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

# Utilizing polydispersity in three-dimensional random fibrous based sound absorbing materials

Quang Vu Tran<sup>a,b,c,*</sup>, Camille Perrot<sup>a,*</sup>, Raymond Panneton<sup>b,*</sup>, Minh Tan Hoang<sup>c</sup>, Ludovic Dejaeger<sup>c</sup>, Valérie Marcel<sup>c</sup>, Mathieu Jouve<sup>c</sup>

## ARTICLE INFO

### Keywords:

Multiscale model

Fibrous materials

Polydispersity

Transport properties

Sound absorption

Optimization

## ABSTRACT

The distribution of fiber diameters plays a crucial role in the transport and sound absorbing properties of a three-dimensional random fibrous (3D-RF) medium. Conventionally, volume-weighted averaging of fiber diameters has been utilized as an appropriate microstructural descriptor to predict the static viscous permeability of 3D-RF media. However, the long wavelength acoustical properties of a 3D-RF medium are also sensitive to the smallest fibers, this is particularly true in the high-frequency regime. In our recent research, we demonstrated that an inverse volume-weighted averaging of fiber diameters can effectively serve as a complementary microstructural descriptor to capture the high-frequency behavior of polydisperse fibrous media. In the present work, we reexamine the identification of two representative volume elements (RVEs) which relies on the reconstruction of 3D-RF microstructures having volume-weighted and inverse-volume weighted averaged fiber diameters, respectively in the low-frequency and high frequency regimes. We investigate the implication of such a weighting procedure on the transport and sound absorbing properties of polydisperse fibrous media, highlighting their potential advantages. Furthermore, we discuss the challenges associated with this research field. Finally, we provide a brief perspective of the future directions and opportunities for advancing this area of study.

## 1. Introduction

Three-dimensional random fibrous (3D-RF) structures have become one of the most widespread technologies in large-scale manufacturing of sound-absorbing materials known as nonwovens or felts, employing various types of fibers to achieve their acoustical constraints. Some commonly used manufacturing processes include the airlay process [1,2] and the carding, cross-lapping, and needling process [3,4]. Traditional thermobonded airlaid nonwoven typically incorporates a mixture of non-homogeneous shoddy fibers (75%) and bicomponent fibers (25%) in the airlay process. The bicomponent fibers have a core made from PET and a surface made from coPET. On the other hand, to facilitate recycling, needlefelt nonwoven use a mixture of PET fibers (60%) and bicomponent fibers (40%). In both cases, in post-processing, the nonwoven materials are reinforced by thermobonding (with a chosen compression ratio), where the bicomponent fibers have an adhesive effect. However,

the use of non-homogeneous fibers in the manufacturing process comes with some drawbacks.

One of the disadvantages is that the transport mechanisms can be significantly impacted by the fiber size variations induced by the manufacturing process. For example, shoddy fibers are obtained after the tearing of textile waste from a mixture of 55% cotton and 45% PET, and cotton fibers can introduce fiber dust particles, which affect the distribution of fiber diameters. The manufacturing process of nonwoven leads to a wide distribution of fiber diameters and fiber orientations, which affects the frequency-dependent response of the fluid within the corresponding 3D-RF structure. Moreover, mixing fibers of different sizes is generally recommended. The largest fibers ensure the rigidity and mechanical stability of the nonwoven material, while the smallest fibers provide sound absorbing capabilities to the fibrous materials.

Therefore, an important research direction in this field is the study of various distributions of fiber diameters and orientations and the

<footnote><sup>a</sup> Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, UMR 8208, MSME, F-77454, Marne-la-Vallée, France</footnote>

<footnote><sup>b</sup> Département de Génie Mécanique, Université de Sherbrooke, J1K 2R1, Québec, Canada</footnote>

<footnote><sup>c</sup> Adler Pelzer Group, Acoustic TechCenter R&D, Z.I. François Sommer - BP13, 08210, Mouzon, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: quang.vu.tran@usherbrooke.ca (Q.V. Tran), camille.perrot@univ-eiffel.fr (C. Perrot), raymond.panneton@usherbrooke.ca (R. Panneton).</footnote>

<footnote>https://doi.org/10.1016/j.matdes.2024.113375</footnote>

<footnote>Received 16 August 2024; Received in revised form 8 October 2024; Accepted 13 October 2024</footnote>

<footnote>Available online 17 October 2024</footnote>

<footnote>0264-1275/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
development of models that account for the corresponding frequency-
dependent response functions. Several approaches have been explored
to achieve this goal. Perhaps the most direct method is to conduct
a series of laboratory measurements on samples of varying fiber size
and orientation (semi-empirical models) [5–7]. Alternatively, in the
quest for theoretical understanding, one may seek to better understand
the mathematical (analytical derivations) [8–13] or physical (semi-
phenomenological description) [14–17] basis of the generalized (dy-
namic) Darcy-scaled equations for macroscopic transport. Lastly, one
can consider studies based on numerical simulations [18–20]. Notably,
Peyrega and Jeulin [21] used a volume-weighted average diameter
$D_v = \sum_{i=1}^{N_f} V_i D_i / \sum_{i=1}^{N_f} V_i$ to successfully predict the static viscous per-
meability of heterogeneous fibrous materials and our group [22] has re-
cently demonstrated that an inverse volume weighted average diameter
$D_{iv} = \sum_{i=1}^{N_f} \frac{D_i}{V_i} / \sum_{i=1}^{N_f} \frac{1}{V_i}$ can serve as a complementary microstructural
descriptor to predict the characteristic lengths and tortuosity of these
polydisperse 3D random fibrous materials. In the previous definitions,
$V_i$ represents the volume of fiber *i* with diameter $D_i$ and $N_f$ the num-
ber of fibers introduced in the RVE. The non-weighted average diameter
is given by $D_m = \frac{1}{N_f} \sum_{i=1}^{N_f} D_i$.

