Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling Ju Hyuk Park a, Sei Hyun Yang a, Hyeong Rae Lee b, Cheng Bin Yu a, Seong Yeol Pak a, Chi Sung Oh c, Yeon June Kang b,n, Jae Ryoun Youn a,nn a Research Institute of Advanced Materials (RIAM), Department of Materials Science and Engineering, Seoul National University, Seoul 08826, Republic of Korea b Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Republic of Korea c NVH Research Lab at the Hyundai Motor Group in Seoul, Republic of Korea

a r t i c l e i n f o

abstract

Article history: Received 8 September 2016 Received in revised form 27 February 2017 Accepted 6 March 2017 Handling editor: O. Ganilova

Sound absorption of a polyurethane (PU) foam was predicted for various geometries to fabricate the optimum microstructure of a sound absorbing foam. Multiscale numerical analysis for sound absorption was carried out by solving flow problems in representative unit cell (RUC) and the pressure acoustics equation using Johnson-Champoux-Allard (JCA) model. From the numerical analysis, theoretical optimum cell diameter for low frequency sound absorption was evaluated in the vicinity of 400 μm under the condition of 2 cm80 K (thickness of 2 cm and density of 80 kg/m3) foam. An ultrasonic foaming method was employed to modulate microcellular structure of PU foam. Mechanical activation was only employed to manipulate the internal structure of PU foam without any other treatment. A mean cell diameter of PU foam was gradually decreased with increase in the amplitude of ultrasonic waves. It was empirically found that the reduction of mean cell diameter induced by the ultrasonic wave enhances acoustic damping efficiency in low frequency ranges. Moreover, further analyses were performed with several acoustic evaluation factors; root mean square (RMS) values, noise reduction coefficients (NRC), and 1/3 octave band spectrograms. & 2017 Elsevier Ltd All rights reserved.

Keywords: Ultrasonication Cell structure control Acoustic damping Poroacoustics modeling Finite element analysis

1. Introduction There are many issues to improve the driving conditions for better quality of life. Among them, noise pollution is a critical problem in daily life [1–4]. In order to solve this problem, many sound proof materials and systems have been developed over a long period of time. There are two kinds of sound proof materials: sound absorption materials and sound insulation materials. Sound insulation materials are generally massive materials having a high surface density [5–7]. These materials having high transmission loss are capable of reflecting the sound energy to the incident direction. Unlike the sound insulation materials, sound absorption materials are much lighter materials with porosity higher than 90% [8–12]. Porous sound absorption materials have two sound damping functions; structural damping in a fluid domain (air) and material n Correspondence to: Institute of Advanced Machinery and Design, School of Mechanical and Aerospace Engineering, Seoul National University, Republic of Korea. nn Corresponding author. E-mail addresses: [email protected] (Y.J. Kang), [email protected] (J.R. Youn).

http://dx.doi.org/10.1016/j.jsv.2017.03.004 0022-460X/& 2017 Elsevier Ltd All rights reserved.

Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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damping in a solid domain (i.e., polymeric part) [12,13]. For the structural damping, acoustic wave propagation in porous media mainly dissipates as viscous friction on interconnected pores and thermal heat exchange on solid-fluid boundary, due to complex microscale porous structures [14–16]. Theoretical models reflect these two mechanisms as complex effective density ( ρ( ω)) and effective bulk modulus (K ( ω)), and these mechanisms will be discussed in Numerical simulation section. In the aspect of the material damping, a transferred acoustic wave from the air to the solid diminishes by molecular friction in vibration modes [17,18]. According to these mechanisms, the energy of offensive noise transforms into heat loss. For the sound absorption, porous polymeric foams are used in buildings, automobiles, and aircraft [19–21]. Polyurethane (PU) foams are the most frequently employed polymeric foams because of its light weight, low cost, and good processability in industrial areas. In the fabrication process of PU foams, Reactant A (composed of polyol, foaming agent, catalyst, surfactants and etc.) and Reactant B (composed of isocyanates) are mixed at high speed, and then the mixture is inserted into a mould to manufacture a desirable shape [22–25]. Two chemical reactions occur simultaneously; a gelling reaction and a blowing reaction. From the gelling reaction, covalent chemical bonds are generated with –OH groups in polyol and –NCO groups in isocyanate, yielding urethane linkages. The blowing reactions are classified into chemical and physical blowing reactions. In the former reaction, the chemical foaming agent (e.g., water) reacts with the isocyanate functional groups (–NCO), and then carbon dioxide gas yields the nuclei when they contain sufficient energy. The generated nuclei are grown up into bubbles with microsize diameter. In the process of physical blowing, the physical blowing agent dissolved in polyol (e.g., HFC, Cyclopentane) acts as a nucleus after forming a cluster by triggering phase change of the dissolved gases under heating or pressure drop process. Damping of the low frequency noise has attracted considerable attentions for many industrial applications, since general sound absorbing materials do not show good performance in low frequencies [26–30]. Moreover, there were only two solutions for the improvement of low frequency sound damping: an increase of overall mass density [31,32] or a utilization of thickness of foams [33]. However, these solutions have limitations for industrial applications. So, the microstructure manipulation is considered as the best strategy to enhance or optimize the acoustic damping properties without changing other intrinsic properties. Several groups have studied the manipulation of the microstructures to improve thermal and mechanical properties. Lee et al. [9], Verdejo et al. [32], Zhai et al. [34], and Wee et al. [35] have fabricated composite foams with nanoparticles to promote heterogeneous nucleation. If nanoparticles were suspended in a polymeric matrix during a foaming reaction, critical energy needed for generation of bubbles would be decreased on the surface of the particles due to lower surface tension [34]. The other methodology for the manipulation of a cellular structure is an ultrasonic foaming method [22,25,36,37]. Irradiation period, timing, and magnitude are most significant processing conditions. Wang et al. [38], Zhai et al. [39], and Gandhi et al. [40] reported the relation between exposure time and the cell size distribution with respect to power of an ultrasonic wave. Torres-Sanchez et al. [41–43] investigated the effect of ultrasonication on porosity of foam by carrying out a simulation of the acoustic environment. Various theoretical analysis methods have been performed to model and predict the sound absorption performance of the porous materials [13,44–51]. Among them, a multiscale modeling using finite element method [13,46,47,52–54] was reported recently to consider not only macroscopic parameters but also microstructural parameters. For microscale simulation, a representative unit cell (RUC) is made by considering a real geometry of the sample. Poroacoustics parameters for macroscale simulation (e.g., flow resistivity) are obtained by solving two flow problems based on the RUC. Sound absorption coefficient is calculated by an analytical or numerical method with such theoretical models as Diphasic models (e.g., Biot’s theory [18]) and Motionless skeleton models (e.g., Johnson-Champoux-Allard model [15,16]). So the attribution of a microcellular structure can be evaluated theoretically and compared with measured data, resulting in the high reliability of the current research. In this study, we theoretically demonstrated effects of the cell size on sound absorption properties from a multiscale numerical analysis using a finite element method. From the theoretical results, the optimum cell diameter of 80 K density and 2 cm thick PU foam was estimated. The 80 K density and 2 cm thickness are specific conditions applied for sound absorbing foams in automobile industries. We generated polyurethane foams with the cell size in the vicinity of the optimum value. And the cell size was manipulated by irradiation of the ultrasonic waves to the resin mixture during the foaming process. It is noted that enhancement of low frequency sound damping could be governed by the mechanical excitation which can increase the bubble nucleation rate. Acoustic absorption coefficients were measured experimentally by employing the B&K impedance tube and additional characterizations, e.g., root mean square (RMS) analysis, noise reduction coefficient (NRC), and 1/3 octave band spectrogram, were conducted.

2. Materials and methods 2.1. Materials NIXOL SA-120 (Reactant A, KPX Chemical, Republic of Korea) is composed of polyether polyol (
94 wt%), water (
2.3 wt%), surfactants, and catalysts. SUPRASECs 2527 (Reactant B, Huntsman Holland BV, Netherlands) is an isocyanatebased compound of diphenylmethane 4,4'-diisocyanate (MDI,
50 wt%) and isocyanic acid (
20 wt%), and other additives. AKO-HM207K (Akochem, Republic of Korea) was used as a releasing agent of PU foams. Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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2.2. Fabrication of polyurethane foam with ultrasonication A foaming reaction was initiated after high speed mixing at around 3000 rpm running a digital overhead stirrer HS-100D (WiseStirs, Germany) with 5:2 mixture of NIXOL SA-120 and SUPRASECs 2527. After the high speed blending for 5 s, the mixture was inserted into a pre-heated mould (at 50 °C) of 15  15  2 cm3. In this step, the ultrasonic wave was irradiated for 7 s at 20 kHz frequency right after the high speed mixing. Ultrasonication power was varied from 0%, 37% to 75% of the maximum power (750 W) to estimate the effect of the intensity of the negative pressure (respectively 0 MPa, -8.5 MPa, and -15 MPa) and they were labeled as 80-1, 80-2, and 80-3 respectively. After the mixture was poured into the mould, the mould was covered to fabricate the samples having constant overall mass density and thickness. Finally, the mixture was cured for 15 min. 2.3. Scanning electron microscopy (SEM) Cell diameters and interconnecting pore diameters were observed by using a field emission scanning electron microscope (FE-SEM, JEOL, Japan). The fabricated PU foams were cut into specimens, after freeze drying using liquid nitrogen to sustain the structure of microscale cells. Each cell diameter was estimated by the average length of long and short axis sizes of the cell and the total mean value of the cell diameter was evaluated from average diameter of each cell in the FE-SEM images. The diameters of the interconnecting pores and the total mean value of the interconnecting pores were also obtained with the previous method. 2.4. Acoustic absorption measurement Normal incidence absorption coefficient was measured by using a B&K two-microphone standing wave tube type 4206 (29 mm diameter) (ASTM E1050 [55], see Fig. S1(a) in Supplementary material). The sound pressure at two microphones is measured and reflection coefficient (R) is calculated by employing the transfer equation which governs the sound pressure ratio between the two microphones. Finally, the absorption coefficient (α ) is obtained by the equation α = 1− R 2 . 2.5. Flow resistivity measurement Flow resistivity is one of the parameters for evaluating the acoustic performance such as sound absorption coefficient and transmission loss. Flow resistance is defined as the ratio of the air pressure differential to the steady state air velocity and the flow resistivity is the flow resistance per unit material thickness. The flow resistivity is defined as Rf = p2 − p1 S /vh , where v is the volumetric airflow rate passing through the material in m3/s, S is the cross sectional area of the material, and h is the thickness of the material. The flow resistivity has the unit of N m4 s or MKS Rayls/m. The flow resistivity of porous materials was measured by using the equipment as shown in Fig. S1(b) (in Supplementary material) according to ISO 9053.

