3D printed sound-absorbing materials with double porosity 1

Tomasz G. Zieli´nski 2

Institute of Fundamental Technological Research, Polish Academy of Sciences ul. Pawi´nskiego 5B, 02-106 Warsaw, Poland

Nicolas Dauchez Thomas Boutin Mikel Leturia Alexandre Wilkinson Universié de Technologie de Compiègne, Alliance Sorbonne Université, CNRS Laboratoire Roberval, Centre de recherche Royallieu, CS 60319, 60203 Compiègne cedex, France

Fabien Chevillotte François-Xavier Bécot MATELYS – Research Lab 7 rue des Maraîchers (bâtiment B), F69120 Vaulx-en-Velin, France

Rodolfo Venegas University Austral of Chile, Institute of Acoustics P.O. Box 567, Valdivia, Chile

ABSTRACT The paper shows that acoustic materials with double porosity can be 3D printed with the appropriate design of the main pore network and the contrasted microporous skeleton. The microporous structure is obtained through the use of appropriate additive manufacturing (AM) technology, raw material, and process parameters. The essential properties of the microporous material obtained in this way are investigated experimentally. Two AM technologies are used to 3D print acoustic samples with the same periodic network of main pores: one provides a microporous skeleton leading to double porosity, while the other provides a single-porosity material. The sound absorption for each acoustic material is determined both experimentally using impedance tube measurements and numerically using a multiscale model. The model combines finite element calculations (on periodic representative elementary volumes) with scaling functions and analytical expressions resulting from homogenization. The obtained double-porosity material is shown to exhibit a strong permeability contrast resulting in a pressure di ff usion e ff ect, which fundamentally changes the nature of the sound absorption compared to its single-porosity counterpart with an impermeable skeleton. This work opens up interesting perspectives for the use of popular, low-cost AM technologies to produce e ffi cient sound absorbing materials.

1 This paper presents preliminary results of an extended comprehensive work submitted to Applied Acoustics .

2 tzielins@ippt.pan.pl

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

Various AM technologies [1–3] are widely used recently to develop new acoustic materials and metamaterials [4–9], despite the fact that popular, inexpensive AM devices allow the formation of designed microstructures in relatively low resolution compared to the characteristic sizes found in e ffi cient, conventional, sound-absorbing materials. Although the reproducibility of 3D printed porous sound absorbers has been recently confirmed through round robin tests involving various technologies, input materials, and 3D printers [10], manufacturing imperfections related to these technologies have been identified as a serious impediment when designing and prototyping such acoustic solutions. However, it has also been shown that some of these imperfections – unless they significantly alter the engineered geometry – may enhance acoustic properties. A good example here is the surface roughness, typical of many AM technologies, which usually increases a little the sound absorption of the 3D printed porous material thanks to the enhancement of dissipation associated with viscous e ff ects [10, 11]. This work shows that another significant imperfection, namely porosity or rather microporosity, common with some AM technologies using powder as raw input material [12–14], can under certain conditions be used to 3D print materials with strongly enhanced sound-absorbing properties due to the double porosity thus realised. The outline of this paper is as follows. The design and manufacturing of samples with a periodic pore network, as well the results of their microscopic examination are discussed in Section 2.1. This section also presents the results of direct measurements of the relevant properties of the 3D printed microporous material. Theoretical considerations and the necessary formulas for rigorous modelling related to the microstructure and sound-absorbing properties of the investigated materials are discussed in Section 2.2. Section 2.3 provides an overview and comparison of the measurement and modelling results. The main findings and conclusions are summarised in Section 3.

