Critical pore dimensions for gases in a BTESE-derived organic-inorganic hybrid silica: A theoretical analysis

Separation and Purification Technology 191 (2018) 27–37

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Critical pore dimensions for gases in a BTESE-derived organic-inorganic hybrid silica: A theoretical analysis ⁎

Xuechao Gaoa, Guozhao Jib, Jiacheng Wanga, Li Penga, Xuehong Gua, , Liang Chenc,

MARK

⁎

a State Key Laboratory of Materials-Oriented Chemical Engineering, College of Chemical Engineering, Nanjing Tech University, 5 Xinmofan Road, Nanjing 210009, PR China b School of Environment, Tsinghua University, Beijing 100084, China c Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Critical pores Gas diffusion Adsorption Dispersive interaction Hybrid silica

This work studied the critical pore dimensions for different light gases in disordered microporous hybrid silica which was derived from 1,2-bis(triethoxysilyl)ethane (BTESE) precursor. The Lennard-Jones (LJ) parameters of the material, obtained from high pressure isotherms of light gases, were employed to evaluate dispersive interaction, diffusivity and adsorption coefficient in the nanopores of the hybrid silica, and the results demonstrated that the entrance barrier was the key factor to break down the separation limits confining the Graham law and the Knudsen model by gas kinetic theory. Finally, the critical pore dimensions for different resistancedominated regions were evaluated, which theoretically explained why the separation capacity of the hybrid silica was less promising than pure silica for the purpose of gas separation.

1. Introduction The design of porous material usually relies on the understanding of molecular movements in the channels, so the proper modelling of fluid transport can boost the practical applications of porous materials in traditional and new-merging technologies, such as nanocatalysis, fuel cells and membrane purification [1–9]. The subject of the fluid transport in narrow channels could be dated back to 1846 when Graham discovered the inverse proportionality of flux to the square root of molecular weight [10–12], which was later refined by Fick in 1855, using a Fickian diffusivity (DAB). For mass transfer in pores, the motionless wall is generally referred as component B, so the diffusivity is re-defined as pore diffusivity. Using a data set of gas permeation [13], Knudsen proposed the wall-mediated diffusion regime for rarefied gases in 1909, and the Knudsen diffusivity (DKn) was widely accepted for the gas transport in porous materials afterwards. However, the influences of solid-fluid interactions were not considered [14–18], since the pores sizes used in Knudsen’s experiments (33–145 μm) were far larger than gas kinetic diameters, which results in the main deficiency of the Knudsen model for diffusion estimation in smaller nanopores [19–22]. However, this issue was not widely concerned due to the limit number of studies of mass transport in micro and meso-porous materials. The rapid development of material synthesis has enabled scientists and engineers to investigate the transport process in much narrower

⁎

Corresponding authors. E-mail addresses: [email protected] (X. Gu), [email protected] (L. Chen).

http://dx.doi.org/10.1016/j.seppur.2017.09.013 Received 3 July 2017; Received in revised form 31 August 2017; Accepted 5 September 2017 Available online 06 September 2017 1383-5866/ © 2017 Elsevier B.V. All rights reserved.

pores, and the direct application of the Knudsen model in microporous media often yielded extremely high tortuosities, indicating the overestimation of diffusivity in such pore dimensions [23–25]. The potential exerted by the pore wall to the gas molecule is supposed to play a central role for the diffusion misinterpretation [26]. To amend the insufficiency of the Knudsen model, an activation energy, Ea, was introduced as a form of Arrhenius type to represent the effect of potentials, and a gas translational (GT) model was thus established [27,28] as follows:

DGT = D Kn e−Ea / Rg T

(1)

The original derivation of GT model was used for transport of heavy hydrocarbons in zeolites. Due to its strong molecule-wall interactions, the interaction was assumed to be governed by a single ring of atoms of pore entrance. The prediction of activation energy in GT model was recently generalized by taking pore length into account, where a suction energy (W) can be evaluated from a discrete integration of axial force (F) over travelling distance (Z) [29–31].

W=

+∞

∫−∞

FZtol (Z )dZ

(2)

Based on the magnitude analysis of W, different transport regimes can be readily distinguished [29]. However, the discrete atom–atom formulation used in the GT model can only be applied for rigid pores with crystallized structures, such as metal organic frameworks (MOFs),

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zeolites and carbon nanotubes (CNT) [32]. For amorphous materials, continuum interaction integration should be employed [26,33–36]. Further, the GT model only considers the velocity loss caused by pore entrance, and the influence of dispersive potentials in pore body is still neglected in accordance with Knudsen’s derivation, so the body resistance is largely overlooked in the later regression, or lumped in the activation energy. Therefore, such an arbitrary treatment is only applicable when pore entrance dominates the transport resistance. For instance, in a silica pore of 3.0 nm, the diffusivity of CH4 as modeled by the GT approach still falls into the Knudsen model due to its negligible entrance barrier. However, such a prediction greatly contradicts the results obtained by molecular dynamic (MD) simulations [27], where a much lower diffusivity was expected. Such an inconsistency was largely due to the fact that the transport resistance is dominated by pore body for such pore dimensions instead of pore entrance. Dispersive interaction considerably increases the colliding frequency of solid-fluid, and its mathematical expression of diffusivity is recently resolved by numerically averaging the travelling time in radial direction [37], where the potential effect is explicitly incorporated in a Lennard-Jones (LJ) equation. The new model is termed as oscillator model, following the form

