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ISSN : 1226-0088(Print)
ISSN : 2288-7253(Online)
Membrane Journal Vol.29 No.1 pp.51-60
DOI : https://doi.org/10.14579/MEMBRANE_JOURNAL.2019.29.1.51

# Gas Permeability through Mixed Matrix Membrane of Poly(dimethylsiloxane) with Aluminosilicate Hollow Nanoparticles

Xiaoyi Fang, Bumsuk Jung†
Department of Environmental Engineering and Energy, Myongji University, 116 Myongji-ro, Yongin, Gyeonggi-do 17058, Korea
Corresponding author(e-mail: bjung@mju.ac.kr)
February 24, 2019 February 26, 2019 February 26, 2019

## Abstract

In order to improve gas separation properties of polymeric membranes which have been widely applied in the industry field, aluminosilicate hollow nanoparticles named as allophanes were synthesized by sol-gel method and formulated in Poly(dimethylsiloxane) (PDMS) matrix to investigate the gas separation properties of PDMS membrane. Transmission electron microscope (TEM), Energy dispersive X-ray analysis (EDX), X-ray diffractometer (XRD), Surface area and pore size analyzer (BET) and Fourier transform infrared spectrophotometer (FTIR) were carried out to characterize the synthetic allophanes. Then the PDMS mixed matrix membranes were prepared by adding different volume fraction of allophanes. To examine the effect of allophanes addition in PDMS matrix using unmodified allophane and modified ones, the gas permeation experiments were performed using oxygen, nitrogen, methane and carbon dioxide. As the volume fraction of modified allophane increased up to 4.05 Vol% the permeability of four test gases through PDMS mixed matrix membranes increased. Also, the selectivity of O2/N2 and CO2/CH4 increased with the contents of the modified allophane. Further improvement of gas separation properties of PDMS mixed matrix membranes containing higher volume percent of allophanes can be expected as long as well dispersion of allophanes in PDMS matrix can be achieved for better PDMS membranes.

# 알루미노규산염 나노입자를 이용한 Poly(dimethylsiloxane) 복합매질 분리막의 기체투과 특성

방 효 질, 정 범 석†
명지대학교 환경에너지공학과

## 초록

분리막 소재의 투과도와 선택도 사이의 trade-off 관계로 인해 여전히 많은 연구가 필요하다. 특히 고분자 분리막에 무기물 나노입자가 들어가 있을 때, 기체투과 거동의 학문적 이해는 여전히 부족하다. 따라서 본 연구에서는 분리막 소재로 가장 많이 사용되는 PDMS에 2~5 nm의 기공을 가지고 있으며 직경이 약 5 nm 크기의 aluminosilicate hollow nanoparticles 인 allophane을 이용하여 복합매질 분리막을 제조하여 기체투과특성을 연구하였다. 대표적인 분리막 소재인 PDMS에 친수성 allophane, 그리고 나노입자에 undecylenic acid로 표면을 개질한 allophane을 막 내부에 고르게 분산시켜 함량 별로 복합매질 분리막을 제조하였다. 나노입자가 분산된 혼합매질 분리막 내에서 기체의 투과 특성을 파악하고, 이에 따른 기체투과 거동과 나노입자가 가지고 있는 기공의 역할을 평가하고자 하였다. 표면개질된 allophane을 첨가함에 따라 기체 투과도와 산소/질소 그리고 이산화탄소/메탄의 선택도가 동시에 점진적으로 향상되는 결과를 얻었다.

## 1. Introduction

Polymeric membrane characterized by commercial cost, simplified operation and lower energy consumption has been wildly used for gas separation in industry field. Rubbery membrane distinguished by its more flexible backbone structure is a quite good candidate for industry application for gas separation, because it typically exhibits higher gas permeability. However, the gas separation properties of polymer membrane are restricted by the trade-off between gas permeability and selectivity[1]. Mixed matrix membranes (MMMs) have been much interested as an effective approach to improve the performance of both gas separation membrane systems. In particular, the potential MMMs can be applied to various industrial gas separation applications such as CO2 capture from flue gases, natural gas purification and biogas upgrading[2-7].

