J. Phys. Chem. B 2004, 108, 19325-19329
19325
Desorption as a Rate Limiting Step for Gas Permeation through a Polymer Membrane Daniel M. Jenkins* Department of Molecular Biosciences and Bioengineering, Room 218, UniVersity of Hawaii, 1955 East West Road, Honolulu, Hawaii 96822 ReceiVed: May 24, 2004; In Final Form: October 8, 2004
Under certain conditions, permeation of gases through polymer membranes is not adequately represented by classic diffusion theory. Specifically, thin films can present stronger resistance to gas transfer than would be suggested by the material “permeability” alone, especially when one side of the membrane is wetted. It is proposed that kinetics of gas transfer at the surface may explain these discrepancies, and a series of experiments is proposed to quantify these effects. By measuring CO2 permeation through silicone membranes of different thicknesses, estimates of the finite rate constants governing gas exchange at the surface of a nonwetted membrane can be obtained along with the diffusivity of the material. In addition, observations from a “relaxation” experiment for CO2 transfer through a silicone membrane from a gas phase into a dissolved phase allows the rate constants at the wetted surface to be estimated. Estimates of diffusivity of CO2 through silicone from these experiments (1.54 × 10-5 cm2 s-1 at 23 °C) are reasonably close to other values inferred from the literature, and finite positive values for the surface rate constants indicate the importance of surface effects, especially for wetted membranes.
Introduction and Theory Review of Gas Transport. Steady-state transport through membranes is typically regarded as a simple diffusion process: 1-3
D dc J ) -D ) (ca - cb) dx ∆x
(1)
where J is flux (mass or molar transport per unit area per unit time), D is the diffusivity of the permeant through the membrane material, and x is the direction of transport (note that ∆x is the membrane thickness). The concentrations ca and cb on the edges of the membrane are related to the partitioning coefficient of the species into the membrane material and the relevant species activity at the interfaces. When the diffusing species is a gas, for example, concentrations on the polymer surface would be related to the gas pressures p on either side of the membrane and the gas solubility S in the polymer
c ) pS
(2)
so that the flux described by eq 1 could be expressed as
J)
P DS (p - pb) ) ∆p ∆x a ∆x
(3)
Here, the product DS is often expressed as a membrane permeability P and is presumed to be independent of membrane thickness and species concentration.2,3 The above analytical treatment works sufficiently well for the transport of certain gases through a membrane when both sides of the membrane are in gaseous media.4 In addition, analogues of eq 3 form part of the basis for determining the electrical potential across living cells.5 However, the author has observed that when one side of a silicone membrane is wetted, * Corresponding author. Phone: (808) 956-6069. Fax: (808) 956-3542. E-mail:
[email protected].
Figure 1. Model for quasi-steady-state gas transport through a polymer membrane, including dynamic partitioning at the membrane surfaces. The variables c2 and c3 represent gas concentrations at the opposing surfaces of the membrane, and c1 and c4 represent the relevant activities (pressure or concentration) of the gas in the adjacent media. The diffusivity of gas in the polymer is represented as D, the membrane thickness as ∆x, and the rate constants for adsorption and desorption into the appropriate media as kai and kdi.