D body =

kB T 〈τ 〉 2m

(3)

in which 〈τ〉 is the numerically mean value of travelling time, varying from solid-fluid interaction. It is worth mentioning that the entrance resistance is neglected in the model, which is critical for sub-nanometer pores with a repulsive entrance. Therefore, the oscillator model only applies for the pores with a body-dominated resistance. Based on the above arguments, it is clear that both dispersive interactions in pore body and entrance should be properly addressed for a complete diffusing process. Recently, organic-inorganic hybrid silica, derived from 1,2-bis(triethoxysilyl)ethane (BTESE), has attracted considerable attention as a replacement to pure silica, and its membrane fabrication not only achieved high permeances and selectivities of gases, but also demonstrated much stronger hydrothermal stabilities in wet conditions [38]. However, the bottom-up transport process under the influence of dispersive interaction in such a hybrid pore still remains unknown, which leads to considerable inconveniences for its membrane design and utilization. Therefore, inspired by a series of previous studies [8,27,29,37,39], the aim of the current work is to theoretically analyze the transport process in the cylindrical pore of a BTESE-derived hybrid silica, where the dispersive interactions in both of pore entrance and body are included. In this work, hybrid silica is firstly synthesized and characterized to derive the LJ parameters for the material, which is subsequently used to show how the dominated resistance varies with pore sizes and gas species. The observed trends are expected to have guiding significance for the exploration of the BTESEderived hybrid silica.

Fig. 1. (a) Thermogravimetric analysis (TGA) for the uncalcined hybrid silica powder in air, and (b) the Fourier transform infrared (FTIR) spectra for the calcined and uncalcined sample.

–OH groups contributed to the corresponding weight loss; the C2 elements began to pyrolyze at temperatures above 420 °C. Fig. 1(b) describes the Fourier transform infrared (FTIR) spectra for the calcined and uncalcined xerogels. As expected, the characteristic peaks of methylene group (–CH2–) in BTESE (around 2900 cm−1) were clearly detected, thereby further confirming the presence of organic groups after calcination [21].

2.2. Low pressure gas adsorption The adsorption isotherm of N2 at 77 K, interpreted by nonlinear density function theory (NLDFT), was used to derive the pore size distribution for the sample. As given in Fig. 2, most of pores vary from 1 nm to 3 nm, and the obtained pore volume (Vp) and surface area (Sg) are 0.29 cm3/g and 713.04 m2/g, respectively. To ensure complete coverage of the obtained pore size distribution, CO2 adsorption at 273 K was further conducted, where the isotherm analysis by the Saito-Foley theory denied the presence of any ultramicropores [19]. The hybrid silica was also examined by High Resolution Transmission Electron Microscopy (HRTEM) at 200 kV (JEOL 2010), and the obtained micrograph (c.f. Fig. 3) indicated that most pore sizes were below 2 nm, which was in good agreement with the mean value estimated by 4Vp/ Sg. The true density of the calcined sample was further extracted from the free space of the sample tube, and the obtained value (ρ = 1.71 g/ cm3) was much smaller than the theoretical value for pure silica (ρ = 2.30 g/cm3) [41], suggesting the hybrid silica matrix was more loosely interconnected. The accessible porosity of the xerogel was estimated by

2. Materials characterization and experiments 2.1. Microporous hybrid silica preparation The microporous hybrid silica was prepared by sol–gel process [40], using the hydrochloric acid (HCl) as catalyst. Firstly, a solution of 6.00 g H2O and 2.06 g HCl was dropwisely dissolved with a mixture of 2.00 g BTESE and 9.84 g ethanol within 2 h, which was stirred for 20 min at room temperature, and then the solution was volumetrically diluted by 10 times with ethanol, before being dried in an oven at 80 °C. Based on the result of the thermogravimetric analysis (TGA) in Fig. 1(a), the obtained sample was calcined at 300 °C for 30 min in the air, which was sufficient to remove residuals, without damaging C2 groups (–CH2–CH2–). As suggested, at temperatures below 100 °C, the weight loss was mainly caused by water and ethanol vaporization; afterwards, silanol condensations occurred, where the elimination of 28

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temperature of 500 °C, so a weight loss of 3.0% from 100 to 500 °C can be assigned to silanol condensation, which led to a low hydroxyl number of 2.8 nm−2. This value is much smaller than the result (7.5 nm−2) reported for pure silica synthesized from a mixture of methyltriethoxysilane (MTES) and tetraethylorthosilane (TEOS) [39]. Since the electrostatic interaction is determined by hydroxyl density, the enhancement of CO2 adsorption due to its high quadrupole moment (−14.27 × 10−40 C·m2) [42] is thus expected to be inhibited for the sample. 2.4. High pressure adsorption equilibrium of single gases High pressure adsorption of gases were carried out in an analyzer (Setaram PCTPro-3000) using a classical volumetric method. Since He and H2 have very weak adsorption, four other gases (CH4, Ar, N2 and CO2) were studied at four temperatures (303, 333, 363 and 393 K). In the experiments, the sample was degassed at 473 K for several hours before being exposed to adsorbate for different pressures up to 3000 kPa, and the isothermal condition was controlled by a heating jacket. The classic Langmuir model was employed to correlate excess adsorption amount, q, with gas bulk pressure, P:

Fig. 2. Pore size distribution of the calcined xerogel, obtained using nonlinear density function theory (NLDFT) interpretation of N2 adsorption isotherm at 77 K.

