# A Novel Time Lag Method for the Analysis of Mixed Gas Diffusion in Polymeric Membranes by On-Line Mass Spectrometry: Pressure Dependence of Transport Parameters

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

_{2}/CH

_{4}mixed gas diffusion coefficients of the spirobisfluorene-based polymer of intrinsic microporosity, PIM-SBF-1. It is shown that the modest pressure dependence of the PIM-SBF-1 permeability can be ascribed to a much stronger pressure dependence of the diffusion coefficient, which partially compensates the decreasing solubility of CO

_{2}with increasing pressure, typical for the strong sorption behaviour in PIMs. The characteristics of the instrument are discussed and suggestions are given for even more versatile measurements under stepwise increasing pressure conditions. This is the first report on mixed gas diffusion coefficients at different pressures in a polymer of intrinsic microporosity.

## 1. Introduction

^{2}s

^{−1}). A practical limitation is that for a correct application of this method, the measurements should obey a number of very strict criteria. One of the most important conditions is that the gas solubility and the diffusion coefficient should be constant over the experimental pressure range [27]. The simplest linear model further requires the total absence of the permeant in the membrane at the beginning of the experiment and negligible concentration of the permeant in the permeate side during the experiment. The latter is fundamentally impossible because a finite pressure increase is needed to measure the permeability, and this may lead to errors if the pressure increase is not kept as low as possible [27,28]. Changing boundary conditions at the feed and/or the permeate side requires a significantly more complex combined experimental and numerical approach [29,30]. Some example solutions for situations with changing boundary conditions are given in the review of Rutherford et al. [31]. Alternatively, fully computational methods may be used [32]. Most membrane materials do not have linear sorption behaviour and Al-Qasas et al. recntly proposed a method for the characterization of membranes with strong dual mode behaviour, taking also into account the changing boundary conditions in the feed and permeate side [29]. They quantified how dual mode sorption parameters affect the correctness of time lag measurements and, although the effect of nonlinear sorption is significant, they concluded that this does not exclude accurate analysis of the transport parameters by the classical time lag method [28]. Thus, the effective transport parameters are obtained, averaged over the thickness of the membrane.

_{t}is the permeate pressure (bar) at time, t (s), R is the universal gas constant (8.314 × 10

^{−5}m

^{3}bar mol

^{−1}·K

^{−1}), T is the absolute temperature (K), A is the exposed membrane area (m

^{2}), V

_{P}is the permeate volume (m

^{3}), V

_{m}is the molar volume of a gas at standard temperature and pressure (22.41 × 10

^{−3}m

^{3}

_{STP}mol

^{−1}at 0 °C and 1 atm), p

_{f}is the feed pressure (bar), S is the gas solubility (m

^{3}

_{STP}m

^{−3}bar

^{−1}), and D is the diffusion coefficient (m

^{2}s

^{−1}). In the presence of minor leaks, p

_{0}would be the starting pressure (bar) and (dp/dt)

_{0}would be the baseline slope (bar s

^{−1}), but normally these terms are negligible.

_{t,STP}, in time:

_{0}+ t·(dV/dt)

_{0}, which should be zero in a leak free system with a defect-free membrane), and that of the cumulative permeate volume versus time at steady state.

^{®}membranes, glassy Hyflon

^{®}perfluoropolymer membranes, and the polymer of intrinsic microporosity, PIM-EA-TB. For a system with on-line measurement of the permeate composition, the determination of the time lags of the individual components in the mixture, Θ

_{i}, must take into account the delay in the response due to the dead volume in the system:

_{0}is the instrumental time lag, related to the tube, cell, and analyser volumes in the instrument, as well as the response of the electronics; and D

_{i}is the diffusion coefficient of the individual gas species, i. As discussed previously, the instrumental time lag can be expressed by the following equation [35]:

_{Feed}, V

_{Downstream}, and V

_{Inlet}are the volume of the feed side, the volume of the permeate side until the sampling point, and the volume of the injection line, respectively. The terms φ

_{Feed}, φ

_{Downstream}, and φ

_{Inlet}indicate the respective total volumetric flow rates in those sections of the setup. In the case of very low permeate fluxes, φ

_{Downstream}, is practically equal to the sweep flow rate. Thus, for the total time lag:

_{Mem,i}represents the time lag of the membrane, equal to l

^{2}/6D. At atmospheric feed and permeate pressure, and with 200 cm

^{3}min

^{−1}feed and 30 cm

^{3}min

^{−1}sweep flow rates, the total instrumental time lag was found to be approximately 20 s, and this value must be subtracted from the total time lag to determine the time lag related to the membrane transport, and thus to calculate the effective gas diffusion coefficient [35].