In this paper, we aim to provide a comprehensive report of research
advancements in employing 3D polydisperse RF microstructures, with
a particular emphasis on using polydispersity as a new lever of opti-
mization [23–25]. Starting from existing specimens of nonwoven fibrous
materials, we focus specifically on three different types of studies: (i)
a parametric study, using the coefficient of variation CV defined as
the ratio of the standard deviation to the mean of the fiber diameters
and used to express the polydispersity degree P<sub>d</sub> of random fibrous me-
dia as input parameter; (ii) optimizing the sound absorption average
at normal incidence over a wide frequency range of a given nonwo-
ven specimen by controlling the degree of polydispersity of the fiber
diameters; (iii) achieving targets of industrial interest consisting of low-
ering the lower frequency from which the sound absorption in a diffuse
field is greater than 80%. The paper is organized in the following se-
quences. In the first section, we delve into the fundamental transport
properties of 3D-RF microstructures used as sound-absorbing materials
that are characterized by a wide distribution of their fiber diameters. We
discuss the unique transport properties and microstructural descriptors
derived from the polydispersity of these materials, highlighting their po-
tential advantages over traditional fibrous materials. This section serves
as a foundation for understanding the subsequent discussions. Building
upon the understanding of 3D microstructure based RF materials, we
dedicate the second section to exploring the advantages offered by em-
ploying specific polydispersity degree as a microstructural optimization
lever. We discuss the benefits associated with tuning the polydispersity
degree of 3D-RF microstructures in terms of their transport proper-
ties, sound absorption average at normal incidence, sound absorption
targets in diffuse field. By doing so, we highlight our contribution to
the field, showing the advancements that could be achieved by lever-
aging the polydisperse feature of the these materials to enhance their
functional properties. Finally, we conclude the article with a brief per-
spective and outlook on current challenges and the future of research in
this area. We discuss emerging challenges, potential directions for fur-
ther explorations, and the significance of continued investigations into
3D-RF microstructured materials as a microstructural optimization lever
of their physical properties. By summarizing the current state of the field
and presenting potential avenues for future research, we aim to inspire
and guide researchers in this exciting area of study.

## 2. Nonwoven fibrous media characterized by local heterogeneity

### 2.1. Predicting transport properties

Martys and Garboczi [19] demonstrated through numerical compu-
tations the important effect spatial randomness in the pore space has

on flow problems. In particular, they established a crucial distinction
between the electric fields and the fluid-flow fields for a given pore struc-
ture. Leveraging this distinction, they identified that in a random pore
structure with a distribution of pore sizes, the fluid-flow will tend to go
more through the largest pore necks; while the electrical current-flow
rate was comparatively more sensitive to the width of narrow necks.
As a result, the fluid velocity fields and the electric fields can sam-
ple the pore space quite differently. Subsequently, Peyrega and Jeulin
[21] made substantial advances in this area, particularly in identify-
ing the volume-weighted average diameter ($D_v$ in our notations) as a
key microstructural descriptor for the determination of the represen-
tative volume element (RVE) of a random fibrous medium. Moreover,
Tran [22] and collaborators delved into the associated microscopic ba-
sis of macroscopic viscous permeability, shedding further light on how
macroscopic viscous permeability depends on the largest pore sizes of
a locally heterogeneous 3D-RF microstructure. The volume-weighed av-
erage diameter $D_v$ of a wide distribution of fiber diameters is larger
than non-weighted average diameter $D_m$, and the number of introduced
fibers in a cubic box of normalized size $L/D_m$ is therefore lower in the
volume-weighted average diameter case ($D_v$) than in the non-weighted
average diameter one ($D_m$) – while the porosity $\phi$ is known and given by
measurements in both cases. As a result, the reconstructed pore sizes of
3D-RF structures having a volume-weighted average diameter $D_v$ match
with the largest pore necks through which the viscous fluid-flow will
tend to go in the corresponding nonwoven specimen. This reconstruc-
tion procedure enables the identification of a RVE having $D_v$ as the
average diameter and the computed viscous permeability $k_0$ to be con-
sistent with the measured values [$L/D_m = 25(-), L = 400 \ \mu m$] in real
samples whose sample diameters [$S_D = 45 \ mm$], are much larger than
those of the RVE [22].

The combined presence of a wide distribution of fiber diameters and
random microstructure promotes pore-size local heterogeneity, lead-
ing to the identification of the volume-weighted average diameter $D_v$,
as being an appropriate microstructural descriptor for the prediction
of the static viscous permeability $k_0$. The use of the finite element
method revealed that velocity fields are consistent with experimental
measurements of viscous permeability using volume-weighted average
diameter $D_v$, as shown in Figs. 1(a), 1(d) and 1(j). However, the elec-
tric current paths are clearly less concentrated and tortuous than do
the fluid-flow paths, so there are significantly fewer stagnant areas for
the electrical current flow than for the fluid flow. The viscous char-
acteristic length $\Lambda$ is defined [26] by the following ratio of integrals:
$\Lambda = 2 \int |\mathbf{E}(\mathbf{r})|^2 dV_p / \int |\mathbf{E}(\mathbf{r})|^2 dS$, where $\mathbf{E}(\mathbf{r})$ is the magnitude of the
electrical field in the pore space, $dV_p$ is the volume element in the pore
space, and $dS$ is the surface element on the pore solid interface. There-
fore, $\Lambda$ can be thought of as a dynamically weighted hydraulic radius
[14], where the weighting procedure substantially favors the smaller
pores because of current conservation. Fig. 1(c) shows a 3D-RF mi-
crostructure where inverse volume-weighted average diameter $D_{iv}$ is
used as the appropriate microstructural descriptor for the prediction
of the viscous characteristic length $\Lambda$. The geometrical reconstruction
method is given in a previous work [22]. The smaller pores of the non-
woven polydisperse fibrous materials were probed by measuring the
experimental value of the $\Lambda$ parameter, a value captured by using in-
verse volume-weighted average diameter $D_{iv}$ during the reconstruction
procedure, as shown in Figs. 1(i) and 1(k). The change in pore size re-
construction induced by fiber diameter polydispersity and volume-based
fiber diameter weighting is shown in Fig. 1(a)-1(c). At the scale of the
reconstructed unit-cell, the non-weighted average diameter $D_m$ was nei-
ther an appropriate microstructural descriptor for describing the largest
pores connected with viscous fluid flow nor a correct one to identify the
smallest dynamically connected pore sizes probed by electrical current
flow; Fig. 1(b), 1(e) and 1(h). $D_v$ based and $D_{iv}$ based RVEs predictions
all showed excellent agreement respectively with static viscous perme-
ability $k_0$ and viscous characteristic length $\Lambda$ measurements over a large
range of porosities $\phi$; Fig. 1(j) and 1(k).
<figure><img src="image_2.png" /><figcaption>Fig. 1. Effect of polydispersity on the transport properties. (a)-(c) Three-dimensional random fibrous microstructures corresponding to the sample denoted F2 using volume-weighted, non-weighted, inverse volume-weighted average diameters, respectively. (d)-(f) Velocity fields expressed as local permeability ($k_{0zz}$[m²]) corresponding to Stokes flow in the z direction with the RVE reconstructed using (a) volume weighted, (b) non-weighted and (c) inverse volume-weighted average diameter. (g)-(i) Scaled potential field ($\phi$[m] corresponding to potential flow in the z direction [-] with the RVE reconstructed using (a) volume weighted, (b) non-weighted and (c) inverse volume-weighted average diameter. (j) The static viscous permeability $k_0(m^2)$ of nonwoven fibrous materials (F1, F2, F3, F4) as a function of porosity $\phi$(-). (k) The viscous characteristic length $\Lambda(\mu m)$ of nonwoven fibrous materials (F1, F2, F3, F4) as a function of porosity $\phi$(-).</figcaption></figure>