The porous inorganic sieve materials possessing of large surface area and tiny pores of a precise and uniform size are capable of absorbing gas molecules which are small enough to pass through (eg: carbon molecular sieves, zeolites). These properties of inorganic sieve materials were utilized to enhance the selectivity of rubbery membranes, whereas the size of sieves materials hindered their applications. To achieve higher flux of polymeric membrane, thinner rubbery membrane is more attractive than other types of membranes. But inorganic sieve materials with micro size easily form pinholes which could ruin the gas separation properties of polymeric membrane. Consequently, inorganic sieve materials with nano-size are quite essential to achieve thinner mixed matrix membranes without any pinholes. Also nano-sized inorganic sieve materials are able to improve the gas separation properties of polymeric membranes by enlarging the free volume available for molecular transport through polymer matrix. As reported before, the accessible free volume increased by addition fumed silica in PDMS membrane matrix. Correspondingly the permeability coefficients in the MMMs increased systematically with increasing concentration of nano-sized fumed silica[6]. For these reasons, nano-sized inorganic sieve materials seem perfect candidates in improving both gas permeability and selectivity of the polymeric membranes.

Proper selections of polymer matrix as well as nano-sized inorganic sieve materials are quite crucial to form high performance mixed matrix membranes[3,4]. To improve the gas separation properties, some kinds of inorganic materials such as zeolites[4], silicas[7,8], carbons[9,10], metal-organic frameworks (MOFs)[11-13], and porous organic polymers (POPs)[14] can be used. However, the gas separation properties of these filled polymeric membranes were not improved much on both the permeability and selectivity of the polymeric membrane. The inherent properties of these inorganic sieve materials hindered their application for attaining high performance of polymeric membrane due to the compatibility with matrix, entry of gas in the pores of fillers, the distribution and orientation of fillers and so forth. Accordingly a new type of nano-sized inorganic sieve material known as allophane was chosen as the dispersed phase. Allophane, which is originally found in volcano ash, is amorphous aluminosilicate hollow nanoparticle with each particle diameter 3~5 nm and inner pore size diameter 2~5 nm. It was also reported by some papers that there were porous openings with diameter of 3.5 Ǻ on the shell of each unit allophane particle[15]. The unique structure and size property make this novel material a competitive candidate for improving the gas separation properties of polymeric membranes.

As a filler for mixed matrix membranes in this study, allophanes were synthesized by sol-gel method suggested by Ohashi et al. was modified here[15]. In addition, the mixed matrix membranes were prepared by filling certain amount of allophane particles based on the desirable contents into the well-known membrane material of poly(dimethylsiloxane) (PDMS) matrix. Furthermore, the gas separation properties of mixed matrix membranes with different fractions of allophanes were investigated by measuring the gas permeability of mixed matrix membranes. Thus, the main objective of this study is to examine the gas permeation properties of PDMS mixed matrix membranes incorporated with different fractions of allophanes and to compare the gas separation performance of PDMS mixed matrix membranes containing allophanes and modified allophanes in terms of permeability and selectivity of gases.

## 2. Experimental

### 2.1. Materials

Sodium orthosilicate (tetrasodium monosilicate) and aluminium nitrate nonahydrate were purchased from Wako and Duksan respectively. Deionized water purified with MilliQ system (Millipore) with resistivity greater than 18 MΩ⋅cm was used throughout the experiment. Poly(dimethylsiloxane) (PDMS) (DC184) were received from Dow corning. Toluene reagent with the purity ≥ 99.5% was obtained from Sigma-Aldrich. In synthesize allophane, dialysis tubing with molecular weight cut off from 12,000 to 14,000 was purchased from Fisher Scientific.

### 2.2. Allophane synthesis and characterization

#### 2.2.1. Allophane synthesis

Allophane was synthesized by the sol-gel method. Three millimoles of both sodium orthosilicate and aluminum nitrate solutions were prepared separately. The aluminum nitrate solution and the aqueous solution of sodium orthosilicate were induced into the reactor simultaneity with ratio of Si/Al = 3/4. Vigorously stirring over night at room temperature was necessary. After stirring had been completed, the clear solution was heated with heating mantle for hydrothermal reaction at 95°C for 5 days. In order to precipitate dispersed allophanes from clear solution, the pH value of solution was adjusted to weak acid around 6. The precipitates of allopanes were collected from suspension by centrifuge at 7,000 rmp. Dialysis membrane was used to remove the byproducts existing in the precipitates until the conductivity less than 2 μs/cm. The precipitates were freeze-dried to collect allophane powder for further characterization and modification.