(

)

3. Numerical calculation method 3.1. Unit cell modeling method Unit cells were constructed using CATIA program by mimicking a real topology of the microstructure of the foams shown in SEM images (Fig. 1(a) and (b)). A microcellular topology of 80 K PU foam is exemplified in Fig. 1(a). Tetrakaidecahedron has been known as the most ideally suitable structure for resembling the microgeometry of polymeric foams [56–58]. In recent reports, a method generating solid walls with a hole at the center of each face was developed, in order to deal with the narrow interconnecting pores of high density foams [13,59,60].

Fig. 1. A process for a construction of RUC. (a) Combination of tetrakaidecahedron mimicking the real topology of PU foam. (b) Constructed RUC after Boolean operation.

Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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The diameter of the tetrakaidecahedron (D ), defined as the perpendicular distance between opposite sides of squares, was set to be the mean cell diameter of each sample. And the length of ligaments (L ) was automatically fixed as D=2 2 L . The mean interconnecting pore diameter (I ) obtained from SEM images was used as the diameter of interconnecting pores of each face which has roughly 1/7 size of the mean cell diameter. The porosity was fixed to be 92% coinciding with the mass density of the foams (80 kg/m3), and it was properly adjusted by manipulating the diameter of balls (R ), diameter of ligaments (r ), and thickness of solid walls (t ). 15 tetrakaidecahedron were combined to construct a cluster representing the microscale pore structure of unit cells (Fig. 1(a)). After imparting periodic hexahedron, representative unit cells (RUC’s) were built by subtracting the combined structure from the hexahedron using Boolean operation in CATIA program (Fig. 1(b)). All RUC’s with various cell diameters were constructed by Scaling operation with the same aspect ratio and used for simulation with periodic boundary conditions on the lateral faces. 3.2. Poroacoustics simulation method Numerical simulation was carried out to predict the effect of the cell size on the sound absorption performance by using a multiscale poroacoustics method. In the multiscale simulation, four flow parameters (flow resistivity (Rf ), thermal characteristic length (TCL, Lth ), viscous characteristic length (VCL, LV ), and tortuosity factor (τ∞)) are calculated by solving two s flow problems using COMSOL Multiphysics based on finite element method (FEM) [13,45–47]. One of the flow problems, a viscous flow problem, is solved using Stokes' equation with boundary conditions as [13,46,47,53,54,61,62]

μ∇2v − ∇p = g in Ω f

(1)

∇·v =0 in Ω f

(2)

v = 0 on Ωsf

(3)

where μ is the viscosity of air, p is the pressure, v is the velocity field, Ωf is the fluid domain, g is the constant vector field of pressure gradient throughout the fluid domain, and Ωsf is the boundary of the solid-fluid domain. By solving Eqs. (1)–(3), the velocity vector field is obtained and the permeability field is defined as k0= − μv / g , where g is the pressure gradient (Pa/m) constant throughout the whole fluid domain for RUC. Thus, the flow resistivity of the foam corresponding to RUC is calculated by Rf =μ/( < k 0 >f ·ϵp), where ϵp is the porosity [46]. f ·ϵp is the permeability of porous medium, calculated as the product of ϵp and the averaged value of permeability field, k0 , of the RUC considering the whole fluid domain. The other flow problem, an inertial flow problem, can be replaced by an electrical conduction since the inertial flow behaves like incompressible-inviscid ideal flow [63]. Equations of the electrical conduction (Laplace problem) with boundary conditions [46] are

E =−∇φ+e in Ω f

(4)

∇·E =0 in Ω f

(5)

E·n=0 on Ωsf

(6)

where E is the scaled electric field, e is the unit vector field, and ∇φ is the fluctuating part with the scalar field φ [53]. From Eqs. (4)–(6), the scaled electric field is obtained and τ∞ and VCL are calculated by using E as τ∞ = f /2f and

VCL=L v=2 ∫ E2dΩf / ∫ Ωf

Ωsf

E2dΩsf [46]. Also, ϵp is obtained from the calculation of a volume ratio of the solid/fluid domain and

Lth is evaluated by a hydraulic radius of RUC defined as TCL=Lth=2 ∫ dΩf / ∫ Ωf

Ωsf

dΩsf .

By using the parameters above, theoretical sound absorption coefficients are obtained by calculating the two frequencydependent complex values: effective bulk density ( ρ( ω)) and effective bulk modulus (K ( ω)) from Johnson-Champoux-Allard (JCA) model equations [15,16].