q = qm

Kl P 1 + Kl P

(6) −1

in which Kl (kPa ) and qm (mmol/g) represent adsorption constant and maximum adsorption capacity, respectively. For a given material, it is generally accepted that Kl depends on both temperature and gases, and qm is exceptionally determined by gases. o o In thermodynamics [43], Kl = (e ΔS / Rg )(e−ΔH / Rg T )/ P o , where ΔH o o Δ S and stand for enthalpy and entropy changes relative to the standard pressure P o (101,325 Pa), respectively. The formula can be rearranged as

ln(Kl ) =

ΔS o −ΔH o 1 + Rg Rg T

(7)

Therefore, −ΔH o can be obtained through the fitting of ln Kl with 1/ T, and the value corresponds to isosteric heat of adsorption, Q in kJ/ mol. 3. Adsorption and LJ parameters in hybrid silica This work aims to theoretically analyze transport behaviors of gases in two different parts of an amorphous hybrid silica pore, so the corresponding LJ parameters should be provided for the estimation of dispersive interaction. Assuming adsorption mainly occurs inside pore body [39], the LJ parameters of the wall can be derived from the experimental adsorption isotherms. For a given pore, continuum LJ interaction of a single layer model is employed to represent the interaction, φsf (r ) , in the pore body as [16]

Fig. 3. Transmission electron microscopy (TEM) image of the calcined xerogel.

ε=

Vp ρ 1 + Vp ρ

(4)

Based on Eq. (4), the accessible porosity of the hybrid silica was 0.34.

16εsf ρs rp2 φsf (r ) = kB kB

2.3. Thermogravimetry

2000 wm 3 Sg

∞

dz

∫0

π

12 6 ⎡ ⎛ σsf ⎞ −⎛ σsf ⎞ ⎤ dα ⎢ ⎝ r′ ⎠ ⎝ r′ ⎠ ⎥ ⎣ ⎦

(8)

; σsf and εsf correspond to in which r ′ = + + rp collision diameter and interaction well depth, respectively; ρs is the wall surface density (atoms per unit area). The Lorentz-Berthelot mixing rule was applied to estimate the solid-fluid parameters using the LJ parameters of gases provided in Table 1. In spirit of the oscillator model [37], equilibrium constant (Kosc)

[z 2

Hydroxyl density on pore wall surface has significant influences on CO2 adsorption due to extra electrostatic interaction besides van der Waals force [39]. For the hybrid silica, hydroxyl number can be characterized through the weight loss (wm) of the calcined powder following the protocol by Markovic et al. [23], where the surface density of hydroxyl (NOH) was given by

NOH =

∫0

r2

rp2−2r

cos(α )]1/2

Table 1 Fluid-fluid Lennard-Jones parameters of gases used in this work.

(5)

In the heating program, the weight loss of the hybrid silica is caused by two parts, i.e., the silanol condensation and organic oxidation, whose latter part should be excluded in the calculation. In N2 environment, ligands of the hybrid silica ring started decomposing at a 29

Parameters

H2

He

CH4

N2

Ar

CO2

σff (nm) εff/kB (K)

0.2915 38.0

0.2551 10.22

0.381 148.2

0.3572 93.98

0.341 120.0

0.3472 221.9

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that the specific compositions and atomic arrangements of the hybrid silica pore are unknown, gas molecules are idealized as imaginary sphere to ignore orientation effects. A continuous integration is used for the LJ equation, and the potential for an axial coordinate of z, Φ(z), can be obtained by integrating the LJ function over the surface elements dA

was employed, which is defined as the density ratio of gas inside pores and free spaces. For a given pore, Kosc is evaluated by radially averaging the dispersive interaction

K osc (rp) =

2 rp2

∫0

rp

e−φsf (r )/ k B T rdr (9)

Similarly, its average over pore volume distribution, fv(rp), provides the apparent value (K osc ) for the hybrid material

K osc =

1 ∞ fv (rp) drp 0

∫

∫0

∞

K osc (rp) fv (rp) drp

4ρ εs f Φ(z ) = s kB kB

(10)

K osc =

Vp

+1

⎣

σ 6 −⎛ sf ⎞ ⎤ dA ⎝ l ⎠⎥ ⎦

(12)

By combining the entropic effect with the pre-exponential constant, the equation for activation energy is eventually provided as

Eq. (10) can be algebraically related to the common-used Langmuir parameters as

Kl Rg T qm

12

∫A ⎡⎢ ⎛⎝ σlsf ⎞⎠

Eamouth = Φ(zip)−Φ(z op)

(13)

where zop and zip represent initial and final coordinates in axial direction, respectively. Once the LJ parameters were derived, activation energy, Ea, in a defined pore can be estimated for the hybrid silica material. It is noted that activation energy is mainly determined by interaction during the pore entry, which is more kinetic; on the contrary, adsorption isosteric heat (Q), is largely related to interaction in pore body, which is more static, so the commonly-used assumption of the linear relationship between the two variables (Ea = −aQ ) lacks physical support [29,39,44].

