J. Phys. Chem. B 2008, 112, 3693-3704
3693
Non-Fickian Water Vapor Sorption Dynamics by Nafion Membranes M. Barclay Satterfield† and J. B. Benziger* Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed: October 25, 2007; In Final Form: December 20, 2007
Dynamics of water absorption from a saturated vapor and water desorption into dry air for Nafion 1100 EW ionomers have been measured for film thicknesses between 51 and 606 µm and at temperatures ranging from 30 to 90 °C. Water absorption and desorption exhibit two distinct non-Fickian characteristics: (1) desorption is 10 times faster than absorption and (2) the normalized mass change does not collapse to a single master curve when plotted against time normalized by membrane thickness squared, t/l 2, for either absorption or desorption. Water desorption data were fit well by a model in which diffusion is rapid and interfacial mass transport resistance is the rate-limiting process for water loss. Water absorption is described by a two-stage process. At early times, interfacial mass transport controls water absorption, and at longer times, water absorption is controlled by the dynamics of polymer swelling and relaxation.
Introduction Nafion is a sulfonated perfluoro-ionomer commonly used as an electrolyte in polymer electrolyte membrane (PEM) fuel cells. Its proton conductivity is strongly dependent on its water content.1-3 Events such as start-up and shut-down, changes in circuit load, and environmental changes affect the balance of water production and removal, which can alter the water concentration inside the fuel cell and the membrane. Step changes in operating conditions have been shown to cause dramatic nonlinear dynamic responses on a range of time scales from seconds to hours,4,5 which are associated with water absorption and desorption from the Nafion electrolyte membrane. Water sorption and transport in Nafion has been examined frequently over the past 30 years.6-16 Experiments have generally focused on determining diffusion coefficients via dynamic water uptake and/or loss,6,9-14 steady-state permeation,8,11 and NMR diffusion measurements.16 Majsztrik et al.17 provides an excellent summary of reported diffusion coefficients and trends. The mass gain by absorption is commonly employed to evaluate the diffusion coefficient of solvents in polymers.18 Assuming that the rate-limiting step for water absorption is Fickian diffusion with a constant diffusivity, the diffusion equation (eq 1) can be solved for the mass gain as a function of time.
∂c ∂2 c )D 2 ∂t ∂x c ) co for all x at t ) 0 c ) c∞ at x ) ( l /2
(1)
Almost every investigator has adopted a boundary condition in which the water activity is constant across the membrane/vapor interface, corresponding to the concentration at the membrane surface being in equilibrium with the vapor (c∞); this boundary * Corresponding author. E-mail:
[email protected]. † Present address: Science and Technology Fellow, American Chemical Society, Washington, D.C.
condition is shown in eq 1. The normalized mass change as a function of time is found by solving eq 1 with constant diffusivity; the solution is given by eq 2.
Mt - M0 M∞ - M 0
)1-
8
m)∞
π2
m)0
∑
1 (2m + 1)2
{
-D(2m + 1)2 π2
exp
4l 2
t
}
(2)
Experimental data for the mass uptake as a function of time can be fit to eq 2, from which the diffusion coefficient, D, may be evaluated. Two defining characteristics of diffusion-controlled water uptake with constant diffusivity are (1) that the absorption and desorption curves should collapse to a universal function of the time normalized membrane thickness squared, t/l 2 19 and (2) that the absorption and desorption curves should be symmetric. Many investigators have measured water uptake into Nafion films and evaluated diffusion coefficients based on the assumption of Fickian diffusion.6,10-12 However, some of these same reports have noticed distinct non-Fickian behavior, including diffusion coefficients that depend on membrane thickness,6,8,9,11 diffusion coefficients that depend on concentration,6,10,11,16 diffusion coefficients that are different for water sorption from liquid or vapor,8,11,16,20 and diffusion coefficients that are different for absorption or desorption.10-12,21 Burnett et al.6 and Morris and Sun10 measured water absorption and concluded that the diffusion coefficient increased with membrane water content, up to a maximum at an intermediate hydration level and then decreased. Zawodzinski16 reported similar findings for diffusion measured through NMR experiments. Rivin11 reported similar behavior for water absorption and desorption experiments, but permeation experiments indicated that the diffusion coefficient increased steadily with increasing water concentration. Ge et al.8 also found that permeation increased with water activity and suggested that a decease in diffusivity at higher water contents was offset by an increase in the rate of interfacial mass transport from the vapor to the membrane. The concentration-dependent maximum in D was invoked to explain the faster desorption rate,10 but, as Rivin11 points out, desorption remains faster than absorption over the entire range of water concentrations.
