## 1. Introduction

Gas separation by polymeric membranes is a well-established industrial process for a number of applications and at the same time it is a vivid research field [

1]. Significant effort is dedicated to the search for novel membrane materials with improved performance with respect to state-of-the-art commercial membranes. For successful development of novel membrane materials, the detailed knowledge of their transport properties is required. In this light, sorption, permeation, and modelling studies are all used to determine the basic transport parameters: the permeability, solubility, and diffusion coefficient. Most of those studies are dedicated to the transport properties of pure gases in novel materials, and much less work is focused on gas or vapour mixtures via experimental permeation [

2,

3,

4] and sorption [

5,

6,

7] measurements or via modelling [

8,

9,

10] of these processes. Whereas the measurement of permeability is a routine technique, only very few reports are available on the diffusion of gas mixtures because this requires more sophisticated methods to study the transient phase in the permeation process. Proposed methods based on NMR spectroscopy to study diffusion are very powerful [

11] and can be applied to mixtures [

12], but are not easily integrated in a permeation process. Selective condensation of one of the components of a gas mixture during the permeation transient [

13] provides valuable insight into the kinetics, but is not suitable for routine analysis of membranes. Instead, on-line mass spectrometry is versatile and has been successfully used to follow transient phenomena during pervaporation [

14,

15,

16,

17] and gas permeation experiments [

18,

19,

20] of relatively slow membranes. We used a similar setup for permeation measurements of polymers of intrinsic microporosity, but since they have very high permeability and very fast kinetics, we have only obtained their steady-state permeation data with this method so far [

21,

22,

23,

24].

Of all methods to determine the transport parameters of nonporous polymeric membranes, the time lag method introduced by Daynes a century ago [

25] remains by far the most popular method because of its experimental simplicity and because of the easy interpretation of the results. The diffusion coefficient is most commonly measured with a so-called fixed-volume/pressure increase instrument, measuring the time lag,

Θ, from the permeation transient and steady state [

26]:

where

l is the membrane thickness (m) and

D is the diffusion coefficient (m

^{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.

For the fixed volume time-lag setup for single gases in the present work, the entire time lag curve and the permeation curve in steady state are given by the equations [

33]:

in which

p_{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.

For a constant pressure/variable volume system, Ziegler et al. [

34] suggested an equation expressing the change in the permeate flow rate upon a step change in the feed concentration. In the present case, where we use the cumulative volume of permeating gas, Equations (2) and (3) can be easily converted to express the total permeate volume,

V_{t,STP}, in time:

where the membrane time lag is then given by the intercept between the extrapolated baseline curve (

V_{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.

Recently, we have optimised our constant pressure/variable volume system and the measurement procedures, based on a mass spectrometric residual gas analyser for the on-line analysis of the permeate gas, even during the transient phase in thin films or highly permeable membranes [

35]. Careful identification of all instrumental parameters and extensive error analysis allowed the accurate calculation of the mixed gas diffusion coefficient in Rubbery Pebax

^{®} 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:

where

Θ_{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]:

where

V_{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:

where

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

l^{2}/6

D. 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].

The scope of the present manuscript is to further improve the method, enabling transient and steady state permeation measurements for subsequent calculation of the mixed gas diffusion coefficients also at higher pressures, without compromising the accuracy of the method. Two different approaches will be presented: The first is based on individual measurements with instantaneous exposure of the membrane to the gas mixture at different pressures; the second is based on measurements with a step-wise increase in the feed pressure. The critical factors in the permeation setup and in the experimental procedures will be carefully investigated with a poly(styrene-

b-butadiene-

b-styrene) block copolymer to allow optimization of the operational parameters. The method is then applied to the polymer of intrinsic microporosity, PIM-SBF-1 (

Figure 1), with the CO

_{2}/CH

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

## 4. Conclusions

On-line measurement of transient and steady state permeation of gas mixtures by mass spectrometry allows the determination of the diffusion coefficient of individual components in gas mixtures. An appropriate correction is needed for the instrumental time lag related to the average residence time of the gases in the tubing and various sections of the instrument. Measurements at elevated pressure require an additional correction for the time needed to reach the setpoint of the feed gas pressure. This correction is shorter than the extra residence time of the gas in the system because permeation already starts as soon as the pressure begins rising. Such gradual exposure of the membrane to increasingly higher pressures would require a complex mathematical adjustment of the time lag model with variable boundary conditions. However, a satisfactory solution is the experimental analysis of the instrumental time lag with a thin film composite membrane under the same conditions, followed by subtraction of this value from the experimental time lag of a thick dense membrane.

With an uncertainty of a few seconds in the time lag, this method can provide highly accurate diffusion coefficients of any common gas in dense membranes. In the example of the polymer of intrinsic microporosity, PIM-SBF-1, CO_{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.

Preliminary measurements with a step-wise increase of the feed pressure allow a similar analysis of the transient phenomena for each pressure step and suggest that this may be used as a quicker and more versatile method to determine the pressure-dependent permeation transient, and thus the mixed gas diffusion coefficients. The different boundary conditions, where the membrane is not free of any gas before the measurement, requires a different mathematical treatment of the data, which will be the subject of further research.