# Assessment of the influence of model powder characteristics on their flowability using correlation matrices

Lina Cayla Arianer, Mikel Leturia, Anne-Charlotte Robisson, Carine Ablitzer, Elias Daouk, Khashayar Saleh

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Lina Cayla Arianer, Mikel Leturia, Anne-Charlotte Robisson, Carine Ablitzer, Elias Daouk, et al. Assessment of the influence of model powder characteristics on their flowability using correlation matrices. Powder Technology, 2025, 453, pp.120636. 10.1016/j.powtec.2025.120636. hal-04961412

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# Assessment of the influence of model powder characteristics on their flowability using correlation matrices

Lina Cayla Arianer<sup>a,b</sup>, Mikel Leturia<sup>b</sup>, Anne-Charlotte Robisson<sup>a,*</sup>, Carine Ablitzer<sup>a</sup>, Elias Daouk<sup>b</sup>, Khashayar Saleh<sup>b</sup>

## HIGHLIGHTS

* Characterization of physical characteristics (size, morphology, density).

* Flow properties in different flow regimes.

* Correlations between particle characteristics and flow properties on sixty powders.

* Extrapolation with integration of literature data.

* Multiple physical characteristics are needed to describe the behavior.

## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

**Keywords:**

Powder flowability

Model powders

Flowability correlations

Powder rheometer

Granular materials

Correlation matrices

## ABSTRACT

This study focuses on the use of correlation matrices as a tool to investigate the relationship between the physical characteristics of powders and their flowability. The correlation coefficients are calculated for a dataset containing various types of powders, allowing the study of the influence of particle size, morphology and density on flow properties. Flowability is estimated through various tests such as the shear test, density ratio, powder rheometer and rotating drum. The analysis revealed that correlations between flowability indices are not strictly associated with specific flow regimes (quasi-static and dense regimes). Instead, the physical mechanisms of interparticle cohesion and friction appear to play a determinant role in these correlations. Further analysis involving the division of the dataset into fine and coarse powders provided additional insights, underscoring the different mechanisms governing the flow behavior of cohesive and non-cohesive powders. In a second step, this research work also incorporated data from existing literature, confirming the general trends observed with the initial dataset. Future research should focus on developing more sophisticated models that can account for multiple input parameters simultaneously and consider non-linear relationships, for example, through semi-empirical models or machine learning techniques.

<footnote><sup>a</sup> CEA, DES, IRESNE, DEC, Cadarache, 13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote><sup>b</sup> Université de technologie de Compègne, ESCOM, TIMR, Centre de recherche Royallieu, CS 60319, 60203 Compègne cedex, France</footnote>

<footnote>* Corresponding author at: CEA, DES, IRESNE, DEC, Cadarache, 13108 Saint-Paul-Lez-Durance, France.</footnote>

<footnote>E-mail address: anne-charlotte.robisson@cea.fr (A.-C. Robisson).</footnote>
# 1. Introduction

The flow behavior of powders is a complex yet crucial phenomenon in various industries, including pharmaceuticals [1], food processing [2–4], powder metallurgy [5], civil engineering [6], and nuclear energy [7,8]. Predicting powder flow is essential, as powder rheological properties significantly impact various manufacturing stages, including storage [9–11], transfer [12,13], die filling [14,15], and final product quality [16]. Inadequate powder flowability can lead to various process-related issues, such as pipeline blockages and non-compliance with the final product specifications. This concern is particularly important in the nuclear industry, where each maintenance procedure poses potential risks to workers.

Powder flow constitutes a multifaceted phenomenon influenced by a multitude of parameters. While particle size is often considered as the main influencing parameter for powder flow properties [6,17], other factors such as particle shape, density [3], and surface roughness [18,19] also play significant roles. Furthermore, certain parameters significantly influence interparticle forces, leading to cohesive flow behavior. Powders are typically classified as cohesive when their particle diameter is below 100 µm [20]. For these fine powders, van der Waals forces are considered the predominant mechanism responsible for cohesion, playing a crucial role in determining the bulk behavior and flow properties of cohesive powders, often outweighing the effects of particle weight [21].

Therefore, it is important to accurately predict how these parameters influence powder behavior. Several approaches can be used to build models to predict powder flow:

- Empirical approach [7]: an empirical model is based on observed data without considering any underlying theory or principles. It relies entirely on experimental evidence or observations to establish relationships between variables;

- Semi-empirical approach [1,21]: a semi-empirical model combines both experimental data and theoretical principle (like Rumpf model [22]) to develop a better knowledge of the studied system.

However, the extrapolation of existing models to various types of powders can be inaccurate.

Correlation analysis is an empirical approach that can be employed to evaluate the linear relationships among all the system variables. For powders, this tool can be employed to establish a link between particle characteristics (size, morphology, density) and their flowability (during testing or directly in the industrial process). However, the literature data often uses small databases on similar samples [23], focuses on relationships between certain parameters chosen for their study [24–26], or for the process used [27,28].

The aim of this study is to provide a better knowledge of the link between the physical characteristics of powders and their flow properties. For this purpose, around sixty inorganic powders with varying size, morphology and density characteristics were studied. Their flowability was evaluated using a large number of tests commonly employed for assessing flow properties, thus obtaining several rheological indicators. Using correlation matrices, the first aim was to identify the most influential physical parameters (size, morphology, density) for each rheological indicator. In a second step, data from existing literature was integrated to determine whether, despite the different testing procedures, it was still possible to deduce trends from the correlation matrices. The aim was to confirm or refute the conclusions drawn from our dataset by including several powders of different types. The general trends obtained from these correlation matrices could be utilized in future work to facilitate the development of more specific correlations, taking into account multiple input parameters simultaneously. This work also aims to provide a more comprehensive understanding of the relationships between rheological indicators, physical characteristics and flow regimes, highlighting the interplay between friction and cohesion in powder behavior.

<table><caption>**Table 1**
Powders used in this study.</caption><thead><tr><th>Material</th><th>Morphology</th><th>Mean diameter (d<sub>3,2</sub>, µm)</th><th>True density (g.cm<sup>-3</sup>)</th><th>Provider</th><th>Notation</th></tr></thead><tbody><tr><td rowspan="3">Glass</td><td>Spherical</td><td>5; 35; 50;<br>85; 140;<br>260; 350;<br>480; 940</td><td>2.5</td><td>Silibeads</td><td rowspan="2">G</td></tr><tr><td>Non-spherical</td><td>75; 80; 110;<br>260; 400</td><td>2.5</td><td>Abralis</td></tr><tr><td>Spherical<br>Non-spherical</td><td>280<br>150; 280</td><td>7.7</td><td>Winoa</td><td>S</td></tr><tr><td>Zirconia</td><td>Spherical</td><td>100; 165;<br>330; 1020;<br>1430</td><td>6</td><td>Silibeads</td><td>Z</td></tr><tr><td rowspan="3">Alumina</td><td>Spherical</td><td>1000; 1200<br>15; 20; 35;</td><td>3.9</td><td>Silibeads</td><td rowspan="3">A</td></tr><tr><td>Non-spherical</td><td>50; 70; 150</td><td>4</td><td>Lamplan</td></tr><tr><td>Non-spherical</td><td>4; 5; 40; 70;<br>110</td><td>3.5–4</td><td>Baikowski</td></tr><tr><td>Ceramic<br>(70 % ZrO<sub>2</sub><br>/ 30 %<br>Al<sub>2</sub>O<sub>3</sub>)</td><td>Spherical</td><td>45; 65; 105</td><td>3.8</td><td>Abralis</td><td>C</td></tr></tbody></table>

# 2. Materials and methods

The studied powders and their main characteristics are summarized in Table 1.