(11) ∞ fv (rp) drp . 0

Combining where Vp can be estimated from fv(rp) by Vp = ∫ Eq. (10) with (11) for all the gases, the LJ parameters of the material can be readily optimized. The program was run in Fortran 90 using Levenberg-Marquardt (LM) algorithm, and the convergence was achieved when the weighted sum of squared errors (χ2) was less than 10−6. 4. Pore entrance and transport in the pore 4.1. Pore entrance

4.2. Transport in the pore

As gas molecules approach to the pore, each atoms of the molecule interacts with the wall components. For an extremely narrow pore entrance, the molecule needs to overcome a repulsion to enter the pore body before performing any collisions with walls. Based on the GT model, activation energy is the work done by axial forces, which can be estimated through the force integration over displacement given in Eq. (2). On the other hand, it is worth to mention that the parameter of work is mathematically scalar, which is independent of trajectories, so the value of work equals to the negative change of the potential at different positions in the molecule-pore system. Fig. 4 illustrates the schematic drawings of a polyatomic gas entering a pore channel. As suggested, the atomic orientation of the molecule (off the axis) has a certain influence on the dispersive interaction. However, considering

For microscale and mesoscale pores, dispersive interactions also play an increasingly important role in determining travel trajectories of gas molecules inside the pore, so the well-known hard-sphere treatment, on which the classic Knudsen model and Fickian diffusion were established, causes considerable deviations to the actual scenario. Instead, a soft adjustment should be used to account for trajectory changes of molecules before wall collision. The recent developed oscillator model can be deployed to estimate the resistance in the pore [37], where the dispersive interaction is explicitly incorporated by the LJ potential according to Eq. (8). For a canonical distribution of kinetic energies, the average travelling time of 〈τ〉 in Eq. (3) can be numerically solved, so the diffusivity of a LJ fluid performing diffusive reflection in a cylindrical pore is given as Fig. 4. Schematic drawing of a gas molecule axially enters a cylindrical pore channel with a single layer of atoms. The experimental pore radius is equivalent to the theoretical pore radius, rp-σss/2.

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Table 2 Langmuir parameters of gases on the calcined hybrid silica xerogel using Eq. (6).

Dosc (rosc ) 2 πmΓ

=

∫0

∞

rcl (r ,pr ,pθ )

∫r (r,p ,p ) co

r θ

e−βφsf (r ) dr

∫0

∞

βpr2 e− 2m

dpr

∫0

2 ∞ − βpθ e 2mr 2 dpθ

dr ′ pr (r ′,r ,pr ,pθ )

(14)

Here 1/2

p2 r 2 ⎛ ⎞ pr (r ′,r ,pr , pθ ) = ⎜2m [φsf (r )−φsf (r ′)] + θ2 ⎡1−⎛ ⎞ ⎤ + pr2 (r ) ⎟ ⎥ r ⎢ ⎣ ⎝ r′ ⎠ ⎦ ⎝ ⎠

(15)

Gas

qm mmol/g

Kl (303 K) 1/kPa

Kl (333 K) 1/kPa

Kl (363 K) 1/kPa

Kl (393 K) 1/kPa

CH4 N2 Ar CO2

2.74 2.18 2.38 5.30

5.22 × 10−4 2.63 × 10−4 2.58 × 10−4 1.55 × 10−3

3.00 × 10−4 1.47 × 10−4 1.55 × 10−4 5.71 × 10−4

1.80 × 10−4 1.01 × 10−4 1.13 × 10−4 2.95 × 10−4

1.14 × 10−4 6.12 × 10−5 8.20 × 10−5 1.60 × 10−4

for all the gases, which indicates that the solid-fluid interaction governs the adsorption under these conditions. Also, the adsorption quantity of CO2 is higher compared to the value for CH4, which was followed by N2 and Ar with comparatively intermediate adsorption. The fitted Langmuir parameters are summarized in Table 2, where both qm and Kl are apparent values of the hybrid silica material. Considering Kl reflects adsorption affinities of gases to the material, it is meaningful to compare these values among gases. As given in Table 2, a systematic pattern is also found for all the temperatures, i.e., CO2 > CH4 > N2 ≈ Ar, which matches well with the isotherm order provided by the experiments. Using the linear correlation of ln(Kl) with 1/T, the isosteric heat of adsorption, Q, is readily estimated, and such a plot is depicted in Fig. 6, where the correlations are in good agreement for all the gases (a high precision of R2 above 0.98). The derived gas order of Q is also consistent with the pattern provided by Kl, i.e., CO2 > CH4 > N2 ≈ Ar.

pr is the radial momentum at radial position, r′, and pθ is the angular momentum. rcl and rco stand for oscillating bounds between two continuous reflections of the moving particle in radial direction. Further, ∞ β = (kB T )−1 and Γ = ∫0 re−βφsf (r ) dr . Details of the model derivation and parameter description could be found elsewhere [45]. For a structureless cylindrical pore with a single layer wall, integration of the solidfluid interaction over pore surface yields the hypergeometric potential [46]. Once the LJ parameters are given for the organic-inorganic hybrid silica, transport properties in pore bodies can be readily investigated. 5. Results and discussion 5.1. High pressure adsorption isotherms Fig. 5(a)–(d) illustrates the experimental results (symbols) and the fitted solid lines using the Langmuir model for gases (CH4, N2, Ar and CO2) at different temperatures, in which qm solely depends on gases and Kl varies with both temperatures and gases. As expected, the classic Langmuir model provides good agreements with the experimental data

5.2. Correlation of equilibrium constant with adsorption Dispersive interactions are decisive factors to gas adsorptions. Here

Fig. 5. Measured adsorption equilibria of (a) CH4, (b)N2, (c) Ar and (d) CO2 at four temperatures (T = 303, 333, 363 and 393 K) fitted with Langmuir isotherm model using Eq. (6). The model and experimental isotherms are represented by solid lines and symbols, respectively.

