# The Role of Surfaces in Gas Transport Through Polymer Membranes

^{*}

## Abstract

**:**

_{1}, thus determining the solubility S and rate of desorption k

_{2}of He, N

_{2}, O

_{2}, CH

_{4}, and CO

_{2}on a polydimethylsiloxane (PDMS) membrane. The described procedure is able to determine experimentally all the physical quantities that characterize the gas transport process through a thin rubber polymer membrane. The experiments were carried out at room temperature and at a transmembrane pressure of 1 atm. The results are in good agreement with the available data in the literature and offer an evaluation of k

_{1}and k

_{2}.

## 1. Introduction

_{2}capture in fuel power plants, gas pure production technology, gas sensing, etc. The complete understanding of gas transport through polymer films is still a challenge. Different models, depending on the gas/membrane system, have been developed to describe this phenomenon: Knudsen or Poiseuille diffusion for porous membranes, molecular sieve effects, solution–diffusion (SD) for dense membranes, with dual-mode sorption theory in case of glassy polymers [1]. The experimental results confirm the models in the range of validity of the theory. Considering the permeation of a gas through a dense polymer, the SD model [2,3] describes quite well this phenomenon and is routinely employed to determine the parameters that characterize a membrane, such as permeability P, diffusivity D, and solubility S. However, the behaviour of polymeric materials with respect to gas permeation depends on a great number of variables, including method of fabrication, composition of the polymeric matrix, glass transition temperature, grade of confinement. Consequently, it can happen that the simple SD model does not describe accurately the experimental results. In fact, works that denote some anomalies in the measurements of these quantities are not rare in the literature. This occurs when the permeability of a polymer to gases is measured by using a constant-volume variable-pressure (manometric) method and a constant-pressure variable-volume (volumetric) method, as Lundstrom reports in a recent review [4]. Several of these discrepancies could be avoided by following the procedure suggested by the American Society for Testing and Materials (ASTM International) in one of their reports on standard test methods to determine gas permeability of plastic films and sheeting [5]. This designation highlights that P should not be used unless its constancy with different membrane thicknesses has been verified, implying the possibility that permeability may be thickness-dependent. However, also following all recommendations of the previous designation to determine the values of the physical quantities that describe the mechanism of gas transfer along the membrane, remain some discrepancies between experimental and theoretical results.

_{1}and k

_{2}to describe the dynamics of the adsorption process (SDk

_{1}k

_{2}model). In the present experiments, we used a dynamic apparatus that measures the gas transmission rate through a membrane of PDMS subjected to a known differential partial pressure of the tracer gas (He, N

_{2}, O

_{2}, CH

_{4}, and CO

_{2}). By following the electrical analogy proposed in reference [12] and by using the experimental set up of reference [6], this paper determined permeability P and rate of adsorption k

_{1}. In addition, a lag-time method measured diffusivity D. The knowledge of D permitted the calculation of solubility S and of desorption rate k

_{2,}allowing to determine all the physical quantities that characterize the permeation process. The paper reports the surface kinetic parameters, k

_{1}and k

_{2}, as a function of Lennard–Jones temperatures for all gas tested. The dependence of solubility and solubility-selectivity on gas critical parameters is also reported and compared with theoretical considerations [13].

#### 1.1. Gas Transport Model

_{1}k

_{2}model considers the permeation of a gas through a dense polymer membrane as a three-step mechanism: (1) sorption of the gas on the upstream surface of the membrane, (2) diffusion through the bulk of the membrane driven by a gradient concentration, (3) desorption through the downstream surface of the membrane. Considering the first step, the molar flux J in mol m

^{−2}s

^{−1}at the interface is given by:

_{u}is the upstream pressure, and C

_{u}the upstream concentration in the membrane at the first interface (see Figure 1); k

_{1}is expressed in mol m

^{−2}s

^{−1}Pa

^{−1}, k

_{2}in m s

^{−1}, and C

_{u}in mol m

^{−3}. Under the hypothesis of Henry’s law, C

_{u}can be expressed as S × p

_{u}*, where S = k

_{1}/k

_{2}is the solubility, and p

_{u}* is the upstream pressure in conditions of equilibrium at the interface. Consequently, Equation (1) can be written as:

_{u}= (p

_{u}– p

_{u}

^{*}). From equation (2), it is possible to write Ohm’s law (by unit of membrane area A) for the case of a gas surface sorption on the polymer membrane:

_{s}is the surface resistance. Recent papers report a complete analysis of the phenomenon considering non-equilibrium at the surfaces [4,6,8]

_{u}– p

_{d}), L/P is the bulk resistance R

_{B}, where P is the product of diffusivity D by solubility S (Fick’s law). For L→0, R

_{B}→0, and the permeation process is limited only by the two upstream and downstream surface resistances r

_{s}. On the contrary, for great thicknesses, r

_{s}is negligible with respect to R

_{B}, and Equation (4) becomes

_{u}and p

_{d}are, respectively, upstream and downstream pressure, and p

_{u}* is the upstream pressure in conditions of equilibrium at the interface. C

_{u}and C

_{d}are the upstream and downstream concentrations in the membrane at the interface. The dotted line in Figure 1 represents the linear variation of the concentration in the bulk (diffusivity is supposed to be independent from concentration). R

_{B}and r

_{s}are bulk and surface resistances, respectively, defined as in Equations (5) and (3).

#### 1.2. Experimental Method

_{1}is reported below. We first measured P for a membrane with L >> 100 µm. In this case, by measuring the quantity ϕ = Δp/J, one obtains:

_{1}is easily determined:

_{1}can be determined by a measurement of ϕ.