#### 2.2.2. X-ray diffraction (XRD)

Powder X-ray diffraction measurement was carried out by X-ray diffractometer (PANalytical, X’per-Pro) with the dry powder of the products. The samples were scanned from 5° to 70° 2θ using a step size of 0.05° 2θ and scanning for 10s at each step.

#### 2.2.3. The transmission electron micrograph (TEM)

The TEMs were obtained using a Philips Model EM-300 microscope. Each allophane sample in suspension was dispersed by ultrasonics and one drop of the solution was deposited on a Cu surface, previously covered with a carbon surface. Finally, the TEMs of the dry sample were recorded.

#### 2.2.4. Energy dispersive x-ray analysis (EDX)

Electronic Microprobe was employed to complete EDXS analyses with additional data on Si/Al ratios.

#### 2.2.5. Brunauer-Emmett-Teller specific surface area measurement (BET)

Brunauer-Emmett-Teller specific surface area of the products was determined by means of nitrogen adsorption at liquid nitrogen temperature (T = -195°C). The pore-size distribution was also measured by the BET method using a nitrogen adsorption isotherm (Micromeritics, ASAP 2020).

### 2.3. Allophane surface modification and characterization of modified allophanes

#### 2.3.1. Allophane surface modification

Both undecylenic acid and freeze-dried allophanes were dissolved into toluene respectively. The two solutions were mixed together and heated at 100°C for 12 h with continuous stirring. The suspension turned to be clear. Purification of modified allophane was necessary in this experiment. First, Ethanol was added to the clear solution to precipitate modified allophanes. Second, the precipitates of modified allophanes were collected by centrifuge. The clear upper part of the suspension which containing the unreacted undecylenic acid was decanted. These two steps were repeated three times to fully remove the unreacted undecylenic acid from modified allophanes. The modified allophanes were freeze-dried to get the allophane powders.

#### 2.3.2. The fourier transform-infrared spectrophotometer (FTIR)

The fourier transform-Infrared spectrophotometer (FTIR) analysis was carried out to confirm the surface modification of allophanes. Infrared spectrum was collected in transmission mode in the 500~4,000 cm-1 range.

### 2.4. Preparation of pure PDMS membranes and PDMS mixed matrix membranes

A silicone PDMS elastomer kit (Sylgard 184) was purchased from Dow Corning Co. composed of parts A (dimethylvinyl-terminated PDMS) and B (Pt-based catalyst cross-linking agent) was diluted in toluene to make casting solutions. Then, a certain amount of allophanes based on the desired content, was added to the PDMS polymer solutions. The final solution was mixed using an ultrasonicator for 2 hours. After allophanes were fully dispersed in the PDMS polymer solution, the mixtures were poured in a Teflon dish and were dried at room temperature. Finally, the Teflon dishes were put into a vacuum oven which was set at 80°C for curing the membrane overnight.

### 2.5. Permeation measurements

The gas permeation of pure PDMS membranes and PDMS mixed matrix membranes containing 1.33, 2.68, and 4.05 vol% allophanes or modified allophanes were determined with a constant pressure/variable volume apparatus. The sequence of gases used for measuring permeability through each membrane was nitrogen, oxygen, methane and carbon dioxide. The purities of all gases used for gas permeation experiments were up to 99.9%. The gas permeation experiment was performed at a constant temperature of 30°C. Pure gas permeability were measured with the constant pressure/ variable volume method (time-lag) designed by Airrane Co. Ltd (Korea) at the constant temperature (30°C). Measuring the thickness of membranes, the single gas permeability were calculated in the unit of Barrer (1 Barrer = 1 × 10-10 cm3 (STP)⋅cm/cm2⋅s⋅cmHg). The ideal selectivity of a membrane for a component A relative to a component B, αA/B, is defined as the ratio of their permeability of gars pairs.

## 3. Results and discussion

### 3.1. Structure and properties of synthetic allophanes

The structure and the size property of allophane particles are quite crucial for preparing successful mixed matrix membranes. Any impurities (e.g. silica gel, boehmite, gibbsite) with different structures and micron sizes existed in allophanes are possible to hinder the effect of allophanes embedded in PDMS membrane matrix. In the study, we synthesized allophane particles by sol-gel method which was suggested by OHASHI and his coworkers[15,16]. Allophane particles synthesized by his method were free of impurities (i.e. silica gel, boehmite, gibbsite). However, to achieve well dispersed system, we modified the synthesis method by reducing the concentrations of two reactants until 3 mM and adjusting the pH value to near neutral to weakly alkaline.

