Gas permeation through porous glass membranes

Journal of Membrane Science 336 (2009) 32–41

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Gas permeation through porous glass membranes Part II: Transition regime between Knudsen and configurational diffusion A. Markovic´ a,∗ , D. Stoltenberg a,b , D. Enke b , E.-U. Schlünder c , A. Seidel-Morgenstern a,d,∗∗ a

Max Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany Institute of Technical Chemistry and Macromolecular Chemistry, University of Halle, D-06108 Halle/Saale, Germany c University Karlsruhe, D-76128 Karlsruhe, Germany d Otto-von-Guericke University, D-39106 Magdeburg, Germany b

a r t i c l e

i n f o

Article history: Received 24 October 2008 Received in revised form 19 February 2009 Accepted 25 February 2009 Available online 13 March 2009 Keywords: Porous glass membrane Activated diffusion Adsorption Surface diffusion Gas separation

a b s t r a c t In Part I of this article an analysis of four different mesoporous glass membranes with pores in a relative narrow size range between 2.3 and 4.2 nm was discussed focusing on the effects of pore diameter and surface properties on membrane performances. It was found that the gas transport through these mesoporous membranes is primarily governed by Knudsen diffusion and viscous flow and if adsorption appears by surface diffusion. Selectivities could be altered to some extent exploiting differences in adsorbability of gases but they were still rather limited. The strategy to further reduce the pore size was employed in this paper in order to improve gas separation. A flat membrane based on phase-separated alkali-borosilicate glass with an average pore diameter of 1.4 nm was prepared in order to study the transport characteristics in the transition regime between Knudsen and configurational diffusion. The mechanisms of gas transport through the membrane were studied performing dynamic permeation measurements for several gases (He, Ar, N2 , CO2 and C3 H8 ) in a modified Wicke-Kallenbach cell in the temperature range from 293 to 433 K using the time lag method for analysis. Additionally, adsorption equilibria of the gases were measured using a standard volumetric technique at three different temperatures (293, 323 and 353 K) and at pressures up to 2.5 bar. The permeability data observed experimentally are described theoretically. Ideal selectivity factors determined as the ratios of the permeabilities of different pairs of pure gases are discussed. © 2009 Elsevier B.V. All rights reserved.

1. Introduction During the last two decades, remarkable separation effects of microporous membranes have been documented in the literature. These are due to the pores sizes, which are small enough to separate gases based on differences in molecular sizes [1–4]. Direct preparation of microporous materials with pore sizes smaller than 2 nm is difficult. Usually modification procedures are required to reduce larger pore sizes. Due to the possibility of optimizing cooling rates for the preparation of alkali borosilicate glasses and heat treatments for phase separation, homogeneous microporous glass membranes can be directly prepared providing interesting objects allowing to

∗ Corresponding author. Tel.: +49 391 6110 401; fax: +49 391 6110 403. ∗∗ Corresponding author at: Max Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany. ´ E-mail addresses: [email protected] (A. Markovic), [email protected] (A. Seidel-Morgenstern). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.02.030

study the mechanisms of permeation and to evaluate their potential for gas separation. However, there is a lack of knowledge regarding the textural properties of these materials, which is an obstacle for the development and validation of more reliable models capable to predict separation properties. Permeability measurements were carried out for several inert and adsorbable gases (He, Ar, N2 , CO2 and C3 H8 ) using a prepared flat membrane with an average pore diameter of about 1.4 nm in a temperature range from 293 to 433 K using a modified Wicke-Kallenbach cell. Additionally, to quantify the equilibrium properties, adsorption isotherms of these gases were measured by a volumetric technique. The membrane structure and textural properties were characterized to support better understanding of the observed permeabilities. In general, it is very difficult to measure diameters of such small pores with standard techniques, e.g. nitrogen low temperature adsorption. For this reason in this study, in addition positronium annihilation lifetime spectroscopy was applied as an alternative method for the estimation of pore diameters. For a quantification of the surface properties, after removing

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

physically adsorbed water from the glass surface, thermogravimetric analysis was used to estimate the number of hydroxyl groups per nm2 surface, which should be related to the adsorption properties of the material. The main objective of this paper is to determine the permeation and equilibrium properties of the microporous glass membrane synthesized for this study. Detailed experimental data are presented regarding single gas permeabilities and ideal selectivity factors for a wide temperature range. These data will be used to analyse possible mechanisms by which gases may be separated using such materials. 2. Theory: mass transfer mechanism For mesoporous membranes Knudsen diffusion, viscous flow and surface diffusion can be considered as the controlling rate mechanisms, as presented in Part I of this article. The question arises, whether this established concept can be also used to describe the mass transport through microporous membranes, where the pore diameters are in the same order of magnitude as the molecule diameters. Typical nominal molecule diameters of gases are in the order of 0.25 nm (He) and 0.5 nm (I2 ) [5]. These diameters are based on the 12-6 Lennard–Jones potentials, which describe van der Waals interactions (induced dipole–induced dipole interactions) only. Hereby, the 12-6 Lennard–Jones force constant, , corresponds to the closest distance when two molecules collide. Attractive forces decrease with the sixth power of the distance and vanish at about two times of  [5]. If the pore diameter is only 1.4 nm, there is a strong overlap of the potential fields between the gas molecules and the molecules at the active sites of the pore walls. In this case it appears to be questionable if it makes sense to distinguish between a free space, where molecules travel with their (somehow corrected) three dimensional Maxwell velocity, and an adsorbed phase at the pore walls, where the molecules travel with a two dimensional creeping velocity. However, in microporous membranes with pore diameters around 1.4 nm there is still enough space to allow for different states of molecules and transport mechanisms. Mass transfer in the gas phase is surely prevailing at high temperatures, while mass transfer in the adsorbed phase could be more predominant at lower temperatures. In general the total flux can be expressed as the sum of both contributions: Jtot = Jg + Jads