10.1021/jp7103243 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/04/2008
3694 J. Phys. Chem. B, Vol. 112, No. 12, 2008
Satterfield and Benziger
Figure 1. Experimental apparatus for mass uptake measurements. The photo to the right shows the balance with the humidity- and temperaturecontrolled chamber positioned below. The photo to the right is a close up of the humidity- and temperature-controlled chamber.
Several researchers have suggested that interfacial mass transport may be responsible for the differences between vapor and liquid water uptake,8,11,16,20 sorption and desorption,9,11 and varying membrane thickness.8,9,11 Ge8 noted the importance of interfacial mass transfer in permeation experiments and modeled results to yield interfacial mass transport rates k dependent on the volume fraction of water fV. They attributed the larger desorption rate to the hydrophobic surface slowing condensation during absorption. Permeation experiments have been performed in our lab with Nafion membranes of different thicknesses that clearly showed that the permeation rate did not scale with membrane thickness, as would be the case for diffusion-controlled transport.17,22 The permeation results suggested that interfacial mass transport across the membrane/vapor interface was the rate-limiting step for thin membranes. Krtil et al.9 noted that a mechanism controlled by diffusion and interfacial mass transport predicts that the mass uptake curve would be a sigmoidal curve, which was reported by Rivin.11 To explain the difference observed between sorption and desorption, Krtil adopted a model using a reversible water immobilization reaction, a bulk mechanism, which accounted for their observed thickness-independent sorption rates. They attributed the faster desorption reaction rate to the expanded pores of the swollen membrane.9 Several other researchers have suggested that structural changes and polymer chain rearrangement mechanisms might be important to water uptake and loss dynamics.9-12 However, there are no quantitative models that have examined the role of polymer dynamics in water absorption. We recently proposed that water sorption and transport may be controlled by interfacial mass transport and polymer relaxation dynamics.17 We present here more detailed studies that explore the roles of temperature and membrane thickness on water absorption and desorption rates in Nafion membranes. We show from dimensional scaling analysis that interfacial transport and polymer relaxation dynamics control water absorption and desorption for thin Nafion films. Experimental Section Extruded Nafion 1100 equiv weight films of 50.8, 127, and 254 µm (DuPont product) were obtained from Ion Power (New
Castle, Del), and a 606 µm film was recast from the Nafion solution Liquion (Ion Power product). We also prepared 127 µm films by recasting from solution. The recast films were annealed to 140 °C for 2 h. All films were cleaned and ionexchanged by boiling for 1 h in 3% hydrogen peroxide in water, soaked for 20 min in deionized water, boiled for 1 h in 1 M sulfuric acid, and last soaked for 20 min in DI water. All membranes were stored in a sealed container at room temperature and 100% relative humidity. Water absorption and desorption experiments were carried out with membranes suspended in a controlled atmosphere container shown in Figure 1. Membranes ∼1 × 3.5 cm2 were hung on a hook into the container from a bottom-weighing balance, Ohaus AR0640, accurate to 10-4 g. The humidity container was a stainless-steel vessel, 15 cm tall and 6 cm in diameter, filled with water to a height just below the membrane. The lid was stainless-steel, 13 mm in thickness with a slot allowing it to slide easily into place around the membrane’s hook. The vessel was heated with heating tape. The lid was heated to with a cartridge heater to ∼2 °C above the temperature of the vessel to prevent condensation. Even with the heated lid, liquid condensation on the support wire was a problem; a heat lamp illuminated the support wire, which prevented condensation effectively. We assumed that 100% relative humidity in the chamber was maintained. The drying chamber consisted of an Erlenmeyer flask partially filled with Drierite and heated with heating tape. The humidity and temperature in the environmental containers were checked offline with combination humidity and temperature sensors (Sensirion