the flux of gas across the membrane is significantly diminished compared to that predicted using analyses such as those above.6 This limitation suggests that a more complete kinetic treatment may be justified for situations where a wetted membrane is involved. Examples of applied systems relying on gas transport through wetted membranes are abundant, including gas exchange in sterile perfusion loop bioreactors, pollution mitigation equipment for combustion exhaust, dissolved gas sensors, and gas exchange across living cell membranes and tissue. Proposed Modification to Membrane Transport Model. A modification is proposed to address the limitations of the classical gas permeation model. In the proposed model, the partitioning of the permeant between the different media is not described merely by an equilibrium solubility (e.g., eq 2) but is governed kinetically by the rates of adsorption and desorption from the membrane (Figure 1). Though mathematical treatments for desorption and adsorption into polymers have been given in the past,3 these are based solely on diffusion through the polymer and not on dynamic limitations occurring at the surface. The newly proposed model is analogous to a kinetic treatment for dissolution of carbon dioxide in water,7,8 in which the flux of CO2 is governed by
10.1021/jp0477553 CCC: $27.50 © 2004 American Chemical Society Published on Web 12/09/2004
19326 J. Phys. Chem. B, Vol. 108, No. 50, 2004
Jenkins
the rate constants for transport across the surface at the interface between water and gas. Theoretical Flux for Compound Model. Using the newly proposed model and the definition of flux given above, the following equations can be written:
J1f2 ) ka1c1 - kd1c2
(4)
Da,b (c - c3) J2f3 ) ∆x 2
(5)
J3f4 ) kd4c3 - ka4c4
(6)
where Jifj is the molar flux from pool i to pool j. Equations 4 and 6 may be simplified by introducing new variables ceq1 and ceq4 that describe the respective concentrations of gas in the polymer that would exist at equilibrium with c1 and c4:
ci )
kdi c kai eqi
(7)
This relationship can be invoked in eqs 4 and 6 to give
J1f2 ) kd1(ceq1 - c2)
(8)
J3f4 ) kd4(c3 - ceq4)
(9)
and
For quasi-steady-state transport across the membrane, where the rate of change of ci is nearly zero for all ci, the flux through any single process Jifj is roughly equivalent to the total flux Jtot through the membrane. Under these conditions, the system can be treated analogously to the steady state heat or charge conduction through a series network.9 For transport through series elements, individual “resistances” (which in our analogy are equivalent to the concentration differences in eqs 5, 8, and 9 divided by flux) simply add to give the total resistance across the network.9 The net result in this case is
Jtot )
kd1kd4D
(c1eq - c4eq)
D(kd1 + kd4) + kd1kd4∆x
(10)
Note that when the desorption rate constants are large, eq 10 is equivalent to eq 1. Estimation of Diffusivity and Desorption Rate Constant into Gas. When both sides of the membrane are in contact with a gas (in contrast to a liquid), kd1 ) kd4 ) kd, and ceqi are related to pressures on the respective sides of the membrane by the solubility constant Sp in the polymer, such that a mass transfer resistance Fd can be defined for the dry membrane as the ratio of pressure drop across the membrane to the molar flux:
Fd )
∆p 2 ∆x ) + Jtot Spkd SpD
(11)
On the basis of this relationship, if the mass transfer resistance Fd is plotted against film thickness ∆x, then the intercept is related to the surface rate limitations, and the slope is simply the inverse of the traditional permeability constant P. Determination of Mass Accommodation Coefficient. The adsorption rate constant into air, which is the product of the desorption rate constant and the gas solubility, can be used to estimate a conceptually important quantity called the mass accommodation coefficient (γ). This quantity represents the
probability that a collision with the membrane results in a gas molecule being adsorbed and is simply the ratio of the rate of adsorbed molecules (kap) to the average number of collisions with the membrane as predicted by kinetic theory:10
γ ) kdSpx2πMRT
(12)
where R is the universal gas constant, M is the molar mass, and T is the temperature in absolute terms. Relaxation Time for Transport Across a Wetted Membrane. When gas transport occurs through a membrane from a finite volume of gas into a finite volume of solvent, a “relaxation” to equilibrium gas tension between the compartments occurs. This relaxation is mediated by gas exchange across the membrane on teh basis of eq 10:
dn A ) (S p - c4eq) dt Fw p 1
(13)
where n is the number of moles of gas transported across the membrane of area A, and the wetted membrane mass transfer resistance Fw follows from eq 10:
Fw )
1 1 ∆x + + kd1 kd2 D
(14)
The rate of change of n can be related to the rate of change of pressure in the gaseous volume by the ideal gas law:
(
)
Sp dp1 RT dn ART ) )Sp - c dt Vg dt F wV g p 1 S s 4
(15)
and to the change in concentration in the solvent by conservation of mass:
(
)
Sp dc4 1 dn A ) ) Sp - c dt Vs dt FwVs p 1 Ss 4
(16)
The coefficient Ss in this case represents the solubility of the gas in the solvent, Vs represents the volume of the solvent, and Vg represents the volume of gas. Equations 15 and 16 represent a system of linear differential equations that may be solved using linear algebra. The solution for the pressure p1 is of the form
p1 ) p1(∞) + (p1(0) - p1(∞))e-t/τ
(17)
where p1(0) is the initial pressure, p1(∞) is the equilibrium pressure, and the time constant τ is the negative inverse of the nonzero eigenvalue r that satisfies the following equation:
(
)
ASp RT 1 1 + r)- )τ Fw Vg VlSs
(18)
Estimation of the Desorption Rate Constant into Solvent. If an experiment is performed to determine the relaxation time constant τ of the closed system with a wetted membrane, eq 18 can be used to give an estimate of the mass mass transfer resistance:
(
Fw ) τASp
RT 1 + Vg VlSs
)
(19)
Given the diffusivity D and desorption rate constant into gas kd1 that can be estimated from previously described analyses,
Desorption Limitations for Gas Permeation
J. Phys. Chem. B, Vol. 108, No. 50, 2004 19327
Figure 2. Experimental setup for determination of kinetics of CO2 transport in silicone.