1 ∂p RT ∂x

(2)

where the gas phase diffusivity Dg follows a modified Knudsen type of equation (compare Eq. (14) in Part I): 1 ε Dg = dp 3 ov



Arrhenius-type of activation factor. This activation factor reflects the fact that the molecules must overcome a potential barrier Eg originating from the molecules immobilized at the wall. Moreover in Eq. (3) the tortuosity factor ov has an extended meaning. It describes not only the geometrical zigzag meandering of the gas molecules due to the porous morphology but also the effect of gasto-wall interactions. In micropores there is always some overlap of the potential fields of both, gas and wall molecules. This overlap causes an internal zigzag motion within the pores themselves which increases the overall length of the diffusional path. This contribution differs from gas to gas and therefore the overall tortuosity, ov too. The length of the internal diffusional path depends on the angle of deflection, when gas molecules collide with the pore walls. Although all types of gas molecules have the same kinetic energy, they have different kinetic momentum. So, one might expect, that light molecules have a lower angle of deflection compared to heavier ones, i.e. lighter molecules oscillate more or less perpendicular to the pore walls instead of moving in the axial direction. This could considerably enlarge the overall tortuosity factor ov . 3. Experimental study 3.1. Preparation of microporous membrane Microporous glass membranes were prepared using an initial glass consisting of 70% SiO2 , 23% B2 O3 and 7% Na2 O. This composition was chosen to inhibit the occurrence of stresses during the cooling process of the glass melt. The glass melt itself was quenched in air to leave the temperature region of the phase separation (500–720 ◦ C) and to create two interconnected phases quickly. During this process the initial glass separated into an acid-soluble sodium-rich borate phase and an insoluble silicate phase. The optimized cooling process led to a very low degree of phase separation. By applying a core drill rods of 15 mm diameter were bored from the initial glass blocks and cut into thin plates of ı = 0.5 mm thickness by an annular saw. The plates were leached with hydrochloric acid (1 mol/l) at 90 ◦ C for 2 h to dissolve the soluble borate phase. The remaining SiO2 framework formed the porous network. The synthesized membranes were finally washed repeatedly with distilled water and dried. One of the prepared samples was used in this study.

(1)

The flux in the gas phase Jg follows from Eq. (2) given below and the flux in the adsorbed phase is described by Eqs. (3)–(12) given in Part I of this paper. As these equations have been discussed in details there, they will not be repeated here. Below, only changes in the gas transport mechanism occurring in micropores compared to mesopores will be discussed. In the mesoporous regime, Knudsen diffusion still dominates and selectivities are proportional to the inverse square roots of molecular weights, while in the microporous range the selectivities are additionally influenced by differences in the potentials between gas molecules and membrane surface molecules. The various theories of microporous diffusion [1,4,6,7] can be summarized by the same Arrhenius type equation. Jg = −Dg

33

8RT − Eg e RT M

(3)

Gilron and Soffer [8] indicated Eq. (3) as an activated Knudsen diffusion where the Knudsen diffusivity is multiplied with an

3.2. Membrane characterization 3.2.1. Positronium annihilation lifetime spectroscopy Positronium annihilation lifetime spectroscopy (PALS) is a sensitive and non-destructive tool to investigate the pore size distribution for microporous materials [9]. This technique measures the lifetime of an ortho-positronium, which is formed by a positron implanted in a dielectric amorphous material with an electron. By measuring the ␥-quants emitted during the implantation of the positron and the annihilation of the positronium one obtains the positronium lifetime. The lifetime of the formed positroniums is 142 ns in maximum (in vacuum) and is reduced markedly by pickoff-annihilation, a quenching process of the ortho-positronium caused by interaction with electrons of suitable spin at the surface of the pore. Hence, the positronium annihilation lifetime depends on the pore size of the porous material. The measurements were carried out using a fast–fast coincidence system (home made) with a time resolution of 250 ps, an analyzer channel width of 121.5 ps and a total of 8000 channels. The spectra contained 4 × 106 coincidence counts. The used positronium source showed a weak activity of 0.12 × 106 Bq to avoid a disturbance of background signals. The sample chamber was evacuated to 10−8 mbar. The temperature was held at 300 K.

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Measurements of low temperature nitrogen adsorption and thermogravimetry analysis were also performed. Details of the corresponding experimental procedures are presented in article Part I.

For slightly adsorbable or more adsorbable gases in the low pressure range (where Henry type of adsorption isotherms hold), the time lag expression takes the following form [11]:

3.3. Adsorption equilibrium measurements

L =

Sorption measurements were carried out using a classical volumetric method operated in the static mode [10]. The experimental procedure was already described in Part I. Single component adsorption isotherms were determined for Ar, N2 , CO2 and C3 H8 at three different temperatures (293, 323 and 353 K) for pressures up to 2.5 bar. Before starting the run the sample was simultaneously heated and outgassed at 393 K for several hours. About 30 min were needed to achieve an equilibrium indicated by constant pressure conditions. Maximal possible uncertainties of ±10 mbar of pressure measurements can cause e.g. 3.25% deviations of N2 among amounts adsorbed at 293 K and 2.11% at 353 K. Deviations for CO2 adsorption are lower, 1% at 293 K and below 0.3% at 353 K. Bigger uncertainties of N2 adsorption have to be accepted because of very low adsorption where small pressure changes can cause bigger deviations. 3.4. Gas permeability measurements The membranes were characterized by pure gas permeation experiments in a temperature range between 293 and 433 K. The experimental setup including the mass balance equations of the membrane and of two chambers were the same as described in Part I. The entire system was degassed with a vacuum pump at 443 K before each run. Then, a gas flow was introduced from one side of the membrane (chamber VI ) at constant pressure, pI while the other side was initially evacuated and closed (chamber VII ). As the gas is permeating through the membrane, the increase of the pressure pII over time was recorded. Relative deviations of the permeation measurements based on pressure transducer uncertainties of ±10 mbar are below 2%. The low permeation through this membrane allowed using the conventional time lag technique to evaluate the diffusion coefficients. Hereby, steady state information is obtained simultaneously with transient information and presented in form of a time lag [11]. The time lag theory delivers a connection between the observed time delay, the adsorption equilibrium and diffusion parameters. Knowing evaluated adsorption isotherm parameters, only the gas and surface diffusivities have to be estimated. In the case of the various mesoporous membranes studied in Part I, the observed time lag values were less than 4 s, what was too short for further evaluation. Gas diffusivities were estimated first through measurements of permeation of the non-adsorbing gas helium through the microporous membranes using Eq. (4). For the time lag,  L of a membrane of thickness ı holds [11]: ı2 L = 6Dg

(4)

More details regarding the procedure for the determination of the time lag parameter is given in Appendix A and only final expressions for the time lags are given in the manuscript itself.

ı2 [ε/(RT ) + (1 − ε)K]

(5)

0 K] 6[Dg /(RT ) + (1 − ε)Ds,c

Thus, the time lag depends besides the gas diffusion coefficient also on two additional parameters: the Henry constant K and the cor0 (wherefore according rected surface diffusivity at zero loadings, Ds,c 0 for = 1 and f(q) = 1). As the to Eqs. (4)–(6) in Part I, Ds = Ds,c adsorption isotherms were determined separately, the unknown gas and surface diffusion coefficients can be calculated from the measured time lag using this equation. It is useful to couple both time lag (Eq. (5)) and steady-state slope measurements (see Eq. (A14) in Appendix A) for the determination of gas and surface diffusivities since consistent values should be expected in the low pressure range studied. For highly adsorbable gases, the adsorption isotherms must be described with nonlinear models, e.g. the Langmuir equation (Eq. (9) in Part I). Then, a full analytical solution of the time lag cannot be obtained because of the nonlinear dependency of the surface diffusivity on the adsorbed amount, accounted by the thermodynamic correction factor, . A constant corrected surface diffusion coeffi0 was assumed to be adequate (Eqs. (4)–(6) with f(q) = 1 in cient, Ds,c 0 ). However, with the initial and boundary condiPart I, Ds,c = Ds,c tions presented by Eqs. (23), (24), and (28) in Part I, one can calculate numerically the pressure profiles. Alternatively, using the asymptotic solution method of Frisch [12] it is possible to determine the time lag analytically by integration of the mass balance with respect to time and pressure after interchanging the order of integration. This asymptotic solution provides time a lag function of constant pressure introduced to volume VI yielding diffusion parameters. The required parameters for mass transport quantification can be extracted from the determined time lag (cf. Eq. (6) or Eq. (A17)) and the steady state slope (cf. Eq. (A18)) according to ref. [13].

L =



Dg 1 +

1  

ı2





× (1 − y) +

y 1+

3

ln(1 + )

0



1+ ln( )  1 + y

1+



0 (1 + y) εDs,c

(1 + y)

2

 dy

(6)

0 RT/D , B = bq where = (1 − ε)BDs,c g sat and  = bpI .

4. Results and discussion 4.1. Evaluation of membrane characterization methods As explained in Part I devoted to characterize membranes with larger pore diameters, the specific surface area, the pore volume and the porosity of the microporous membrane were measured by nitrogen adsorption. The values for the pore volume and surface area were calculated according to Dubinin–Radushkevich [14] and Dubinin–Radushkevich–Kaganer [15], respectively. The obtained parameters are given in Table 1.

Table 1 Structural and surface properties of the investigated membrane. Surface area As a (m2 g−1 )

Pore volume Vp b (cm3 g−1 )

Porosity, ε

Pore diameter dp c (nm)

Concentration of hydroxyl groups (nm−2 )

398

0.142

0.237

1.4/2.18 (PALS)

3.8

a

Dubinin–Radushkevich–Kaganer [15]. b Dubinin–Radushkevich [14]. c Pore diameter according to 4Vp /As . PALS (Positronium annihilation lifetime spectroscopy).

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

35

Fig. 1. Nitrogen-adsorption isotherm at 77 K of the microporous glass membrane. Open points—adsorption; filled points—desorption. Fig. 3. Weight loss of the membrane during the thermogravimetry analysis.

The mean pore size of 1.4 nm was calculated from the following ratio: dp = 4Vp /As . The main assumption in this calculating procedure is that all pores are cylindrical, open ended and nonintersecting. As the often used Barrett–Joyner–Halenda method has well-known limitations when the pore sizes approaches molecular diameters (pores are too small for capillary condensation to occur, hysteretic effects are not noticeable [16], see Fig. 1) an alternative Non-local Density Functional Theory method (NLDFT) [17] was used to analyse the nitrogen isotherm. This method is based on given intermolecular potentials which allowed the construction of adsorption isotherm in model pores. A bimodal pore size distribution was obtained. The majority of pores were in the range between 1.2 and 1.5 nm although some pores in the mesoporous range (mean pore diameter ≈2.2 nm) were also identified using this method. Furthermore, positronium annihilation lifetime spectroscopy (PALS) provided a pore size distribution for the studied membrane which is presented in Fig. 2. The resulting pore diameter (Table 1), calculated using the extended Tao-Eldrup model with cylindrical pores [18] gave a mean pore diameter of approximately 2.2 nm, which is in the range but slightly larger than the values based on analysing low temperature nitrogen adsorption. Possible reasons for this discrepancy can be explained with slight gradients in the pore size over the cross-section of the membrane as supported from NLDFT analysis of the nitrogen adsorption isotherm. A characterization of the silica surfaces and an evaluation of the thermal stability of the membrane were performed using ther-

