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Journal of Membrane Science 301 (2007) 93–106

Water sorption, desorption and transport in Nafion membranes

Paul W. Majsztrik b, M. Barclay Satterfield a, Andrew B. Bocarsly b, Jay B. Benziger a,∗a Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, United States

b Chemistry Department, Princeton University, Princeton, NJ 08544, United States

Received 6 February 2007; received in revised form 15 May 2007; accepted 10 June 2007Available online 16 June 2007

bstract

Water sorption, desorption, and permeation in and through Nafion 112, 115, 1110 and 1123 membranes were measured as functions of temperatureetween 30 and 90 ◦C. Water permeation increased with temperature. Water permeation from liquid water increased with the water activity differencecross the membrane. Water permeation from humidified gas into dry nitrogen went through a maximum with the water activity difference acrosshe membrane. These results suggested that the membrane was less swollen in the presence of water vapor and that a thin skin formed on the dryide of the membrane that reduced permeability to water. Permeation was only weakly dependent on membrane thickness; results indicated thatnterfacial mass transport at the membrane/gas interface was the limiting resistance. The diffusivity of water in Nafion deduced from water sorptionnto a dry Nafion film was almost two orders of magnitude slower than the diffusivity determined from permeation experiments. The rate of waterorption did not scale with the membrane thickness as predicted by a Fickian diffusion analysis. The results indicated that water sorption wasimited by the rate of swelling of the Nafion. Water desorption from a water saturated film was an order of magnitude faster than water sorption.

ater desorption appeared to be limited by the rate of interfacial transport across the membrane/gas interface. The analysis of water permeation andorption data identifies different regimes of water transport and sorption in Nafion membranes corresponding to diffusion through the membrane,nterfacial transport across the membrane–gas interface and swelling of the polymer to accommodate water.

2007 Elsevier B.V. All rights reserved.

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eywords: Water sorption; Water permeation; Nafion; Diffusion; Interfacial ma

. Introduction

The sorption and transport of water in polymer membraneslays an essential role in the operation of polymer electrolyteembrane (PEM) fuel cells [1–6]. Water is frequently intro-

uced in the feed stream to the fuel cell and water is producedt the cathode. The sorption and diffusion of water in the mem-rane will determine the distribution of water throughout theuel cell, and in turn affect the local proton conductivity. Nafions the most common polymer membrane material used in PEMuel cells. Water transport in Nafion has been the subject ofumerous investigations over the past 40 years [3,6–24]. Despitehe numerous studies of water sorption and transport in Nafion

here are significant differences in the values for the diffusivityf water in Nafion, determined by various methods.

∗ Corresponding author. Tel.: +1 609 258 5416; fax: +1 609 258 0211.E-mail address: benziger@princeton.edu (J.B. Benziger).

376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.memsci.2007.06.022

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Water sorption and transport in Nafion has been studied byeveral different methods and differences in the diffusivity ofhree orders of magnitude have been reported. Three basic meth-ds have been used to measure water sorption and transport.

1) Mass uptake [3,8,12,13,16–18,22,25,26]. The mass uptakeof water by a polymer film is measured as a function oftime. An effective rate constant for mass gain is determined.Assuming that the water sorption is controlled by Fickiandiffusion, the sorption diffusivity, Dsorption, is determinedby the product of the rate constant and the square of thecharacteristic dimension of the polymer (film thickness, Lm)as given by the following equation:

Dsorption = kuptakeL2m (1)

The rate of water loss by the polymer may also be measured

to determine the rate constant for desorption, which can beused to determine a desorption diffusivity, Ddesorption:

Ddesorption = kdesorptionL2m (2)

94 P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106

Table 1Water sorption and transport measurements in Nafion

Experimental technique Diffusivity result Diffusivity at 25 ◦C Reference

Water sorption from vapor 2 × 10−8 Takamatsu et al. [8]Water sorption from liquid 6.3 × 10−3 e−2400/T 1.8 × 10−6 Takamatsu et al. [8]Desorption 1.1 × 10−7 Takamatsu et al. [8]NMR 6.3 × 10−3 e−2400/T 1.8 × 10−6 Yeo and Eisenberg [7]Water sorption (0.1–1) × 10−7 increasing with � Morris and Sun [12]Desorption (0.25–4) × 10−7, increasing with � Morris and Sun [12]Water sorption 10−8 to 10−7 to 10−9 goes through max with � Legras et al. [19]NMR 2.5 × 10−6 Zawodzinski et al. [10]Streaming potential in liquid water 1.25 × 10−5 at 80 ◦C 6 × 10−6 Okada et al. [14]Permeation through an MEA 2.1 × 10−3λ e−2436/T 2.2 × 10−6 at λ = 4 Motupally et al. [15]Water sorption from vapor 5 × 10−8 at 60 ◦C Gates and Newman [31]NMR 10−7 at low λ, 10−6 at high λ, D increases with T Gong et al. [32]Water sorption 7.7 × 10−9 at 32 ◦C Rivin et al. [17]Desorption 2.6 × 10−8 at 32 ◦C Rivin et al. [17]Permeation 1.3 × 10−7 to 3.1 × 10−6 increasing with water activity Rivin et al. [17]Water sorption ksorption = 0.03–0.04 s−1 at λ = 2–3, 23 ◦C;

ksorption = 0.01–0.02 s−1 at λ = 8–10, 23 ◦CKrtil et al. [18]

