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Proceedings of the Institute of Acoustics

WHAT IS THE ACTUAL INFLUENCE OF A NANO-FIBROUS MEMBRANE ON THE ACOUSTICAL PROPERTY OF POROUS SUBSTRATE? Kirill V. Horoshenkov Alistair Hurrell Mohan Jiao Michael T. Pelegrinis

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Department of Mechanical Engineering, University of Sheffield, Sheffield, South Yorkshire, S1 3JD, UK Department of Mechanical Engineering, University of Sheffield, Sheffield, South Yorkshire, S1 3JD, UK Department of Mechanical Engineering, University of Sheffield, Sheffield, South Yorkshire, S1 3JD, UK John Cotton Group Ltd., Nunbrook Mills, Mirfield, West Yorkshire, WF14 0EH, UK

INTRODUCTION

There is a general lack of publications on the acoustical and related non-acoustical properties of nano-fibrous media. Existing publications typically present scanning electron microscope images of these media together with the acoustic absorption coefficient or transmission loss data (e.g. [1,2]). These works often discuss the nano-fibre production process, quote data on the fibre diameter and surface density for nano-fibrous membranes produced as a result of the reported process. However, little or no information is usually presented on the material pore structure, membrane thickness or bulk density. No effort is made to explain the observed acoustical performance of nano-fibrous membranes using a valid theoretical or semi-empirical model [1,2]. This paper describes the problems associated with acoustic and related non-acoustic characterisation of these materials. It attempts, probably for the first time, to use a Biot- and Darcytype mathematical models to explain the observed acoustical and related non-acoustical behavior of nano-fibres. It identifies theoretical gaps related to the physical phenomena which can be responsible for the observed acoustical behaviour of nano-fibrous membranes and it makes recommendations to fill these gaps.

2 2.1

NANO-FIBROUS MEMBRANES Structure

A nano-fibrous membrane is usually a thin layer of fibrous media which consists of fibres which diameter is less than one micron. These fibres are commonly produced using an electro-spinning process which enables to draw a molten or dissolved in a solvent polymer through a spinning narrow needle or Taylor cone with a help of high pressure and high voltage differential [3]. As a result of this process it is possible to obtain an entangled network of fibres such the one shown in Figure 1. A key characteristic of this network of fibres is that it is highly dense and highly resistive to a flow of air. These two properties make this material highly attractive to noise control engineers because it enables then to increase the air flow resistance of a thin layer of a typical porous absorber (e.g. fiberglass or polyester) to make it much more efficient in terms of its acoustical absorption [4].

2.2

Non-acoustical characterisation

The first question is to ask before attempting any prediction of the acoustical properties of a nanofibrous membrane is “What is its thickness?” This seemingly a simple question is rather difficult to answer. These membranes are relatively soft so that the use of a standard micrometer yields a rather large error because it will compress fibre inevitably. Another method is to take a SEM photo of a layer edge. Figure 2 shows an example SEM photograph of an edge of the same membrane

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which fine structure is depicted in Figure 1. This photograph was taken at an angle of 49 degrees which is optimal to capture a representative section of the membrane’s edge and to avoid any possible collision of the SEM electron beam head with the membrane’s edge. The material shown in this Figure was frozen in a cryostat and then its edge was cut with a sharp guillotine to avoid fibre delamination. Despite all the care taken to ensure a clean cut, it is not obvious from the image shown in Figure 2 what is the actual thickness of this membrane. This means that depending where on the photo is a representative measure of the average thickness we can run into 30-40% errors in the membrane thickness estimate.