Fig. 2. Pore size distributions obtained by Positronium Annihilation Lifetime Spectroscopy (PALS).

mogravimetry. The observed weight loss of the membrane during temperature increase is illustrated in Fig. 3. By excluding the physically adsorbed water on the silica surface the results allow estimating the concentration of the hydroxyl groups as approximately 3.8 nm−2 (Table 1). This value is smaller in comparison with the concentrations of hydroxyl group obtained for mesoporous membranes (Part I). This should be due to sterical hindrances on the micropores surface having less space available for hydroxyl groups. 4.2. Evaluation of adsorption isotherm measurements The adsorption isotherms obtained for carbon dioxide and propane at three different temperatures 293, 323 and 353 K and for argon and nitrogen at T = 293 K are presented in Fig. 4. The adsorbed amounts observed for nitrogen and argon reached values up to 0.08 mmol/cm3 at 293 K while for all mesoporous membranes presented in paper Part I, no adsorption of these gases could be quantified. It can be seen that the observed adsorption isotherms for N2 and Ar are almost linear. From these data Henry constants were fitted (Eq. (11), Part I). The obtained parameters are presented

Fig. 4. Measured adsorption equilibria of argon and nitrogen at T = 293 K fitted with Henry isotherm (parameters given in Table 2) and of propane and carbon dioxide at three different temperatures fitted with Langmuir isotherm (parameters given in Table 3). Calculated isotherms are presented with lines and experimental data with symbols.

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A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

Table 2 Adsorption equilibrium parameters for argon and nitrogen on the membrane at three different temperatures (Eq. (11) in Part I). Henry isotherms parameters Argon gas

Nitrogen gas

T (K)

K (mol cm−3 bar−1 )

q (%)

K (mol cm−3 bar−1 )

q (%)

293 323 353

2.92e−5 1.78e−5 1.17e−5

1.15 2.24 4.12

3.49e−5 2.21e−5 1.51e−5

2.84 2.34 3.41

Table 3 Adsorption equilibrium parameters for the carbon dioxide and propane on membrane at three different temperatures (Eq. (9) in Part I). Langmuir isotherms parameters Carbon dioxide gas

Propane gas

T (K)

qsat (mol cm−3 )

b (bar−1 )

 q (%)

qsat (mol cm−3 )

b (bar−1 )

 q (%)

293 323 353

1.69E−03

0.604 0.229 0.100

2.41 1.54 1.34

1.46E−03

1.152 0.364 0.163

2.08 3.34 4.04

in Table 2. Propane is stronger adsorbed on this membrane than carbon dioxide, what is true also for higher temperatures (Fig. 4). The Langmuir isotherm (Eq. (9), Part I) was used for description of the observed adsorption equilibria of carbon dioxide and propane with the two parameters: the saturated concentration qsat and the temperature dependent parameter b. The fitted parameters are given in Table 3. Further, isosteric heats of adsorption estimated from these isotherms [19,20] were found for both gases not to depend on loadings in the range from 0.22 to 0.40 mmol/cm3 . For carbon dioxide holds (−Hiso ) = 27.35 kJ/mol and for propane (−Hiso ) = 32.09 kJ/mol. These are slightly higher values then for the mesoporous membranes studied in Part I. 4.3. Evaluation of single gas permeability and selectivity The proposed simple theoretical description of gas permeation in microporous membranes discussed above was tested based on the experimental results obtained.

Fig. 5. The molar gas amounts which pass the membrane in dependency of time for measuring the diffusion coefficients with the time lag method. Data are shown for carbon dioxide and propane at three different temperatures. Steady state flux determined from the slopes and time lag obtained as an extrapolated intercept are presented with dashed lines (see Eqs. (A8) and (A9) in Appendix A).

Table 4 Time lags,  L (s) obtained as an intercept of the linear dependency of a gas flow, VII /RT(pII (t) − pII (t = 0)) vs. time, for carbon dioxide and propane at five different temperatures (Eqs. (A8) and (A9) in Appendix A). T (K)

Carbon dioxide

T (K)

Propane

431 410 394 355 331

102 194 293 473 603

431 402 370 347 324

252 466 908 1050 1100

Fig. 5 demonstrates typical time lag measurements for carbon dioxide and propane permeation at three different temperatures, presented as gas amounts which passed the membrane vs. time. The slopes of these curves gave steady state fluxes (S = Jss A) and the ratio of the extrapolated interceptions of the curves and the respective slopes, Jss A L /S gave the time lags  L (see Eqs. (A8) and (A9)). To show the influence of temperature on time lags for the adsorbable gases, carbon dioxide and propane, the obtained values are summarized in Table 4. They are strongly decreasing with temperature. The larger values were obtained for propane, indicating again that propane has a stronger affinity to the membrane in comparison with carbon dioxide. Overall permeability coefficients Ptot can be extracted from the observed steady-state fluxes Jss , (determined from the slope of the linear dependency of the gas flow, VII /(RT)(pII (t) − pII (t = 0)) vs. time, see Fig. 5) divided by the pressure drop across the membrane and multiplied with the membrane thickness, ı: Ptot =

Jss ı pI − pII

(7)

Fig. 6 shows temperature dependence of determined permeability data for individual gases. Similar to the mesoporous membranes described in Part I, reproducibility of measurements were assured using He as a test gas. Small deviations of ∼5% were observed between different runs. The lowest permeability coefficient was observed for N2 (1.19e−14 mol/Pa/s/m) at T = 294 K and the largest one was observed for He (3.38e−13 mol/Pa/s/m) at T = 433 K. These values are located in the range reported for molecular-sieving membranes [21]. It can be seen that the permeability of all examined gases are increasing with increasing temperature. This is in clear contrast to the trend valid for Knudsen permeability (which is decreasing with rising temperature). Nitrogen showed the largest temperature dependence with a factor of 5.57 for the drop in permeability between 433 and 294 K. The permeabilities of He and N2

Fig. 6. Single gas permeabilities as a function of temperature.