## 2.2. Polydispersity effect on the transport properties of three-dimensional random fibrous microstructures

Polydispersity effect occurs when particles of varied sizes in the dispersed phase modifies the expected behavior of the corresponding system. The wide distribution of fiber diameters changes the transport parameters of a 3D-RF microstructure characterized by a sharply peaked distribution of fiber diameters, resulting is a non-conventional RVE. Previous works describe the Gamma-based distribution of fiber diameter configuration and confirm this effect in glass wool, cotton felts and PET felts' composites. [27,22] Fig. 2(a) illustrates the probability density function (PDF) of the Gamma distribution (Γ) of fiber diameters according to the equation

$$f(D; a, b) = \frac{1}{\Gamma(a)b^a} D^{a-1} e^{-\frac{D}{b}}, \quad (1)$$

where D is the fiber diameter of the probability distribution function parameterized with the shape a ($a > 0$) and scale b ($b > 0$) parameters. Fig. 2(b) shows the geometry of 3D-RF structures with an increasing coefficient of variation $CV = 1/\sqrt{a}$ used as a measure of dispersion;

at $CV = 20\%$, $CV = 50\%$, and $CV = 100\%$. The corresponding porosity is equal to $\phi = 94.1\%$ and the mean fiber diameter is given by $D_m = ab = 14\mu m$. The RVEs based on $D_v$ and based on $D_{iv}$ are presented in Fig. 2(c) and 2(d), respectively. Coefficient of variation was around 40% in the cotton felts case (F1-F4) and 30% in the PET felts one, and the orientation of fibers of the 3D-RF microstructures was parameterized assuming transverse isotropy according to an additionally needed factor, the angular orientation parameter $\Omega_{zz}$ ($0 \le \Omega_{zz} \le 1$) [28,29]. Therefore, assuming a Gamma-based distribution of fiber diameters, geometrical parameters of the model include porosity $\phi$, mean fiber diameter $D_m$, coefficient of variation of the fiber diameters CV and angular orientation of fibers $\Omega_{zz}$. There exist a strong analogy between the visco-inertial frequency-dependent response function of a Newtonian fluid-filled rigid porous medium [14], and its frequency-dependent thermal counterpart [30,15]. Therefore, the $D_{iv}$-based cell was simultaneously utilized as the RVE to compute the static viscous permeability $k_0$ and the static thermal permeability $k'_0$. On the other hand, on the asymptotic high frequency range, the $D_{iv}$-based cell was used as the RVE for the calculation of the viscous $\Lambda$ and thermal $\Lambda'$ characteristic lengths together with the tor-
<figure><img src="image_3.png" /><figcaption>Fig. 4. Dimensionless transport parameters as a function of the coefficient of variation CV of fiber diameters for cotton felts (F1-F4) and PET felts (B1-B2). The porosities $\phi$ and orientation of fibers $\Omega_{zz}$ of each felt are unchanged with CV and taken from measurement. [22] The dots correspond to the experimentally determined values of CV in the initial state of polydispersity and are reported along with the corresponding error bars used to indicate the uncertainty determined in the experimental characterization with available samples. The envelope in the model originates from the uncertainty in the porosity determination.</figcaption></figure>

incidence ($SAC_{NI}$) at 16 one-third octave bands $f_i$ from 125 to 4500 Hz with

$$SAA_{NI}^{125-4000} = \frac{1}{16} \sum_{f_i=125}^{4500} SAC_{NI}(f_i), \qquad (2)$$

Here, the $SAC_{NI}$ is calculated with the material of thickness $L_s$ placed on a hard reflecting backing. Consequently, $SAA_{NI}$ is a measure of the energy dissipated by visco-thermal losses in the material for a unit incident acoustic power. This single number rating is an important coefficient that can be used for characterizing the sound absorbing properties of porous materials in a large frequency range, including low frequencies. We have previously shown that using polydispersity can significantly modify the transport properties of 3D-RF microstructures at constant porosity $\phi$ and angular orientation $\Omega_{zz}$ (Sec. 2.3.1). Fig. 5 shows how the sound absorption average at normal incidence $SAA_{NI}^{125-4000}$ can be maximized by optimization of the CV from the initial set of studied materials (F1-F4; B1-B2). The increase in $SAA_{NI}^{125-4000}$ in 3D-RF microstructures is due to a larger fiber diameter polydispersity ($CV \approx 75\%$) that allows for a strong contrast between the largest fibers that dominate transport phenomena in the low frequency regime ($k_0; k'_0$) and the smallest fibers that govern transport phenomena in the high frequency regime ($\Lambda, \alpha_\infty; \Lambda'$); see Table 1. The results show that the optimized

3D-RF materials (controlled by polydispersity) exhibit lower viscous $f_v$ and thermal $f_t$ transition frequencies than the initial materials – which allowed visco-thermal dissipation mechanisms to occur in a lower frequency range. Another striking result emerging from these simulations is to find that $M = 8k_0\alpha_\infty/\Lambda^2\phi \gg 1$ in the polydisperse cases. We recall that we used the volume-weighted fiber diameter $D_v$ to compute the low-frequency transport parameters of the polydisperse microstructure, and the inverse volume-weighted fiber diameter $D_{iv}$ to obtain the high-frequency ones. It follows that the dimensionless shape factor $M$ introduced by Johnson et al. [14] is evaluated by combining transport parameters corresponding to different moments of the weighted fiber diameters distribution, and thus to different microstructural features of the same complex microgeometry. This approach suggests that a polydisperse 3D-RF medium is a heterogeneous material where coexist at a mesoscale fibers with different and contrasted characteristic sizes. Then the viscous transport will take place mostly along the “subnetwork” formed by the wider pores, while the electrical or inertial transport at high-frequencies will take place over the entire heterogeneous porous structure due to the absence of viscosity in the electric case, and will be able to probe the narrowest pores.

