi_{bs} = \Delta V_{body}/\Delta V_{solid}$, the analysis revealed a periodic evolution with the swelling factor, giving birth to regular local minimisation points of the *breathing coefficient* (when the disc packing becomes hexagonal). This periodicity occurs every $2^n \cos(\alpha_0)$ and $2^{n+1} \cos(\alpha_0)\sqrt{3}/3$, by considering the value of the swelling factor, where $\alpha_0$ is the initial packing angle. The position of the local minimisation points of the *breathing coefficient* depends on $\alpha_0$. This observation constitutes an interesting lead for the design of porous material that would self-absorb the swelling of its solid phase. At high swelling factor ($\ge 2^4$), the limit of minimal local point for the *breathing coefficient* is found to be $\sqrt{12}/\pi$, which correspond to the reverse of the maximal solid fraction for a monosized disc packing.

## Nomenclature

### 4.1 Main symbols

$\Xi$ Breathing coefficient, –

$A$ Area, m$^3$

$d$ Distance, m

$n$ Transition index, –

$R$ Radius, m

$V$ Volume, m$^3$

$\alpha$ Packing angle, –

$\beta$ Intermediary angle, –

$\gamma$ Undefined variable $\gamma \in [0, 1]$, –

$\lambda$ Noticeable factor $|\lambda| \gg 1$, –

$\mu$ Negligible factor $|\mu| \ll 1$, –

$\xi$ Swelling factor, –

$\varepsilon$ Porosity, –

$\sigma$ Volume uncertainty, m$^3$

$\chi$ Solid fraction, –
## 4.2 Subscripts

<table><tr><td>0</td><td>Initial value (at ξ = 1)</td></tr><tr><td>1</td><td>Numerator term</td></tr><tr><td>2</td><td>Denominator term</td></tr><tr><td>body</td><td>Porous body</td></tr><tr><td>bs</td><td>Body over solid</td></tr><tr><td>bv</td><td>Body over void</td></tr><tr><td>h</td><td>Horizontal</td></tr><tr><td>max</td><td>Maximum value</td></tr><tr><td>n</td><td>Required to reach n − 1 full transitions</td></tr><tr><td>solid</td><td>Solid phase</td></tr><tr><td>void</td><td>“Void” phase (with gas, liquid, or void)</td></tr><tr><td>vb</td><td>Void over body</td></tr><tr><td>vs</td><td>Void over solid</td></tr><tr><td>r</td><td>Remaining swelling</td></tr><tr><td>sb</td><td>Solid over body</td></tr><tr><td>sv</td><td>Solid over void</td></tr><tr><td>π/3</td><td>Required to reach α = π/3 for the first time</td></tr></table>

## 4.3 Operations

<table><tr><td>Δy</td><td>Variation, [y]</td></tr><tr><td>y'</td><td>Secondary value, [y]</td></tr><tr><td>|y|</td><td>Absolute value, [y]</td></tr><tr><td>ŷ</td><td>Nondimensionalised equivalent, –</td></tr><tr><td>∂y₁/∂y₂</td><td>Partial derivative, [y₁/y₂]</td></tr></table>

## Acknowledgments

The authors acknowledge the collaboration of the team from Umicore company composed of Ducheng Ma, Michal Tulodziecki and Jacob Locke. They also gratefully thank Benoît Mathieu and Willy Porcher for their rich involvement in the thesis project.

## Credit authorship contribution statement

T.B. designed the figures, drafted and wrote the manuscript, O.G. supervised the work, T.B. and O.G. developed the analytical model, discussed the results and reviewed the manuscript.
# Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, T.B. occasionally used “DeepL” as a French-to-English translator for unique words or short expressions (up to five words). T.B. also used “LanguageTool” for grammar / spelling check. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

# Declaration of competing interest

The authors declare no conflicts of interest regarding this manuscript.

# Data availability

The data that support the findings of this study are openly available at the Gitlab repository [13].

# References

[1] Nahia Sassine. "Study of the Thermo-Mechanical Behavior of Granular Media and Interactions Medium-Tank". PhD thesis. Université Grenoble Alpes, Nov. 2018. URL: https://theses.hal.science/tel-02043455.

[2] O. Gillia. "Hydride Breathing and Its Consequence on Stresses Applied to Containers: A Review". In: *International Journal of Hydrogen Energy* (2021), pp. 35594-35640. DOI: 10.1016/j.ijhydene.2021.07.082.

[3] Théo Boivin et al. "Breathing of a Silicon-Based Anode: Mechanical Discrete Approach Using DEM". In: *Journal of The Electrochemical Society* 171.1 (2024), p. 010505. DOI: 10.1149/1945-7111/ad14cf.

[4] D. J. Cumberland and R. J. Crawford. *THE PACKING OF PARTICLES*. Elsevier Science Pub. Co. Inc., Amsterdam, NY, 1987. URL: https://openlibrary.org/books/OL15084426M/The_packaging_of_particles.

[5] Philip L. Bowers. "Introduction to Circle Packing: A Review". In: *Circle Packing: The Theory of Discrete Analytic Functions*. 2008, pp. 1-16. URL: https://api.semanticscholar.org/CorpusID:16173679.

[6] Mhand Hifi and Rym M'Hallah. "A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies". In: *Advances in Operations Research* 2009 (2009), p. 150624. DOI: 10.1155/2009/150624.

[7] Boris D. Lubachevsky and Ronald L. Graham. "Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles". In: *Discrete Mathematics* 309.8 (2009), pp. 1947-1962. DOI: 10.1016/j.disc.2008.03.017.

[8] E. Specht. "High Density Packings of Equal Circles in Rectangles with Variable Aspect Ratio". In: *Computers & Operations Research* 40.1 (2013), pp. 58-69. DOI: 10.1016/j.cor.2012.05.011.

[9] Mihály Csaba Markót. "Improved Interval Methods for Solving Circle Packing Problems in the Unit Square". In: *Journal of Global Optimization* 81.3 (2021), pp. 773-803. DOI: 10.1007/s10898-021-01086-z.

[10] Paolo Amore. "Circle Packing in Regular Polygons". In: *Physics of Fluids* 35.2 (2023), p. 027130. DOI: 10.1063/5.0140644.

[11] Parthasarathy M. Gomadam and John W. Weidner. "Modeling Volume Changes in Porous Electrodes". In: *Journal of The Electrochemical Society* 153.1 (2005), A179. DOI: 10.1149/1.2136087.
[12] Hai-Chau Chang and Lih-Chung Wang. *A Simple Proof of Thue's Theorem on Circle Packing*. Cornell University, 2010. DOI: 10.48550/arXiv.1009.4322.

[13] GitLab – Théo Boivin – Articles – Breathing Coefficient. Version 1.0.0. URL: https://gitlab.com/TheoBoivin/articles/Breathing-Coefficient.