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

37

Table 6 Determined pre-exponential factor ov and activation energy Eg of gas diffusion, Eq. (3).

Fig. 7. Single gas permeability as a function of Lennard–Jones 6-12 force constant (molecular diameter), at T = 433K.

differ at T = 294 K by a ratio of 25, while according to Knudsen diffusion this ratio would be inversely proportional to the square root of their molecular masses and, thus, only 2.63. Another analysis was done to compare permeability coefficients with kinetic diameters of the molecules. Obviously permeation depends strongly on this diameter indicating a similarity to the trends of diffusion in zeolites [1,22]. Fig. 7 represents the permeabilities in correlation with the molecular diameters (as 6-12 Lennard–Jones force constants, ) at T = 433 K. For small inert gases, molecular diameters of spherical molecules can be accounted well with this constant . However, in case of larger molecules this is not the best way of estimating the molecular diameters [5]. In order to evaluate this effect in a wider range, additional measurements were carried out with noble gases namely Ne, Kr, Xe. In Table 5 are given Lennard–Jones 6-12 force constants, , and molecular masses for each of the examined gases. It is interesting to note that the permeabilities of lighter gases with the larger molecular diameters are lower than of heavier gases with smaller molecular diameters (e.g. N2 vs. Ar). Observed results for noble gases showed a clear trend of decreasing permeability coefficients with increasing molecular diameters, confirming a molecular sieving effect of the glass membrane applied. Only carbon dioxide does not follow the trend described above. Apparently, besides molecular interactions captured by Lennard–Jones potential, additional interactions between pore walls and gas molecules occur. For carbon dioxide these are, electrostatic interactions originating from quadrupole contributions [19,23]. In case of the nonpolar propane only non-specific terms like for inert gases contribute to the potential energy (dispersion, repulsion and polarisation). Thus, it was expected and confirmed that propane behaves in this respect not far from noble gases.

Table 5 Molecular pore diameters given as a Lennard–Jones force constanta  and molecular masses of series of examined gases. Gas

 a (Å)

M (g mol−1 )

He Ne Ar Kr N2 CO2 Xe C3 H8

2.58 2.79 3.42 3.49 3.68 4.00 4.06 5.06

4 10 39 84 28 44 131 44

a

Obtained from viscosity data based on reference [5].

Gas

ov

Eg (kJ mol−1 )

He Ar N2 CO2 C3 H8

63.70 6.48 5.91 6.44 3.87

2.43 11.52 13.89 8.11 12.42

In order to verify the simple mathematical model presented above theoretical predictions based on the diffusivity coefficients given by Eq. (3) for gas diffusion and by Eqs. (4)–(8) and (12) from Part I for surface diffusion of adsorbable gases were compared with the experimental observations. Therefore, for He the gas diffusivity was calculated from observed time lags using Eq. (4). For adsorbable gases (N2 , Ar, CO2 and C3 H8 ), as the adsorption isotherms were determined independently, from permeability measurements using the volumetric method, the required gas and surface diffusivities were calculated from measured time lag values and steady state slopes at different temperatures determined for N2 , Ar from Eqs. (5) and (A14) and for CO2 , C3 H8 from Eqs. (6) and (A18). Consistent values were observed for diffusivities determined from steady state slopes and time lag values. Gas diffusivity for N2 and Ar can be also determined from time lag defined with Eq. (4) at temperatures higher than 393 K, because in this range gas diffusion is the main contribution to mass transfer. Even at 293 K the contribution of adsorbed phase transport for these two gases is very low (less than 10%) in comparison with gas diffusion. The activation energies, Eg and pre-exponential coefficients, ov were calculated for the examined gases from the slopes and intercepts of Arrhenius plots, ln Dg vs. 1/T. The parameters estimated and applied subsequently are given in a Table 6. It can be seen that for nonpolar molecules the activation energy is increasing with increasing molecular diameter, diminishing the permeabilities. The pre-exponential factors, i.e. overall tortuosity factors ov are difficult to predict. The fitted values for Ar/N2 /CO2 /C3 H8 lie between 3.8 and 6.5 which is the same order of magnitude as has been found for the mesoporous materials [24]. An exception shows He, where ov is much larger (64). As mentioned above light molecules, like He, have a much lower momentum than heavier ones, like e.g. Ar. Consequently, He molecules have a much smaller angle of deflection when they collide with the pore walls compared to heavier gases. This enlarges the length of the internal diffusional path way. 0 depend on the temperature, two As surface diffusivities Ds,c 0,0 and Es , defined with Eq. (7) in Part I) had to be parameters (Ds,c estimated. The obtained results are given in Table 7. The surface diffusivity coefficients of N2 and Ar are nearly identical and 10 times smaller in comparison with the values for the strongly adsorbable gases CO2 and C3 H8 . Due to the small contributions of the fluxes in the adsorbed phase the total permeabilities of Ar and N2 can be described in a reduced way as activated gas diffusion. Comparing the surface diffusivities determined for CO2 and C3 H8 , CO2 molecules move faster. The experimentally observed permeability data are compared with theoretical values, presented in

Table 7 Estimation of surface diffusion parameters using Eqs. (6) and (7) in Part I (for f(q) = 1 in Eq. (6)). Gas

0,0 Ds,c (m2 s−1 )

Es (kJ mol−1 )

Argon Nitrogen Carbon dioxide Propane

1.48e−11 1.21e−11 2.04e−9 6.46e−10

6.96 5.29 13.04 12.13

38

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

Fig. 8. Comparison between experimentally observed and theoretically obtained permeability data (described by activated gas diffusion using Eq. 3 and by surface diffusion using Eqs. (4)–(12) in Part I) through membrane for series of investigated gases in dependency of temperature.