ρ( ω ) =

⎛ 4iωτ∞2μρf τ∞ρf ⎜ R f ϵp 1 1 + + ϵp ⎜ iωρfτ∞ R2f L v2ϵ2p ⎝

⎡ ⎛ γf PA ⎢ 8μ ⎜ K ( ω) = ⎢ γf −(γf −1)⎜ 1+ 2 ϵp ⎢ iωL th Prρf ⎝ ⎣

⎞ ⎟ ⎟ ⎠

(7) −1 −1⎤

1+

2 ⎞ iωρf PrL th ⎟ 16μ ⎟ ⎠

⎥ ⎥ ⎥ ⎦

(8)

In the Eqs. (7) and (8), ρf is the density of air, ω is the angular frequency, γf is the heat capacity ratio for air, PA is the ambient pressure, Pr is the Prandtl number, and μ is the viscosity of air. JCA model is a rigid frame poroacoustics model, i.e., five parameter semi-empirical equivalent fluid model which does not consider a solid frame damping. Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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Fig. 2. Microscale numerical simulations for obtaining poroacoustics parameters. (a) 400 μm RUC designed. (b) Meshes generated on fluid domain. (c)– (d) A scaled velocity field from a viscous flow problem at (111) plane and (100) plane. (e)–(f) A scaled electric field from an inertial flow problem at (111) plane and (100) plane. s

Sound propagation behaviors were numerically simulated by using a pressure acoustics module in COMSOL Multiphysics with 2-D geometry which has the same dimension of the specimen used for measurement of the sound absorption coefficient. In the measurement of absorption coefficient, normal incident sound field without transverse mode in the tube hits the porous layer. This sound propagation behavior is applied to establish a background incident pressure field (pi ) of the air domain within the tube having a linear elastic fluid property. At the top of the geometry, there is a perfectly matched layer which is used for reflecting an infinitely large air domain. And the sound hard boundary condition is applied to the bottom. The thickness (20 mm) and width (29 mm) of the foam layer are the same as the sample height and the radius, as shown in Fig. 3(a). Used normal incident background pressure field is in a plane wave condition up to 6400 Hz, so it is given as pi = exp (−i(k ·x )), where pi is the pressure of incident wave, k is the wavenumber. By solving Simple Helmholtz equation with pi , ρ( ω), and K ( ω), the scattered pressure field (ps ) is obtained and the total pressure field (pt ) is also calculated from the definition of pt = pi + ps . Consequently, the sound absorption coefficient (α ) is calculated by the following equation:

α = 1−R2, where R=ps /pi at the boundary interface between the air domain and the foam domain.

4. Results and discussion 4.1. Theoretical analysis of optimum cell size The JCA model, a rigid frame poroacoustics model, was employed for numerical simulation, in order to track the optimum cell size and understand the improvement of an acoustical damping behavior in the view of microscale. RUC’s were constructed by mimicking the real microstructure of the foams; average cell diameters, average pore diameters, and mass density (80 K (¼kg/m3)) (Fig. 2(a)). Tetrahedron meshes containing 264,867 domain elements, 61,286 boundary elements, and 9004 edge elements were put into each RUC (Fig. 2(b)). Since the interconnecting pore diameters of high density foams did not coincide with a membrane diameter of the conventional ideal tetrakaidecahedron, Hoang et al. [13] tried to construct thin solid films with a hole at the center for more applicable modeling. To reflect the mass density of the foams on RUC, a volume fraction of empty space was set to be 0.92 with corresponding to that of 80 K foam. Seven RUC’s with 100, 200, 400, 600, 800, 1000, and 1200 μm of the cell diameter were constructed to figure out the best cell size.

Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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For each RUC, two transport problems were solved to obtain the viscous and the inertial transport flow properties (Table 2). At first, in the viscous flow problem, solid domains were assigned to the empty space and the remained region in RUC was the fluid domain. A uniform pressure gradient in z-direction, no-slip boundary conditions, and periodic boundary conditions were applied. By solving Stokes’ equation, velocity profiles in the fluid domain were calculated. The scaled velocity field, in other words, “static viscous permeability field” was obtained by dividing the velocity profile by the pressure gradient (Fig. 2(c) and (d)). The streamlines indicate the velocity field and the multislices represent the magnitude of permeability in z-direction. Consequently, the flow resistivity (Rf ) was obtained by using the viscosity of air and the porosity with the velocity field calculated from the simulation of the viscous flow problem in the RUC [46]. To obtain the inertial flow properties, a uniform voltage gradient was assumed in z-direction and then the scaled potential profiles were represented by the streamlines and the multislices (Fig. 2(e) and (f)). From the solution of the inertial flow problem, viscous characteristic length (VCL, LV ) and tortuosity (τ∞) were obtained by the equations as mentioned in the numerical method section [46]. VCL is a weighted hydraulic diameter related to energy damping of air flow in the complex porous media. The energy damping is caused by viscous friction due to relatively narrower paths than surrounding structures [15]. VCL is calculated by solving the inertial flow problem only, because the inertial flow is not coupled with the viscous friction [63]. By the same logic, tortuosity was calculated to reflect the complexity of sound propagating paths in the porous media. And it was 1.83 for all RUC’s, since the tortuosity depends on shape of the foam only and scaling does not change a shape of RUC. The last parameter, thermal characteristic length (TCL, Lth ), was easily obtained by calculating the conventional hydraulic radius of RUC [15].