An overview of the different types of morphology observed for these powders is presented in SEM images shown in Fig. 1.

Some mixtures of these powders were prepared and characterized, to enlarge our database. The mixture compositions are described in Table 2. Each of these six mixtures was prepared by blending the two components using a Turbula mixer (T2C model, Wab) filled to 50 %, for 15 min at 32 rpm, to ensure good homogeneity. Samples were collected using a sample divider (Retsch PT100) to ensure representativity.

## 2.1. Physical characteristics evaluation

The particle size distribution of all materials was determined using laser diffraction techniques, specifically Mastersizer 2000 and 3000 (Malvern Instruments). This method gives a volume distribution and measures the volume-equivalent sphere diameter (diameter of a sphere having the same volume as the particle). From these measurements, it is possible to identify different characteristic diameters ($d_{10}$, $d_{50}$, $d_{90}$, $d_{3,2}$, $d_{4,3}$) and a dispersion index called “Span”:

$$Span = \frac{d_{90} - d_{10}}{d_{50}} \qquad (1)$$

The parameters $d_{10}$, $d_{50}$ and $d_{90}$ denote the diameters below which 10 %, 50 % or 90 % in volume of particles are found, respectively. The two parameters $d_{3,2}$ and $d_{4,3}$ are mean diameters. The parameter $d_{3,2}$ corresponds to the average diameter weighted by particle surface area. It is therefore more sensitive to fine particles. The diameter $d_{4,3}$, more rarely used, is more sensitive to larger particles, and corresponds to the average diameter weighted by particle volume.

Particle morphology was assessed using a Morphologi G3 (Malvern Instruments), and true density was measured through helium pycnometry using an Ultrapyc 5000 instrument by Anton Paar. These measurements allow different morphological parameters to be determined,
<figure><img src="image_5.png" /><figcaption>Fig. 1. SEM pictures of the studied powders, such as glass beads (a), metal beads (b), zirconia beads (c), glass fragments (d), metal fragments (e), alumina fragments (f), alumina agglomerates (g), alumina particles (h,i).</figcaption></figure>

<table><caption>**Table 2**
Presentation of the prepared mixtures.</caption><thead><tr><th>Mixture</th><th>Component 1 (d<sub>3,2</sub> in µm)</th><th>Component 2 (d<sub>3,2</sub> in µm)</th><th>Composition (% of powder bed volume)</th></tr></thead><tbody><tr><td>1</td><td>G - Spherical (260)</td><td>G - Non-spherical (250)</td><td>20/80<br>40/60<br>60/40<br>80/20</td></tr><tr><td>2</td><td>G - Spherical (50)</td><td>G - Spherical (5)</td><td>20/80<br>40/60<br>60/40<br>80/20</td></tr><tr><td>3</td><td>A - Non-spherical (35)</td><td>G - Non-spherical (2)</td><td>20/80<br>40/60<br>60/40<br>80/20</td></tr><tr><td>4</td><td>G - Non-spherical (10)</td><td>G - Spherical (5)</td><td>20/80<br>40/60<br>60/40<br>80/20</td></tr><tr><td>5</td><td>G - Spherical (85)</td><td>G - Spherical (50)</td><td>20/80<br>40/60<br>60/40<br>80/20<br>20/80</td></tr><tr><td>6</td><td>A - Non-spherical (150)</td><td>A - Non-spherical (35)</td><td>40/60<br>60/40<br>80/20</td></tr></tbody></table>

as detailed in Table 3.

From the distribution of the shape parameters, and similarly to granulometric parameters, several indices can be defined [29]:

* Circularity: $c_{10}, c_{50}, c_{90}, c_{mean}$

* Convexity: $cx_{10}, cx_{50}, cx_{90}, cx_{mean}$

* Elongation: $e_{10}, e_{50}, e_{90}, e_{mean}$

The mean parameters ($c_{mean}, cx_{mean}, e_{mean}$) were calculated using the arithmetic mean.

For each of the mixtures described in Table 2, the size and morphological distributions were determined based on the particle size distributions (volume-based) and morphological distributions (number-based) of the mixture components, using eqs. (2) and (3).

The volume-based fraction of particles of size $d_{p_j}$ in the binary mixture, noted $F_{v_m}(d_{p_j})$, was calculated based on the mass fraction of each component in the mixture (noted $x_{m_i}$):

$$F_{v_m}(d_{p_j}) = \left( \frac{\frac{x_{m_1}}{\rho_{p_1}}}{\frac{x_{m_1}}{\rho_{p_1}} + \frac{x_{m_2}}{\rho_{p_2}}} \right) F_{v_1}(d_{p_j}) + \left( \frac{\frac{x_{m_2}}{\rho_{p_2}}}{\frac{x_{m_1}}{\rho_{p_1}} + \frac{x_{m_2}}{\rho_{p_2}}} \right) F_{v_2}(d_{p_j}) \quad (2)$$

with $F_{v_i}$ the volume-based particle size distribution of component i, and $\rho_{p_i}$ the particle density of component i.

The number-based fraction of particles with property $p_k$ in the binary mixture, noted $F_{N_m}(p_k)$, was calculated as follows:
<table><caption>**Table 3**

Definition of the morphological parameters.</caption><thead><tr><th>Parameter</th><th>Definition</th><th></th></tr></thead><tbody><tr><td>Circularity (c)</td><td>c = 2√π × A<sub>p</sub>/P<sub>p</sub><br>where A<sub>p</sub> is the projected area of the particle and P<sub>p</sub> its actual perimeter</td><td></td></tr><tr><td>Convexity (cx)</td><td>cx = P<sub>c</sub>/P<sub>p</sub><br>where P<sub>p</sub> is the actual perimeter of the particle, while the convex perimeter P<sub>c</sub> corresponds to the perimeter of the smallest convex domain that is capable of entirely containing the original particle (this can be visualized as the domain that is defined by an elastic band surrounding the particle)</td><td></td></tr><tr><td>Elongation (e)</td><td>e = 1 - w/l<br>where l and w are the length and width of the particle, respectively. The length l is the longest projection of lines between any two points of the particle perimeter onto its major axis (line passing through the center of mass of the particle at an orientation corresponding to the minimum rotational energy). The width w is the longest projection of lines between any two points of the particle perimeter onto its minor axis (line passing through the center of mass of the particle and perpendicular to the major axis)</td><td></td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