Fig. 8 for different temperatures in an extended range aiming to see general trends. The determined diffusion parameters obtained by time lag analysis and adsorption equilibrium parameters obtained by volumetric measurements were used to predict the corresponding permeabilities. A relative good agreement is observed for all gases considered. An evaluation of the accuracy of the determined parameter sets and predicted permeabilities in the extrapolated temperature range are given in Appendix B. In order to evaluate the applicability of this type of membrane for gas separation, ideal separation factors, calculated as the ratio of the permeabilities of a pair of two individual gases (defined with Eq. (32) in Part I), are presented in Fig. 9. Using He as a reference gas the observed selectivities are for this microporous glass membrane much higher compared with the Knudsen ratio. It can be seen that the selectivity factors are decreasing with increasing temperature, but still at 433 K, the values are higher than the Knudsen selectivity ratios. The pairs of gases CO2 /N2 (Fig. 10a) and CO2 /C3 H8 (Fig. 10b) were further analysed. The pair CO2 /N2 provides an important separation problem in medical applications and is of interest for the development of CO2 sensors [25]. The maximum ratio of the permeances

Fig. 10. Comparison between experimentally and theoretically evaluated ideal selectivity factors (ratio of single permeability coefficients, Ptot,i /Ptot,j ) as a function of temperature: (a) CO2 /N2 and (b) CO2 /C3 H8 .

CO2 /N2 ≈ 15 was achieved at T = 293 K, what is considerably higher than the corresponding value for Knudsen diffusion (0.80). The pair CO2 /C3 H8 is significant because no separation by Knudsen diffusion is possible. The largest selectivity factors (≈2.5) were obtained at lower temperatures due to different activation energies and strong sorption effects (Fig. 10b). A similar selectivity factor of the gases C3 H8 /CO2 (∼1.95) was observed for the modified membrane M1mod exploiting only selective surface flow as presented in Part I. Increasing the difference between adsorbability of CO2 and C3 H8 by appropriate surface modification, which provides e.g. bigger affinity of CO2 to the microporous membrane (e.g. by amino-silanization), it should be feasible to further increase the observed ratio of 2.5. 5. Conclusion

Fig. 9. Comparison between experimentally and theoretically evaluated ideal selectivity factors (ratio between permeabilities of investigated gases with respect to helium, Ptot,He /Ptot,i ) as a function of temperature.

In Parts I and II of this paper are described the synthesis and characterization of mesoporous (dp between 2.3 and 4.2 nm) and microporous (dp ≈ 1.4 nm) glass membranes. Both articles offer a large data basis regarding gas transport and equilibrium properties. Conclusions regarding possible transport mechanisms using established theoretical approaches are given. To quantify mass transport through these membranes the following information was needed: (i) gas and surface diffusivities and (ii) adsorption equilibrium isotherms. These data were determined from permeation experiments and independent volumetric equilibrium measurements. The microporous membrane analysed in this Part II was characterized by activated diffusion with significant selectivities but relative low permeabilities. The latter aspect allowed using the time lag analysis of permeation providing estimates of the coefficients describing the activated transport. A mathematical model taking into account the combined process of activated gas diffusion and

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

surface diffusion coupled by adsorption equilibria was applied and found to match well the experimental observations for microporous membranes discussed in this Part II. In contrast the experimental findings and the theoretical analysis revealed that Knudsen and viscous flow coupled with surface diffusion of adsorbable gases are essential mechanisms for the gas transport in the mesoporous membranes, discussed in Part I. Comparing the selectivities of the microporous membrane with the selectivities of the mesoporous membranes (based on the values of single gas permeabilities) larger values were obtained for the microporous membrane. This is due to the fact that the separation is based more dominantly on differences in molecular diameters. Small changes in the molecular diameters gave large differences in permeabilities and separation factors. Hereby, larger selectivity values were obtained in the lower temperature range investigated. Open questions remaining after this study are in particular, whether the achieved selectivities can be further enhanced by more specific tailor made surface modifications. In a more detailed study also the real selectivities of separating mixtures need to be evaluated. Finally, the aspect of reproducibility of membrane preparation needs to be addressed in more depth. Since the theoretical concepts applied, based e.g. on assuming uniform pore sizes and using macroscopic diffusion models, are still basic tools for membrane characterization, there is surely a need in developing more detailed models using molecular modelling concepts. The experimental data generated in the course of this study provide useful information for validating such more detailed models. Acknowledgments

The flux at any point along the axis can be obtained by applying Fick’s law.