DD 0%RH

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2

2

rbsipidtciRsac

esorption kdesorption = 5ksorption

esorption 5 × 10−7 at 40%RH, 2 × 10−6 at 1

2) NMR relaxation [10,16,23,27,28]. The relaxation timesfrom a pulsed gradient NMR experiment can be used todetermine the self-diffusion coefficient of water in Nafion.

3) Permeation experiments [16,17,20,29]. Steady-state waterpermeation through membranes has been measured. Assum-ing a linear concentration gradient across the membrane theintegrated permeation at different flow rates can be used todetermine the diffusivity of water.

Table 1 summarizes most of the measured diffusivities ofater in Nafion reported in the literature. As mentioned above

he values vary by more than three orders of magnitude. Someeneral conclusions that can be drawn from these measurementsre:

i. NMR self-diffusion measurements give the highest valuesof diffusivity with values of ∼10−6 cm2/s.

ii. Permeation measurements give slightly lower diffusivitiesthan NMR in the range of 10−7 cm2/s.

ii. Water uptake measurements give the lowest diffusion coef-ficients typical values are ∼10−8 cm2/s.

iv. Water desorption measurements give intermediate diffusioncoefficients of ∼10−7 cm2/s.

v. Water diffusivity increases with temperature.i. Water diffusivity increases with membrane water content at

low water content. Several studies have suggested the dif-fusivity goes through a maximum with water content anddecreases as the membrane becomes saturated with water[3,6,19,30].

As part of a broader effort to understand the mechanical and

ransport properties of polymer electrolytes employed in fuelells we re-examined water sorption and diffusion in Nafions a function of temperature and water activity using waterptake and permeation. Water uptake (sorption), water loss (des-

is

w

Krtil et al. [18]Damay and Klein [21]

rption) and water permeation were measured as functions ofhe membrane thickness, temperature and water activity gra-ient. Interfacial transport across the membrane/gas boundary,iffusion and membrane swelling all contribute significant resis-ances to water transport and sorption. The limiting resistance toater uptake and transport depends on the sample dimensions,

he temperature and water activity. We also present modificationso water uptake and loss analysis that accounts for the interfacial

ass transport and polymer swelling. The results and analysisresented here help to explain the range of values reported inhe literature.

. Experimental

.1. Water sorption/desorption

Dynamic water sorption/desorption experiments were car-ied out by hanging a Nafion sample from a bottom weighingalance into a temperature and humidity controlled containerhown in Fig. 1. For water sorption the membranes were driedn a vacuum oven at 70 ◦C for 2 h. The dry samples were sus-ended on a fine metal wire from a bottom weighing balancento a temperature-controlled stainless steel container (6 cmiameter × 16 cm tall) with a removable slotted top. The con-ainer was filled 1/3 with water, and the relative humidity in theontainer was checked periodically with a capacitive humid-ty sensor (Sensiron Model SHT75) and found to be 95–100%H at the temperature of interest. The top of the stainless

teel container was heated to 3–5 ◦C higher than the temper-ture in the container to avoid liquid condensation, which couldause liquid drops to condense and fall onto the sample, giv-

ng rise to large fluctuations in the weight measured by thecale.

Desorption experiments were carried out by replacing theater container with a container filled with desiccant. The rel-

P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106 95

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ig. 1. Water sorption/desorption apparatus. The sample membrane is sus-ended in the temperature-controlled stainless steel container.

tive humidity in the desiccant chamber was <2% RH at theemperature of interest.