2. INVESTIGATIONS

2.1. Design and manufacturing A simple periodic network of spherical pores was designed and used to 3D print acoustic samples in two di ff erent AM technologies: one technology produces a microporous skeleton leading to double porosity, while the other provides a single-porosity material with an impermeable skeleton. In this way, the double-porosity material should have the same main pore network as its single-porosity counterpart, allowing for a comparative analysis of acoustic e ffi ciency, quantified through their sound absorbing properties. The representative elementary volume (REV) containing the periodic main pore network is presented in Figure 1. It is a cubic cell of size ℓ p = 4 mm, containing a single spherical pore with a diameter d sp = 0 . 9 ℓ p = 3 . 6 mm. The pore is directly connected with six neighbouring pores from the adjacent cells by two vertical and four horizontal channels, all of them cylindrical with the same

pore network cubic cell skeleton

ℓ p

pores

d sp

d ch

fillets channels

Figure 1: Periodic REV in the form of a cubic cell composed of complementary parts of the periodic pore network (blue) and microporous or solid skeleton (orange).

diameter d ch = 0 . 4 ℓ p = 1 . 6 mm. Note that the periodic cubic cell shown in Figure 1 is periodically shifted so that it actually contains two separate pore halves connected by the vertical channel. The connections between each pore and the channels are rounded with a fillet radius of r f = 0 . 25 d ch . The porosity associated with such a main pore network will be denoted by ϕ p . Its nominal, i.e. designed, value ϕ p = 44 . 1% will be corrected for actual 3D printed samples. A computer-aided design (CAD) model of a cylindrical sample with a diameter of 31 mm (i.e. 2 mm larger than the target value) and height H = 9 ℓ p = 36 mm was virtually cut out from a three- dimensional array of 8 × 8 × 9 representative skeletal cells, see Figure 2. This model was used to 3D print samples using two di ff erent AM technologies: (i) Binder Jetting 3D printing (BJP) [2] – from a gypsum-based powder bounded with butyrolactam, and (ii) Stereolithography (SLA) [3] – from a photopolymer resin. The diameter of each 3D printed sample was tightly trimmed to perfectly fit the impedance tube of diameter D t = 29 mm, see Figure 2.

D t

D t

H

CAD model uncut sample trimmed sample

Figure 2: CAD model used to 3D print samples and the corresponding gypsum sample before and after it has been trimmed to fit perfectly into the impedance tube.

The flat surfaces of both samples were examined under a microscope (see Figure 3) to find that the quality of the resin sample is very good with all shapes very well mapped and all surfaces smooth, while the surfaces of the gypsum sample are rather rough and in the photography it is even di ffi cult to

Channel C=4.777 mm .766 mm*2 130,750 mm 1.945 mm*2 -950 mm Cell L=4.003 mm

gypsum sample resin sample

Figure 3: Microscopically examined surfaces of the 3D printed double-porosity gypsum sample (left) and single-porosity resin sample (right).

5003 mm Creagial C=4. 76a 1=0.750 mm: Cell L=4.003 mm - —- -

precisely define the actual shape and size of the pore. Nevertheless, in both cases the actual diameter of the channel can be reliably determined as d ch = 1 . 50 mm, which means that it is reduced by 0 . 1 mm from the value present in the CAD model used for 3D printing. The pore size in the resin sample is also accurately determined as d sp = 3 . 56 mm, so it is only slightly smaller than the nominal pore diameter in the CAD model, while the pore in the gypsum sample appears to be slightly larger and its actual diameter can be estimated as d sp ≈ 3 . 9 mm. These actual values were used to update the designed geometry of the periodic REVs that are used for numerical calculations during the modelling (see Sections 2.2 and 2.3). The results of preliminary experiments suggested that the skeleton of the gypsum sample was microporous. Therefore, homogenous (i.e. without the main pore network) disc-shaped specimens were 3D printed from the gypsum-based power in BJP technology, using the same 3D printer, binder, and process parameters as those used for the periodic cylindrical sample. These specimens were used for direct measurements of their microporosity and airflow resistivity of this 3D printed material. The measured microporosity is ϕ m = 42 . 6%, while the airflow resistivity is 32 . 24 MPa · s / m 2 . The latter means that the viscous permeability of the microporous material is K 0m = 5 . 7 · 10 − 13 m 2 .

2.2. Modelling Acoustic waves penetrating air-saturated porous media with open pore networks and rigid frames induce harmonic flow and pressure fluctuations in the pores [15–18]. These phenomena are described by the complex amplitudes of the flux velocity V and pressure p . Assuming the time-harmonic convention exp( + i ω t ), where i is the imaginary unit, ω is the angular frequency, and t denotes time, the wave propagation in such media can be described by the e ff ective model comprising the macroscopic mass balance equation and dynamic Darcy’s law [15], viz.