In the previous calculations, the sound absorbing coefficient at normal incidence, $SAC_{NI}$, derives from the knowledge of the intrinsic transport parameters of the 3D-RF materials ($k_0, k'_0, \Lambda, \Lambda', \alpha_\infty$). The
<figure><img src="image_4.png" /><figcaption>Fig. 5. The sound absorption coefficient at normal incidence, SAA<sub>NI</sub>, as a function of frequency for the two families of felts studied in this work: Cotton felts (F1-F4) and PET felts (B1-B2). The SAA<sub>NI</sub> responses obtained prior to modification of the polydispersity (coefficient of variation CV) are shown with a blue line. The red line represents the SAA<sub>NI</sub> obtained after optimization of the polydispersity by maximizing the SAA<sub>NI</sub> between the third octave bands from 125 to 4500 Hz. The two dotted lines correspond to the associated single number ratings before (blue) and after (red) optimization.</figcaption></figure>

frequency-dependent response functions, $\tilde{\rho}_{eq}(\omega)$ and $\tilde{K}_{eq}(\omega)$ are described as [14,30,15]

$$ \tilde{\rho}_{eq}(\omega) = \frac{\alpha_{\infty} \rho_0}{\phi} \left[ 1 + \frac{\phi \eta}{i \omega k_0 \alpha_{\infty} \rho_0} \sqrt{1 + i \frac{4 \alpha_{\infty}^2 k_0^2 \rho_0 \omega}{\eta \Lambda^2 \phi^2}} \right], \quad (3) $$

and

$$ \tilde{K}_{eq}(\omega) = \frac{\gamma P_0 / \phi}{\gamma - (\gamma - 1) \left[ 1 - i \frac{\phi \kappa}{k_0^2 C_p \rho_0 \omega} \sqrt{1 + i \frac{4 k_0^2 C_p \rho_0 \omega}{\kappa \Lambda^2 \phi^2}} \right]^{-1}}, \quad (4) $$

where $\tilde{\rho}_{eq}(\omega)$ is the equivalent dynamic density, $\tilde{K}_{eq}(\omega)$ is the equivalent bulk modulus, $\omega = 2\pi f$ is the angular frequency, $i$ is the imaginary unit, $\rho_0$ is the mass density of air, $\eta$ is the dynamic viscosity of air, $P_0$ is the atmosphere pressure, $\gamma = C_p/C_v$ is the ratio of heat capacities at constant pressure and volume, and $\kappa$ is the heat conductivity of air.

By knowing the visco-inertial $\tilde{\rho}_{eq}(\omega)$ and thermal $\tilde{K}_{eq}(\omega)$ response functions of the equivalent fluid of the porous medium, one can determine the wave number $\tilde{q}_{eq}(\omega)$ and the characteristic impedance $\tilde{Z}_{eq}(\omega)$ of the material,

$$ \tilde{q}_{eq}(\omega) = \omega \sqrt{\tilde{\rho}_{eq}(\omega)/\tilde{K}_{eq}(\omega)}, \quad (5) $$

$$ \tilde{Z}_{eq}(\omega) = \sqrt{\tilde{\rho}_{eq}(\omega)\tilde{K}_{eq}(\omega)}. \quad (6) $$

At normal incidence, the surface impedance $\tilde{Z}_s(\omega)$ and sound absorption coefficient $SAC_{NI}(\omega)$ of a porous material of thickness $L_s$, backed by a rigid and impervious wall, for any angular frequency $\omega$, are provided by

$$ \tilde{Z}_s(\omega) = -i \tilde{Z}_{eq} \cot(\tilde{q}_{eq} L_s), \quad (7) $$

$$ SAC_{NI} = 1 - \left| \frac{\tilde{Z}_s - Z_0}{\tilde{Z}_s + Z_0} \right|^2, \quad (8) $$

where $Z_0 = \rho_0 c_0$ is the impedance of the air and $c_0$ is the sound speed in air.

In Fig. 5, it can be observed that there is a significant change in the wavelength-to-thickness ratio $\lambda_0/L_s$, where $\lambda_0 = \omega_0/c_0$, and $\omega_0$ denotes the angular frequency at the first peak absorption. In fact, optimized polydispersed-based 3D-RF microstructures show ratios $L_s/\lambda_0$ that can achieve values lower than 1/32 (F4 and B2) - note that the typical ratio for conventional porous materials is around 1/4 (for low resistive material, for which the sound speed is close to that observed in air). This result can be seen as the signature of a certain class of metamaterials called subwavelength material (or slow-sound material). [31] In fact, the optimized properties increase $\tilde{\rho}_{eq}$ and decrease $\tilde{K}_{eq}$. Consequently, the sound speed in the equivalent medium is reduced. These results demonstrate that the use of controlled fiber diameter polydispersity, such as that provided with a gamma distribution shape and characterized with a CV ~ 80%, has the potential to enhance the performance of sound-absorbing nonwoven materials.