Appendix A. Summary of the time lag permeation technique for determination of the diffusion coefficients (based on ref. [11]) The procedure for determination of diffusion parameters using the time lag method is described here in more details. The time lag is evaluated from the interception of linearized dependency of the amounts of gas which passed the membrane versus time while the steady state fluxes were evaluated from respective slopes. The solution of the mass balance for inert gases (e.g. for helium) which accounts only for transport in the gas phase, so that permeation is governed by a linear differential equation (Eq. (A1)) is given by Daynes (1920) [26,27] using constant diffusion coefficients and slab geometry of the membrane: Dg ∂2 p ε ∂p = RT ∂t RT ∂x2

(A1)

Using initial and boundary conditions, expressed with Eq. (A2), p(t, x = 0) = pI = patm p(t, x = ı) = 0 p(t = 0, x) = pin ≈ 0

(A2)

∞

× cos

p(x, t) = pI

x 1− ı

× sin

where ˛ =

2pI  (1 + ˛) cos(n) − ˛ − ×  n

nx ı

pin − pI pI

∞

 exp

n=1

−

Dg

n2 2 t ı2

nx cos ı



n=1

−n2 2 Dg t

 (A5)

ı2

By integrating the flux with respect to time we can obtain the amounts which pass the membrane as a function of time given as:



ADg pI ı2 ı2 2ı2 Qı = × t+˛ + (1 + ˛) − 2 6Dg 3Dg ıRT  Dg ∞  [(1 + ˛) cos(n) − ˛]

×

n2

n=1

 × exp

−

Dg n2 2 t



ı2

(A6)

This permeate flow (Eq. (A6)) becomes asymptotic to a linear function as t → ∞: Qı =



ADg pI ıRT

t+˛

ı2 ı2 + (1 + ˛) 6Dg 3Dg



(A7)

Because in the experiments membrane was initially evacuated (pin ≈ 0), it turns out that ˛ = −1 (Eq. (A4)) what reduces Eq. (A7) to the following simple relation: ADg ıRT



t−

ı2 6Dg



= AJss (t − L )

(A8)

By measuring the slope (S = AJss ) and the time axis interception (AJss  L ) of this response using the following Eq. (A9) the diffusion coefficient can be analytically determined from Eq. (A10). The determined diffusivities from time lags and steady-state slopes measurements are expected to be consistent in the low pressure range studied. Qı =

VII (pII (t) − pII (t = 0)) RT

L =

ı2 6Dg

(A9) (A10)

The mass balance for the diffusion process when adsorption and diffusion of the adsorbed phase-surface diffusion occur simultaneously with gas phase diffusion is represented as: ε

∂q ∂p ∂ ∂p + (1 − ε) =− ∂t ∂p ∂t ∂x



Dg 0 ∂ ln p ∂q + Ds,c RT ∂ ln q ∂p



∂p ∂x

(A11)

The derivative of the adsorbed amount to a gas pressure is described for slightly adsorbable gases or also adsorbable gases in low pressure range with the linear Henry adsorption isotherm. In this case the gas diffusivity Dg in Eq. (A10) is replaced by the effective diffusivity which accounts to the Henry parameter, K, and the 0 , with the followcorrected surface diffusivity at zero loadings, Ds,c ing relation:

the solution follows:



 2Dg Dg [(1 + ˛) cos(n) − ˛] pI + pI × RT RT

J(x, t) =

Qı =

The authors would like to thank Mr. S. Thränert (MLU, Halle) for PALS measurement and the Fond der Chemischen Industrie for financial support.

39

Deff =

0 Dg /RT + (1 − ε)KDs,c

(A12)

ε/RT + (1 − ε)K

Therefore the time lag and the steady state slope are:

 (A3)

(A4)

L = S=

ı2 [ε/RT + (1 − ε)K] 0 K] 6[Dg /(RT ) + (1 − ε)Ds,c

pI A ı

D

g

RT

0 K + (1 − ε)Ds,c

(A13)

(A14)

40

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

For highly adsorbable gases, the derivative of the adsorbed amount with respect to the pressure is described by the Langmuir isotherm, (Eq. (9), Part I). A fully analytical solution cannot be obtained because of the nonlinear dependency of surface diffusion of the adsorbed amount and additionally of the nonlinear adsorption isotherm. But with using the asymptotic solution method of Frisch [12] we can determine the time lag analytically and therefore extract the required diffusion coefficients from the respective time lag: L =

ı2

 pI 0

 pI

pG(p)H(p)(

p

(

0

p

different temperatures relying on the parameters of the Langmuir isotherm determined independently. In addition to the time lags, the steady state slopes also give a measure of the mobility of the combined gas and surface diffusion based on the following relation: S=

H(p) dp)

(A15)

Dg 0 ∂ ln p ∂q + (1 − ε)Ds,c RT ∂ ln q ∂p

∂q ε + (1 − ε) RT ∂p

G(p) =

and

Using surface diffusivity equation, (Eqs. (4)–(6) and (12), Part I) and Langmuir isotherm (Eq. (9), Part I), the following relation for the time lag is obtained:

1  

ı2



Dg 1 +



× 1+



y 1+

3

ln(1 + )

(1 + y)



0

(1 − y) +



0 (1 + y) εDs,c

2



1+

ln( )  1 + y

H(p) dp = 0

A ı

D

g

RT



0 qsat ln(1 + ) pI + (1 − ε)Ds,c

(A18)

As in Part I [28], in order to see how the estimated parameters effect the total permeabilities of N2 and CO2 relative hypothetical errors of 3% and 10% were included with respect to each parameter (or parameter sets) at four different temperatures (120, 200, 300 and 700 K). Biggest deviations were caused by uncertainties of activation energies and heats of adsorption. For N2 permeabilities the biggest deviation of 54.1% was identified for 10% ␦Eg at 300 K, whereas gas diffusion is the only transport mechanism. Permeability deviations of CO2 due to activation energy uncertainties are more pronounced at lower temperatures. At 120 K small uncertainties of 3% ␦Eg resulted in 23.1% of permeability deviations and 10% ␦Eg resulted in even 77.2% of deviations. Maximal uncertainties of N2 permeability are expected in region of measurements and maximal deviations of CO2 at temperature closed to 100 K (Table B1). The uncertainties of the selectivity ratios estimated taking into account relative deviations are much smaller in comparison with respective permeability uncertainties. E.g. for CO2 /N2 selectivities, relative uncertainties with including 10% uncertainties of CO2 and N2 parameters sets are approx. 35% at 120 K.