Quasi-static

<figure><img src="image_9.png" /></figure>

Dense

<figure><img src="image_10.png" /></figure>

Collisional

<figure><img src="image_11.png" /><figcaption>**Fig. 2.** Different powder flow regimes.</figcaption></figure>

$$F_{N_m}(p_k) = \left( \frac{\sum_{j} N_{1j}}{\sum_{j} N_{1j} + \sum_{j} N_{2j}} \right) F_{N_1}(p_k) + \left( \frac{\sum_{j} N_{2j}}{\sum_{j} N_{1j} + \sum_{j} N_{2j}} \right) F_{N_2}(p_k) \quad (3)$$

with $N_{ij} = \frac{F_{vi}(d_{pj})}{\frac{1}{6} \frac{d_{pj}}{p_{pi}} 3} \frac{x_{mi}}{p_{pi}}$ the number of particles of component i and of diameter $d_{pj}$, $F_{N_i}(p_k)$ the number-based fraction of particles of property $p_k$ of component i, $F_{vi}(d_{pj})$ the volume-based fraction of particles of component i and size $d_{pj}$. It should be noted that the property $p_k$ can be any morphological property, such as circularity, elongation or convexity.

## 2.2. Flow properties evaluation

Various tests can be conducted to assess the flowability of powders under different configurations and stresses. Consequently, different flow regimes can be described [30]:

- Quasi-static regime: the stresses are mainly transferred by enduring frictional contacts between particles, and the powder exhibits a solid-like behavior.

<table><caption>**Table 4**

Summary of conducted flowability tests classified by flow regime.</caption><thead><tr><th>Quasi-static regime</th><th>Dense regime</th></tr></thead><tbody><tr><td><strong>Shear cell:</strong> ([31]; [32]; [19,33]; [34])</td><td>Rheometer - Variable Flow Rate: ([35]; [36]; [37]; [38])</td></tr><tr><td><strong>Density ratio:</strong> ([39]; [40]; [41])</td><td><strong>Rotating drum:</strong> ([42]; [43]; [44])</td></tr></tbody></table>

<table><caption>**Table 5**

Flowability indices resulting from the shear cell characterization [19].</caption><thead><tr><th>Flowability indices</th><th>Definitions</th></tr></thead><tbody><tr><td>Cohesion c</td><td>ordinate at the origin of the yield locus</td></tr><tr><td>Unconfined Yield Stress UYS</td><td>stress at bed failure occurs</td></tr><tr><td>Major Principal Stress MPS</td><td>maximum stress to which the divided solid was subjected during the test</td></tr><tr><td>Angle of Internal Friction AIF</td><td>corresponds to the angle of the yield locus</td></tr><tr><td>Flow factor ff<sub>c</sub></td><td>UYS<br>MPS</td></tr></tbody></table>

- Collisional regime: the stresses are mainly transferred by quasi-instantaneous collisions between particles, and the powder exhibits a gas-like behavior.

- Dense regime: it corresponds to an intermediate regime between the quasi-static and collisional regimes, where the flow of particles is driven by a combination of frictional contacts and collisions.

The different flow regimes are schematically represented in Fig. 2.

The flowability tests used in the present study are summarized in Table 4 and a description of each test is provided in the following sections. In order to accommodate specific experimental constraints, the number of samples subjected to analysis in this study was adjusted for each of these tests.

### 2.2.1. Shear cell

The flowability was evaluated for 52 powders using an 85 mL shear cell available on the Freeman Technology FT4 rheometer. A pre-consolidation step is necessary to guarantee test reproducibility. This stage involves applying a selected normal stress ($\sigma_{pre}$ equal to 9 kPa in this study), then shearing the powder or granular medium until the critical consolidation state is reached. Then, during the shearing step, a normal stress $\sigma_{sh} < \sigma_{pre}$ is applied. An increasing shear stress $\tau$ is applied until the rupture of the powder bed at a shear stress $\tau = \tau_{sh}$. These two steps are repeated for several values of the normal stress $\sigma_{sh}$. For each value of $\sigma_{sh}$, the pre-consolidation step ($\sigma_{pre}$) is reproduced beforehand. Several couples ($\sigma_{sh}, \tau_{sh}$) are thus obtained, enabling the yield locus to be determined.

Once the yield locus curve has been established, the Mohr circles can be drawn from it. These circles are both tangent to the yield locus, with the larger circle passing through the pre-consolidation point and the smaller circle passing through the origin. Different flowability indices can be obtained through the analysis of Mohr circles, as detailed in Table 5.

The maximum normal stress $\sigma_{sh}$ is frequently set at 80 % of the pre-consolidation stress. The minimum normal stress $\sigma_{sh}$ is adapted

<table><caption>**Table 6**

Minimum normal stress according to expected flowability [19].</caption><thead><tr><th>Expected flowability</th><th>Minimum normal stress</th></tr></thead><tbody><tr><td>Free flow</td><td>10 to 20 % of σ<sub>pre</sub></td></tr><tr><td>Easy to slightly cohesive flow</td><td>20 to 30 % of σ<sub>pre</sub></td></tr><tr><td>Cohesive flow</td><td>30 to 40 % of σ<sub>pre</sub></td></tr><tr><td>Very cohesive flow</td><td>Superior to 40 % of σ<sub>pre</sub></td></tr></tbody></table>
<caption>**Table 7**</caption>

Flowability indices resulting from the Variable Flow Rate characterization [45].

<table><thead><tr><th>Flowability indices</th><th>Definitions<sup>*</sup></th></tr></thead><tbody><tr><td>SE</td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">E<sub>test 6</sub> (↑) + E<sub>test 7</sub>(↑)</div><div style="padding-top: 1px;">2 × Powder mass (g)</div></div></td></tr><tr><td>SI</td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">E<sub>test 7</sub> (↓)</div><div style="padding-top: 1px;">E<sub>test 1</sub>(↓)</div></div></td></tr><tr><td>BFE</td><td>E<sub>test 7</sub>(↓)</td></tr><tr><td>FRI</td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">E<sub>test 11</sub>(↓)</div><div style="padding-top: 1px;">E<sub>test 8</sub>(↓)</div></div></td></tr><tr><td>CBD</td><td>Conditioned Bulk Density</td></tr></tbody></table>

<caption>**Table 8**</caption>

Flowability indices resulting from the rotating drum characterization [43].