Figure 1. A SEM photograph of a nano-fibrous membrane with mean fibre diameter of 277 nm. The uncertainty in the measurement of the nano-fibrous membrane’s thickness affects the accuracy with which we can estimate its bulk density and porosity of nano-fibres. Even if the mass of a representative sample of nano-fibrous membrane, ms , can be measured accurately to 4-6 decimal places, the bulk density,

b , in the equation

b  ms / Vs

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(1)


is prone to large errors because of the 30-40% uncertainty in the calculation of the sample volume, Vs . This means that the accuracy in estimation of the porosity is also affected. A simple way to determine the porosity,

nf , of a fibrous material is to use the following equation

nf  1  b /  f , where

(2)

 f is the density of the fibre material (e.g. polyester). Let us give an example. The mass of a

45 mm diameter sample of the material shown in Figure 1 is 8.1 mg. From the image of its edge shown in Figure 2 we estimate that its mean thickness is 28 m. We assume that the error in this estimate can be 30%. For this material eq. (1) gives the following mean, minimum and maximum values of the bulk density: 182 kg/m 3, 140 kg/m3, and 236 kg/m3, respectively. If these fibres are polyester for which we assume the fibre density of 1370 kg/m 3, then the corresponding values of the membrane’s porosity predicted with eq. (2) are: 0.87, 0.90 and 0.83, respectively.

Figure 2. A SEM photograph of the edge of the nano-fibrous membrane shown in Figure 1. The above data variance has a strong implication to the value of the flow resistivity we can estimate. A common way to estimate the flow resistivity of fibrous media is to use the KozenyCarman equation [5] or its various derivatives which are summarised in ref. [6]. According to the Kozeny-Carman model the flow resistivity of a random network of fibres can be estimated as Vol. 40. Pt. 1. 2018


 nf  where

180 (1  nf ) 2 d 2f nf3

,

(3)

 is the dynamic viscosity of air and d 2f is the fibre’s diameter. Substituting the above values

of porosity in eq. (3) yields the following values of the flow resistivity: 1.149 GPa s m -2, 2.875 GPa s m-2, and 612.7 MPa s m -2, respectively. Because of the non-linearities in the porosity term in eq. (3) the 30% error in the membrane thickness measurement has grown into 4.7 fold difference (470% error!) between the minimum and maximum flow resistivity values estimated for this kind of nanofibre. Can we actually measure the flow resistivity of this membrane and compare it with the estimated value? In principle, yes, but it is not obvious how. According to the BS 29053 method to measure the flow reisistivity [7] the job is relatively simple. We stretch a sample of the membrane in a flow resistivity tube in the way of a slow, direct air flow and measure the pressure drop across it. There are at least two problems which we may encounter. The first one is how to make sure that there is no circumferential air gap between the sample and the tube? For a material with such a massive flow resistivity the error due to the presence of even a tiny air gap will be incredibly large being in tens of event hundreds percent [8]. One possibility is to stretch the membrane across the air flow and seal it well between the flow resistivity tube and sample holder as shown in Figure 3 (left). This arrangement can potentially work, but stretching a thin, highly flexible membrane is a delicate process which can result in the alteration to the material density and pore size. One alternative to this is to sandwich the membrane between two samples of high-porosity foam (e.g. melamine) that fit tightly in the tube as it is shown in Figure 3 (right). In this way the membrane can be wrapped around the bottom foam sample so that the membrane is resting naturally on the foam substrate and the air gap between the membrane and tube wall is carefully sealed by the bottom foam sample pressing well against the tube wall.

Figure 3. Ways to measure the flow resistivity of a thin, nano-fibrous membrane in a flow resistivity tube. By stretching (left). By sandwiching between two foam samples (right). The membrane sample arrangement shown in Figure 3 (right) was tested in the Jonas Laboratory at the University of Sheffield. Large enough samples of the nano-fibrous membrane shown in Figure 1 were cut and placed on a 15 mm thick, 100 mm diameter sample of melamine foam. This sample of melamine foam served as the bottom substrate. Another 15 mm thick sample of melamine foam was placed on the top of the nano-fibrous membrane and this sandwich structure was placed in the sample holder of a standard AFD300 AcoustiFlow device (The Gesellschaft für Akustikforschung Dresden mbH, Dresden, Germany). A separate measurement was taken on the two melamine sample sandwiched together without the membrane to measure the controlled value of the flow resistivity of melamine foam. The flow resistivity of this sandwich structure,  s , is a combination of Vol. 40. Pt. 1. 2018


the pressure drop across the two samples of melamine foam,

2Pm and the pressure drop across

the nano-fibrous membrane, Pnf i.e.