_{2}/CH

_{4}mixture as a demonstration of the suitability of this method, even in the case of strongly pressure dependent transport parameters.

## 2. Materials and Methods

#### 2.1. Materials

^{−1}) was purchased from Sigma-Aldrich. Toluene (reagent grade), used for dissolving the SBS without further purification, was purchased from Carlo Erba. The porous support poly (vinylidene fluoride) (PV350) 75 kDa was supplied by Nanostone Water. The Polymer of Intrinsic Microporosity, PIM-SBF-1, was synthesised as described previously [36]. Pure gases (>99.99% purity) and a certified mixture (N

_{2}/CO

_{2}/O

_{2}with 79.88 mol %, N

_{2}10.10 mol %, CO

_{2}and 10.02 mol % O

_{2}) were supplied by SAPIO (Italy).

#### 2.2. Membrane Preparation

#### 2.2.1. Dense SBS Film Preparation

#### 2.2.2. SBS Thin Film Composite Membrane Preparation

#### 2.2.3. Dense PIM-SBF-1 Film Preparation

#### 2.3. Pure Gas Permeation Measurements

_{2}, N

_{2}, O

_{2}, CH

_{4}and CO

_{2}), although repeated measurements usually showed that the gas order was irrelevant at such low pressures. Before the first test, the membrane was evacuated for at least 1 h, or more if necessary, to guarantee the complete removal of all gases dissolved in the membrane. The correct starting condition of the membrane was checked and confirmed by the absence of significant baseline drift. To guarantee a complete evacuation of the membrane during the test cycle, it was kept under high vacuum for at least ten times the time lag of the previous gas tested. The permeabilities, P, are reported in Barrer (1 Barrer = 10

^{−10}cm

^{3}

_{STP}cm cm

^{−2}s

^{−1}cm Hg

^{−1}) and the values are calculated from the steady state of the permeation curve. The detailed measurement procedure, experimental setup, and data elaboration procedure were reported recently [35].

#### 2.4. Mixed Gas Permeation Measurements

^{3}min

^{−1}argon at ambient pressure. The feed pressure (0–5 bar(g)) was controlled by an EL-PRESS electronic back pressure controller (Bronkhorst High-Tech, Ruurlo, The Netherlands) and the feed (200–500 cm

^{3}min

^{−1}) and the sweep (typically 30 cm

^{3}min

^{−1}) flow rates were controlled by EL-FLOW electronic Mass Flow Controllers (Bronkhorst High-Tech, Ruurlo, The Netherlands). The measured mass spectrometer data were recorded with the MASsoft 7 software (Hiden Analytical, Warrington, UK) package supplied with the mass spectrometer, while the pressure and flow rates were recorded with the FlowPlot software (Bronkhorst High-Tech, Ruurlo, The Netherlands) supplied with the pressure and mass flow controllers. Data elaboration was carried out by a custom-written macro in MS Excel. The development, error analysis and validation of this method with a detailed description of the procedures were reported previously [35].

## 3. Results and Discussion

#### 3.1. System Response and Setup

^{®}2533 and Hyflon

^{®}AD60X membrane samples, as well as an aluminium film with a pinhole, and resulted to be approximately 20 s [35]. In this work, further evaluation and optimisation of the setup is carried out with a thin film composite (TFC) membrane with an effective thickness of 5 µm of the rubbery block copolymer SBS and a thick dense film (159 µm) for comparison (Figure 2). The active layer of the TFC membrane is thin enough to have a negligible time lag, and therefore it is used to verify the integrity of the system and the membrane, through measurement of the selectivity, and to determine the instrumental time lag, through measurement of the permeation transient and steady state. Both the TFC membrane and the thick SBS film appear completely dense by the SEM analysis (Figure 2), without visible defects, as later confirmed by their permselectivity.

#### 3.1.1. Instrumental Time Lag at Atmospheric Feed Pressure

_{2}in the TFC as the only exception. The reason for the lower slope of CO

_{2}is not fully understood, but since the permeance (SI Figure S1b) and the permselectivity (SI Figure S1c) are nearly constant, and since the permeance of the most permeable gas, CO

_{2}, does not decrease with decreasing sweep flow rate, it may be assumed that there are no significant polarization phenomena at the downstream side of the membrane. This confirms SBS to be a suitable material for further optimization studies. The thin film composite SBS membrane has a negligible time lag, and the observed values correspond to the instrumental time lag, whereas the difference with the thicker membrane accounts for the time lag of the SBS membrane itself. The slope in the curves of SI Figure S1a allows for the determination of V

_{Downstream}, which is one of the volumes that contribute to the instrumental time lag.