<table><thead><tr><th>Flowability indices</th><th>Definitions</th><th>Number of characterized samples</th></tr></thead><tbody><tr><td>AOR</td><td>angle formed by the surface of a powder bed with the horizontal plane</td><td>58</td></tr><tr><td>CI</td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">1/D &sum;<sub>x</sub> &sigma;(x)</div><div style="padding-top: 1px;">&sigma;(x)</div></div></td><td>60</td></tr><tr><td>1st_Av</td><td>Angle of the first avalanche</td><td>53</td></tr></tbody></table>

according to the flowability of the powder, in a similar way to the characterization protocol with a Schulze shear cell [19]. Indeed, the minimum stress value $\sigma_{sh}$ is chosen such that it lies to the right of the point where the yield locus is tangent to the small Mohr circle. The criteria for selecting the minimum normal stress $\sigma_{sh}$ are detailed in Table 6.

### 2.2.2. Density ratio

The packing test compares the densities before and after packing of a powder. Density ratio measurements give the Carr Index IC, defined in eq. (4) (Carr Jr. 1965). Determinations on 52 powders were conducted using an automated machine (Densitap, Granuloshop) equipped with a 100 mL cell.

$$IC = 100 \left( 1 - \frac{\rho_0}{\rho_N} \right) \qquad (4)$$

with $\rho_0$ the aerated density and $\rho_N$ the tapped density.

This test was carried out using a Granuloshop Densitap volumeter. The aerated density $\rho_0$ is first measured with a graduated cylinder filled with 100 mL of sample, using a funnel to avoid compacting the sample. The mass before tapping is recorded. The specimen is then subjected to 2400 shocks to compact the sample. The tapped volume is then

measured to determine the tapped density $\rho_N$.

### 2.2.3. Variable flow rate (VFR)

The Freeman Technology powder rheometer (FT4) can be used to characterize different flow regimes, including the dense regime. The dynamic test consists in traversing a bed of powder with a blade following a helical path at controlled speed. It is composed of 11 elementary tests. The first 7 elementary tests are performed at constant blade tip speed, to characterize the stability of the powder or granular medium (Stability). The last 4 elementary tests are performed at decreasing speed (Variable Flow Rate, VFR), to characterize the sensitivity of the powder to flow rate. In the present study, this test was performed with a 25 mL cell on 61 samples. It yields a range of flowability indices, presented in Table 7.

### 2.2.4. Rotating drum

A rotating drum (Granudrum, provided by Granutools) was used to characterize the powder flow behavior by rotation of a half-filled cylinder [46]. Consequently, the 110 mL cell was first filled to 40–60 % of its volume with the sample. As the rotating drum test is based on image analysis, parameters for interface detection (exposure, gain) needed to be set in the software. When analyzing cohesive powders, particles may stick to the drum walls, impeding interface detection. To solve this issue, it is then possible to exclude part of the drum to analyze only the surface in the center. The interface between the air and the powder was evaluated at different predefined rotation speeds.

From the measurements, the dynamic Angle of Repose (AOR), Cohesive Index (CI), and the first avalanche angle (1st_Av) can be evaluated (definitions provided in Table 8). To determine the Angle of Repose (AOR), the bed surface is determined for each image. The angle is then calculated at the center of the interface, and averaged over all images. It should be noted that the angle is calculated only on a circle with a diameter corresponding to one-fifth of the diameter of the drum. The cohesive index (CI) is calculated from the fluctuations of the interface. Then, the angle of first avalanche is determined by setting the drum in motion at low speed until the first avalanche. These flowability indices were determined on a variable number of samples due to the presence of agglomeration phenomena, which made some results difficult to interpret.

Nearly all the samples presented in section 2.1 have been characterized with these techniques. Only the powders with a mean diameter higher than 300 µm could not be characterized with a shear cell (particles too large for the shear cell). Some difficulty also appeared to measure the first avalanche angle for non-cohesives powders. A summary of all the performed characterizations is given in Table 9.

In this paper, the expression “Flowability index” (or “Flowability indices”) is used to refer to the quantities resulting from the rheological tests described in the present section.

<caption>**Table 9**</caption>

Table summarizing the flowability indices and the associated flow regimes.

<table><thead><tr><th>Flow regime</th><th colspan="2">Quasi-static</th><th colspan="2">Dense</th></tr><tr><th>Test</th><th>Shear cell</th><th>Density ratio</th><th>VFR</th><th>Rotating drum</th></tr></thead><tbody><tr><th scope="row">Schematic</th><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">&tau;<sub>sh</sub></div><div style="padding-top: 1px;">&sigma;<sub>sh</sub></div></div></td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">&sigma;<sub>sh</sub></div><div style="padding-top: 1px;">&sigma;<sub>sh</sub></div></div></td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">&sigma;<sub>sh</sub></div><div style="padding-top: 1px;">&sigma;<sub>sh</sub></div></div></td><td><div style="text-align: center; line-height: 1.2em;"><div style="border-bottom: 1px solid black; padding-bottom: 1px;">&sigma;<sub>sh</sub></div><div style="padding-top: 1px;">&sigma;<sub>sh</sub></div></div></td></tr><tr><th scope="row">Flowability indices</th><td>c, UYS, MPS, AIF, ff<sub>c</sub></td><td>IC</td><td>SE, SI, BFE, FRI, CBD</td><td>AOR, CI, 1st_Av</td></tr><tr><th scope="row">Number of samples</th><td>52</td><td>52</td><td>61</td><td>~60</td></tr></tbody></table>

<footnote>* Upward (↑) or downward (↓) blade displacement.</footnote>
<figure><img src="image_12.png" /><figcaption>Fig. 3. Simplified linear correlation matrix between input parameters (particle physical properties).</figcaption></figure>

## 2.3. Correlation matrix

Correlation matrices can be a valuable tool for detecting relationships among parameters. They can be useful as a first approach in various statistical studies, such as PCA (Principal Component Analysis) or Machine Learning techniques. The correlation coefficients are calculated between the particle physical characteristics (input parameters) and the flowability indices (output parameters) obtained from the previously described rheological tests (section 2.2).

Correlation is indicated by a coefficient, with +1 or -1 representing a strong correlation (positive or negative), while 0 indicates no correlation. The Pearson correlation coefficient r between two parameters x and y is calculated as [47]:

$$r = \frac{\text{cov}(x, y)}{\sigma_x \sigma_y} = \frac{\frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})^2} \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i - \bar{y})^2}} = \frac{\sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{N} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{N} (y_i - \bar{y})^2}} \quad (5)$$

with $\bar{x}$ and $\bar{y}$ the arithmetic means of the two parameters (x and y), $\sigma_x$ and $\sigma_y$ their standard deviations, cov(x,y) the covariance between x and y, and N the number of studied samples.