Fig. 3. Macroscale numerical simulations for analyzing sound wave propagation. (a) 2-D geometry with meshes consisted of 1527 domain elements and 228 boundary elements. Simulated acoustic pressure fields for 400 μm RUC (left: background sound field, right: scattered sound field) at (b) 1000, (c) 2000, and (d) 4000 Hz. Sound pressure field profiles in the tube for 100, 400, 1000 μm RUC cases at (e) 1000 Hz, (f) 2000, and (g) 4000 Hz. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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Fig. 4. Parametric sweep analysis with respect to cell diameter of RUC's and samples. (a) Sound absorption coefficient curves with the variation of cell diameters from 100 to 1200 μm. Root mean square of sound absorption coefficient curves in the frequency range (b) from 0 to 2000 Hz ( αrms,1000) , (c) from 0 to 1000 Hz (αrms,2000 ), (d) from 5000 to 6000 Hz (αrms,5000 − 6000 ), and (e) from 0 to 6400 Hz ( αrms,6400) (Black line: simulated curve, blue X: measured value of 80-1, 80-2 ,and 80-3, orderly). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Sound propagation behaviors were investigated using finite element method in 2-D geometry for a foam sample of 2 cm thickness. By using the obtained four parameters ( ρf , Lth , LV , and τ∞) and porosity (ϵp ) for JCA model, effective bulk density ( ρ( ω)) and effective bulk modulus (K ( ω)) of each RUC were obtained by substituting the variables into Eqs. (7) and (8). The background acoustic pressure fields and scattered fields were simulated for the sound frequency from 0 to 6400 Hz and the representative results of 400 μm RUC at 1000, 2000, and 4000 Hz were shown in Fig. 3(b)–(d). It was observed from the color contour plot that the sound wave affected by the porous layer has a relatively lower amplitude of the pressure field than that of the background pressure field, especially at 2000 Hz. Sound damping property depending on the alteration of the cell diameters was confirmed from the sound absorption coefficient equation defined by the ratio of the scattered pressure field to the incident pressure. Sound absorption coefficient curves for each RUC is shown in Fig. 4(a). All curves have distinct shapes with respect to the variation of the cell size. Root mean square (RMS) values of sound absorption curves from 0 to 2000 Hz (αrms, 2000 ) were evaluated for the quantitative comparison. The RMS values were plotted as a function of the cell diameter (Fig. 4(b)). This frequency interval was chosen as a representative range of the low frequency damping since it contains most of the low frequency noises generated by automobiles. From the curve, 400 μm cell size was determined as the best cell diameter of RUC with a RMS value of 0.66. The reason why low frequency sound absorption performance drops in the RUC smaller than 400 μm is the excessive increase in surface impedance. High flow resistivity in very small RUC’s hinders the penetration of sound wave into the foam and this phenomenon induces the increase of the surface impedance. At the surface of the foam, most incident sound wave is reflected to the incident media rather than transmitted and then diminished. The cell larger than 400 μm, on the other hand, is not able to dissipate sound energy sufficiently due to low flow resistivity with the same tortuosity. Therefore, the average cell diameter of the PU foam should be controlled to about 400 μm to deliver the best performance for the 2 cm-80 K foam in the frequency range from 0 to 2000 Hz. The RMS analysis in different frequency ranges was carried out to examine the cell size effect more widely, as shown in Fig. 4(c)–(e); αrms,1000 (from 0 to 1000 Hz), αrms,5000 − 6000 (from 5000 to 6000 Hz), and αrms,6400 (from 0 to 6400 Hz). At first, from Fig. 4(c), we could find that the optimum cell size is not always 400 μm. The αrms,1000 value indicates 200 μm is the best condition for the lower frequency range below 1000 Hz. Generally, the sound absorption curve is shifted to the lower frequency ranges when Rf and τ∞ increase [10,64]. For this reason, more complex RUC with smaller cell size must show the advanced sound absorption performance at low frequency. Meanwhile, Fig. 4(d) and (e) show that 400 μm is the best cell size for αrms,5000 − 6000 and αrms,6400 , representing sound absorption performance on high frequency range and entire frequency Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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range, respectively. Consequently, it could be figured out that the cell size inducing moderately high Rf enables the optimized sound absorbing performance, except for a few cases. 4.2. Cell structure modulation Cell structure modulation for smaller cell size was conducted using an ultrasonic foaming method. Since the optimum cell size of 2 cm thickness-80 K density PU foam was predicted as 400 μm from the numerical simulation, the mean cell diameter was adjusted in the vicinity of that range. In the cluster nucleation theory, the equation of the nucleation rate is J =C ( N )exp −∆Fn*/kT , where J is the nucleation

(

n

)

n

rate, C ( N ) is the number of gas molecules in polymeric resin, ∆Fn* is the critical free energy needed to form critical clusters, k is

Boltzmann 5

3

constant,

and

T

is

the

absolute

temperature.