### 2.3.3. Enhancing diffuse field sound absorption

Sound absorbing targets present in the industry for a porous material generally corresponds to a diffuse field (DF) acoustical excitation, satisfying $SAA_{DF} \ge 0.8$, for $f_t \ge f_t$, where $f_t$ is a given one-third octave band. Lowering $f_t$ can lead to various issues in designing sound absorbing materials, as absorbing low frequencies is a challenge that suggests targets of increasing difficulty: $f_t = 1600Hz$ (Target 1), $f_t = 1250Hz$ (Target 2), $f_t = 1000Hz$ (Target 3), $f_t = 800Hz$ (Target 4) and $f_t = 630Hz$ (Target 5). These targets are defined without air gap behind the materials (hard-backed configuration), and the recommended or targeted thickness and mass are usually not mentioned, suggesting that the targets need to be reached with the lowest weight and cost. The sound absorption coefficient in diffuse field, $SAC_{DF}$, is described as:
<table><caption>**Table 1**
Transport parameters evolution ($\phi$, $k_0$, $k'_0$, $\Lambda$, $\Lambda'$, $\alpha_\infty$) before (Init.) and after (Opt.) optimization of polydispersity (CV) in the three-dimensional random fibrous microstructures of cotton felts (F1-F4) and PET felts (B1-B2). $f_v$ and $f_t$ represent the visco-inertial and thermal transition frequencies which have been significantly reduced after optimization, which consists in maximizing the sound absorption average at normal incidence across the 16 one-third octave bands ($f_i$) ranging from 125 to 4500 Hz, $SAA_{NI}^{125-4000}$ [relative increase: F1(14%), F2(23%), F3(26%), F4(50%), B1(60%), B2(114%)].</caption><thead><tr><th></th><th></th><th>CV (%)</th><th>Φ (-)</th><th>k<sub>0</sub> (×10<sup>-10</sup>m<sup>2</sup>)</th><th>k'<sub>0</sub> (×10<sup>-10</sup>m<sup>2</sup>)</th><th>α<sub>∞</sub> (-)</th><th>Λ (μm)</th><th>Λ' (μm)</th><th>f<sub>v</sub> (Hz)</th><th>f<sub>t</sub> (Hz)</th><th>M (-)</th><th>SAA<sub>NI</sub><sup>125-4500</sup> (%)</th></tr></thead><tbody><tr><td rowspan="2">F1</td><td>Init.</td><td>40.3</td><td>0.948</td><td>4.5</td><td>12.1</td><td>1.022</td><td>48</td><td>84</td><td>4668</td><td>2557</td><td>1.7</td><td>42</td></tr><tr><td>Opt.</td><td>73.3</td><td>0.948</td><td>11.1</td><td>27.8</td><td>1.022</td><td>6.9</td><td>12</td><td>2018</td><td>1112</td><td>201</td><td>48</td></tr><tr><td rowspan="2">F2</td><td>Init.</td><td>39.8</td><td>0.941</td><td>3.9</td><td>9.9</td><td>1.026</td><td>42</td><td>74</td><td>5684</td><td>3103</td><td>1.9</td><td>35</td></tr><tr><td>Opt.</td><td>74.6</td><td>0.941</td><td>10.0</td><td>27.1</td><td>1.026</td><td>5.3</td><td>9.4</td><td>2213</td><td>1209</td><td>311</td><td>43</td></tr><tr><td rowspan="2">F3</td><td>Init.</td><td>41.9</td><td>0.914</td><td>2.1</td><td>5.1</td><td>1.042</td><td>25</td><td>47</td><td>10102</td><td>5850</td><td>3.1</td><td>27</td></tr><tr><td>Opt.</td><td>72.1</td><td>0.914</td><td>4.7</td><td>13.1</td><td>1.042</td><td>3.3</td><td>6.1</td><td>4533</td><td>2631</td><td>394</td><td>34</td></tr><tr><td rowspan="2">F4</td><td>Init.</td><td>38.9</td><td>0.856</td><td>0.7</td><td>1.6</td><td>1.077</td><td>15</td><td>28</td><td>25286</td><td>17463</td><td>3.1</td><td>14</td></tr><tr><td>Opt.</td><td>73.8</td><td>0.856</td><td>1.9</td><td>5.1</td><td>1.077</td><td>1.3</td><td>2.4</td><td>9998</td><td>7073</td><td>1132</td><td>21</td></tr><tr><td rowspan="2">B1</td><td>Init.</td><td>26.6</td><td>0.888</td><td>2.7</td><td>6.3</td><td>1.056</td><td>42</td><td>79</td><td>7215</td><td>4601</td><td>1.5</td><td>20</td></tr><tr><td>Opt.</td><td>75.5</td><td>0.888</td><td>9.6</td><td>23.1</td><td>1.056</td><td>3.3</td><td>6.1</td><td>2116</td><td>1356</td><td>839</td><td>32</td></tr><tr><td rowspan="2">B2</td><td>Init.</td><td>33.1</td><td>0.76</td><td>0.7</td><td>1.2</td><td>1.144</td><td>15</td><td>29</td><td>21495</td><td>21023</td><td>3.7</td><td>7</td></tr><tr><td>Opt.</td><td>77.6</td><td>0.76</td><td>2.4</td><td>4.3</td><td>1.144</td><td>0.8</td><td>1.6</td><td>6690</td><td>6518</td><td>4516</td><td>15</td></tr></tbody></table>

$$ SAC_{DF} = \frac{\int_{\theta_{min}}^{\theta_{max}} SAC(\omega, \theta) \cos \theta \sin \theta d\theta}{\int_{\theta_{min}}^{\theta_{max}} \cos \theta \sin \theta d\theta}, \qquad (9) $$

where $\theta_{min} = 0^\circ$ and $\theta_{max} = 90^\circ$ are the selected diffuse field integration limits.

$SAC(\omega, \theta) = 1 - \left| \frac{\tilde{Z}_s(\omega, \theta) \cos \theta - Z_0}{\tilde{Z}_s(\omega, \theta) \cos \theta + Z_0} \right|^2$ is the sound absorption coefficient at oblique incidence, $\tilde{Z}_s(\omega, \theta) = -i \tilde{Z}_{eq}(\tilde{q}_{eq}/\tilde{q}_x) \cot(\tilde{q}_{eq} L_s)$ is the surface impedance at oblique incidence with $\tilde{q}_x = \sqrt{\tilde{q}_{eq}^2 - q_t^2}$ the longitudinal wave number and $q_t = q_0 \sin \theta$ the transverse wave number.

Our simulations suggest that 3D-RF materials, specifically polydisperse with controlled $CV \sim 75\% - 79\%$ can serve as effective sound-absorbing materials in diffuse field to achieve the required industrial targets. By simulating the $SAC_{DF}$ from the initial felts (F1-F4; B1-B2), we identified the optimal coefficient of variation $CV$ allowing to reach the targets of increasing level of difficulty with the minimal thickness. The results, shown in Fig. 6a for the case of cotton felt F1, revealed that none of the targets can be reached from the initial microstructure ($CV = 40.3\%$) and the thickness of the sample ($L_s = 20$ mm). Fig. 6b demonstrated an achievement of Target 1 by using a $CV$ of 78% with a significant reduction of the sample thickness ($L_s = 11$ mm). We obtained similar polydispersity results for the other targets of increasing difficulty (Target 2, $CV = 79\%$, Fig. 6c; Target 3, $CV = 78\%$, Fig. 6d; Target 4, $CV = 77\%$, Fig. 6e; Target 5, $CV = 75\%$, Fig. 6f). However, the sample thicknesses had to be increased (Target 2, $L_s = 13$ mm, Fig. 6c; Target 3, $L_s = 16$ mm, Fig. 6d; Target 4, $L_s = 20$ mm, Fig. 6e; Target 5 $L_s = 27$ mm, Fig. 6f. To better appreciate the performance that has been obtained by increasing and controlling the degree of polydispersity $P_d$ through the coefficient of variation $CV$ of the Gamma law, the results can be plotted in terms of the wavelength-to-thickness ratio, $\lambda_0/L_s$. Using a controlled degree of polydispersity allowed us to reach a so-called sub-wavelength sound absorption in the diffuse field, by lowering the maximum sound absorption peak below the quarter-wavelength resonance frequency ($L_s/\lambda_0 = 1/4$), indicating a greater ability to absorb low-frequency sound. Therefore, using 3D-RF materials with controlled polydispersity to efficiently absorb low-frequency sound is an important approach as it a priori does not require substantial modifications of the manufacturing processes.