In this study, a correlation will be considered significant when the correlation coefficient r is greater than 0.5 (r > 0.5). Several correlation matrices will be investigated to achieve a range of objectives. The main analyses will include:

* **Linear correlation matrix between physical characteristics:** to identify linear relationships within the input data, offering insights into how inherent properties of powders (e.g., size, shape, and density) interact;

* **Linear correlation matrix between flowability indices:** to detect linear relationships between output data, highlighting connections between different measurements of flow properties;

* **Linear correlation matrix between physical characteristics and flowability indices:** to explore linear correlations between the input physical characteristics and the output flowability indices, helping to identify general trends and determine which physical characteristics most strongly influence the flow behavior of the powders;

* **Logarithmic correlation matrix between physical characteristics and flowability indices:** to examine power-law relationships between the input physical characteristics and the output flowability indices. It could help to better capture complex interactions that may not follow simple linear trends;

* **Separated correlation matrices for fine powders (< 75 µm) and granular media (> 75 µm):** to provide the ability to differentiate between cohesive powders and coarser granular media, reflecting the unique behavior of these two categories in terms of flowability;

* **Correlation matrix using bibliographic data:** combine experimental data with additional data from the literature to extrapolate the models and evaluate their robustness. This approach ensures that the primary trends observed in the experimental results are maintained, despite the different testing procedures, supporting the broader applicability of the findings.

By conducting these analyses, the article aims to provide a detailed framework for understanding how various factors influence bulk powder behavior, paving the way for better predictive modeling.
<figure><img src="image_13.png" /><figcaption>Fig. 4. Linear relationship between mean elongation and mean circularity for the studied powders.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 5. Linear correlation between flowability indices.</figcaption></figure>

### 3. Results and discussion

#### 3.1. Correlation between particle physical characteristics (input parameters)

Before studying the correlation between particle physical characteristics (input) and flowability indices (output), we constructed the correlation matrix relative to particle physical characteristics. This matrix enables the verification of linear correlations between input

parameters, to identify potential biases among different parameters (correlation between several input parameters for example). The correlation matrix between input parameters is illustrated in Fig. 3 (only the mean morphological parameters are presented for better visualization).

Results show that a strong correlation exists between all the parameters related to the particle size distribution ($d_{10}$, $d_{50}$, $d_{90}$, $d_{3.2}$, $d_{4.3}$). Indeed, the studied powders are characterized by monomodal distributions with a relatively low Span (i.e., low dispersion). Consequently,
<figure><img src="image_15.png" /><figcaption>Fig. 6. Linear correlation matrix between particle physical characteristics and flowability indices.</figcaption></figure>

these characteristic diameters exhibit a consistent correlation, indicating that they vary in a coordinated manner when comparing different powders (e.g., if one powder has a higher $d_{10}$ than another, it will also have higher values of $d_{50}$, $d_{90}$, $d_{3,2}$, $d_{4,3}$).

Additionally, a relatively strong correlation can be observed between the morphological parameters ($c_{mean}$, $c_{xmean}$ and $e_{mean}$). For example, as presented in Fig. 4, a strong relationship between circularity ($c_{mean}$) and elongation ($e_{mean}$) is visible. More precisely, from our data obtained on 62 different powders, the following equation is obtained:

$$e_{mean} = -2.34 c_{mean} + 2.39 \quad (6)$$

This relation comes from the proximity in the definition of these two morphological values. It is also linked to the morphology of the chosen powders. In fact, our non-spherical powders are more elongated. This relation is only valid for the chosen powders. The extrapolation to other types of powders needs to be investigated.

## 3.2. Correlation between flowability indices (output parameters)

As for the correlation between input parameters, the correlation between flowability indices was also examined. The outputs correlation matrix is presented in Fig. 5 to determine whether there is any relationship between the flowability indices, obtained through the rheological tests described in section 2.2.

The analysis of this correlation matrix reveals several significant findings. First, it can be observed that correlations are not entirely associated with the powder flow regime (quasi-static or dense). Indeed, flowability indices derived from the same rheological test (i.e., representing the same flow regime), are not necessarily correlated. For instance, in the quasi-static regime, the cohesion strength (c), Unconfined Yield Stress (UYS) and flow factor (ffc), obtained from the shear

test, show no significant correlation with the Angle of Internal Friction (AIF) and Major Principal Stress (MPS) from the same test. Similarly, in the dense regime, the Flow Rate Index (FRI) obtained from the dynamic test (with the FT4 powder rheometer) does not correlate with the Basic Flow Energy (BFE) and Specific Energy (SE) from the same test. Conversely, some indicators from tests corresponding to different flow regimes exhibit significant correlation levels. For example, the cohesion strength (c) from the shear test (quasi-static regime) clearly correlates with the Flow Rate Index (FRI) from the powder rheometer (dense regime), as well as the Cohesive Index (CI) from the rotating drum (dense regime).

It is noteworthy that some results of the present study align well with existing literature (despite using different types of powders). Particularly, the relationship between the Carr Index (IC) and the Flow Rate Index (FRI) has been previously observed in the literature [48]. In the same way, Marchetti and Hulme-Smith [23] found correlations between the cohesion strength (c) and Unconfined Yield Stress (UYS), the Major Principal Stress (MPS) and Specific Energy (SE), as well as the flow factor (ffc) and dynamic Angle of Repose (AOR). Tan et al. [25] also found a correlation between the flow factor (ffc) and dynamic Angle of Repose (AOR) for their twelve samples.

In-depth analysis of the outputs correlation matrix (Fig. 5) suggests that the physical mechanisms of interparticle cohesion and friction seem to be determinant in the correlation between different flowability indices. Indeed, as the cohesive strength (c) from the shear test is governed by interparticle cohesion, it can be assumed that the predominant mechanisms in the tests correlated to the cohesive strength (c) are also driven by cohesive forces rather than friction. This is the case of the Unconfined Yield Stress (UYS) and flow factor (ffc) from the shear test, the Carr Index (IC) from the tapping test, the Flow Rate Index (FRI) from the powder rheometer, and the dynamic Angle of Repose (AOR),
<figure><img src="image_16.png" /><figcaption>Fig. 7. Linear relationship between the Specific Energy (SE) and circularity for our powders.</figcaption></figure>

Cohesive Index (CI) and first avalanche angle (1st. Av) from the rotating drum. Conversely, the Angle of Internal Friction (AIF) from the shear test reflects the importance of friction phenomena within a particulate medium. As the Basic Flow Energy (BFE) and Specific Energy (SE) from the powder rheometer appear to correlate with the Angle of Internal Friction (AIF), it suggests that these indices are particularly sensitive to frictional forces.

This discussion in terms of cohesive and frictional forces requires further investigation, particularly by analyzing the influence of the particle physical properties. This analysis will be the focus of the following section.

### 3.3. Correlation between particle physical characteristics (input) and flowability indices (output)

The aim of this section is to investigate the link between the inputs (particle physical characteristics) and the outputs (powder flowability indices). It is expected that the correlation matrix will help identify general trends and determine which physical characteristics most strongly influence the flow behavior of the powders.

To obtain precise correlations, it is necessary to maximize the number of studied materials. However, the characterizations performed on each sample are not uniform (the number of samples for each rheological test varies, as shown in Table 9). Thus, for the correlation matrix presented in Fig. 6, the correlations are calculated separately for each output variable.