#### 3.1.1. X-ray diffraction (XRD)

X-ray powder diffraction patterns of synthetic allophane were performed at room temperature (Fig. 1), exhibited broad reflections centered at 0.34 and 0.22 nm, typical of X-ray amorphous aluminoslicates. Amongst these peaks, is one which is specific to spherical hollow particles and which originates in the SiO4 tetrahedral sheet of allophane particles especially the dif frac tion area of 0.33~0.35 nm[16]. Furthermore, the diffraction peaks of the silica gel and aluminum hydroxides (boehmite, gibbsite) appearing nearby at 0.39 nm and 0.64~0.69 nm, respectively, were not observed[17], which indicated that the synthetic allophanes were free of impurities which always exist in natural allophane clusters.

#### 3.1.2. The transmission electron micrograph (TEM)

By means of the transmission electron micrograph, the proper ratio for synthesizing allophanes can be determined. As can be seen in Fig. 2, synthetic ratio of Si/Al was about 3/4, the unit particle size can be recognized around 3~5 nm. However, allophanes were observed only in agglomerations instead of single particles. The beam damage cannot be ignored, satisfactory TEM image is difficult to obtain. The inset in Fig. 2 shows a schematic picture of allophane. At the same time Energy Dispersive X-ray Spectrometer (EDX) analysis was also taken in site of the allophane sample. The atom ratio of Si/Al of synthetic allophanes was calculated as about 0.67 as for the natural allophanes, the ratio of Si/Al is in the range of 0.5~1[17].

#### 3.1.3. Brunauer-Emmett-Teller specific surface area measurement (BET)

As shown in Fig. 3, the pore-size distribution curves of the synthetic allophanes revealed that the synthetic allophanes have a peak centered at 3.5 nm which results from a fraction of nanopores inside of the hollow spheres. Also there is much larger peak illuminating that much more pores exist among the nanoparticles. This can be validated by the TEM image in which the dry hollow nanoparticles were aggregated together. The Brunauer-Emmett-Teller (BET) specific surface area of synthetic allophane is 444 m2/g in the middle of the range of specific surface area of nature allophanes (200 ~700 m2/g).

From the characterization results and the description for allophanes from some references[15,16], it can be recognized that single allophane particle is a shell-structure spherical particle with outer diameter around 3~5 nm and the diameter of hollow is around 2~5 nm. There are many accessible opening pores existing on the shell.

### 3.2. Characterization of surface modified allophanes

By using undecylenic acid, it was possible to get a clear dispersion of allophanes in toluene. As can be seen in Fig. 4, FT-IR spectra for the synthetic allophane, modified allophanes and undecylenic acid noted as “S”, “M”, and “U”, typical absorption bands based on allophanes were clearly observed. IR spectra display a broad vibration band in the high wavenumber centered at 3,423 cm-1 that can be assigned to hydration water. The corresponding bending vibration was observed at 1,633 cm-1. At higher energy, one can observe an intense vibration band at 1,022 cm-1, followed by a shoulder at ~900 cm-1 that can be assigned to Si-O-(Si) and Si-O-(Al) vibrations. Similar bands were reported in most studies dealing with natural allophanes and synthetic allophanes[ 15-17]. Likewise, FT-IR spectra for the surface modified allophane, the C-Hx bonds at 2,922, 2,856 cm-1 appeared after surface modification with undecylenic acid. The existence of such bonds confirms surface modification of the allophane particles.

### 3.3. Gas separation properties of PDMS mixed matrix membranes

#### 3.3.1. Gas separation properties of PDMS and mixed matrix membranes incorporated with unmodified allophanes

As shown in Fig. 5, the gas permeability of MMMs are illustrated as a function of the amount of unmodified allophanes. When adding 1.33 vol% of allophanes in PDMS matrix, it is obvious that the gas permeability of oxygen, carbon dioxide and nitrogen through mixed matrix membranes were improved except for methane. As adding more than 1.33 vol% of allophanes in PDMS matrix, the trend of gas permeability of four gases de crease with increasing volume fraction of allophanes filled.