Model SHT75). The humidity was >95% at the temperature of the humidified container and 0.1 cm/s, which is much larger than the values measured in the experiments reported here, kint < 0.001 cm/s. Hence, we conclude that the primary interfacial mass transport resistance is not the result of a gas-phase boundary layer. Permeation studies from Majsztrik et al. were able to extrapolate to infinite gas flow where the gas boundary layer was eliminated and showed that an interfacial mass transfer resistance existed associated with the interface of the Nafion membrane.17,22 Nafion is known to microphase separate into hydrophobic (teflonic) domains and hydrophilic domains. When exposed to a vapor the low-energy hydrophobic domains will phase separate to the membrane/vapor interface (similar to a layer of oil on the surface of water). Water molecules must pass through this teflonic layer at the vapor/membrane interface during both absorption and desorption. We suggest that the interfacial transport coefficient we report here corresponds to this membrane surface interfacial resistance. The Biot number signals the relative importance of interfacial mass transport to diffusion. We have evaluated the Bi numbers for water desorption based on the parameters for the interfacial mass transport coefficient in Table 1 and the temperaturedependent diffusion coefficient reported by Takamatsu,12D ) 8 × 10-3 exp(-20.5kJ/RT). For thin membranes at low temperatures, Bi ,1 (0.06 for Nafion 112, l ) 51 µm, at 30 °C). This is clearly in the region where interfacial transport is the rate-limiting step to water absorption or desorption. The thickest membranes and highest temperatures studied showed Bi ∼ 1 (1.85 at 90 °C for Nafion 1123, l ) 606 µm), meaning that diffusion is comparable to interfacial mass transport so that the analysis for determining the interfacial transport coefficient will be an underestimate. When analyzing water desorption data at higher temperatures and with thicker membranes, the functional form given by eq 4 should be applied because it combines interfacial transport resistance and diffusion effects. However, for membranes e127 µm thick at temperatures relevant to fuel cell operation (e90 °C), the Biot number is small; interfacial mass transport is the
3700 J. Phys. Chem. B, Vol. 112, No. 12, 2008
Satterfield and Benziger
Figure 11. Comparing the rates for water absorption and desorption at 50 °C for different thickness Nafion membranes.
rate-controlling process for water desorption, and the concentration of water inside the membrane will be nearly uniform. Water Absorption Water absorption is an order of magnitude slower than water desorption. If both absorption and desorption were limited by only diffusion or interfacial mass transport, then the two rates would be the same. Furthermore, if absorption/desorption were an equilibrium-limited process then the rate constant for absorption would be greater than that for desorption. (Kequilibrium ) kads/kdes > 1 when absorption is thermodynamically favored.) We conclude that the rate of water absorption into Nafion films is not controlled entirely by transport (diffusion or interfacial transport) or by thermodynamics. The unusual result is that the rate of absorption shows universal scaling with time normalized by the membrane thickness, the same as seen for desorption. Figure 11 is a comparison of the rate of water absorption and water desorption scaled at time normalized by membrane thickness; the slope of these curves is equal to the rate constant (kint for desorption). At short times, water desorption and absorption rates appear to be the same. The slopes of the absorption and desorption lines are the same for ln(1 - ∆M(t)/ ∆Mmax) > -0.45 in Figure 11, which corresponds to ∆M(t)/ ∆Mmax < 0.35. For larger mass changes, desorption rates stay the same while absorption rates decrease. Water desorption data can be fit well by eq 6 with a single interfacial transport coefficient, kint. The fit to the water absorption data with a single rate parameter is poor, but the data can be fit very well by introducing a two-step process (eq 7). Uptake for the initial 35% of water absorption is described by the same interfacial transport rate constant determined from desorption data. For water uptake >35%, a second slower rate constant must be invoked, giving the two-term expression for water absorption, where φ ≈ 0.35.