The analysis of this correlation matrix confirms the findings presented in section 3.2, but also reveals new insights. Indeed, the granulometric properties ($d_{10}$, $d_{50}$, $d_{90}$, $d_{3.2}$, $d_{4.3}$ or Span) appear to have a significant influence on the aforementioned correlated flowability indices: cohesion strength (c), Unconfined Yield Stress (UYS) and flow factor (ffc) from the shear test, Carr Index (IC) from the tapping test, Flow Rate Index (FRI) from the powder rheometer, and dynamic Angle of Repose (AOR), Cohesive Index (CI) and first avalanche angle (1st. Av) from the rotating drum. It is well known that flowability is complicated by the presence of interparticle cohesive forces, which become predominant compared to gravitational forces for fine powders. Indeed, for two given interacting particles, the dimensionless granular Bond number $Bo_G$ accounts for the relative influence of interparticle forces ($f_{IP}$) compared to gravitational forces ($W_p$). Particles with high granular Bond number $Bo_G$ values generally exhibit a cohesive behavior, while particles with low $Bo_G$ values are considered as free-flowing. It should be noted that van der Waals interparticle forces ($f_{vdw}$) are always present (regardless of the particle chemical nature and environmental

conditions) and constitute the main interparticle interaction in many situations. Considering the simplified situation of two particles of the same diameter $d_p$, perfectly spherical and smooth, the granular Bond number $Bo_G$ is then given by the following equation [49]:

$$Bo_G = \frac{f_{vdw}}{W_p} = \frac{\frac{H_A d_p}{24 z_0^2}}{\frac{\pi g \rho_p d_p^3}{6}} = \frac{H_A}{4\pi g \rho_p z_0^2 d_p^2} \quad (7)$$

where $H_A$ is the Hamaker constant, $d_p$ the particle diameter, $\rho_p$ the particle density, and $z_0$ the distance between the two particles. This simplified model shows that a smaller particle diameter $d_p$ leads to a higher granular Bond number $Bo_G$, which generally corresponds to a more cohesive behavior. Consequently, as particle granulometric properties play a major role in influencing cohesive forces, they are correlated with the flowability indices primarily governed by cohesion and less sensitive to friction.

Additionally, it should be noted that most of these flowability indices (c, UYS, IC, FRI, CI) are mostly influenced by the spread of the particle size distribution (Span), which is also known to degrade powder flowability. Indeed, a wider particle size distribution allows for more efficient packing, as smaller particles can fill the voids between larger ones. This increased packing efficiency leads to higher compactivity and higher number of contacts, affecting flowability. The Span parameter also captures the presence of fines, which are well known to increase cohesive forces, also degrading powder flow properties [50].

On the other hand, other flowability indices appear to be mainly correlated to morphological properties (c, cx and e), such as the Angle of Internal Friction (AIF) and Major Principal Stress (MPS) from the shear test, and the Specific Energy (SE) from the powder rheometer. As explained previously, this can be interpreted as a greater significance of friction phenomena and interlocking between particles. Compared to spherical particles, the contact surface area is more important between two non-spherical particles, drastically increasing frictional forces. Also, elongated or irregular particles are more prone to mechanical interlocking, increasing resistance to flow and degrading flowability. Generally speaking, particle shape influences stress transmission through the powder bed, which affects the macroscopic flow behavior. This explains why, in contrast to the previous tests which were correlated with cohesion and therefore with particle granulometric properties, the flowability indices presented here (AIF, MPS, SE) appear to be more dependent on the friction and interlocking phenomena occurring during powder flow, and consequently on particle morphology.

Furthermore, it can also be observed that several flowability indices depend on both granulometric and morphological properties, such as
<figure><img src="image_17.png" /><figcaption>Fig. 8. Logarithmic correlation matrix between particle physical characteristics and flowability indices.</figcaption></figure>

the Carr Index (IC) and the dynamic Angle of Repose (AOR). This in-
dicates that both cohesive and frictional forces intervene during these
rheological tests, highlighting the complexity of the phenomena gov-
erning powder flowability. Regarding the Carr Index (IC) from the tap-
ping test, it is influenced by particle shape, as friction and interlocking
phenomena affect both the initial packing (before tapping) and particle
rearrangement (during tapping). It is also strongly influenced by particle
granulometric properties, as more cohesive powders tend to form looser
initial packings, leading to higher compressibility and higher Carr Index
(IC) values. As for the dynamic Angle of Repose (AOR) from the rotating
drum, it is well known that for cohesive powders, the macroscopic
behavior is complex, with intermittencies, cracks and avalanches
observable over a wide range of rotating speeds [46]. Particle shape also
affects rolling and sliding behavior, explaining the correlation of the
dynamic Angle of Repose (AOR) with both granulometric and morpho-
logical properties. Overall, these results highlight the complex interplay
between cohesive and frictional forces occurring during powder flow.

Regarding the Conditioned Bulk Density (CBD), it appears to be
primarily correlated with the true density of the particles, which is to be
expected. The granulometric properties also seem to have a slight in-
fluence, which could likely be explained by the effect of cohesive forces
on bulk density. This last aspect will be detailed in section 3.5, where the
dataset will be divided between fine cohesive powders and coarser
granular materials.

Finally, it should also be noted that the obtained correlations are not
sufficient to precisely predict the powder flow behavior from particle
physical characteristics. While they indicate global trends, they do not
allow for the establishment of mathematical relationships between
particle characteristics and flowability indices. For example, the Specific
Energy (SE) is primarily influenced by morphological properties,

particularly circularity (c<sub>mean</sub>) and elongation (e<sub>mean</sub>), which are inter-
related. However, establishing a direct and precise relationship between
circularity (c<sub>mean</sub>) and Specific Energy (SE) remains challenging, as
illustrated in Fig. 7. Additionally, the correlations observed in this ma-
trix only reflect linear relationships and do not account for power-law or
polynomial relationships.

3.4. Logarithmic correlation

The logarithm is applied to each value (after scaling) before the
correlation coefficient is calculated to reveal potential non-linear re-
lationships, particularly power laws found in the literature for certain
flowability indices [1]. The obtained correlation matrix is presented in
Fig. 8. The results undergo a slight modification when the logarithm is
applied, as the application of a logarithm can help the data normaliza-
tion, the variance stabilization and reduces the sensitivity to extreme
values [51].