Samples of Nafion 112 (51 �m thick), Nafion 115 (127 �mhick), Nafion 1110 (254 �m thick) and Nafion 1123 (600 �mhick) were tested. The Nafion 112, 115 and 1110 samples werextruded Nafion obtained from Ion Power. The Nafion 1123ample was cast from 5% Nafion solution (Ion Power), dried invacuum oven at 60 ◦C, annealed at 140 ◦C in a vacuum oven

or 2 h and then subjected to the standard treatment listed below.he dry sample thicknesses were all checked with a micrometer.amples were cut into strips with an area of ∼7 cm2. Each sam-le was initially boiled for 1 h in hydrogen peroxide solution,ransferred to boiling water for 20 min, then placed in boil-ng 1 M sulfuric acid for 1 h followed by boiling in de-ionizedater for 20 min and lastly dried at 70 ◦C for 2 h. The initialry weight of the sample was determined before each experi-ent using the same balance used for the sorption/desorption

xperiments. During sorption/desorption the samples were sus-ended half way down into the controlled temperature container.he gas phase in the container was stagnant. The temperatureas recorded by computer every second for a period of 4000–0,000 s.

.2. Water permeation

Water permeation through Nafion membranes was measuredn a custom built permeation cell shown schematically in Fig. 2.he cell was designed to be a one-dimensional differentialermeation cell. The flow pattern in the space above the mem-rane should have negligible lateral gradients, so transport isearly one dimension across the membrane. The driving forceor permeation is uniform, which makes the data easy to ana-yze.

The cell was made from two polycarbonate plates with

cm2 × 0.5 cm deep cavities. One plate had channels machinedlong two sides of the cavity fitted with stainless steel bars, whicherved as electrodes that permitted in situ resistance measure-ents. Nafion membranes were clamped between the two plates.

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ig. 2. Permeation cell for water through Nafion membranes. Mixing is goodn the gas cavities so the cell is one-dimensional.

umidity and temperature sensors (Sensirion Model SHT75)ere placed in tees at the outlets from both cavities. The gashase in the cavities is well mixed, so the compositions in theutlets are the same as the composition in the cavities.

The entire permeation cell was placed inside a temperature-ontrolled insulated box. Permeation was measured by flowingiquid water or humidified nitrogen through one cavity of theell (referred to as the wet side) and dry nitrogen through thether side (the dry side). A humidifier bomb was placed insidehe temperature-controlled box and nitrogen from a mass flowontroller was flowed through the humidifier. This humidifiedtream was mixed with dry nitrogen to achieve the desired rela-ive humidity. The flow rate of the humidified gas stream was seto between 2 and 3 L/min on the wet side of the cell. The relativeumidity at both the inlet and outlet of the humidified nitrogenas measured to verify that the RH was maintained constant toithin 2% RH at the wet side of the cell.Permeation was measured for Nafion 112, 115 and 1110

or temperatures 30–90 ◦C. The nitrogen flow was varied from0 to 1500 mL/min using a mass flow controller (Aalborg)nd the relative humidity was logged by computer. We waitedntil steady-state relative humidity values were obtained over a0 min period before changing the nitrogen flow rate.

The lateral membrane resistance was measured at 100 Hz byaking the membrane one leg of a voltage divider. We have

reviously shown that the ac impedance of Nafion membraneseasured in the lateral dimension is insensitive to frequency

bove 2 Hz [33]. A 1 Vrms driving voltage of 100 Hz was appliedcross the membrane in series with a fixed 1 k� resistor. A pairf ac voltmeters measured the voltage drop across both legs ofhe voltage divider, from which the membrane resistance coulde determined.

The one-dimensional permeation cell makes the analysis ofermeation rates much simpler than using a system with flowhannels where the driving force changes along the length of theow channel. The plenums on both sides of the membrane permit

ood mixing so there are no lateral gradients. Fig. 3 illustrates thearious mass transfer resistances. The total permeation is deter-ined from the nitrogen flow rate and outlet relative humidity

96 P.W. Majsztrik et al. / Journal of Memb

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bw

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3

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ig. 3. Schematic of simplified water concentration profile in the one-imensional permeation cell.

y a simple mass balance:

Permeation rate = Q′Pw

AmRT

= Q[(RH × P0w(T ))/100]

AmRTref[1 − (RH × P0w(T ))/100P]

mol

cm2 s(3)

At steady-state the permeation rate is equal to the mass trans-er rate across the membrane fluid interface at the wet side, theater diffusion rate across the membrane as well as the mass

ransfer across the membrane fluid interface at the dry side. Foronvenience an overall mass transfer coefficient, or permeabil-ty, is defined based on the water activity at the wet and dry sidesf the membrane:

ermeation rate = kL(aLw − amL

w ) = c0wD

Lm(amL

w − amgw )

= kg(amgw − ag

w) = ko(aLw − ag

w) (4)

For simplicity the mass transfer resistance at the wet side ofhe membrane will be neglected. Since all the experiments wereun with a constant water activity at the liquid side (either liq-id water or fixed relative humidity) this will at most contributeconstant systematic offset to the determination of the diffu-

ivity and mass transfer resistance at the dry side. All the ratesave been expressed in terms of activity of water. The diffusioncross the membrane is based on simple Fickian diffusion withdiffusivity that is concentration independent.