#### 3.1.2. Instrumental Time Lag at Variable Feed Pressure

^{3}min

^{−1}. This means that the total dead volume in the system that must be pressurised is ca. 20 cm

^{3}. Inversely, the time needed for a certain pressure increase, Δp, is given by:

_{2}, N

_{2}, and CO

_{2}is most likely due to the different flow regime in the narrow sampling capillary and in the low pressure quadrupole inlet, where Knudsen diffusion plays a role. This difference is negligible compared to the membrane-related time lag (Figure 5a,b), and since it falls more or less within the range of the experimental error, it does not affect any of the conclusions.

^{3}

_{STP}min

^{−1}, the filling-up time is ca. 25 s at 5 bar (SI Figure S2a), however, over this pressure interval the instrumental time lag increases only 10–12 s (SI Figure S2b). Thus, for the total time lag:

_{Δp}is the additional time lag induced by the slow pressure increase, and:

^{3}

_{STP}min

^{−1}.

#### 3.2. Mixed Gas Diffusion in the Polymer of Intrinsic Microporosity PIM-SBF-1

#### 3.2.1. Individual Pressure Steps

_{2}/CH

_{4}mixtures in PIM-SBF-1 at different pressures, and the results are given in Figure 5. All experiments were performed on a well-aged sample (2088 days) to avoid time-dependent phenomena. At high feed flow rates of 500 cm

^{3}min

^{−1}, the feed pressure during permeation experiments at 1–4 bar still requires 10–15 s to reach the setpoint value (SI Figure S4a). This time is not negligible and contributes, to some extent, to the overall time lag. Figure 5a shows the measured overall individual time lags for CO

_{2}and CH

_{4}, and the instrumental time lag, measured with the TFC membrane of SBS, is plotted for comparison as the lower blue line in Figure 5a. For a similar problem in gravimetric sorption measurements, where significant time is needed to fill the sorption chamber, Vopička et al. proposed a mathematical correction of the model [40]. In the present work, we assume additivity of each characteristic time in the procedure. Therefore, we subtract the time lag for the TFC membranes from that of CO

_{2}and CH

_{4}at the same pressure. Rearranging of Equation (12) then allows for the calculation of the diffusion coefficient (Figure 5c) of each gas according to:

_{F,i}/p

_{P,I}> 40–50 the error in D is less than 5%. In the present work, this ratio was generally higher so that the inaccuracy in the determination of D due to the changing boundary conditions in the downstream side is even lower than 5%.

_{2}and CH

_{4}show a very distinct and nearly exponential trend with increasing feed pressure. Both increase two-fold in the range from 1 to 4 bar(a), with only a marginal increase in the diffusion selectivity. In the same pressure interval, the CO

_{2}/CH

_{4}permselectivity decreased about 20% from ca. 40 to 32 as a result of a 10% decrease of the CO

_{2}permeability and a 10% increase of the CH

_{4}permeability. An increase in diffusivity at nearly constant permeability means that the solubility, S, decreases with increasing pressure, considering that permeability is the product of solubility and diffusivity:

#### 3.2.2. Incremental Pressure Steps

_{4}, but nearly identical for CO

_{2}(Figure 5a). This gives an excellent qualitative impression of the pressure dependence of the diffusion coefficients and the diffusion selectivity (Figure 5c,d). However, since the boundary conditions for which Equations (2)–(5) are valid, namely, that the membrane is penetrant-free at the beginning of the experiment, are not satisfied in the second and subsequent pressure steps, quantitative interpretation of these data must be done with care. This procedure would require a correction for the initial concentration of the penetrant [42] that takes into account the finite non-zero initial gas concentration in the membrane before each pressure step. After the first step, there is already a concentration gradient across the membrane, corresponding to the steady state permeation at the previous pressure, as schematically displayed in SI Figure S5. Theoretically, also the desorption kinetics would allow the calculation of the diffusion coefficient. Simple application of Equations (2)–(5), without appropriate corrections for the boundary conditions, gives the same trend with shorter time lags and, thus, apparently faster diffusion than for the pressure increase run (SI Figure S6).

## 4. Conclusions

_{2}demonstrated to have an approximately five times higher diffusion coefficient than CH

_{4}, which both increase by a factor of two when increasing the feed pressure from 1 to 4 bar(a), with a marginal increase in the diffusion selectivity. Instead, in the same pressure interval, the CO

_{2}/CH

_{4}permselectivity decreased about 20% from 40 to 32, as a result of a 10% decrease of the CO

_{2}permeability and a 10% increase of the CH

_{4}permeability. This means that the solubility must show the opposite effect and thus decrease with increasing pressure, as generally observed for PIMs with a strong dual mode sorption behaviour.