In order to analyze gas transport behaviors in PDMS MMMs, the incorporation of a small volume fraction of allophanes into PDMS matrix can result in a sig- nificant effect on overall separation performance, as predicted by the so-called Maxwell model[6,18,20]. This model was originally derived for the estimation of the dielectric properties of composite materials[19], but has been widely accepted as a simple and effective tool for estimating mixed matrix membranes properties. The Maxwell model equation for mixed matrix membranes with dilute suspension of impermeable spherical particles can be written as follows:

$P e f f = P c [ P d + 2 P c − 2 ϕ d ( P c − P d ) P d + 2 P c − ϕ d ( P c − P d ) ]$
(1)

where Peff is the effective composite membrane permeability, ϕ the volume fraction, P the single component permeability and the subscripts d and c refers to the dispersed and continuous phases, respectively.

In order to obtain a general behavior of the PDMS MMMs, effective gas permeability were illustrated as a function of the volume fraction of allophanes into PDMS matrix in Fig. 6.

As show in Fig. 6, the dashed line represents the prediction of the Maxwell model for comparison. When the P/P0 value of a certain gas species is bigger than the value predicted by the Maxwell equation, this gas species can pass through the fillers so the gas permeability of this gas species increased. When the P/P0 value of a certain gas species is smaller than the value predicted by the Maxwell equation, this gas species cannot pass through the fillers so the fillers effect as barriers for this certain gas species. If the fillers can be permeable for some gases and block some other gases when the gases pass through the membranes, the fillers effect as sieve materials which are capable to enhance the gas separation performance of polymer membranes.

As can be seen in Fig. 6, all values of P/P00 for both nitrogen and oxygen are bigger than the values predicted by the Maxwell equation. Values of P/P0 for methane almost agree with the values predicted by the Maxwell equation. However, values of P/P0 for carbon dioxide are smaller than the values predicted by the Maxwell equation. This is due to the sieving effect of allophanes. Due to the different molecular diameter of gases, allophanes are permeable for smaller gases such as oxygen and nitrogen, and not permeable for bigger gases such as methane and carbon dioxide. However, because of large surface area of allophanes, the solubility of the carbon dioxide in membrane increased, which compensated its decreasing diffusivity due to the sieving effect of the nanoparticles. In effect, the gas permeability of carbon dioxide did not decrease too much. For these reasons, the selectivity of carbon dioxide/methane increased obviously from 3.3 to 7.7 as shown in Fig. 7. The selectivity of oxygen/nitrogen also slightly increased especially when filling 1.33 vol% of allophanes into PDMS matrix.

As mentioned above, the PDMS mixed matrix membranes should be greatly increased due to the large surface area of allophanes. All the gas permeability should be improved by filling allophanes in PDMS matrix. However, because of the large degree of aggregation of allophane particles, which makes local impermeable domains, the barrier effect of allophanes in PDMS matrix is also quite huge, which hindered the effect of allophanes in improving the gas separation properties of PDMS membranes in terms of permeability.

#### 3.3.2. Gas separation properties of PDMS mixed matrix membranes incorporated with modified allophanes

The surface modification of allophanes for achieving well dispersed system is quite essential to form good performance mixed matrix membranes. As can be seen in Fig. 8, the gas permeability of all gas species such as oxygen, nitrogen, methane and carbon dioxide increased. Compared with the gas permeability of the other gases, the gas permeability of carbon dioxide through PDMS mixed matrix membranes increased more with the addition of modified allophanes. It seems that this is probably due to the permeable openings in PDMS mixed matrix membranes by adding modified allophanes. The effect of modified allophanes into PDMS mixed matrix membranes can be obviously ob served from Fig. 9.