s  where

1 2Pm  Pnf , U 2xm  xnf

(4)

2xm = 30 mm is the combined thickness of two samples of melamine foam and xnf = 28

m is the mean membrane thickness. The measured mean flow resistivity of melamine foam used in this experiment was 16.3 kPa s m -2. This means that 2Pm was 0.2445 Pa. The value air flow resistivity of the sandwich structure shown in Figure 3 (right) varied from 382.7 to 549.2 kPa s m-2 in 10 separate experiments with the mean value of 481.6 kPa s m-2. Substituting the knowns in eq. (4), noting that xnf  xm and taking u  0.5 mm/s as recommended in [7] suggest that the pressure drop across the nano-fibrous membrane in this flow resistivity experiment was Pnf  6.956 Pa which corresponded to the flow resistivity value of nano-fibres of

 nf  498.8

close to the 612.7 MPa s m-2 we estimated by taking the higher porosity value of

2.3

MPa s m-2. This is

nf  0.87 .

Acoustical characterisation

Let us now try to measure the acoustical properties of these membranes. We need these data for two reasons. The first one is to determine how good these materials are in terms of their acoustic absorption. The second one is to use acoustical data to invert their non-acoustical (intrinsic) properties such as mean pore size, tortuosity and flow resistivity. We can then compare these properties against those which we were able to measure non-acoustically.

Figure 5. The absorption coefficient spectra for three different material arrangements in the 100 mm diameter impedance tube. For this purpose it is common to perform a simple experiment in the impedance tube in accordance with ISO 10534-2 [9]. The question is now “How to fix the membrane sample in the tube?” If we simply place the sample on the rigid bottom of the sample holder in the impedance tube, then effectively we would measure the residual absorption in the tube because the sample is so thin and so flow resistive that it would unlikely to add much of acoustic absorption when tested on its own. Vol. 40. Pt. 1. 2018


This is easy to demonstrate with data. Figure 5 shows the acoustic absorption coefficient of the empty 100 mm diameter impedance tube in our Jonas Laboratory and absorption coefficients of different material arrangements. Clearly, the data shown in this figure suggest that absorption coefficient of the nano-fibrous membrane is almost identical to that of the empty tube and that it is rather noisy for some reason. The effect of a thin, nano-fibrous membrane becomes pronounced when it is added to a porous substrate, e.g. to a 15 mm thick, hard-backed layer of melamine foam. This effect is illustrated in Figure 5 which shows the absorption coefficient of the 15 mm thick layer of melamine foam alone and that with the nano-fibrous membrane added to its top. The presence of the 28 m thick nano-fibrous membrane on the 15 mm thick, hard backed layer of melamine foam almost doubles the absorption coefficient across the considered frequency range. This effect is explained by the increase in the real part of the surface impedance of the melamine foam when the thin, nano-fibrous layer is added on the top it. This effect is well explained in [4] (eq. (7) in ref. [4]). Effectively, the surface impedance of a stack of these two layers is

Z s   nf xnf  Z m coth(ikm xm ) , where

(4)

Z m and km are the characteristic impedance of and wavenumber in melamine foam,

respectively. This effect is illustrated in Figure 6 which shows the measured real and imaginary parts of the surface impedance of the 15 mm, hard-backed layer of melamine foam and the layer of foam covered with the 28 m layer of nano-fibrous membrane. Figure 6 also presents the predicted surface impedance using the model for the acoustical properties of porous media proposed in [9]. The following non-acoustical parameters were used to obtained the best fit (within 2.6% error) for the measured properties of melamine foam (the index m stands for melamine):  m  16.3 kPa s m2,

m  1.0 ,   ,m  1.77 (tortuosity)

and

 s ,m  0.388 (standard

deviation in pore size). For the

nano-fibrous membrane these parameters were (within 9% error between measured and predicted impedances):  nf  6.51 MPa s m-2, nf  0.89 ,   , nf  1.00 and  s , nf  0 , respectively. We note that the deduced value of the flow resistivity for melamine foam is remarkably close to the measured value. The same cannot be said about the measured and inverted porosities of nanofibrous membrane. We will discuss the deduced value of the flow resistivity for this membrane in the following paragraph.