## Supplementary Materials

^{−2}). Feed flow rate 200 cm3STP min

^{−1}and sweep flow rate 30 cm3STP min

^{−1}; SI Figure S3. Profile of the feed flow rates and feed pressure (A), and the permeate flow rates (B) as a function of time for the SBS thin film composite membrane with a 80/10/10 vol% N2/CO2/O2 mixture; SI Figure S4. Comparison of the increasing feed pressure as a function of time for individual pressure increase ramps (A) and for stepwise increasing pressure (B) for 2088 days aged sample SBF with the gas mixture CO2/CH4 35/65 vol%; SI Figure S5. Schematic representation of the development of the concentration profiles in the membrane after its first exposure to the gas and during two subsequent pressure increase steps; SI Figure S6. Individual time lag for CO

_{2}and CH

_{4}in a 2088 days aged sample of PIM-SBF with a stepwise pressure increase (full circles). The empty circles show the corresponding points for the pressure decrease steps. Points of the instrumental time lag Θ

_{0}for N

_{2}, O

_{2}and CO

_{2}overlap.

## Author Contributions

## Funding

^{4}CO

_{2}. This work was further supported by the CNR-CAS bilateral agreement 2016–2018 “Innovative polymeric membranes for pervaporation and advanced gas and vapour separations”.

## Acknowledgments

^{®}1657 samples, and Phenom-World B.V., Eindhoven (NL), is gratefully acknowledged for lending a Phenom Pro X desktop SEM to ITM for evaluation of its use in advanced membrane characterization.

## Conflicts of Interest

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**Figure 2.**SEM image of the 159 µm thick dense SBS membrane (

**a**); the 5 µm thin film composite SBS membrane (

**b**).

**Figure 3.**Details of the experimental setup, highlighting with the thick shaded grey line the feed section, which determines the pressure increase rate (adapted from Fraga and Jansen et al. [35]).

**Figure 4.**Plot of the increase of the feed pressure as a function of time upon switching of the six-way feed valve in Figure 3 from Argon purge gas to the 80/10/10 vol % N

_{2}/CO

_{2}/O

_{2}mixture, mimicking CO

_{2}-poor flue gas (

**a**), and corresponding instrumental time lag (

**b**) determined with an SBS TFC membrane (area 1.77 cm

^{−2}). Examples of the individual time lag curves for N

_{2}(

**c**) and CO

_{2}(

**d**). Feed flow rate 500 cm

^{3}

_{STP}min

^{−1}and sweep flow rate 30 cm

^{3}

_{STP}min

^{−1}. The numbers on the curves in (

**a**,

**c**) and (

**d**) indicate the feed pressure in bar(a); the intermediate curves are the half-integer values.

**Figure 5.**Individual time lag for CO

_{2}and CH

_{4}in a 2088 days aged sample of PIM-SBF-1 with individual pressure steps (

**a**) and a stepwise incremental pressure increase (

**b**). The instrumental time lag at the bottom of the graphs (

**a**,

**b**) needs to be subtracted for the calculation of the corresponding diffusion coefficients (

**c**,

**d**), showing strong pressure dependence. The permeability and selectivity show a much lower pressure dependence (

**e**,

**f**). Gas mixture 35/65 vol % CO

_{2}/CH

_{4}.

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**MDPI and ACS Style**

Monteleone, M.; Esposito, E.; Fuoco, A.; Lanč, M.; Pilnáček, K.; Friess, K.; Bezzu, C.G.; Carta, M.; McKeown, N.B.; Jansen, J.C. A Novel Time Lag Method for the Analysis of Mixed Gas Diffusion in Polymeric Membranes by On-Line Mass Spectrometry: Pressure Dependence of Transport Parameters. *Membranes* **2018**, *8*, 73.
https://doi.org/10.3390/membranes8030073

**AMA Style**

Monteleone M, Esposito E, Fuoco A, Lanč M, Pilnáček K, Friess K, Bezzu CG, Carta M, McKeown NB, Jansen JC. A Novel Time Lag Method for the Analysis of Mixed Gas Diffusion in Polymeric Membranes by On-Line Mass Spectrometry: Pressure Dependence of Transport Parameters. *Membranes*. 2018; 8(3):73.
https://doi.org/10.3390/membranes8030073

**Chicago/Turabian Style**

Monteleone, Marcello, Elisa Esposito, Alessio Fuoco, Marek Lanč, Kryštof Pilnáček, Karel Friess, Caterina Grazia Bezzu, Mariolino Carta, Neil Bruce McKeown, and Johannes Carolus Jansen. 2018. "A Novel Time Lag Method for the Analysis of Mixed Gas Diffusion in Polymeric Membranes by On-Line Mass Spectrometry: Pressure Dependence of Transport Parameters" *Membranes* 8, no. 3: 73.
https://doi.org/10.3390/membranes8030073