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quadruple moment and F ̇ is the electric potential gradient) due to the presence of hydroxyls [39,47]. However, in current study, the obtained hydroxyl number of the hybrid silica (2.8/nm2) is remarkably smaller than the value reported for the pure silica in our previous work (7.5/ nm2), so the corresponding enhancement coefficient is expected to be diluted. As suggested in Fig. 7, good agreements for CO2 are still obtained even only van der Waals interactions are taken into account, suggesting the electric potential gradient (F ̇ ) generated by hydroxyls is very weak. Nevertheless, to quantitatively analyze the influence of hydroxyl number on extra electrostatic interaction, ξe (r ) , the enhancement coefficient (B) is also estimated by an empirical equation, as [37]

ξe (r ) =

B (rosc−r )2

(16)

where B (K·nm ) is a fitted constant and is exclusively determined by walls and gases, with a higher value for a stronger enhancement. For CO2, the newly-fitted apparent equilibrium constant is plotted by red dash line in Fig. 7, where a slight improvement over the previous fitting is obtained. The electrostatic coefficient is found to be −10.57 K nm2 (the negative sign stands for an attraction). It is clearly seen that although the current coefficient is much smaller than the value (−37.51 K·nm2) for the pure silica [39], the proposed assumption is confirmed, that is, the electrostatic interaction enhancement is stronger for higher hydroxyl numbers [39]. Such a discovery is also supported by the work of Al-Maythalony et al., where a reversed CO2/H2 permeation occurred to the ZMOF-1 membrane due to the extra electrostatic interactions via quadrupole moment of CO2 with imidazoledicarboxylate linker (containing carboxyl groups) [48]. 2

Fig. 6. Variation of ln(Kl) versus 1/T, and the isosteric heat, Q (kJ/mol). The symbols are the experimental values based on the parameters in Table 2, and the fitted solid lines were obtained by Eq. (7).

5.3. Correlation of LJ parameters and computational chemistry The best fitted LJ parameters of the hybrid silica in Fig. 7 are given as ρs = 7.85/nm2, σss = 0.4735 nm and εss/kB = 277.64 K. It is noted that the dimension of solid-solid collision diameter is far above the kinetic diameter of oxygen atom for pure silicas (σss = 0.28), indicating carbon and silicon atoms in the hybrid material play an important role in the interaction. Apart from that, the well depth (εss/kB) of solid-solid potential also becomes smaller, which can be attributed to two main reasons, i.e., the weak potential of carbon atoms, and repulsive effect by silicon atoms. To explore the chemical composition for the interaction unit of the pore wall, several van der Waals volumes of relevant small clusters are estimated by Gaussian 09 [49], where the Becke3lyp hybrid functional with a 6-311G(2d,p) polarized basis set is employed to optimize the cluster structures [50]. The extracted parameters (volume, kinetic diameter and bond length) of related structures for the hybrid silica are listed in Table 3. It is seen that the van der Waals diameters are systematically larger than the LJ value by a factor of around 1.15. The mismatch between the two diameters is largely caused by differences in the unit boundaries, i.e., the LJ diameter is the distance having the minimum potential between two spherical units, while the volume diameter (van der Waals) is defined based on the electron cloud inside a contour of 0.001 electrons/Bohr3 density. The above observation suggests that the dispersive interaction is caused by the electron cloud,

Fig. 7. Comparison of the model prediction with experimental data for the apparent equilibrium constants. The symbols represent the experimental values from gas adsorption based on Eq. (11), and the solid lines are the model results, using Eq. (10). The red dashed line is the fitted result of CO2, after an empirical addition of the electrostatic interaction in Eq. (16). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a cylindrical pore of a single layer structure was employed to estimate the potential to derive the predicted apparent equilibrium constant (K osc ), which was later compared with the experimental value derived from the fitted Langmuir parameters in Eq. (11). Fig. 7 depicts the variation of ln(K osc ) with 1000/T for the adsorption data (symbols) and the model predictions (solid lines). Good agreements occur to all the gases, and the obtained gas order of the apparent equilibrium constants regularly follows: CO2 > CH4 > N2 ≈ Ar. As mentioned above, CO2 adsorption can be considerably enhanced 1 by extra electrostatic interaction (φ FQ ̇ = 2 QF ̇ , where Q represents the

Table 3 Optimized molecular parameters (volume, kinetic diameter and bond length) of related structural units of the wall in the organic-inorganic hybrid silica. Atoms and molecules

Volume (cc/mol)

Volume diameter (nm)

LJ diameter σss (nm)

Bond length Si–O (nm)

Bond length Si–C (nm)

Bond length C–H (nm)

C O CH4 H4SiO4 H3SiO3–O–SiO3H3 H3SiO3–CH3 H3SiO3–CH2–CH3

17.6990 13.0450 25.8900 57.5060 66.3790 80.9010 95.9150

0.3829 0.3459 0.4347 0.5672 0.5949 0.6355 0.6726

0.3400 0.2800 0.3810 – – – –

– –

– – – – – 0.1856 0.1861

– – 0.1070 – – 0.1093 0.1096

0.1636 0.1638 0.1642 0.1644

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respectively. As given in Fig. 8(a), the diffusivities monotonically increase with pore size; the equilibrium constants in Fig. 8(b), initially increase for pores below 0.5 nm, and then follow a consistently decreasing mode. When the temperature goes up to 423 K in Fig. 8(c) and (d), a similar tendency with pore size is obtained except for higher diffusivities and weak equilibrium constants. The equilibrium constant during the transport related to the gas molecule-wall interaction, equals to the Henry law constant during the gas adsorption [53]. The gas order of diffusivities and equilibrium constants in pore body depends on pore dimensions and temperatures, rather than exclusively determined by molecular weight as predicted for the classic Knudsen model. This observation is consistent with the model prediction, i.e., the tangled intricacy between interaction potential and gas kinetic energy leads to a much more complex relationship with gas species and pore sizes. To determine the temperature dependence of the diffusivity in the pore, we investigate the apparent activation energy, Eabody , defined by [45]