Compared with Maxwell model, the addition of modified allophanes gives rise to well permeable sites for all permeation gases. In particular, the enhancement of permeation can be observed for CO2 gas. It seems that the acidic CO2 gas is quite prone to reside and move freely in the hollow of allophane due to a smaller kinetic diameter or a specific interaction between hydroxyl groups and carbon oxide. Compared with the gas selectivity of PDMS mixed matrix membranes incorporated with allophanes as shown in Fig. 7, the gas selectivity of PDMS mixed matrix membranes incorporated with modified allophanes is not significantly, but it seems that the selectivity were well kept, and even rather increased by the addition of modified allophanes As well known, due to a poor interfacial adhesion between fillers and matrix polymer, nano-sized void can be formed. The formation of voids at the interface is made as sieve-in-a-cage morphology, which affect considerably membranes performance. As can imagine the interfacial voids can offer the path of the least resistance, so gas molecules move through them rather through sieve particles, which results in much higher permeability as compared to that of the pure membrane matrix, while the selectivity can be higher than, nearly equal to or lower than the pure membrane matrix, which is also dependent on the size and the number of voids. It can be also expected that when the void size is larger than the permanent gas molecule, the flux increases significantly but the selectivity decreases. Fig. 10

As for the unmodified allophane MMMs, it can be explained that the aggregated allophanes as mentioned in Fig. 6, were easily formed continuous path ways for smaller gas molecules to pass through due to the formation of voids, but the aggregate domains are formed as barrier domains to block bigger gas molecules to detour the barrier domains. However the well-dispersed system of modified allophane MMMs, allophanes were separated enough with each other. PDMS matrix governs the gas permeability of all the gas species as well as the sieve effect of allophanes was not obvious if only a little volume fraction of allophanes added. Thus, the addition of modified allophane in PDMS membranes behaves extrinsic microporosity membrane being with nano-sized voids in the MMMs, which cause Kudsen transport or surface diffusion through the pores of allophanes in the membrane matrix. In particular, compared with other gas permeations, the transport of CO2 shows quite enhancement. Thus, it can be expected that the modified allophane MMMs would be a useful materials for fabricating mixed matrix membranes for CO2 separation.

## 4. Conclusions

Using hollow nanoparticles (allophanes), we have prepared defect-free self-standing MMMs from different nanoparticles embedded in PDMS membranes, fabricated by the solvent casting method. PDMS mixed matrix membranes were also achieved by adding volume fraction of allophanes and modified allophanes. Gas permeability of all MMMs were shown to increase with increasing filler loading without any reduction in the gas selectivity. The effect of allophanes and modified allophanes in PDMS mixed matrix membranes were examined by the gas permeation experiments. It can be seen that allophanes play the important role of a permeable openings rather than molecular sieves in the PDMS mixed matrix membranes. It seems that the addition of modified allophane in PDMS membranes offers as extrinsic microporosity membrane being with nano-sized voids in the MMMs, which cause Kudsen transport or surface diffusion through the pores of allophanes in the membrane matrix. For the PDMS mixed matrix membranes incorporated with modified allophanes, the permeability of all the testing gases were improved. The further improvements of gas separation properties of PDMS mixed matrix membranes containing more modified allophanes can be expected.

## Acknowledgements

This work was supported by the technology development program (S2273230) funded by the ministry of SMEs and startups (MSS, Korea).

## Figure

X-ray powder diffraction patterns of synthetic allophane.

Transmission electron micrograph of synthetic allophanes from aqueous solution (synthetic ratio of Si/Al = 3/4) and the inset of schematic drawing (cross-section) of synthetic allophane unit particle.

Pore-size distribution curves of the synthetic allophanes.

FTIR spectra of the synthetic allophanes, surface modified allophanes and undecylenic acid.

Permeability of N2, O2, CH4, and CO2 in pure PDMS membranes and PDMS mixed matrix membranes containing unmodified allophanes with different fractions.

Ratio of N2, O2, CH4, and CO2 permeability in PDMS mixed matrix membranes, P, to that in the pure PDMS membranes, P0, as a function of volume fraction of allophanes. The dashed line represents the prediction of the Maxwell model.

Selectivity of O2/N2 and CO2/CH4 in pure PDMS membranes and PDMS mixed matrix membranes containing 1.33, 2.68, and 4.05 vol% allophanes.

Permeability of N2, O2, CH4, and CO2 in pure PDMS membranes and PDMS mixed matrix membranes containing 1.33, 2.68, and 4.05 vol% modified allophanes.

Ratio of N2, O2, CH4, and CO2 permeability in PDMS mixed matrix membranes, P, to that in the pure PDMS membranes, P0, as a function of volume fraction of modified allophanes. The dashed line represents the prediction of the Maxwell model.

Selectivity of O2/N2 and CO2/CH4 in pure PDMS membranes and PDMS mixed matrix membranes containing 1.33, 2.68, and 4.05 vol% modified allophanes.

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