For certain flowability indices, the correlation coefficients from
logarithmic matrix are significantly increased compared to those in the
linear correlation matrix (Fig. 6). This is particularly evident for flow-
ability indices related to granulometric properties, which are governed
by cohesive forces (i.e., c, UYS, ffc, IC, FRI, CI, 1st_Av). These correla-
tions reflect a power-law relationship between particle physical char-
acteristics and flow properties, which have been previously reported in
the literature, especially for the flow factor ffc [31,52,53]. Indeed, it has
been shown that the granular Bond number B<sub>oc</sub> (ratio of cohesive forces
to gravitational forces) depends notably on the particle size distribution
(eq. (8)), and that it can also be mathematically related to the flow factor
(ffc) through a power-law (eq. (9)). Consequently, it is also possible to
demonstrate a power-law relationship between the particle diameter
<figure><img src="image_18.png" /><figcaption>Fig. 9. Logarithmic relationship between SE and circularity for our powders.</figcaption></figure>

(the mean diameter $d_{3,2}$ is frequently used) and the flow factor (ffc), as described in Eqs. (8) to (10):

[49]

$$B_{oG} \propto \frac{1}{(d_{3,2})^{2,6}} \qquad (8)$$

[1]

$$ffc = \alpha B_{oG}^{-\beta} \qquad (9)$$

Then:

$$\log(ffc) \propto \log(d_{3,2}) \qquad (10)$$

Regarding the flowability indices mostly related to morphological properties, which are primarily governed by frictional forces (i.e., AIF, MPS and SE), the differences between the logarithmic correlation matrix and the linear correlation matrix are generally subtle, with similar trends and correlation levels. Similarly to the linear regression illustrated in Fig. 7, a power-law regression between the Specific Energy (SE) and circularity ($c_{mean}$) can be performed, as shown in Fig. 9. Compared to the linear regression, this correlation is slightly improved and reflects the relationship between morphology and flowability: the more spherical the particle, the lower the Specific Energy (SE). As explained previously, elongated or irregular particles are more prone to mechanical interlocking, and the contact surface area is larger between two non-spherical particles, increasing frictional forces. These mechanisms increase resistance to flow, leading to higher Specific Energy (SE) values.

In summary, these results highlight the value of examining logarithmic correlation matrices, particularly for capturing the influence of cohesion through granulometric properties. Nevertheless, as already commented for the linear correlation matrix, it should be recalled that the correlation matrix can only be used to describe general trends.

### 3.5. Dividing the dataset

As show, in the previous sections (3.2, 3.3 and 3.4), the physical mechanisms of interparticle cohesion seem to be determinant in explaining why granulometric properties significantly influence certain flowability indices (which are primarily sensitive to cohesion). Conversely, the mechanisms associated with friction help to explain why morphological properties have a significant impact on other flowability indices (which are primarily sensitive to friction). It is well known that interparticle cohesive forces become predominant compared to gravitational forces for fine powders. Upon examination of the characteristics of the studied powders, it appears that for a mean diameter $d_{3,2}$ =

75 µm, the ratio of cohesive forces to gravity (corresponding to the granular Bond number $B_{oG}$) is approximately equal to 2. Therefore, for powders with a mean diameter $d_{3,2} < 75$ µm, cohesive forces are significantly higher than gravity, and are expected to have a significant influence on flowability.

Since the main governing phenomena are expected to differ when considering the flow of fine powders (mainly driven by interparticle forces) compared to coarse powders (mainly driven by gravity and friction), the dataset has been split into two parts. The objective is to study separately the influence of particle physical characteristics on the flow properties of these two categories of powders. The linear correlation matrices for fine ($d_{3,2} < 75$ µm) and coarse powders ($d_{3,2} > 75$ µm) are presented in Fig. 10.

The results show that the linear correlations differ significantly between the two datasets. For coarse powders (Fig. 10-a), the majority of the flowability indices are primarily correlated with morphological properties. As discussed in section 3.3, particle shape has a strong influence on frictional forces, as non-spherical particles exhibit larger contact surface areas and are more prone to mechanical interlocking, which increases resistance to flow. When cohesive forces are negligible, as is the case for the coarse powders considered here, the flow behavior is mainly governed by friction and interlocking phenomena, which explains why the majority of the flowability indices are primarily correlated with morphological properties.

Additionally, it should be noted that the Angle of Internal Friction (AIF) from the shear test, the Carr Index (IC) from the tapping test, and the flowability indices from the rotating drum (AOR, CI, 1st_Av), appear to be also influenced by granulometric properties, in particular the spread of the particle size distribution (Span). As explained in section 3.3, a wider particle size distribution allows for more efficient packing, leading to higher compactivity and a higher number of contacts, affecting frictional forces and consequently flowability. This indicates that particle size distribution still has a moderate influence on powder flow behavior, even in the absence of cohesive forces.

Regarding the correlation matrix for fine cohesive powders (Fig. 10-b), the results can also be compared to those detailed in section 3.3 for the global correlation matrix (Fig. 6). In general, the correlations between flowability indices and granulometric properties that were previously present are reinforced when considering only cohesive powders (i.e., c, UYS, ffc, IC, FRI, CI, 1st_Av). As discussed in section 3.3, particle size plays a major role in influencing cohesive forces, with a smaller particle diameter $d_p$ leading to a higher granular Bond number $B_{oG}$ (more cohesive behavior). This explains why particle granulometric properties are correlated with the flowability indices primarily governed by cohesion.
<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 10. Linear correlation matrices between particle physical characteristics and flowability indices for coarse (a) and fine powders (b).</figcaption></figure>
<table><caption>**Table 10**
Number of materials used for each output (flowability index).</caption><thead><tr><th>Output</th><th>BFE</th><th>SI</th><th>FRI</th><th>SE</th><th>CBD</th><th>c</th><th>UYS</th><th>MPS</th><th>ffc</th><th>AIF</th><th>IC</th><th>AOR</th><th>CI</th><th>1st_Av</th></tr></thead><tbody><tr><td>Our dataset</td><td>61</td><td>61</td><td>61</td><td>61</td><td>61</td><td>52</td><td>52</td><td>52</td><td>52</td><td>52</td><td>52</td><td>58</td><td>60</td><td>53</td></tr><tr><td>From literature</td><td>11</td><td>6</td><td>6</td><td>0</td><td>0</td><td>23</td><td>7</td><td>7</td><td>33</td><td>18</td><td>41</td><td>18</td><td>16</td><td>0</td></tr><tr><td>Complete dataset</td><td>72</td><td>67</td><td>67</td><td>61</td><td>61</td><td>75</td><td>59</td><td>59</td><td>85</td><td>70</td><td>93</td><td>76</td><td>76</td><td>53</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**Fig. 11.** Linear correlation matrices between particle characteristics and flowability with literature data.</figcaption></figure>

Other flowability indices, which were not correlated with granular properties before, are now correlated with these properties, such as the Angle of Internal Friction (AIF) or the Basic Flow Energy (BFE). However, it should be noted that morphological properties still have a significant influence on some flowability indices (AIF, BFE, SE, AOR). Indeed, even for fine cohesive powders, frictional forces are still present and may significantly impact flow behavior.

In the specific case of the Conditioned Bulk Density (CBD), the analysis of the correlation matrices highlights that for fine powders (Fig. 10-b), the CBD is more strongly correlated with particle size distribution, while for coarse powders (Fig. 10-a), it is primarily influenced by particle density. This distinction indicates the different mechanisms that govern the packing behavior of fine powders compared to coarse powders. Indeed, the Conditioned Bulk Density can be calculated as:

$$ CBD = (1 - \varepsilon)\rho_p \qquad (11) $$

where $\varepsilon$ is the interparticle porosity and $\rho_p$ the particle density.