∇· V + i ω C e ( ω ) p = 0 , V = − K e ( ω )

η ∇ p , (1)

where η is the dynamic viscosity of air saturating the pores, while C e and K e are the e ff ective compressibility and dynamic viscous permeability, respectively. Isotropic porous media are considered here, so C e and K e are complex-valued scalar functions of frequency. Other useful e ff ective properties can be introduced, in particular the e ff ective density ϱ e and speed of sound c e , i.e.

s

i ω K e ( ω )

ϱ e ( ω ) = η i ω K e ( ω ) , c e ( ω ) =

η C e ( ω ) . (2)

Eliminating V from Equation 1 leads to the classic Helmholtz equation for time-harmonic acoustics, viz. ∇ 2 p + k 2 e ( ω ) p = 0 , where k e ( ω ) = ω/ c e ( ω ) is the complex wave number in the e ff ective fluid equivalent to the porous medium. Single- and double-porosity materials are studied in this work and their e ff ective properties (denoted above by the subscript ‘e’) as well as some specific parameters are distinguished below with the subscripts ‘p’ or ‘db’, respectively. The main di ff erence between these two material types is the skeleton which can be impervious or microporous, and in the latter case, together with the main pore network, creates a double porosity material with highly contrasted permeabilities. The parameters and e ff ective properties associated with the microporous material of the skeleton are denoted by the subscript ‘m’. The dynamic viscous permeability K p for a single-porosity medium with open porosity ϕ p can be determined using the well-known scaling function of the so-called Johnson-Champoux-Allard- Lafarge-Pride (JCALP) model [15,19–22], viz.

− 1

s

 i ω ω vp + 1 −P vp +



P 2 vp + M vp

2 i ω ω vp

, ω vp = ϕ p ν K 0p α ∞ p , (3)

K p ( ω ) = K 0p

where K 0p ≡K p (0) is the static viscous permeability, ω vp is the Biot frequency, while M vp = 8 K 0p α ∞ p / ϕ p Λ 2 vp
and P vp = 0 . 25 M vp α ∞ p / α 0vp − α ∞ p
are the corresponding shape factors. Moreover, ν is the kinematic viscosity of air saturating the pores, and the remaining parameters are the (kinematic) tortuosity α ∞ p , static viscous tortuosity α 0vp , and viscous characteristic length Λ vp . Five real-valued parameters associated with the pore network are required to calculate the complex- valued function K p ( ω ), i.e. ϕ p , K 0p , Λ vp , α ∞ p and α 0vp . These can be determined directly from the pore geometry ( ϕ p ) or by solving the specific Stokes’ flow ( K 0p , α 0vp ) and Laplace’s problem ( Λ vp , α ∞ p ) formulated in the pore space of the periodic REV [18,23,24] The e ff ective compressibility C p for single-porosity media is calculated as

γ Θ p ( ω )

i ω ω tp

1 − γ − 1

C p ( ω ) = ϕ p

! , (4)

P 0

Θ 0p

where P 0 is the ambient mean pressure and γ is the adiabatic index for air, while the so-called dynamic thermal permeability Θ p can be determined from the scaling function of the JCALP model, i.e.

− 1

s



 i ω ω tp + 1 −P tp +

P 2 tp + M tp

2 i ω ω tp

, ω tp = ϕ p τ

Θ 0p , (5)