Figure 6. The real (left) and imaginary (right) parts of the normalised surface impedance of the 15 mm hard-backed layer of melamine foam and layer of foam with the nano-fibrous membrane on the top. The most noticeable effect of nano-fibrous membrane is in the real part of the surface impedance. The application of the nano-fibrous membrane increases the real part of the normalised surface impedance of melamine foam by approximately 0.350 . The increase in the measured imaginary part is relatively small and the predicted imaginary parts are almost identical (see Figure 4 (right)). Let us assume that the sound speed in air and equilibrium density of air were c0  343 m/s and Vol. 40. Pt. 1. 2018


0  1.20 kg/m3 (at 20oC), respectively. Then according to eq. (4) the increase

in the real part of

the normalised surface impedance due to the presence of the membrane should be

 nf xnf /  0 c0   0.397 , which is within 15% from that estimated from the data shown in Figure

6. This is a sufficiently close result. One problem here is that the flow resistivity value that is required to explain the observed increase in the real part is 76 times smaller than the one measured with the BS 29053 method [7] and 94 times smaller than that estimated from the Kozeny-Carman expression [5]. This phenomenon does not have an answer at the moment and it deserves further investigation.

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CONCLUSIONS

This paper presented new experimental data which illustrate the effect of a nano-fibrous membrane attached to the surface of a layer of melamine foam on the acoustical impedance and absorption coefficient spectra. An attempt was made to explain this effect with theoretical modelling which treated the acoustical properties of the membrane and substrate as a classical problem of sound propagation in rigid frame porous media. This research showed that the prediction of the acoustical properties of this seemingly simple system is far from easy. Firstly, there was a problem of characterising the non-acoustical properties of nano-porous membrane with standard laboratory methods. Secondly, there was a high uncertainty in the value of the membrane thickness which could lead to a substantial variance in its predicted acoustical properties. Thirdly, there was a question if sound propagation in a nano-porous membrane can actually be treated as that in a classical porous media. There is a clear need for more research into this effect. A classical, Biot-type model cannot explain why the directly measured value of the flow resistivity of a nano-fibrous membrane appears so high and why does not it explain the measured increase in the real part of the surface impedance. Strangely, the flow resistivity measured with the BS 20953 [7] compares well with that predicted using the Kozeny-Carman equation [5]. Obviously, neither of these two flow resistivity values can fit the model proposed in [4] because they are far too high. There are likely to be a number of reasons for this discrepancy. Firstly, one can question if the nano-fibrous membrane vibrates and interacts mechanically with the melamine foam substrate in some complex manner which is not accounted for by the theoretical model used in this work. It is easy to show that is not possible to explain this effect by simply adding a surface mass term to eq. (4) to account for membrane vibration. This term will be complex and it will affect the imaginary part of the surface impedance leaving the real part completely unchanged. Secondly, there is a question whether or not a classical, Biot-type model will work when the thickness of the viscous boundary layer becomes comparable to the membrane thickness. For example, the boundary layer thickness in the frequency range presented in Figures 6 (200 – 1800 Hz range) will vary from 110 m to 37 m. This is comparable with the nano-fibrous membrane thickness of 28 m which can make the interaction of the oscillatory flow with nano-fibres a more complex process than the one assumed in the adopted acoustical model. Thirdly, the fibre diameter in this kind of media (277 nm) becomes comparable to the mean free path in air (68 nm). Therefore, it is unclear if a Kozeny-Carman-type model that makes use of the Darcy law is actually valid to predict accurately the resistance of these tiny fibres to the direct flow of air.

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REFERENCES

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