### 3. Outlook and perspective

In this section, we briefly discuss the challenges of this research area and a possible roadmap for a future work.

#### 3.1. Challenges from the manufacturing process

3D-RF microstructures, including cotton felts and PET felts, are commonly manufactured by techniques coming from the textile industry. After their selection, the recycled (shoddy, PET) or noble (bicomponent) fibers are provided within different fineness specified by the *dtex* unit which enables a linear density estimates of fiber size, such that a corresponding fiber diameter can be determined. However, the model we developed from which we evidenced a strong potential of optimization is not without drawbacks, as it relies on continuous distribution of fiber diameters. Fig. 7 provides the distribution of fiber diameters obtained through SEM images that corresponds to the F1 cotton felt sample and compares this distribution with the one required to reach Target 4 ($CV = 77\%$, same sample thickness). On the left side of the distribution, the fibrous material presents a relatively high number of very thin fibers. The same is also true for the right side of the distribution, where the fibrous material presents a significant amount of large fibers which were merely absent in the initial sample ($2r_f > 30\mu m$). The asymmetry of the probability distribution function suggests a mix of at least six initial families of fibers of different linear densities (*dtex*) with lower $CV$ (15%) to achieve the required distribution. However, the fiber distribution of the corresponding manufactured material may exhibit mismatch compared to the targeted Gamma law. This mismatch is not necessarily problematic if, as expected, the microstructural descriptors governing the physics of the transport parameters are $D_v$ and $D_{tv}$. Also, the experimental validation of the model is still incomplete, since it relies on samples already manufactured which do not correspond to the expected optimal behavior ($CV \sim 30\% \neq CV \sim 77\%$).

#### 3.2. Future roadmap

In order to address the challenges associated with manufacturing and validation of the model in highly polydispersed 3D-RF microstructures, a crucial aspect of our future roadmap involves the development of a manufacturing method that enables the precise and controlled elaboration of a targeted distribution of polydisperse fiber diameters with $CV \sim 77\%$ from a chosen reduced set of fiber families with $CV \sim 15\%$.
<figure><img src="image_5.png" /><figcaption>**Fig. 6.** Diffuse field sound absorption coefficient SAC<sub>DF</sub> as a function of frequency for cotton felt F1: (a) initial configuration with CV = 40.3% and L<sub>s</sub> = 20mm. The next panels (b) to (f) represent the optimized configurations satisfying targets of increasing difficulty when the polydispersity degree (CV) is allowed to be increased as a microstructural optimization lever, for the targets listed in the legend.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 7.** Comparison of the Gamma distribution of fiber diameters obtained through SEM images (CV = 40.3%), cotton felt F1 and the Gamma distribution of fiber diameters required to reach Target 4 (CV = 77%).</figcaption></figure>

Here, we show that combining several fiber families with moderate polydispersity degree (CV ≈ 15%) and appropriate ratios (Table 2) can exhibit reconstructed fiber distributions of increasing polydispersity degree (CV = 30%, CV = 60%, CV = 75%) with an improved level of sound absorption (Class 3, Class 2, Class 1); Fig. 8. While the reconstructed fiber distributions may not perfectly match the targeted ones due to a limited number of existing fiber families, the associated discrepancies in sound absorption coefficients are minor. This reconstruction method offers a promising solution for achieving a highly polydisperse fiber distribution with controlled characteristics.

Another promising and complementary approach for achieving a controlled fiber diameter distribution with a high degree of polydispersity is to consider emerging techniques for the fabrication of nanofibers [32,33]. In our forthcoming work, we aim to leverage nanofiber fabrication techniques on a chosen basis of polydisperse fiber diameters in our experiments to achieve improved 3D-RF structures with controlled distribution of fiber diameters.

<caption>**Table 2**</caption>

Fiber families with a moderate polydispersity degree (CV ≈ 15%) and the corresponding ratios determined in order to reconstruct distribution of fiber diameters corresponding to classes of fibrous media with increasing sound absorption and polydispersity degree (CV ≈ 30%, Class 3; CV ≈ 60%, Class 2; CV ≈ 75%, Class 1) at constant mean fiber diameter D<sub>m</sub> and layer thickness (L<sub>s</sub> = 20 mm).

<table><thead><tr><th>Fiber</th><th>Dimensionless diameter (D<sub>f</sub>/D<sub>m</sub>)</th><th>Class 3</th><th>Class 2</th><th>Class 1</th></tr></thead><tbody><tr><td>1</td><td>0.28</td><td>-</td><td>-</td><td>28%</td></tr><tr><td>2</td><td>0.34</td><td>-</td><td>20%</td><td>-</td></tr><tr><td>3</td><td>0.64</td><td>-</td><td>25%</td><td>24%</td></tr><tr><td>4</td><td>0.72</td><td>24%</td><td>-</td><td>-</td></tr><tr><td>5</td><td>0.91</td><td>43%</td><td>18%</td><td>10%</td></tr><tr><td>6</td><td>1.30</td><td>33%</td><td>24%</td><td>21%</td></tr><tr><td>7</td><td>1.35</td><td>-</td><td>-</td><td>-</td></tr><tr><td>8</td><td>1.41</td><td>-</td><td>-</td><td>-</td></tr><tr><td>9</td><td>1.69</td><td>-</td><td>-</td><td>-</td></tr><tr><td>10</td><td>1.98</td><td>-</td><td>14%</td><td>15%</td></tr><tr><td>11</td><td>4.30</td><td>-</td><td>-</td><td>3%</td></tr></tbody></table>