Θ p ( ω ) = Θ 0p

where Θ 0p ≡ Θ p (0) is the static thermal permeability, ω tp is the thermal characteristic frequency, while M tp = 8 Θ 0p / ϕ p Λ 2 tp
and P tp = 0 . 25 M tp / α 0tp − 1
are the corresponding shape factors. Moreover, τ is the thermal di ff usivity of air, while the remaining parameters are the static thermal tortuosity α 0tp and thermal characteristic length Λ tp . Four real-valued parameters associated with the pore network are required to calculate the complex-valued function Θ p ( ω ), i.e. ϕ p , Θ 0p , Λ tp , and α 0tp . These can be determined directly from the pore geometry ( ϕ p , Λ tp ) or by solving the specific Poisson’s problem ( Θ 0tp , α 0tp ) formulated in the pore space of the periodic REV [18,23,24]. The microporous material is a specific single-porosity case. Therefore, its e ff ective properties can determined from the formulas presented above, where the subscript ‘p’ is replaced with ‘m’, which means that all macro-parameters are determined for the microporous network. However, due to the fact that the micropores are by definition very small, one can expect that in the frequency range of interest, e.g. up to 6 . 4 kHz, the dynamic viscous permeability K m and e ff ective compressibility C m are practically constant and equal to their (real-valued) static counterparts, i.e. K m ( ω ) ≈K m (0) ≡K 0m and C m ( ω ) ≈C m (0) ≡C 0m = ϕ m / P 0 . It is therefore su ffi cient to determine only the static viscous permeability K 0m and porosity ϕ m of the microporous material, and these properties were measured directly for disc-shaped material specimens 3D printed from the gypsum powder (see Section 2.1). In this study, high permeability contrast is assumed between the main pore network and the microporous skeleton, i.e. K 0p ≫K 0m , which means that visco-inertial flow is restricted mainly to the main pores and the dynamic viscous permeability K db for such double-porosity materials can be determined as for their single-porosity counterparts, i.e. K db ( ω ) ≈K p ( ω ). Still, the e ff ective compressibility C db depends not only on the air compression and the associated heat dissipation e ff ects in the main pore network, but also on those in the microporous skeleton. Moreover, an additional dissipation e ff ect may occur due to the fact that two local acoustic pressure fields can coexist, namely a locally constant pressure field in the more permeable pore network and a pressure field that varies locally in the less permeable microporous skeleton, which ultimately leads to pressure di ff usion that provides additional sound energy dissipation. Therefore, the e ff ective compressibility C db for a double-porosity material [16,17] is calculated as

i ω ω d . (6)

C db ( ω ) = C p ( ω ) + ϕ d C 0m F d ( ω ) , F d ( ω ) = 1 − B d ( ω )

B 0d

where ϕ d = 1 − ϕ p is the volume fraction of the microporous skeleton and F d is the ratio of the averaged pressure locally fluctuating in the microporous domain to the pressure in the main pores [16,17]. The

latter depends on the so-called dynamic pressure di ff usion function B d which can be determined using the scaling function as

− 1

s

 i ω ω d + 1 −P d +



2 i ω ω d

, ω d = ϕ d D 0m

P 2 d + M d

B d ( ω ) = B 0d

B 0d , (7)

where B 0d ≡B d (0) is the static pressure di ff usion, ω d is the pressure di ff usion characteristic frequency, while M d = 8 B 0d / ϕ d Λ 2 d
and P d = 0 . 25 M d / α 0d − 1
are the corresponding shape factors. Here, the so-called static pressure di ff usivity of microporous material D 0m = K 0m / η C 0m
is introduced, and two additional parameters related to the pressure di ff usion occurring in the microporous domain are the corresponding static tortuosity α 0d and characteristic length Λ d . Four real-valued parameters associated with the microporous domain are required to calculate the complex-valued function B d ( ω ), i.e. ϕ d , B 0d , Λ d , and α 0d . These can be determined directly from the domain’s geometry ( ϕ d , Λ d ) or by solving the specific Poisson’s problem ( B 0d , α 0d ) formulated in the microporous domain of the periodic REV [16,17]. When the dynamic viscous permeability and e ff ective compressibility as well as other useful e ff ective properties (see Equation 2) have been determined for porous media with micro-, single- or double-porosity, the sound absorption coe ffi cient A at normal incidence can be calculated (from the solution of the relevant Helmholtz problem) for a porous layer of a specified thickness H , made of one of the determined materials and backed by a rigid wall or, alternatively, by an air gap of a thickness H g between the wall and porous layer. For this purpose, the surface acoustic impedance [15] is first determined as