the observed mass transfer resistance can be used to solve for the desorption rate constant into solvent:
kd4 )
Dkd1 D + kd1∆x + Fwkd1D
(20)
Experimental Methods Determination of the Diffusivity and the Desorption Rate Constant into Gas. The permeability of a commercial silicone material was tested using membranes of two thicknesses. The membranes were composites made up of a thin film (0.5, 4.2, or 10 µm) of Poly(dimethyl siloxane) on a porous backing designed to give the film mechanical strength (CAPSUM Technologie GmbH, Trittau, Germany). Acrylic housings were machined to seal approximately 1.9 cm diameter membranes in the flow path of CO2. Pressurized CO2, typically at less than 60 kPa (gage), was applied to the membranes until a steady state flow was established. The flow rate and pressure were then recorded, typically for several minutes to ensure steady state flow. For improved accuracy at the relatively low flow rates, an improvised flow meter was made by weighing water displaced from a small flask (Figure 2). Weight from the scale (Adventurer Model AR 2140, Ohaus Corp., Pine Brook, NJ) was communicated to a PC via serial communication. Mass flow rates of water were converted to volume flow rates of gas, which were then converted to molar flux using the ideal gas law. For use in this relationship, ambient temperature and pressure were recorded using a digital barometer (BAR888A, Oregon Scientific Co., Denver, CO). Hydrostatic head in the improvised flow meter was measured with a ruler to correct for the pressure of the flowing gas. To correct for evaporation from the flow meter during the experiments, evaporation rates were estimated by recording the mass of the receiving beaker for about 10 min without the addition of water. The pressure drop across the membranes was recorded using a commercially available pressure sensor with a nominal range of 100 kPa (MPX2100DP, Motorola Semiconductor, Phoenix AZ). The output of this sensor was amplified with a homemade instrumentation amplifier and digitized with a commercial data acquisition system. To reduce the effects of other ambient gases, the system was purged with CO2 before each experiment by loosening the seal around the acrylic membrane housing and flask stopper. To minimize the effects of CO2 dissolution into the water in the flask, CO2 saturated water was obtained from the sparger.