where repulsion and attraction are due to short-range electron-cloud overlapping and mediate-range electron-cloud dispersion effect, respectively [51]. Based on the relationship of the van der Waals volume and LJ diameters, the interaction unit of the wall can be estimated to exist between [–(SiO4)–] and [–(SiO3–CH2)–], which is in coincidence with the assumption for the weak adsorption in the hybrid silica, i.e., the presence of carbon and silicon atoms in the interaction unit. Since Si–C bond of the hybrid silica (0.1861 nm) is considerably longer than Si–O length of the pure silica (0.1638 nm), a more loosely connection in the hybrid matrix can be expected. Compared with Si–O length, a shorter bond length of C–H (0.1096 nm) in the hybrid silica enables a closer contact of molecular particles with silicon and carbon atoms, so the corresponding repulsive effects of silicon and carbon atoms have to be taken into account of the effect for the hybrid silica. The explanation about the shield weakening effect of oxygen atoms is further supported by our previous work [52], where an interaction unit of σss = 0.303 nm is used for gas adsorption in the unconsolidated gamma-alumina pores. The high value of LJ diameter for alumina is caused by the presence of Al atom in the dispersive interaction, and the shield effect is weakened due to a smaller stoichiometry number between O and Al atoms (O: Al = 1.5) in comparison with the value for pure silica (O: Si = 2.0). Due to the dominance loss of oxygen interaction on the pore surface, a low well depth of energy interaction (εss/kB = 108.47 K) also occurs.

Eabody = −

d lnDosc dβ

(17)

The activation energy for the pore body is only proposed as a virtual parameter based on the Arrhenius type of the temperature dependence in each pore (D body = Do e−Ea / Rg T ), as no real energy barrier occurs for the axial movement in the cylindrical pore used here; this can only be regarded as an apparent energy barrier suitably averaged over the molecular axial trajectories arising from wall collisions. Fig. 9 illustrates the correlation for different gases at the temperatures of 303 and 423 K, where the Arrhenius form accurately describes the diffusivity for all the used pore sizes and temperatures. Fig. 10 provides the variation of apparent activation energies with pore diameters and gases, determined in Fig. 9. As suggested, the apparent activation energy

5.4. Transport activation energy in pore body Since the LJ parameters of the material have been obtained, it is of interest to examine the activation energies for different pores (pore entrance vs. pore body), using above established LJ models. In the pore channel, Fig. 8 illustrates the variation of diffusivities and equilibrium constants with pore diameters for the temperatures of 303 and 423 K,

Fig. 8. Pore diffusivity and equilibrium constant of light gases vary with pore diameters of BTESE-derived organic-inorganic hybrid silica at different temperatures: pore body diffusivity (a) and equilibrium constant (b) versus pore diameter at 303.15 K; pore body diffusivity (c) and equilibrium constant (d) versus pore diameter at 423.15 K.

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Fig. 9. Correlation of body diffusivities with temperatures. Symbols represent theoretical results and solid lines are model correlation on the semilogarithmic coordinates.

constitutes the other part of the total transport resistance in a cylindrical pore, and this can be quantitatively examined by the changes of dispersive interactions at different axial coordinates. Based on the combination of the work in the literatures [27,29,31], activation energy of the pore entrance in the hybrid silica can be specifically evaluated in Eq. (13), which can be positive, zero, and negative. Since activation energy varies with gases and pores, different transport regions can be distinguished, using its relationship with the energy barrier in the pore body. Instead of ignoring the body resistance as for the work of Thornton et al. [29], three diffusion regions are classified, i.e., entrance dominated region (RI), coupled region between entrance and body resistance (RII), and the body dominated region (RIII). As depicted in Fig. 11, the variation of energy barriers at the pore entrance with pore sizes typically follows an interatomic LJ pattern, where the potential goes through a minimum attractive force well (dopt) before reaching up to its repulsive zone. An appropriate pore dimension can promote entry velocities, and favors the mass transfer inside the pore. To compare

decreases for pores exhibiting repulsions (below 0.5 nm), which turns into an increasing mode for intermediate pores with attractions, until eventually flattening out for large pores, with negligible interactions. Strongly-adsorbed gases generally experience higher barriers in the pore body. Besides, the obtained activation energy for micropores (below 2 nm), slightly varies between 2.0 and 4.0 kJ/mol for all the gases, so the transport resistance in the pore is not the decisive factor for molecular sieves, as a much higher activation energy (above 10 kJ/ mol) is generally required to break down the gas selectivity set by kinetic velocity [33,36,54,55]. The obtained activation energy in the pore will be applied to compare with energy barriers of the pore entrance, and to further quantitatively examine the decisive resistance in molecular sieves in the next part. 5.5. Transport activation energy in pore entrance Besides, the resistance in the pore body, the pore entrance 34

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Fig. 12. The transient interaction energy profile of gases (Ar and H2) versus z-axial coordinates in a cylindrical hybrid silica pore with different pore sizes, in which the framework origin is the surface center of the pore surface.