_{1.}

_{SS}is the gas flow at steady state, and t

^{*}is greater than or equal to the time to reach the steady state. Considering that S = k

_{1}/k

_{2}= P/D:

## 2. Materials and Methods

_{2}, N

_{2}, He, O

_{2}, and CH

_{4}. The experimental setup employed to measure ϕ and diffusivity was the same reported in references [6,12], as was the assembling of the sample that guaranteed the same exposed area on the upstream and downstream sides. The apparatus was a high-vacuum chamber, where J and Δp were measured by means of pressure gauges with high accuracy. The method measured transient fluxes J directly as the product of the pressure of the vacuum chamber by the pumping speed of the system to the tracer gas. The tracer gases were tested without a systematic order because none of them created irreversible effects on PDMS (CO

_{2}plasticization occurs only for glassy polymers, for example). However, we waited for a complete degassing of the polymer between two measurements with different tracer gases. The membranes tested had two different thicknesses, i.e., 2 mm and 10 µm, to measure P in the first case and k

_{1}from Equation (7).

_{SS}and I

_{0}are the ion currents of the tracer gas at steady state and at the background, respectively, Δt is the RGA sampling time, and n is the number of samples of ion currents equal to t*/Δt. Measurements of permeability and diffusivity were always taken for pure gases (purity grade N5.0, minimum purity 99.999%).

## 3. Results and Discussion

_{1}measured following the previously described procedure. The values of P were in good agreement with those reported in the literature [6,15,16].

_{2}were obtained by using Equations (8) and (9).

_{1}and k

_{2}, it is interesting to report this parameter as a function on ϵ/k. Figure 3 shows this relationship.

_{2}in Figure 3. The results were in accord with the thermodynamic approach developed by Teplyakov et al. [17] which gives the following expression for the solution of a gas in polymers:

## 4. Conclusions

_{1}, k

_{2}. The method was carried out for simple molecules, such as He, N

_{2}, O

_{2}, CH

_{4}, and CO

_{2}permeating a PDMS membrane. The results are in excellent agreement with those reported in the literature. The values of k

_{1}and k

_{2}depend on Lennard–Jones temperatures of the studied gases, showing that a high value of Lennard–Jones temperature facilitates the binding of gas molecules with the membrane surface and, consequently increases gas condensability and, finally, solubility.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Diffusivity measurements. (

**a**–

**e**) Ion currents vs time of the tracer gases after filling the upstream chamber a t = 0 (p

_{u}= 10

^{5}Pa); (

**f**) representation of the parameters of Equation (10); in our case Δt = 2 s.

**Figure 3.**Surface kinetics parameters vs Lennard–Jones temperature. Graphs show the dependence of k

_{1}and k

_{2}on Lennard–Jones temperature ϵ/k.

**Figure 4.**Solubility vs ϵ/k. The graph reports solubility as a function of Lennard–Jones temperatures.

**Table 1.**Permeability P, sorption rate k

_{1}, diffusivity D, solubility S, desorption rate k

_{2}, Lennard–Jones temperature ϵ/k. The transmembrane pressure was Δp = 1.013 × 10

^{5}Pa. The temperature was T = 293 K; *data were taken from reference [13].

Gas | P × 10^{13}(mol m ^{−1} s^{−1} Pa^{−1}) | k_{1} × 10^{8}(mol m ^{−2} s^{−1} Pa^{−1}) | D × 10^{9}(m ^{2} s^{−1}) | S × 10^{4}(mol m ^{−3} Pa) | k_{2} × 10^{4}(m s ^{−1}) | ϵ/k* (K) |
---|---|---|---|---|---|---|

He | 2.3 ± 0.1 | 2.7 ± 0.4 | 16.8 ± 0.3 | 0.14 ± 0.01 | 20 ± 4 | 10.2 |

N_{2} | 1.8 ± 0.1 | 3.9 ± 0.7 | 3.5 ± 0.2 | 0.52 ± 0.06 | 8 ± 2 | 71 |

O_{2} | 2.8 ± 0.2 | 4.5 ± 0.7 | 3.1 ± 0.2 | 0.9 ± 0.1 | 5 ± 1 | 107 |

CH_{4} | 4.6 ± 0.3 | 7 ± 1 | 2.2 ± 0.2 | 2.1 ± 0.3 | 3 ± 1 | 149 |

CO_{2} | 17 ± 1 | 17 ± 2 | 2.4 ± 0.2 | 7.3 ± 0.9 | 2.3 ± 0.6 | 195 |

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**MDPI and ACS Style**

Firpo, G.; Angeli, E.; Guida, P.; Pezzuoli, D.; Repetto, D.; Repetto, L.; Valbusa, U.
The Role of Surfaces in Gas Transport Through Polymer Membranes. *Polymers* **2019**, *11*, 910.
https://doi.org/10.3390/polym11050910

**AMA Style**

Firpo G, Angeli E, Guida P, Pezzuoli D, Repetto D, Repetto L, Valbusa U.
The Role of Surfaces in Gas Transport Through Polymer Membranes. *Polymers*. 2019; 11(5):910.
https://doi.org/10.3390/polym11050910

**Chicago/Turabian Style**

Firpo, Giuseppe, Elena Angeli, Patrizia Guida, Denise Pezzuoli, Diego Repetto, Luca Repetto, and Ugo Valbusa.
2019. "The Role of Surfaces in Gas Transport Through Polymer Membranes" *Polymers* 11, no. 5: 910.
https://doi.org/10.3390/polym11050910