## 4. Summary

In summary, this article highlights the potential of using three-dimensional random fibrous microstructures with various distributions of fiber diameters as a way to control visco-thermal dissipation mechanisms in sound absorbing materials. This approach offers numerous advantages including tuning transport parameters across a broad range of values, enhancing the sound absorption average and achieving targets of sound absorption in diffuse field with increasing levels of difficulty. However, challenges persist, such as manufacturing a wide distribution of fiber diameters with prescribed properties. To overcome this challenge, we suggest reconstructing the prescribed distribution with a limited set of six or more families of fibers carefully selected, and exploring the synthesis of nanofibers through emerging techniques such as the electro-spinning method. These perspectives offer promising avenues for further advancements in the field of utilizing 3D-RF microstructures for advanced sound absorbing materials.
<figure><img src="image_7.png" /><figcaption>**Fig. 8.** Gamma distribution of fiber diameters at constant mean fiber diameter $D_m$ for an increasing polydispersity degree (CV = 30%, CV = 60%, CV = 75%); left panel. The dotted lines represent the distributions of each fibers population taken from an existing database (Table 2) from which a targeted degree of polydispersity (solid line) can be approximately reconstructed (dashed line). The sound absorbing coefficient at normal incidence ($SAC_{NI}$) as a function of frequency for each distribution of fiber diameters is plotted for a sample thickness $L_s$ of 20 mm in the two cases listed in the legend; right panel.</figcaption></figure>
<table><caption>**Table A.3**
Key parameters of the differential evolution (DE) algorithm used in the optimization studies.</caption><thead><tr><th></th><th>Section 2.3.2</th><th>Section 2.3.3</th></tr></thead><tbody><tr><td>Fixed parameters</td><td>L<sub>s</sub>, φ, D<sub>m</sub>, β</td><td>φ, D<sub>m</sub>, β, f<sub>i</sub></td></tr><tr><td>Optimized parameters</td><td>CV</td><td>L<sub>s</sub>, CV</td></tr><tr><td>Constraints</td><td>10% ≤ CV ≤ 300%</td><td>1 mm ≤ L<sub>s</sub> ≤ 100 mm<br>10% ≤ CV ≤ 300%</td></tr><tr><td>Fitness function (cost function)</td><td>Minimize: 1 - SAA<sup>125-4500</sup><br>SAA<sup>125-4500</sup> given in Eq. (2)</td><td>Minimize: 1 - SAC<sub>DF</sub><br>with SAC<sub>DF</sub> ≥ 0.8 for f<sub>i</sub> ≥ f<sub>i</sub><br>SAC<sub>DF</sub> given in Eq. (9)</td></tr><tr><td>Population size</td><td>10</td><td>10</td></tr><tr><td>Maximum number of generations</td><td>100</td><td>100</td></tr><tr><td>Mutation control parameter</td><td>0.5</td><td>0.5</td></tr><tr><td>Crossover control parameter</td><td>0.9</td><td>0.9</td></tr><tr><td>Mutation strategy</td><td>DE/rand/1</td><td>DE/rand/1</td></tr></tbody></table>

## CRediT authorship contribution statement

**Quang Vu Tran:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Camille Perrot:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Raymond Panneton:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Minh Tan Hoang:** Writing – review & editing, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Conceptualization. **Ludovic Dejaeger:** Writing – review & editing, Validation, Supervision, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Valérie Marcel:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Mathieu Jouve:** Writing – review & editing, Validation, Resources, Methodology, Investigation.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Quang Vu Tran reports financial support, administrative support, and article publishing charges were provided by Gustave Eiffel University (Ref. EIFFEL 2021-00120). Quang Vu Tran reports financial support and administrative support were provided by Natural Sciences and Engineering Research Council of Canada. Quang Vu Tran reports financial support was provided by Association Nationale de la Recherche et de la Technologie. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was supported by the Association Nationale de la Recherche et de la Technologie and Adler Pelzer Group under convention Cifre No. 2020/0122 (Ref. EIFFEL 2021-00120) and the Natural Sciences and Engineering Research Council of Canada (Ref. RGPIN-2018-06113). We greatly acknowledge the fruitful discussions with Jean-Yves Curien during the course of this work.

## Appendix A. Key parameters of the differential evolution (DE) algorithm used in the optimization studies

The optimizations in Sec. 2.3.2 and Sec. 2.3.3 used a differential evolution (DE) algorithm [34,35]. This evolutionary algorithm works

by iteratively improving a population of potential solutions over multiple generations. Each solution is represented by a set of parameters. The algorithm uses mutation to create variations, crossover to combine features, and selection to choose the best individuals based on their performance in minimizing a fitness function. Based on Sec. 2.3.2 and Sec. 2.3.3, Table A.3 summarizes the key parameters used for both optimizations.

## Data availability

Data will be made available on request.

## References

[1] G. Bhat, S. Malkan, Polymer-laid web formation, in: S. Russell (Ed.), Handbook of Nonwovens, in: Woodhead Publishing Series in Textiles, Woodhead Publishing, 2007, pp. 143–200.

[2] S. Gramsch, A. Klar, G. Leugering, N. Marheineke, C. Nessler, C. Strohmeyer, R. Wegener, Aerodynamic web forming: process simulation and material properties, J. Math. Ind. 6 (2016) 1–23, https://doi.org/10.1186/s13362-016-0034-4.

[3] A. Ahmed, Nonwoven fabric finishing, in: S. Russell (Ed.), Handbook of Nonwovens, in: Woodhead Publishing Series in Textiles, Woodhead Publishing, 2007, pp. 368–400.

[4] W. Albrecht, H. Fuchs, W. Kittelmann, Nonwoven Fabrics: Raw Materials, Manufacture, Applications, Characteristics, Testing Processes, John Wiley & Sons, 2006.

[5] L. Lei, N. Dauchez, J. Chazot, Prediction of the six parameters of an equivalent fluid model for thermocompressed glass wools and melamine foam, Appl. Acoust. 139 (2018) 44–56, https://doi.org/10.1016/j.apacoust.2018.04.010.

[6] X. Tang, X. Yan, Acoustic energy absorption properties of fibrous materials: a review, Composites, Part A, Appl. Sci. Manuf. 101 (2017) 360–380, https://doi.org/10.1016/j.compositesa.2017.07.002, https://www.sciencedirect.com/science/article/pii/S1359835X17302609.