In

the

equation

above, ∆Fn*

is

defined

as

2

∆Fn* = fL (kT ) /486vm (Ps − P0), where P0 is the environmental pressure, fL is the lost degree of freedom in the process of dissolution, vm is the equilibrium molecular volume of a gas when it loses all of the translation kinetic energy, and Ps is the constant saturation pressure [25]. Therefore, ∆Fn* is inversely proportional to the magnitude of the negative environmental pressure, −P0 . The ultrasonic wave from an ultrasonication tip induces the negative pressure in the polymeric matrix by a cavitation effect. It reduces the critical energy for the bubble nucleation, and then the nucleation rate is increased. This phenomena lead to smaller cell size and more uniform cell size distribution by nucleating more clusters. In addition, the ultrasonication process helps two resins to be dispersed well and also promotes the generation of more nuclei. Irradiation time, period, and power are the most important processing conditions in the ultrasonic foaming method. To reduce cell size by increasing nucleation rate, it has been known that the ultrasonic irradiation should be employed only in the bubble nucleation step [25]. If the irradiation were kept until the end of the cell growth state, the generated bubbles would be collapsed since cell walls cannot be stabilized due to the excessive vibration [38]. Therefore, we applied the ultrasonic irradiation to the resin mixture only for the cream stage before foam rising. Because the amplitude of the ultrasonic wave is determined by the power of the ultrasonicator, high power can induce higher nucleation rates. In the our previous work, the proportionate relationship between the ultrasonic power and the nucleation rate was already proven. And 4 times higher magnitude of negative pressure is achieved by increasing the ultrasonic power from 10% to 20% [65]. Schematic illustration (Fig. 5) shows the underlying mechanism of the increase in nucleation rate during foaming. And the gradually increased amplitude of ultrasonic wave is represented as the standing wave mode shape. The enhanced nucleation rate is corresponding to the increase in the number of grey dots in Fig. 5(a) and the smaller cell size is depicted in Fig. 5(b). In the present work, three different foams (labeled as 80-1, 80-2, and 80-3, respectively), having different cell size distribution, were manufactured by applying the ultrasonication with the power of 0%, 37%, and 75% of 750 W. To verify the internal microstructure of each sample, a field emission scanning electron microscope (FE-SEM) was employed and the images were shown in Fig. 6(a)–(c). The average cell diameter of the samples became smaller as the

Fig. 5. Schematic illustration of ultrasonic PU foaming. (a) Generation of nuclei (gray dots), wave front of ultrasonication (black curves), and the amplitude of wave (red standing wave mode shape) were shown. (b) Microcellular structures in each ultrasonic condition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 6. Observation and analysis of microcellular structure. Inner cell structures of (a) 80-1, (b) 80-2, and (c) 80-3 observed by SEM. Alteration of (d) mean cell diameter, (e) mean pore diameter, and (f) cell number density.

sonication power is increased. The average cell diameter of the sample 80-1 was 723 μm (standard deviation, 7159 μm), that of the sample 80-2 was 588 μm (7108 μm), and that of the sample was 445 μm (786 μm). The average interconnecting pore diameters were 102 μm, 91 μm, and 69 μm, respectively (Fig. 6(d) and (e)). The total volume of inner cells is the same for all samples, since the identical amount of the blowing agent (water) was added to the Reactant A. This is the reason why the increase of the nucleation rate induces the smaller volume of each cell. The cell diameter of 80-3 (445 μm) correlates to the optimum value of the simulation results, and it is worthwhile to manipulate the cell size up to 60% of the initial size (723 μm) without any further treatment. Smaller interconnecting areas that are determined by the average pore diameter were generated at the interfaces between adjoining cells in the ultrasonicated foams. Cell number density (number of cells per mm2) was specifically identified by the number of cells in SEM images divided by the total area of image (Fig. 6(f)). It was inversely proportional to the mean cell diameter, so the higher values of the cell number density were obtained at the ultrasonicated foams. Compared with 80-1, 80-3 showed 38% reduction in the mean cell diameter and 32% in the mean pore diameter, while the cell number density was increased by about twice from 3.75 to 7.81. Information about these structural changes were listed in Table 1. Actually, the cell size of the fabricated foam had to be made smaller than 445 μm, in order to clearly prove the optimum cell size. However, yielding the foam with the cell size smaller than 400 μm was impossible due to limitation of the employed process condition. In the foaming process using physical blowing agent, a sufficiently small cell size can be achieved by generating more instantaneous nucleation of bubbles under the ultrasonic wave irradiation. In our previous study, the minimum cell diameter of the foam obtained was 50 μm employing the ultrasonic foaming method [25]. When the physical blowing agent (i.e., carbon dioxide gas) is already incorporated in polyol resin with high saturation pressure, high degree of nucleation rate is achieved with ultrasonic excitation upon release of the saturation pressure. Consequently, the foam containing small closed cells can be created by using the supersaturated resin. In the chemical blowing method, carbon dioxide gases is not saturated in the resin, but rather the blowing gases are generated in the reaction between the chemical blowing agent and isocyanate for a certain period of time. Therefore, at the beginning of ultrasonic excitation, the resin mixture was not saturated by the gas sufficiently. Since the cells are nucleated through the chemical reaction of the blowing agent and continue to grow for some time, it is hard to obtain the cell size less

Table 1 Cell structures: mean cell diameter and pore diameter with standard deviation, and cell number density.