For coarse powders, the interparticle porosity is generally close to $\varepsilon \approx 0.4$, which means that the CBD is almost entirely dependent on the particle density $\rho_p$. This results in a nearly linear relationship between the CBD and the particle density (correlation coefficient very close to 1

in Fig. 10-a). In contrast, fine powders behave differently due to the influence of interparticle cohesive forces, which create relatively large voids within the powder bed, increasing interparticle porosity $\varepsilon$, and leading to a lower bulk density. Consequently, as decreasing particle size leads to a higher granular Bond number $Bo_G$ (eq. (7)), it also leads to a lower powder bulk density. This provides an explanation for the observed correlation differences between fine and coarse particles for CBD.

In summary, the observed trends confirm the differences in the phenomena governing the flow of cohesive powders compared to non-cohesive powders.

### 3.6. Correlation using bibliographic data

To complete the dataset, data from several publications were collected [1,6–8,21,24,25,28,31,45,48,54–58]. The number of materials used for each flowability index to establish the new correlation matrix is presented in Table 10. The objective is to investigate whether, despite the different testing procedures, it is still possible to deduce trends from the correlation matrices. In this way, this extended study can confirm or refute the conclusions drawn from our dataset by including several powders of different types.
<figure><img src="image_22.png" /><figcaption>Fig. 12. Logarithmic correlation matrices between particle characteristics and flowability with literature data.</figcaption></figure>

The main issue with this approach is that each sample has not un-
dergone the same characterizations (physical characteristics or flow
properties), particularly regarding morphology or density, which are not
always measured. Even if the particle physical properties and powder
flow properties are measured with the same apparatus, the measurement
methods can vary significantly. Consequently, to incorporate literature
data, the problem had to be simplified. The parameter *d*<sub>4,3</sub> was not
considered, as it is not frequently used in the literature. A variable is
introduced to attempt to incorporate a morphological description
applicable to all samples. The variable *S* is equal to one if the particles
are spherical and zero otherwise. As density is not a main parameter for

most of the flowability indices, this parameter was not considered.

Linear and logarithmic correlations that takes into account the
literature data are presented in Fig. 11 and Fig. 12, respectively.

Even with the increase in the number of materials used for the cor-
relation matrix, the majority of the general trends observed in the pre-
vious sections are still observed. This is particularly true for flowability
indices related to granulometric properties, which are governed by
cohesive forces (i.e., *c*, *UYS*, *ffc*, *IC*, *FRI*, *CI*, 1st\_Av). The same applies to
the flowability indices primarily related to morphological properties,
which are mainly governed by frictional forces (i.e., *AIF*, *MPS* and *SE*).
Overall, the results confirm the trends already observed in our initial

<figure><img src="image_23.png" /><figcaption>Fig. 13. Logarithmic relationship between d3,2 and ffc for our powders.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 14. Logarithmic relationship between $d_{3,2}$ and ffc with literature data.</figcaption></figure>

dataset.

However, it is clear that the expansion of the dataset can also result in lower correlation coefficients for many flowability indices. Indeed, the use of very different powders, along with varying experimental equipment and protocols, can lead to greater dispersion in the results, resulting in lower correlation coefficients. To provide a more concrete comparison, we can for example examine the relationship between the flow factor *ffc* (shear cell) and the mean diameter $d_{3,2}$. As discussed in section 3.4, a power-law relationship is generally observed between these two variables, as shown in eqs. (8) to (10). A comparison of the power-law regressions between *ffc* and $d_{3,2}$, for our initial dataset and for the complete dataset (with literature data), is presented in Fig. 13 and Fig. 14, respectively. The greater dispersion of the literature results can be attributed to variations in the experimental protocols used to determine the flow factor *ffc*. Indeed, various shear cells can be employed to characterize the powder flow properties in the quasi-static regime [59], contributing to the observed dispersion in the results. This increase in dispersion underlines the importance of taking rheological property measurement protocols into account when building predictive models.

## 4. Conclusions and perspectives

This study aimed to investigate the relationship between physical characteristics of powders and their flow properties. Around sixty inorganic powders with varying size, morphology and density were examined using multiple flowability tests. Pearson correlation matrices were employed to identify influential physical parameters for each rheological indicator. The ultimate goal was to provide a comprehensive understanding of the relationships between rheological indicators, physical characteristics and flow regimes, highlighting the interplay between friction and cohesion in powder behavior.

The analysis revealed that correlations between flowability indices are not strictly associated with specific flow regimes (quasi-static and dense regimes). Instead, the physical mechanisms of interparticle cohesion and friction appear to play a determinant role in these correlations. Granulometric properties significantly influence flowability indices that are mostly governed by cohesive forces, such as the cohesion strength (c), Unconfined Yield Stress (UYS) and flow factor (ffc). Conversely, morphological properties impact indices that are primarily sensitive to friction and interlocking, like the Angle of Internal Friction (AIF) and Specific Energy (SE). The research also highlighted the complex interplay between cohesive and frictional forces in some flowability indices, such as the Carr Index (IC) and dynamic Angle of Repose (AOR).

Logarithmic correlations mainly revealed power-law relationships between granulometric properties and cohesion-dominated flowability indices, for which the correlations were reinforced.

Further analysis involving the division of the dataset into fine and coarse powders provided additional insights. For coarse powders, morphological properties predominantly influenced flowability indices, while for fine powders, granulometric properties showed stronger correlations. This distinction underscores the different mechanisms governing the flow behavior of cohesive and non-cohesive powders. The study also revealed interesting trends in the relationship between Conditioned Bulk Density (CBD) and particle properties for different powder types. In a second step, this research work also incorporated data from existing literature to validate trends across different testing procedures and powder types. The study confirmed that general trends for flowability indices related to granulometric and morphological properties persisted.

Despite the valuable insights gained, this study has also several limitations. The correlations observed are insufficient for precise prediction of powder flow behavior from particle physical properties, as they only indicate general trends. The incorporation of literature data, while expanding the dataset, introduced additional variability due to differences in experimental protocols and equipment. This highlights the need for standardized measurement techniques in powder characterization. Future research should focus on developing more sophisticated models that can account for multiple input parameters simultaneously and consider non-linear relationships. Additionally, investigating the impact of environmental conditions, such as humidity and temperature, on powder flowability could provide a more comprehensive understanding of powder behavior in real-world applications. Lastly, exploring machine learning techniques to analyze complex relationships between particle properties and flow behavior could lead to more accurate predictive models for powder flowability.

## CRediT authorship contribution statement

**Lina Cayla Arianer:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis. **Mikel Leturia:** Writing – review & editing, Supervision. **Anne-Charlotte Robisson:** Writing – review & editing, Supervision, Conceptualization. **Carine Ablitzer:** Writing – review & editing, Supervision. **Elias Daouk:** Supervision. **Khashayar Saleh:** Supervision.
# Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

# Data availability

Data will be made available on request.

# References

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