{
[ ]}
kintt Mt - M 0 ) φ 1 - exp M∞ - M 0 l
+
{
[ βtl ]} (7)
(1 - φ) 1 - exp -
This two-stage solvent absorption has been reported in other polymer systems, and the parameter φ is referred to as the “quasi-equilibrium” point in those models.20,24-26 The two rate constants are largely separated in time, so the first mechanism
predominates at short time and goes to equilibrium before the next mechanism takes over for larger values of φ at longer times. We posit that after an initial amount of water uptake the rate of water sorption becomes controlled by polymer chain rearrangement and relaxation associated with polymer swelling. The molecular details of the structural changes accompanying water absorption are not fully understood, but we can obtain a semiquantitative model of this phenomena based on the cluster model of Gierke.27 Water absorption expands the hydrophilic domains that stretch the “teflonic” continuum, creating stress that subsequently relaxes by viscous flow of the teflonic continuum. The stress induced by water absorption increases with the amount of water absorbed. The swelling stress (or swelling pressure) and the subsequent stress relaxation has been measured as described by us previously.28 Water absorption, which solvates the sulfonic acid groups, is exothermic. The swelling of the polymer matrix is endothermic. Water absorption occurs only when the exothermic solvation energy is greater than the endothermic swelling energy. At early times of water uptake, the energy of water absorption is greater than the energy required for elastic deformation and water uptake is limited only by interfacial transport Water absorption will continue as long as the solvation energy is greater than the energy of elastic deformation. At longer times, the stress of swelling is reduced by viscous flow of the polymer. The reduced stress after polymer relaxation permits more water to absorb. The model for water uptake presented in eq 7 relies on three parameters, the interfacial mass transfer coefficient, kint, the relaxation rate, β, and the quasi-equilibrium coefficient, φ. We have fit the experimental data using the interfacial mass-transport coefficients, kint, calculated from desorption data and fixing φ ) 0.35, so the only fitting parameter is β. Examples of the data fits of water absorption curves at 80 °C appear in Figure 12. The values of β range from 10-2 to 1 cm/s, increasing with temperature; these rate coefficients are 10-100 times larger than the interfacial mass transport coefficients, which is why absorption is much slower than desorption. We associate the rate constant β with the relaxation of the stresses produced from the membrane swelling due to water absorption. Because of Nafion’s structure, it is subject to a range of relaxation frequencies, and the single frequency used in this
Non-Fickian Water Vapor Sorption Dynamics
J. Phys. Chem. B, Vol. 112, No. 12, 2008 3701
Figure 12. Modified two-stage sorption model fit to Nafion water vapor uptake curves at 80 °C.
model is a significant simplification. The increase of β with temperature is consistent with faster stress relaxation at higher temperature. We have successfully fit uniaxial stress relaxation of Nafion with three relaxation times, which average ∼3000, ∼200, and ∼10 s.29,30 The shorter relaxation times are comparable to or less than the time for interfacial transport (tint ) lmembrane/kint > 100 s). The average stress relaxation time was observed to increase with increasing membrane water content.29 The increase in relaxation time with water content creates a situation in which the faster relaxation frequencies fade during the early sorption and long-term sorption is controlled by the slower frequencies. The quasi-equilibrium uptake is frequently considered to be equivalent to an elastic response of the membrane while the second-stage rate is relaxation-controlled.20,24-26 Newns equated the osmotic stress to the change in volume at quasi-equilibrium uptake multiplied by the bulk modulus.24 In the Nafion water system examined here the change in volume at quasi-equilibrium (taken as 0.35 of equilibrium uptake) is roughly 10%, which, assuming isotropic swelling, imposes a strain of 3-4% in each dimension. This strain is in the vicinity of the proportional limit measured from independent tensile tests: 2-3%, a good indication that after the quasi-equilibrium point the change in volume is no longer an elastic response.29 Long and Richman31 developed a similar model to account for stress relaxation. They also introduced the concept of twostage water uptake by designating a critical concentration below which there was no stress relaxation in the polymer and above which the stress relaxation was first-order. The primary difference between this model and theirs is that they only included diffusion and did not consider vapor/membrane interfacial transport. They did not realize that the boundary condition they wrote removed any spatial dependence of the diffusion equation, and so it was functionally equivalent to eq 7. The abrupt change in the rate-controlling process at φ ∼ 0.35, as well as the rate of water absorption can be accounted for in a semiquantitative model that includes the time-dependent energy of swelling. The total energy change due to water sorption, Esorption, is the sum of the chemical energy of water solvating the sulfonic acid groups and the mechanical energy of swelling the membrane, as expressed in eq 8. The first term