Fig. 10. Variation of activation energies in pores with pore sizes for different gases.

that only a single atom ring of the mouth is used to estimate the interaction, and it is expected that the diffusivity estimated by the GT model is overestimated for zeolites due to its underestimation of repulsive interaction. Further, it is seen that the entrance barrier is much more sensitive to pore dimensions and gas species than the body barrier provided in Fig. 10. A much stronger variation of activation energy can provide significantly different barrier for diffusion of gases (high selectivities), so it is reliable to conclude that the decisive factor in molecular sieves is mainly from the entrance repulsion, where the separation limit, established by gas kinetic theory for the Graham law and the Knudsen model, is broken down in subnanometer pores. To further investigate the energy barrier dependence on pore sizes and gas species, Fig. 13 illustrates the pore entrance activation energy of several light gases in the pores from 0.2 to 10 nm. For the repulsive region of RI zone (0.2–0.3 nm), the entrance resistance is more dominant over the pore body, and the gas order of activation energy follows: CH4 > CO2 > N2 > Ar > H2 > He, demonstrating that the corresponding activation energy is a function of gas adsorption intensity (LJ energy parameter), instead of molecular size (LJ collision parameter). Further, for the attractive zone (0.3–2.0 nm) of the coupled region (RII), both the entrance and body resistances should be appropriately taken into account, where the obtained activation energy is constantly negative and generally correlates with the molecular adsorption intensity, with a gas order: CO2 > CH4 > Ar ≈ N2 > H2 > He. As

Fig. 11. Schematic drawing of pore entrance barriers of argon with pore size. The plot is divided three regions according to the dominance of the transport resistance, i.e., (I) entrance region in which the pore entrance resistance overweight the body resistance, (II) coupled region in which both the pore entrance and body resistance should be considered, and (III) body region in which the body resistance governs the diffusing process. The diameter, yielding identical activation energies of pore entrance and body, is labelled as deq; and the diameters, yielding zero and minimum dispersive interactions correspond to dzo and dopt, respectively.

activation energies, the critical pore dimension with an equivalent activation energies between the pore entrance and body is defined as deq (|Eamouth | = Eabody ), which is followed by dzo1 (|Eamouth | = 0 ) and dzo2 (10|Eamouth | = Eabody ) for the pores with zero and negligible activation energies, respectively. As molecules approaching pore mouth from outside, the transient interaction can be repulsive, attractive or negligible. In accordance with the work by Thornton et al. [29], only z-axis coordinate along the cylindrical center is considered. By taking the mouth center as the coordinate origin and idealizing the polyatomic gases as spheres in the continuum LJ model to neglect the orientation effects, Fig. 12 depicts the transient interaction energy profile along z-axis for Ar and H2, with two different pore dimensions. As suggested, the interaction energy for both gases is constantly negligible, until considerably close to the mouth center (z = −0.85 nm), where a repulsion interaction occurs; afterwards, the repulsion sharply increases even after passing the mouth surface. The repulsion eventually flattens out at a z-coordinate of 1.08 nm, which demonstrates the maximum interaction is not located just right at the entrance surface, but positioned inside the pore body. It is pertinent to recall the derivation in the work of Xiao and Wei [27,28]

Fig. 13. Pore entrance activation energy of light gases versus pore size for a BTESEderived organic-inorganic hybrid silica.

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the hybrid silica matrix. The obtained LJ parameters are subsequently used to estimate gas diffusivities, equilibrium constants and activation energies of gases in both pore bodies and mouths. For pore bodies, the obtained apparent activation energies varies between 2 and 7 kJ/mol, with slight dependence on pore dimensions and gas species, so the result denies the pore body as the decisive factor to sieve gases. For pore entrances, the interaction increases as gases approaching the mouth surface, which continues even after the molecule passing through the mouth crosssection before eventually becoming a constant over a short distance inside the pore. The entrance activation energy not only significantly depends on gases, but also is far above the value for the pore body in the repulsion region, so it is justifiable to argue that the separation limit confined by the gas kinetic theory is broken down by entrance barriers. The results suggest that the common-used activation energy in GT model is generally underestimated due to an inappropriate determination of maximum repulsion. In the end, several diameters are defined based on the ratio of activation energies between pore entrance and body, and it is discovered that the featured diameters of the hybrid silica are constantly narrower than that for the pure silica, suggesting the hybrid silica-based materials are physically less effective to separate gas molecules than pure silica. In summary, the proposed approach estimates the coupled effect of resistance for the pore body and entrance based on the adsorption isotherm, and the results facilitate the prediction of gas transport in the hybrid silica nanopores by deriving the dominated transport resistance. The method is expected to quantitatively predict the separation capabilities for well-known pore dimensions and wall compositions.

Table 4 Characterized parameters of pore diameters and energy barriers defined according to its activation energy relationship between the pore entrance and body, using the region divisions in Fig. 11. Gases

deq (nm)

Eeq (kJ/ mol)

doz1 (nm)

dopt (nm)

Eopt (kJ/ mol)

doz2 (nm)

H2 He CH4 N2 Ar CO2

0.23 0.19 0.31(−) 0.29(−) 0.28 0.28(−)

2.83 2.78 2.93 2.97 2.93 2.95

0.24 0.21 0.32(+) 0.30(+) 0.29 0.29(+)

0.36 0.32 0.46 0.42 0.42 0.42

−12.53 −5.90 −30.87 −23.20 −25.20 −34.86

1.85 1.55 2.45 2.25 2.25 2.45

(−) indicates a marginal value between 0.005 and 0.009 nm. (+) indicates a marginal value between 0.001 and 0.004 nm.