[7] J. Ning, Y. Li, Dynamic flow resistivity and sound absorption of compressed fibrous porous materials: experimental and theoretical, Phys. Fluids 32 (2020) 127103, https://doi.org/10.1063/5.0029879.

[8] J. Auriault, Dynamic behaviour of a porous medium saturated by a newtonian fluid, Int. J. Eng. Sci. 18 (1980) 775–785.

[9] R. Burridge, J.B. Keller, Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am. 70 (1981) 1140–1146.

[10] V. Tarnow, Airflow resistivity of models of fibrous acoustic materials, J. Acoust. Soc. Am. 100 (1996) 3706–3713, https://doi.org/10.1121/1.417233.

[11] O. Umnova, K. Attenborough, K.M. Li, Cell model calculations of dynamic drag parameters in packings of spheres, J. Acoust. Soc. Am. 107 (2000) 3113–3119, https://doi.org/10.1121/1.429340.

[12] B.P. Semeniuik, E. Lundberg, P. Göransson, Acoustics modelling of open-cell foam materials from microstructure and constitutive properties, J. Acoust. Soc. Am. 149 (2021) 2016–2026, https://doi.org/10.1121/10.0003824.

[13] G. Núñez, R. Venegas, T.G. Zieliński, F.-X. Bécot, Equivalent fluid approach to modeling the acoustical properties of polydisperse heterogeneous porous composites, Phys. Fluids 33 (2021) 062008, https://doi.org/10.1063/5.0054009.

[14] D. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech. 176 (1987) 379–402, https://doi.org/10.1017/s0022112087000727.

[15] D. Lafarge, P. Lemarinier, J.-F. Allard, V. Tarnow, Dynamic compressibility of air in porous structures at audible frequencies, J. Acoust. Soc. Am. 102 (1997) 1995–2006, https://doi.org/10.1121/1.419690.

[16] V. Langlois, High-frequency permeability of porous media with thin constrictions. I. Wedge-shaped porous media, Phys. Fluids 34 (2022) 077119, https://doi.org/10.1063/5.0086257.
[17] V. Langlois, C.T. Nguyen, C. Perrot, High-frequency permeability of porous media with thin constrictions. II. Porous media containing thin holed membranes, Phys. Fluids 34 (2022) 077120, https://doi.org/10.1063/5.0093672.

[18] A.M. Chapman, J.J.L. Higdon, Oscillatory Stokes flow in periodic porous media, Phys. Fluids A, Fluid Dyn. 4 (1992) 2099-2116, https://doi.org/10.1063/1.858507.

[19] N. Martys, E. Garboczi, Length scales relating the fluid permeability and electrical conductivity in random two-dimensional model porous media, Phys. Rev. D 46 (1992) 6080, https://doi.org/10.1103/physrevb.46.6080.

[20] A. Koponen, D. Kandhai, E. Hellen, M. Alava, A. Hoekstra, M. Kataja, K. Niskanen, P. Sloot, J. Timonen, Permeability of three-dimensional random fiber webs, Phys. Rev. Lett. 80 (1998) 716, https://doi.org/10.1103/physrevlett.80.716.

[21] C. Peyrega, D. Jeulin, Estimation of acoustic properties and of the representative volume element of random fibrous media, J. Appl. Phys. 113 (2013) 104901, https://doi.org/10.1063/1.4794501.

[22] Q.V. Tran, C. Perrot, R. Panneton, M.T. Hoang, L. Dejaeger, V. Marcel, M. Jouve, Effect of polydispersity on the transport and sound absorbing properties of three-dimensional random fibrous structures, Int. J. Solids Struct. 296 (2024) 112840, https://doi.org/10.1016/j.ijsolstr.2024.112840, https://www.sciencedirect.com/science/article/pii/S0020768324001999.

[23] V. Trinh, J. Guilleminot, C. Perrot, On the sensitivity of the design of composite sound absorbing structures, Mater. Des. 210 (2021) 110058.

[24] A. Mousavi, M. Berggren, E. Wadbro, Extending material distribution topology optimization to boundary-effect-dominated problems with applications in viscothermal acoustics, Mater. Des. 234 (2023) 112302.

[25] Z. Guo, Z. Li, K. Zeng, X. Lu, J. Ye, Z. Wang, Hierarchical-porous acoustic metamaterials: a synergic approach to enhance broadband sound absorption, Mater. Des. 241 (2024) 112943.

[26] D.L. Johnson, J. Koplik, L.M. Schwartz, New pore-size parameter characterizing transport in porous media, Phys. Rev. Lett. 57 (1986) 2564, https://doi.org/10.1103/physrevlett.57.2564.

[27] M. He, C. Perrot, J. Guilleminot, P. Leroy, G. Jacqus, Multiscale prediction of acoustic properties for glass wools: computational study and experimental validation, J. Acoust. Soc. Am. 143 (2018) 3283-3299, https://doi.org/10.1121/1.5040479.

[28] S.G. Advani, C.L. Tucker III, The use of tensors to describe and predict fiber orientation in short fiber composites, J. Rheol. 31 (1987) 751-784, https://doi.org/10.1122/1.549945.

[29] H.T. Luu, C. Perrot, R. Panneton, Influence of porosity, fiber radius and fiber orientation on the transport and acoustic properties of random fiber structures, Acta Acust. Acust. 103 (2017) 1050-1063.

[30] Y. Champoux, J.-F. Allard, Dynamic tortuosity and bulk modulus in air-saturated porous media, J. Appl. Phys. 70 (1991) 1975-1979, https://doi.org/10.1063/1.349482.

[31] J.-P. Groby, N. Jiménez, V. Romero-García, Acoustic Metamaterial Absorbers, Springer, Cham, 2021, pp. 167-204.

[32] G. Chang, X. Zhu, A. Li, W. Kan, R. Warren, R. Zhao, X. Wang, G. Xue, J. Shen, L. Lin, Formation and self-assembly of 3d nanofibrous networks based on oppositely charged jets, Mater. Des. 97 (2016) 126-130.

[33] I. Alghoraibi, S. Alomari, Different Methods for Nanofiber Design and Fabrication, Springer International Publishing, 2019, pp. 79-124.

[34] R. Storn, K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. 11 (1997) 341-359, https://doi.org/10.1023/A:1008202821328.

[35] K. Price, R. Storn, J. Lampinen, Differential Evolution-A Practical Approach to Global Optimization, vol. 141, 2005.