in the integral is the heat of solvation of the ionic groups, and the second term is the work of polymer swelling. The solvation energy decreases with water content; we have approximated the solvation energy with a simple linear function of the water content per sulfonic acid residue, λ. The initial heat of water solvation of the sulfonic acid groups, ∆Ho, is approximately -10 kJ/mol, and the energy of solvation falls to zero at the maximum water uptake λmax ) 15 water molecules/sulfonic acid group. The second term on the right-hand side of eq 8 represents the work from straining the polymer due to swelling. For a purely elastic material the strain energy would be the product of the elastic modulus, Eo, and the change in volume. It has been assumed that the volume of swelling is linear with the molar volume of water, Vw. Previous measurements showed the elastic modulus of Nafion decreases with relative humidity at constant temperature.28 Here, we approximate the elastic modulus as decreasing linearly with water activity from Eo at aw ) 0 to REo at aw ) 1. The elastic modulus of dry Nafion is also a function of temperature; at room temperature Eo ) 300 MPa and REo ) 100 MPa at aw ) 1. Nafion is viscoelastic, so the strain energy is time-dependent. Figure 13 contains data for the stress in Nafion due to a fixed strain of 3% applied in 0.5 s with varying water content.32 When the Nafion is first strained, the stress increases to approximately 150-300 MPa. Higher stresses were observed for low water content. When straining is stopped and held fixed, the stress relaxes. The time constant for stress relaxation is found from the slope of the modulus versus time in Figure 13. The stress relaxation time constant increases from 800 s for dry Nafion to 10 000 s for Nafion equilibrated at 100% relative humidity. As the water content in the membrane increases during absorption, there will be a decreasing rate of stress relaxation. For simplicity, we assume a constant stress relaxation time constant of 3000 s. The elastic modulus decrease with relative humidity is also evident in Figure 13; the stress required to strain Nafion to 3% decreases with relative humidity. The final assumption in eq 8 is that there is no time dependence for the heat of solvation.
3702 J. Phys. Chem. B, Vol. 112, No. 12, 2008
Satterfield and Benziger
Figure 13. Stress relaxation of Nafion 115 at different water activities to a fixed strain of 3%. The sample was strained to 3% in 0.5 s and then held at that fixed strain. The relative humidity of the vapor was controlled by use of various salt solutions (from ref 30).
Esorption(λ) )
∫λλ o
( ( -∆Ho
)
λmax - λ + λmax
(
h w exp EoV
))
)(
t λ 1-R dλ (8) 3000 λmax
Figure 14 illustrates the contributions to the energy of water absorption based on eq 8. The solvation energy decreases with increasing water uptake, while the elastic energy increases with water uptake. The total energy goes through a minimum, and that minimum is a function of time. The equilibrium water content, λe, is expressed in eq 9.
t 3000 λmax λe ) t -∆Ho + R EoV h w exp 3000
(
-∆Ho + EoV h w exp -
(
)
)
(9)
Equation 9 predicts an instantaneous water uptake to the quasiequilibrium point given by eq 10, followed by an exponential approach to the maximum water uptake.
φ)
-∆Ho + EoV hw λ(t ) 0) ) λmax -∆Ho + R EoV hw
(10)
The effect of stress relaxation on water uptake is illustrated graphically in Figure 14. Initial water absorption will occur spontaneously, limited only by mass transport from the vapor into the Nafion membrane. Because interfacial mass transport is fast relative to stress relaxation, the elastic energy contribution will be approximately at the time ) 0 limit. The total energy of water absorption at short times is the sum of the elastic energy at t ) 0 and the solvation energy. Using the experimental values presented above in eqs 9 and 10, we find that the minimum in the energy at t ) 0 is λ ) 5.5, corresponding to φ ) 0.36. This zero time limit is the quasi-equilibrium point identified in eq 7. Water absorption would stop at this point if not for stress relaxation of the Nafion. As the polymer relaxes, more water can be associated with the sulfonic acid groups with the solvation energy overcoming the elastic deformation energy. The water uptake beyond the quasiequilibrium point as a