Mean cell diameter (μm) Mean pore diameter (μm) Cell number density (#/mm2)

80-1

80-2

80-3

723 ( 7 159) 102 ( 7 79) 3.75

588 ( 7108) 91 ( 7 66) 5.63

445 (7 86) 69 ( 7 54) 7.81

Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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Table 2 Calculated poroacoustics parameters. 80-1

80-2

80-3

Porosity, ϵp

0.92

0.92

0.92

Flow resistivity, Rf (N m4 s)

6998

10,467

16,514

Tortuosity, τ∞ Viscous characteristic length, Lv (μm) Thermal characteristic length, Lth (μm)

1.94 65 219

1.94 49 169

1.94 39 135

Table 3 Acoustic damping evaluators.

Frequency at max peak αrms, 2000

αrms,

2000 − 6400

Noise reduction coefficient

80-1

80-2

80-3

2544 0.457

2272 0.543

1920 0.643

0.777

0.802

0.791

0.354

0.408

0.444

than 400 μm in processing of open cell PU foams. For this reason, we just predicted the optimum cell size theoretically, without producing a PU foam with smaller cells than the optimum cell size. From the empirical cell diameter modulation, three samples were manufactured and three RUC’s were constructed by reflecting their structural information. The poroacoustics parameters are derived by studying the multiscale simulation for each RUC and the complex values were calculated. One of the poroacoustics parameters, the Rf indicating the resistance of air flow in a sample, was increased by increasing the power of ultrasonic waves and decreasing the cell size. The normalized Rf is defined by the ratio of the Rf in a non-treated foam to that in the ultrasonicated foam. The normalized Rf predicted in the simulation is plotted and compared with the empirically measured Rf in Fig. 7(a). Both measured and simulated values of the Rf for the 80-3 sample are larger than those for 80-1 by more than two times. The reason why the measured values are much higher than simulated values is the skin layer on the surface of the sample inducing friction and the simulation error. τ∞ was the same for all RUC’s but TCL and VCL had lower values in 80-3 because of its much smaller cell size and interconnecting pore size (Fig. 7(b)). These alteration of the poroacoustics parameters induced the variation of the frequency-dependent complex values and the normalized parameters by air properties are plotted in Fig. 7(c)–(e). Effective bulk density ( ρ( ω)) is the specific air density in porous media corrected by the poroacoustics parameters. And ρ( ω) is proportional to the Rf and inversely

Fig. 7. Changes of poroacoustics parameters and complex values by decrease of cell diameter. (a) Simulated flow resistivity with empirical value, (b) simulated VCL and TCL for each RUC. Simulated (c) effective bulk density, (d) effective bulk modulus, and (e) complex phase speed curves normalized by air properties, with respect to frequency in the foam domain.

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Fig. 8. Sound absorption coefficients curves. (a) Measured (solid lines) and simulated (dash lines) sound absorption coefficients from 0 to 6400 Hz. (b) A tendency of max peak shift. (c) Noise reduction coefficient curves. Root mean square values (d) from 0 to 2000 Hz, and (e) from 2000 to 6400 Hz.

proportional to VCL (Eq. (7)). Therefore, 80-3 RUC revealed the maximum value among the samples. Meanwhile, effective bulk modulus (K ( ω)) is the specific air modulus affected by the complicated microstructure of the foam. And K ( ω) is inversely proportional to the increase of TCL and decreased by the ultrasonication. From the two complex effective values above, complex phase speed in porous media defined as cc =