The data recorded during these experiments was used to estimate the diffusivity of CO2 in the silicone and the rate constant for desorption of CO2 from silicone into gas using the relationships shown above. The solubility of the gas in the polymer, which was necessary to calculate the rate constants from the experimental data, was provided by the manufacturer. In addition, the experimental data were used to determine the mass accommodation coefficient γ for CO2 into silicone. Determination of Relaxation Time and Desorption Rate Constant into Water. A membrane housing was custom-made for this experiment that was similar to the housings used in the previous experiment (Figure 2). In this case, the two chambers on either side of the membrane were asymmetrical: the water side holding approximately 180 mL and the gas side holding about 18 mL. These volumes were chosen to give a reasonably large change in gas pressure during the relaxation while the scale of the experiment was kept easily manageable. The volumes were verified both by weighing the amount of water required to fill the chambers and by recording the isentropic compression of gas in the chambers with a syringe.11 To ensure complete mixing and a uniform oxygen concentration in the water side of the system, the water was continuously stirred with a 3.175 cm Teflon-coated magnetic stir bar. Before each experiment, the system was purged with CO2 to reduce the effects of other ambient gases. To do this, water was drained from the housing, the bolts on the housing were loosened, and CO2 was allowed to blow through the system, including the line to the pressure sensor. When the system was sufficiently purged, the bolts on the housing were tightened and carbonated water was quickly added to the water side through valves d and g. Care was taken to minimize the amount of bubbles caught in the system. Carbon dioxide was then charged into the gas side of the housing, and gas pressure was recorded during the transient ensuing the closing of valves e, f, and g. To saturate the CO2 gas with water vapor (and thus prevent competing effects of water vapor transport through the membrane), gas for this experiment was taken from the sparger (valve c). The valves used were Hoffman type clamps and were operated manually. The relaxation time constant τ was estimated using the Guggenheim method,12 which allowed the value to be estimated without always waiting for complete equilibrium to be achieved. To summarize this method, data from transient first-order systems such as that described by eq 17 were paired at fixed time intervals δt:
∆p ) |p(t) - p(t+δt)|
(21)
19328 J. Phys. Chem. B, Vol. 108, No. 50, 2004
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Figure 3. Typical relaxation curve (continuously recorded data) for CO2 transfer between gaseous and dissolved states through a 0.5 µm thick silicone membrane.
and these data were linearized by taking the logarithm
t ln(∆p) ) - + constant τ
Figure 4. Guggenheim plot for relaxation data from Figure 3 (y ) 2.74-0.028t; R2 ) 0.9993). From these data, the relaxation time constant is approximately 35.6 min, and the observed desorption rate constant for CO2 from silicone into water is approximately 1.11 × 10-3 cm s-1.
(22)
The value of τ was estimated by performing a linear regression of ln(∆p) against time and comparing the observed slope to that predicted by eq 22. The desorption rate constant for CO2 from silicone into water was estimated from observed relaxation time constants using eqs 19 and 20. For use in these relationships, the gas solubility in water was adapted from readily available tables.13 Results and Discussion Determination of Diffusivity and Desorption Rate Constant into Gas. With two observations each at three different film thicknesses (0.5, 4.2, and 10 µm), mass transfer resistance was observed to be approximately linearly related to film thickness (Fd ) 6.69 × 1013x + 4.63 × 109 Pa s cm2 mol-1; R2 ) 0.95). The product of diffusivity and solubility for CO2 in silicone observed in these experiments (1.50 × 10-14 mol cm-1 s-1 Pa-1) was similar to the reported permeability of CO2 through vulcanized silicone rubber2 (1.07 × 10-14 mol cm-1 s-1 Pa-1) at the same temperature (23 °C). No reported values for diffusivity of CO2 in silicone polymers were available, but the observed diffusivity of CO2 through the silicone film (1.54 × 10-5 cm2 s-1) was similar to that for nitrogen and oxygen gas through vulcanized silicone rubber (1.25 × 10-5 and 1.63 × 10-5 cm2 s-1, respectively) at 23 °C. The intercept of the resistance against membrane thickness was not significantly different from zero at the 5% level (p ) 0.30), supporting the classic permeability model where surface resistance is negligible. Although the intercept was not statistically significant, it may be used to give a best estimate for the desorption rate constant for CO2 from silicone into air as 0.45 cm s-1. Using the nonzero estimate for the intercept of the resistance plot, the mass accommodation coefficient γ can be calculated as 1.1 × 10-4. This value is about 2-4 orders of magnitude greater than reported values for the dissolution of several different gases into water7,8 and may be partly reconciled with the fact that these gases are more soluble in silicone than in water. Determination of Relaxation Time and Desorption Rate Constant into Water. Relaxation time constants for the systems described above typically ranged from 30 to 36 min (Figures 3 and 4). To ensure that diffusion through a boundary layer of water was not dynamically limiting, the relaxation experiment was repeated under varying levels of agitation using a magnetic stir bar. The slope of the observed mass transfer rate against
Figure 5. Observed desorption rate constants for CO2 from silicone into water, as a function of speed of magnetic stirring (R2 ) 0.05; average ) 1.18 × 10-3 cm s-1).