function of time is given by eq 9. This closely corresponds to the second term in eq 7. In summary, the minimum in the total energy at time ) 0 corresponds to the first stage of water absorption, φ ) 0.36. The stress from elastic deformation associated with water absorption does not have time to relax, and the rate of water absorption is controlled by interfacial mass transport. Water uptake beyond the first stage is then limited by stress relaxation, which shifts the minimum in total energy to higher water content; this gives rise to the longer time constant in the second exponential term in eq 7. The model presented here does not explain why the second stage of water uptake, controlled by stress relaxation, scales with the membrane thickness. We speculate that water absorption scales with time normalized by membrane thickness because the swelling and relaxation occurs with a front that moves from the membrane surface into the membrane.17 Water absorption in the second stage would still be limited to interfacial mass transport, but this time the interface would be moving. This mechanism would be similar to CaseII diffusion, though we do not see other characteristics of CaseII diffusion in the shape of water absorption curves. The model presented here shows that the rate of water absorption will increase with temperature because the stress relaxation rate increases. Equation 10 for the model also suggests that the quasi-equilibrium point will shift to higher values of λ, (larger φ) with increasing temperature. By fitting the experimental data with eq 7, we were unable to get a significant improvement in the fit by varying both the quasi-equilibrium concentration and the relaxation time constant, β. Comparison with other Non-Fickian Mechanisms NonFickian sorption behavior has been reported for many polymer systems.19,20,24-26,31,33 Solvent absorption tends to be controlled by Fickian diffusion in rubbery polymers, but absorption is generally not described properly by Fickian behavior in glassy polymers because the solvent plasticizes a glassy polymer.19,25 Non-Fickian behaviors are characterized by sorption and desorption curves that are not symmetric, sorption curves for different thicknesses that do not collapse to a single curve with t/l 2, and sorption curves that display an inflection point when plotted against either time or t1/2.25,34,35
Non-Fickian Water Vapor Sorption Dynamics
J. Phys. Chem. B, Vol. 112, No. 12, 2008 3703
Figure 14. Energetics of water absorption. The solvation energy is the energy for solvating the sulfonic acid groups and is time-independent. The elastic energy is the stress induced by swelling to take up water. The stress relaxes with a time constant τrelax. As the polymer stress relaxes, more water can be absorbed.
Typical non-Fickian sorption mechanisms are usually categorized as two-stage, Case-II, or super Case-II.19,35,36 In CaseII-type behaviors, the solvent profile in the absorbing polymer is discontinuous and appears as a propagating front into the dry polymer, different from the continuous parabolic concentration profile predicted by Fick’s law. This class of behaviors produces sorption curves that are linear when plotted against tn, where n ) 1 for Case II (linear with time), n > 1 for super-Case II (curves upward with time), or 0.5 < n < 1 for behavior between Fickian (or Case I in which n ) 0.5) and Case II. None of the Case-II characteristic behaviors are exhibited in the Nafionwater system, which maintains a downward curvature when plotted as functions of t or t1/2. Two-stage absorption, which is proposed here, is distinct from Case II in that it is not necessarily characterized by a sharp advancing front. Instead, there is an initial diffusion-controlled solvent uptake up to a quasi-equilibrium concentration followed by stress-relaxation as the polymer chains rearrange to accommodate more solvent.19,24-26,31,37 In the second stage, diffusion is fast compared to relaxation, so the concentration profile through the sample is essentially flat.31 One of the earliest treatments of this behavior came from Crank and Park33 who proposed and solved a time-dependent boundary condition similar to that proposed later by Long and Richman.31 Crank and Park noted that desorption can be faster than absorption because chain rearrangement and relaxation is not necessary for penetrant to leave the polymer. Deborah and Biot Numbers The Deborah number, De ) 4D/βl 2, compares the characteristic time for relaxation, 1/β, to 4D/l 2, the time for diffusion.19,25,31,38 In earlier models that have considered polymer stress-relaxation, the Deborah number was used to predict the behavior.31,39 For De , 1 or β . l 2/4D, any changes in the polymer are fast compared to diffusion, and water uptake is diffusion-controlled. For De . 1 or β , l 2/D, the uptake resembles a diffusing solvent front, moving from a rubbery medium into a glassy medium. For a Deborah number on the order of 1, β ∼ l 2/D, polymer relaxation is on a time scale relevant to diffusion and leads to classic two-stage sorption described by Long and Richman.31 However, in the Nafion-water vapor system the Deborah number is .1 and behavior is not diffusion-controlled. Because
the Biot number is