− i Z e ( ω ) cot Hk e ( ω )
for a porous layer backed by a rigid wall ,

Z s ( ω ) =

Z g ( ω ) − i Z e ( ω ) cot Hk e ( ω )
for a porous layer backed by an air gap . (8)

Z e ( ω ) Z e ( ω ) − i Z g ( ω ) cot Hk e ( ω )

Here, Z e ( ω ) = ϱ e ( ω ) c e ( ω ) is the e ff ective characteristic impedance of the porous material of the layer and Z g ( ω ) = − i Z 0 cot H g ω/ c 0
is the impedance of the air gap, where c 0 and Z 0 are the speed of sound and characteristic impedance of air, respectively. Then, the acoustic reflection coe ffi cient R ( ω ) = Z s ( ω ) − Z 0
/ Z s ( ω ) + Z 0
and sound absorption A ( ω ) = 1 −  R ( ω )  2 are calculated [15].

2.3. Results and discussion To discuss the nature of the manufactured materials, the experimental results will be compared with the corresponding predictions obtained from numerical modelling. Therefore, the air properties used in calculations were determined for the ambient conditions of pressure ( P 0 = 996 . 3 hPa), temperature (27 . 6 ◦ C), and relative humidity (53%), found during the experimental tests in the impedance tube. Table 1 shows parameters calculated numerically from the periodic REVs with geometries updated thanks to microscopic examination of real samples. These updates were a little di ff erent for the gypsum sample and for the resin sample, so the visco-inertial and thermal parameters calculated for

Table 1: Parameters calculated numerically for the gypsum (BJP) and resin (SLA) samples.

Visco-inertial e ff ects Thermal e ff ects Pressure di ff usion

ϕ p K 0p Λ vp α 0vp α ∞ p ϕ p Θ 0p Λ tp α 0tp ϕ d B 0d Λ d α 0d % 10 − 8 m 2 mm – – % 10 − 8 m 2 mm – % 10 − 8 m 2 mm –

BJP 50.7 2.66 0.862 2.65 1.88 50.7 14.4 1.56 1.36 49.3 11.9 1.52 1.36 SLA 42.3 2.10 0.850 2.5 9 1 . 8 7 42.3 9.86 1.40 1.38 — — — —

one sample di ff er slightly from those calculated for the other. The parameters associated with the pressure di ff usion were calculated only for the gypsum sample which has a microporous skeleton. The parameters listed in Table 1 were used to determine the e ff ective properties for both samples as discussed in Section 2.2. In particular, to compute the e ff ective compressibility for the double-porosity gypsum sample, the pressure ratio function F d ( ω ) was determined, see Figure 4. This function indicates whether pressure di ff usion takes place and over what frequency range this e ff ect is strong. The pressure di ff usion is present at frequencies where the function F d is essentially complex-valued, which means that there is a phase shift between the pressure in the main pore network and the micropores, leading to increased dissipation of acoustic energy. Figure 4 shows that pressure di ff usion becomes significant around 1 kHz and above, reaching its maximum around the characteristic frequency f d = ω d / (2 π ) = 4 . 75 kHz. The pressure di ff usion becomes weak below 1 kHz and very weak below 600–700 Hz.

f d

1

+ Re F d − Im F d − phase( F d ) /π

0 . 9

0 . 8

0 . 7

Pressure ratio function

0 . 6

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

10 1 10 2 10 3 10 4 10 5 10 6 0

Frequency (Hz)

Figure 4: The frequency-dependent ratio F d of the averaged pressure in the micropores to the pressure in the main pores of the periodic cell, responsible for pressure di ff usion in the gypsum sample with double porosity.