stirring speed (Figure 5) was not significantly different from zero at the 95% level (p ) 0.56), suggesting that the water could be considered well mixed at all of the speeds tested. To determine whether gas exchange through the tubing and acrylic housing played an important role in the dynamics of the relaxation experiment, the membrane was removed and the dry housing was pressure tested at 50 kPa, with valves e, f, and g closed. Aside from thermal compression and decompression, no appreciable loss of pressure was observed over 10 h. All tubing used in these experiments was Tygon laboratory type tubing. On the basis of these experiments, the average desorption rate for CO2 from the silicone membrane into distilled water was observed to be about 1.18 × 10-3 cm s-1, a value about 3 orders of magnitude slower than the best estimates of the desorption rate into air. Interestingly, the values for the desorption rate constant into water change by less than 1 part in 1000 if the desorption rate constant into air is considered infinite (no surface resistance), demonstrating that the mass transfer resistance through a wetted membrane occurs predominantly through the dynamic limitations at the wetted surface. These simple experiments demonstrate that wetted membranes can present a much stronger resistance to gas permeation than classic diffusion theory suggests and help explain deviations from predicted dynamic responses in real systems.6 Conclusions Gas transfer through wetted membranes is an important process in many applied engineering, chemical, and biological systems, and it is demonstrated here that kinetic limitations at the surface of a wetted membrane can greatly diminish the rate of gas permeation. This has important implications in the design of systems where gas transfer across gas permeable membranes
Desorption Limitations for Gas Permeation is necessary, including the fact that mechanical strength and durability may be gained with only a nominal sacrifice of dynamic performance by increasing membrane thickness. Also, containment of the solvent by surface forces behind a porous membrane14 may yield improvements in dynamic response when significant kinetic limitations exist at the surface between the solvent and the membrane. Acknowledgment. I thank Charles Nelson for building the membrane housings and USDA-TSTAR agreement 2003-3413513980 for providing funds for this research. References and Notes (1) Tyrrell, H. J. V. J. Chem. Educ. 1964, 41, 397-400. (2) Pauly, S. Permeability and Diffusion Data. In Polymer Handbook, 4th ed.; Brandup, J., Immergut, E. H., Grulke, E. A., Eds.; Wiley: New York, 1999; pp VI/543-VI/569. (3) Diffusion in Polymers; Crank, J., Park, G. S., Eds.; Academic: London, 1968; pp 5, 15-16.
J. Phys. Chem. B, Vol. 108, No. 50, 2004 19329 (4) Long, L. J. Chem. Educ. 1944, 21, 139-141. (5) Hodgkin, A. L.; Katz, B. J. Physiol. 1949, 108, 37-77. (6) Jenkins, D. M.; Krishnan, A. Improving Dynamic Performance and Specificity in Immersible Dissolved Gas Biosensors. ASAE International Meeting, Las Vegas, July 27-30, 2003, Paper 037001. (7) Noyes, R. M.; Rubin, M. B.; Bowers, P. G. J. Phys. Chem. 1996, 100, 4167-4172. (8) Bowers, P. G.; Rubin, M. B.; Noyes, R. M.; Andueza, D. J. Chem. Educ. 1997, 74, 1455-1458. (9) Welty, J. R.; Wicks, C. E.; Wilson, R. E. Fundamentals of Momentum, Heat, and Mass Transfer, 3rd ed.; Wiley: New York, 1984; pp 254-256. (10) Atkins, P. W. Physical Chemistry, 4th Ed.; Freeman: New York, 1990; p 733. (11) C¸ engel, Y. A.; Boles, M. A. Thermodynamics: An Engineering Approach; McGraw-Hill: New York, 1989; pp 282-283. (12) Moore, J. M.; Pearson, R. G. Kinetics and Mechanism, 3rd ed.; Wiley: New York, 1981; pp 70-71. (13) Gevantman, L. H. Solubility of Selected Gases in Water. In CRC Handbook of Chemistry and Physics, 74th Ed.; Lide, David, Ed.; CRC, Boca Raton, 1993; p 6-4. (14) Jenkins, D. M.; Delwiche, M. J. Biosens. Bioelectron. 2003, 18, 1085-1093.