(A16)

L =

pI

Appendix B. Estimation of errors of permeability data

The functions H(p) and G(p) are defined as follows: H(p) =



0 RT/D ), B = bq where = ((1 − ε)BDs,c g sat and  = bpI .

H(p) dp) dp 3

A ı

dy

(A17)

Since the experiments were performed at pI = patm and as the surface diffusion dependency of adsorbed amount was described with Darken equation (Eqs. (4)–(7) with f(q) = 1, Part I) the gas and surface diffusivities can be estimated from measured time lags at

Table B1 Relative permeability deviations (in %) at four different temperatures (120, 200, 300 and 700 K) for N2 and CO2 considering 3% and 10% of uncertainties of selected parameters or parameter sets. Temperature, K

120

Uncertainties of selected parameters or parameter sets, %

3

200 10

3

300 10

3

700 10

3

10

0,0 N2 : Ptot = f(ov , Eg , K0 , Q, Ds,c , Es )

ıPtot =

∂Ptot ∂ov

ıov

0.39

ıPtot =

∂Ptot ∂Eg

ıEg

5.6

ıPtot =

∂Ptot ∂K0

ıK0 a

0.46

ıPtot =

∂Ptot ∂Q

ıQ

6.2

23.7

2.7

ıPtot =

∂Ptot ∂Es

ıEs

3.20

10.7

2.1

7.1

8.9

21.82

11.6

39.7

16.5

2.4

ıPtot =



∂Ptot ∂ov

ıov

2  ∂Ptot +

∂Eg

ıEg

2

 ∂Ptot

+2

∂K0

ıK0

2

+ · · ·b

1.46 18.2 1.53

1.4

6.9

2.9

9.8

3.0

10

11.1

36.9

16.5

54.1

0.50

1.67 10

7.2

23.9

0.12

0.0

0.0

0.18

0.59

0.0

0.1

0.10

0.34

0.01

0.05

0.04

55.8

7.8

25.9

8.0

2.9

9.9

22.9

4.1

13.6

0,0 , Es ) CO2 : Ptot = f(ov , Eg , qsat , b0 , Q, Ds,c

ıPtot =

∂Ptot ∂ov

ıov

ıPtot =

∂Ptot ∂Eg

ıEg

ıPtot =

∂Ptot ∂qsat

ıqsat c

0.11

0.36

ıPtot =

∂Ptot ∂Q

ıQ

0.0

0.0

ıPtot =

∂Ptot ∂Es

ıEs

1.43

4.79

6.31

ıPtot = a b c



∂Ptot ∂

ıov

2  ∂Ptot +

∂Eg

ıEg

2

0,0 Ds,c

 ∂Ptot

+3

∂qsat

ıqsat

2

+ · · ·b

2.9

9.6

2.3

7.7

23.1

77.2

12.8

35.2

6.80

0.81

3.3

1.03

2.9

0.02

0.07

0.13

0.45

6.2

17.8

0.1

0.31

4.48

14.9

0.04

0.15

35.2

5.06

23.4

77.9

12.5

21 41.9

Error uncertainties of K0 and have same effect on relative deviation of N2 permeability causing the factor 2. Relative deviation of total permeability with including uncertainties of the parameter sets. 0,0 Error uncertainties of qsat , Ds,c and b0 have same effect on relative deviation of CO2 permeability causing the factor 3.

10.5

16.9

A. Markovi´c et al. / Journal of Membrane Science 336 (2009) 32–41

Nomenclature As b Dg dp Ds,c 0 Ds,c 0,0 Ds,c

Eg Es Jads Jg Jtot Jss K M Ptot Qs q qsat R S t T

specific surface area (m2 g−1 ) parameter of Langmuir adsorption isotherm (bar−1 , cf. Eqs. (9) and (10) in Part I) diffusion in the gas phase (m2 s−1 , Eq. (3)) pore diameter (m) corrected surface diffusivity (m2 s−1 , cf. Eqs. (4) and (6) in Part I) corrected surface diffusivity at zero loadings (m2 s−1 , cf. Eq. (6) in Part I) temperature independent corrected surface diffusivity at zero loadings (m2 s−1 , cf. Eq. (7) in Part I) gas activation energy (J/mol, Eq. (3)) surface potential energy (J/mol, cf. Eq. (7) in Part I) flux of adsorbed phase (mol m2 s−1 ) gas diffusion flux through membrane (mol m2 s−1 ) total permeation flux (mol m2 s−1 ) steady state flux (mol m2 s−1 , Eq. (A8), cf. in Appendix A) Henry law adsorption constant (mol cm−3 bar−1 , cf. Eq. (11) in Part I) molecular mass (kg mol−1 ) total permeability (mol/m/s/Pa, Eq. (7)) gas amount which passed the membrane (mol, cf. Eqs. (A8) and (A9) in Appendix A) adsorbed phase concentration (mol/m−3 ) total saturation capacity of adsorbed species (mol/m−3 , cf. Eq. (9) in Part I) universal gas constant (J/mol/K) steady state slope (mol s−1 , cf. Eq. (A8) in Appendix A) time (s) temperature (K)

Greek letters ı membrane thickness (m) ε porosity of the membrane  Lennard–Jones force constant (Å) ov pre-exponential factor (overall tortuosity factor), Eq. (3) L time lag (s, cf. Eq. (A8) in Appendix A)  standard deviation (%) ortho-positronium (o-Ps) lifetime 4 4 mean dispersion of the o-Ps lifetime distribution

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