The sound absorption coe ffi cient of both samples was measured in an impedance tube, for frequencies up to 6 . 4 kHz. The samples were backed by a rigid piston closing the tube, but in addition, the measurements were also carried out with an air gap of thickness H g = 44 mm between the rear face of the sample and the piston. The results are compared in Figure 5 with the corresponding predictions calculated as explained in Section 2.2. Sound absorption predictions were also shown for a microporous layer of the same thickness H = 36 mm as the height of the samples, to show that this absorption is very poor and that such microporous materials can only be acoustically e ffi cient when used as constituents in double-porosity solutions. This is actually demonstrated here by comparison of the corresponding absorption curves measured (and, in fact, very well predicted) for the double-porosity gypsum sample and the single-porosity resin sample. It becomes evident that the presence of micropores in the skeleton of the gypsum sample produces strong additional dissipation e ff ects related mainly to pressure di ff usion. In general, the sound absorption in the double-porosity material – when compared with the results obtained for the single-porosity material with the same (or very similar) main pore network – is much better in almost the entire frequency range: the absorption peaks are shifted to lower frequencies and the sound absorption between them is significantly increased. Specifically, the results found for porous layers (samples) backed by a rigid

1

H : 36 mm

0 . 9

0 . 8

Sound absorption coe ffi cient

0 . 7

0 . 6

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 0

Frequency (Hz)

Measurements: gypsum sample resin sample impregnated gypsum Calculations: double porosity single porosity microporosity

1

H : 36 mm + air gap: 44 mm

0 . 9

0 . 8

Sound absorption coe ffi cient

0 . 7

0 . 6

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 0

Frequency (Hz)

Figure 5: Sound absorption measured and calculated for the gypsum sample with double porosity and for the single-porosity resin and impregnated gypsum samples, backed by a rigid wall (upper graph) or by an air gap (lower graph), also the corresponding sound absorption predictions for a microporous layer of the same thickness as the 3D printed samples.

wall (piston), shown in the upper graph in Figure 5, reveal that the first absorption peak has been improved: from 0 . 6 at about 1 . 7 kHz (single-porosity material) to 0 . 73 at 1 . 4 kHz (double-porosity material). However, when each porous layer is backed by the air gap (so that the total thickness of such a double layer is 80 mm), the first absorption peak is very similar for both samples (see the lower graph in Figure 5), as it has been shifted to much lower frequency of about 600 Hz, where the pressure di ff usion e ff ect is weak (see Figure 4).

e ~ es e e e @ =

To show that such a significant improvement in sound absorption is mainly achieved due to the double porosity, the gypsum sample was impregnated with cyanoacrylate in order to close the micropores in its skeleton, making it a single-porosity material. Then, the impregnated sample was tested again to find that the corresponding absorption curves are very similar to the ones obtained for the single-porosity resin sample, although with noticeably, but not significantly, greater absorption between the peaks (see Figure 5). The latter can be attributed to the fact that the skeleton surfaces of the gypsum sample are rough as opposed to the perfectly smooth surfaces of the resin sample, and thus, the viscous dissipation e ff ects are slightly increased in the gypsum specimen.

3. CONCLUSIONS

Prototypes of e ffi cient acoustic materials with double porosity can be designed and easily manufactured with some low-cost AM technologies using powders as raw input materials and allowing inherent microporosity. This is because the appropriate high permeability contrast between the 3D printed microporous skeleton and the designed main pore network can be readily achieved, leading to the phenomenon of pressure di ff usion which is an additional, e ffi cient sound energy dissipation mechanism. These observations have been proven in dedicated experimental tests combined with rigorous multi-scale modelling which gives accurate predictions provided that the microporosity and airflow resistivity of the microporous skeleton have been determined, e.g. with direct measurements, and the real sizes and shapes of the designed pore network are known. In practice, the designed geometry can be updated on the basis of microscopic examination of samples or – in some cases – from the 3D printer user’s experience acquired in this way. All this opens up great prospects for the design and prototyping of innovative acoustic materials.

ACKNOWLEDGEMENTS

T. G. Zielinski acknowledges the financial support from the project “Sound-absorbing composites: coupled acoustic energy dissipation mechanisms, multiscale modelling and prototyping”, financed under Grant Agreement No. 2021 / 41 / B / ST8 / 04492 by the National Science Centre (NCN), Poland. R. Venegas acknowledges support from the Chilean National Agency for Research and Development (ANID) through FONDECYT Grant No. 1211310.

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