mentioned in the literature [29], negative activation energy corresponds to suction energies, which converts dispersive potentials into motion energies. Provided no thermal dissipation effect occurs, the entry velocity is increased before performing collisions on the wall, so transport in the pore body becomes faster in such cases. The enhancement of entry velocity in RII region indicates that the commonly-used combination of Knudsen flow and surface diffusion may quantitatively correlate the experimental data [56,57]. For pores larger than 2.0 nm, the entry barrier eventually turns to be negligible, and the transport is governed by the pore body in such circumstances. Table 4 summarizes the defined critical pore dimensions and the corresponding activation (suction) energies. As suggested, the first barrier-free pore, doz1, representing the boundary where repulsion starts to take into effect, is very similar to the dimension of gas kinetic diameters. The second barrier-free pore, doz2, representing the boundary where attraction starts to play a role by increasing the gas kinetic energy, is 6–8 times the size of the gas diameters. The optimal diameter corresponds to dopt, and the optimized suction energy is represented by Eopt, where the corresponding gas order follows: CO2 > CH4 > Ar ≈ N2 > H2 > He. The last featured pore, deq, stands for the size when the entrance resistance overweights the body resistance, where the gas order of activation energy is: CH4 > N2 > CO2 > Ar > H2 > He. It is evident that deq is only marginally smaller than doz1, indicating the sharp enhancement of entrance resistance. As given in the Table, all the featured pore diameters for the hybrid silica are smaller than the values proposed in the literature for pure silicas [29] due to a much weaker interaction effect, so it is reasonable to argue that the hybrid material is physically less promising in terms of separation capabilities than pure silica, due to a much higher difficulty of size tuning for smaller pores (below deq). The modeled critical pore dimensions and activation energies adequately facilitate the understanding of the bottom-up transport mechanism in the hybrid silica membrane.

Acknowledgment This research has been supported by National Science Foundation of China (Grant No. 51502311), the program for Ningbo Municipal Science and Technology Innovative Research Team (2015B11002 and 2014B81004) and the China Postdoctoral Science Foundation (Grant No. 2017M610910). Dr. Gao acknowledges the generosity of Prof. Bhatia in The University of Queensland to share his oscillator model code. References [1] S. Smart, C.X.C. Lin, L. Ding, K. Thambimuthu, J.C. Diniz da Costa, Ceramic membranes for gas processing in coal gasification, Energy Environ. Sci. 3 (2010) 268–278. [2] S.K. Bhatia, M.R. Bonilla, D. Nicholson, Molecular transport in nanopores: a theoretical perspective, Phys. Chem. Chem. Phys. 13 (2011) 15350–15383. [3] S. Hwang, Fundamentals of membrane transport, Korean J. Chem. Eng. 28 (2011) 1–15. [4] Y. Wan, D. Zhao, On the controllable soft-templating approach to mesoporous silicates, Chem. Rev. 107 (2007) 2821–2860. [5] G. Ji, G. Wang, K. Hooman, S.K. Bhatia, J.C. Diniz da Costa, The fluid dynamic effect on the driving force for a cobalt oxide silica membrane module at high temperatures, Chem. Eng. Sci. 111 (2014) 142–152. [6] K.E. Gubbins, Y. Liu, J.D. Moore, J.C. Palmer, The role of molecular modeling in confined systems: impact and prospects, Phys. Chem. Chem. Phys. 13 (2011) 58–85. [7] D. Ishihara, K. Tao, G. Yang, L. Han, N. Tsubaki, Precisely designing bimodal catalyst structure to trap cobalt nanoparticles inside mesopores and its application in Fischer-Tropsch synthesis, Chem. Eng. J. 306 (2016) 784–790. [8] J. van den Bergh, S. Ban, T.J.H. Vlugt, F. Kapteijn, Diffusion in zeolites: extension of the relevant site model to light gases and mixtures thereof in zeolites DDR, CHA, MFI and FAU, Sep. Purif. Technol. 73 (2010) 151–163. [9] T. Van Gestel, F. Hauler, M. Bram, W.A. Meulenberg, H.P. Buchkremer, Synthesis and characterization of hydrogen-selective sol–gel SiO2 membranes supported on ceramic and stainless steel supports, Sep. Purif. Technol. 121 (2014) 20–29. [10] D. Duong, Absorption Analysis: Equilibria and Kinetics, Imperial College Pr, London, 1998. [11] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, New York-Chichester-Weinheim-Brisbane-Singapore-Toronto, 2006. [12] J.R. Welty, C.E. Wicks, G. Rorrer, R.E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer, John Wiley & Sons, 2009. [13] M. Knudsen, W.J. Fisher, The molecular and the frictional flow of gases in tubes, Phys. Rev. 31 (1910) 586–588. [14] S.K. Bhatia, D. Nicholson, Transport of simple fluids in nanopores: Theory and

6. Summary and conclusions Here we have reported a general methodology to explore gas transport behaviors in amorphous hybrid silica pores, using adsorption data of gases in a synthesized sample. In this work, the adsorption of the hybrid silica is found to be adequately represented by a single layer model of structure, and the fitted LJ parameters are given as εss/ kB = 227.6 K, σss = 0.47 nm, with ρs = 7.86 nm−2. The electrostatic interaction coefficient for CO2 is found to be −10.57 K·nm2, being much less than the value reported for the pure silica derived from the copolymerization of TEOS and MTES (−37.51 K·nm2). The obtained LJ parameter (σss) is further explored by ab initio computation to calculate the van der Waals volumes of several related molecular clusters. The results predict an interaction unit between [–(SiO4)–] and [–(SiO3–CH2)–] functional groups, which confirms the presence of Si and C atoms in the interaction due to a much looser interconnectivity in 36

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