K ( ω) /ρ( ω) was also evaluated. Consequently, the complex

phase speed of sound in the smaller microstructure foam has a lower value than that of the others in all frequency ranges and affects the sound absorption performance. 4.3. Sound absorption performance Sound absorption properties were experimentally characterized by using a B&K impedance tube (Table 3). Sound absorption coefficients measured were plotted from 100 to 6400 Hz in a log scale with solid lines in Fig. 8(a). Maximum peaks of the empirical curves were shifted to lower frequency from 2544 to 1920 Hz by increasing the ultrasonic wave power for foam processing from 0% to 75%. And the simulation results were similar to the measurement curves (Fig. 8(b)). Cell diameter reduction was presumed to be the dominant factor for determining the position of maximum peak because of induced lower phase speed, and it was theoretically explained in the previous section (Fig. 7(e)). The shift of the peak frequency indicated that noise in lower frequency ranges from 0 to 2000 Hz can be facilely damped in the ultrasonicated foams, but that in higher frequency region over 2000 Hz was not effectively absorbed. The only difference among the samples is the internal microscale cell structures since all samples had the same mass density, chemical composition, and fabrication conditions except for mechanical excitation. The difference between the experimental and numerical results was observed especially in the high frequency region. This difference might stem from experimental trial error and the theoretical assumption of the rigid frame that ignores the damping effect on the solid frame. Among the five poroacoustics parameters, Rf , Lth , and LV showed distinguished values depending on the different sizes of RUC’s (Fig. 7(a) and (b)). It was found that among these parameters, the changes of Rf and LV affect the enhancement of low frequency sound damping while the change of Lth had a negative influence. A sensitivity analysis was conducted by varying one poroacoustics parameter only (see Fig. S2(a)–(c) in Supplementary material). As shown in Fig. S2(a), the alteration of the curves mainly originates from the Rf , as reported in several studies [10,64]. Further empirical characterizations were performed to compare the experimentally measured Rf , with the numerically obtained data to support our argument. Both normalized lines had the same ascending trends from 80-1 to 80-3 (Fig. 7(a)). Relatively high Rf in the ultrasonicated foams was induced from narrower and more complicated sound propagating paths by reducing the cell diameter. High effective bulk density and low effective bulk modulus achieved in the foam having high Rf would make the sound speed slower. Moreover, other researchers proved that the slow sound speed can induce the high sound damping efficiency in low frequency noise [26,27]. Consequently, the foams with higher Rf would be more suitable where the low frequency noise control is required. Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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Phenomenologically, the sound energy of the low frequency could be dissipated better in the foams with small cell diameters. For further investigation of sound absorption performances, root mean square (RMS), noise reduction coefficient (NRC), and 1/3 octave band spectrogram were analyzed. NRC is the standard of the sound damping materials that is calculated by obtaining the average value of the sound absorption coefficients at 128, 256, 512, 1024, 2048, and 4096 Hz. And it refers to the sound damping capability in overall frequency region. In Fig. 8(c), the ultrasonicated foam samples, 80-2 and 80-3, showed more advanced damping performances than 80-1 in overall frequencies although they had similar sound absorption ability in high frequency ranges. RMS value from 0 to 2000 Hz (αrms,2000 ) (Fig. 8(d)) had a distinguishable ascending slope with respect to the reduction of the cell size. But there was no remarkable enhancement in residual ranges from 2000 to 6400 Hz (αrms,2000 − 6400 ) (Fig. 8(e)). Comparing the measured data of the ultrasonicated foams with the theoretical parametric sweep results (Fig. 4(b), blue X label), it is worth noting that the fabricated 80-3 foam is close to the optimum sound absorber in the perspective of αrms,2000 . Octave band spectrogram is often used for the analysis of acoustical data. It is characterized by dividing measured frequencies into one octave band range. One octave range is a frequency range where the highest frequency is two times higher than the lowest frequency. In the octave band spectrogram, some acoustics evaluation indices (e.g., sound absorption coefficient) are substituted into the average value of each band representing the central frequency ( fc ). There are many kinds of octave bands, but 1/3 octave band spectrogram is widely used among them because its frequency range of each band well

Fig. 9. 1/3 octave band spectrograms. (a) 1/3 octave bands from #1 band to #25 band. (b)–(d) Comparisons between measured and simulated values for each sample. Absorption coefficient of each sample for (e) #17 band, (f) #20 band, and (g) in #23 band.

Please cite this article as: J.H. Park, et al., Optimization of low frequency sound absorption by cell size control and multiscale poroacoustics modeling, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.004i

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corresponds to the audible frequency range of human. 1/3 octave band spectrograms were plotted by using both the empirical and the numerical results of sound absorption coefficient curves to evaluate the noise damping capability from the human perspective. Each band was numbered from #1 to #25 and their central frequencies were listed in Table S1 (in Supplementary material). The ultrasonicated foams revealed the enhanced sound absorption performances for most bands from #1 to #22, but the foams had lower performances than the non-treated foam for the other three bands #23-25. The large number of high 1/3 octave bands means that the foams with reduced cell sizes are more efficient for reduction of the noise pollution. The measured 1/3 octave band values were also in good agreement with simulated values, so the measured values were acceptable for theoretical octave band analysis. In #17 ( fc ¼1000 Hz) band, 1/3 octave band value of the sample 80-3 was increased compared with the sample 80-1, in both measured (increasing rate,
56%) and simulated results (increasing rate,
55%). And also, in #20 ( fc ¼2000 Hz) band, a distinction was observed for both measured (increasing rate,
16%) and simulated results (increasing rate,
10%). However in high frequency 1/3 octave bands, e.g., #23 ( fc ¼4000 Hz), it had a descending slope with the decrease of the cell size (Fig. 9(g)).

5. Conclusions In this study, we develop a strategy to design and manufacture the advanced sound absorbing foam. It was shown that the enhanced noise damping at low frequency was achieved by structural modulation employing the ultrasonic foaming method. And the feasibility of this assertion was supported by the theoretical simulation. An internal cell structure of the foams irradiated by the ultrasonic wave had about 40% smaller dimensions than the non-irradiated foams. And the yielded mean cell diameter was close to the optimum value predicted in the numerical simulation for the open cell foam of 2 cm thickness-80 K density. With this strategy modulating cell size in the desired range, even a small space is enough for a control of noise at low frequency. And increase of density is also not necessary inducing fuel economy of transportation vehicles. Furthermore, greater cell size variation will be achieved if limitation on fabrication is overcame by diversifying the ultrasonication amplitude.

Acknowledgements This study was supported by the Hyundai Motor Company through the contract with the Hyundai NGV (R155309.0001). The authors are grateful for the financial support.

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jsv.2017.03.004.

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