There is a concentration jump in water between the mem-rane and liquid or gas; c0

w is the concentration of water in theembrane equilibrated with water activity of unity. The diffu-

ion resistance is always coupled with water concentration inhe membrane at unit activity. Nafion is unusual in that the equi-ibrium water concentration is multi-valued when equilibratedith liquid water or saturated vapor (this is the phenomenon ofchroeder’s paradox that has been discussed by several different

nvestigators [34–36]). We will assume c0w is single valued for

ur analysis, but Eq. (4) anticipates that there are differences inhe permeation of water between liquid water and humidifiedas because of the difference in equilibrium water content of

afion equilibrated with liquid versus with vapor.The overall mass transfer coefficient, ko, or permeability, can

e expressed in terms of resistances in series—the membraneater diffusivity and the interfacial mass transfer coefficient.

tf

p

rane Science 301 (2007) 93–106

he interfacial resistance we refer to is the transport of waterrom outside the membrane to inside the membrane:

o = (c0wD/Lm)kg

(c0wD/Lm) + kg

(5)

To distinguish between the two resistances, interfacial trans-ort (kg) and cross-membrane diffusion (cD/Lm), experimentsere carried out with different membrane thicknesses and dif-

erent nitrogen flow rates on the dry side of the membrane. Therere two contributions to the mass transfer coefficient; a gas phaseoundary layer whose resistance decreases with increasing nitro-en flow rate on the gas side, and an interfacial resistance that isersistent at all gas flow rates. The Fickian diffusive resistanceo mass transport is independent of the gas flow rate and willcale with the membrane thickness.

. Results

.1. Water sorption/desorption

A typical set of water uptake (sorption) experiments for aafion 1110 membrane as a function of temperature are sum-arized in Fig. 4 (only a limited number of temperatures were

hown for the sake of clarity). Mass uptake is normalized to thenal mass uptake at very long time (∼25,000 s). The rate of waterptake increases with temperature. Results for other membranehickness show similar trends. Occasionally large mass fluctu-tions were observed as the sample approached its final mass;hese fluctuations resulted from liquid drop condensation on theample (or sample support).

The water uptake data was fitted to various diffusion mod-ls. Diffusion into a slab with constant diffusivity is the mostommonly applied analysis (see Takamatsu et al. [8] and Morrisnd Sun [12]); the slope of ln{(M(t) − M0)/(M∞ − M0)} versusime gives an effective rate constant for diffusion, ksorption. Thets are not very good—the values of the diffusivity changes by aactor 3–4 over time, even when the long time approximation isalid. The best data fits are obtained using a growth model withWeibull relaxation parameter. The Weibull model introducesstretched exponential, where ζ is related to the visco-elastic

elaxation time for the polymer [37,38]:

M(t) − M0

M∞ − M0

}∼ exp[−(ksorptiont)

ς] (6)

Independent of a specific model for diffusion, the approxi-ate rate constant for water sorption can be extracted based on

he exponential increase in the mass. The simplest approxima-ion of this rate constant is the time to approach 1/e of the final

ass gain. This approximate rate constant is accurate to withinfactor of 2–3. The diffusivity of water can be obtained from

he effective rate constant for water sorption by use of Eq. (1);ssuming a constant diffusivity, independent of water content,

he water diffusivity in Nafion from the sorption data rangesrom 3 × 10−9 cm2/s at 30 ◦C to 1 × 10−7 cm2/s at 80 ◦C.

Fig. 5 shows the water uptake for different thickness sam-les at 60 ◦C. The obvious result is that the normalized water

P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106 97

F unction of temperature. The mass uptake is normalized by the final mass gain of thes

ucwntunwtTttatiatS

f

Fu

ig. 4. Water sorption in Nafion 1110 from saturated vapor (RH = 100%) as a fample.

ptake is slower for thicker samples. If the water sorption wasontrolled by diffusion, the time for water sorption should scaleith the sample thickness squared. Fig. 6A replots the data asormalized mass uptake as a function of t/(Lm)2; this normaliza-ion is a poor fit to the data. Fig. 6B plots the normalized massptake as a function of time normalized by the sample thick-ess, t/Lm. The water sorption data shows a good correlationith time normalized by membrane thickness which suggests

hat the rate of water sorption scales with the area of the sample.here are two physical models that are consistent with the sorp-

ion rate scaling with the area of the sample. One possibility ishat the gas/membrane interfacial mass transport is rate limitingnd diffusion in the membrane is fast, so the water concentra-ion is uniform across the membrane. The second possibilitys a shrinking-core model where water diffuses rapidly through

hydrated shell and the slow step is water incorporation into

he dry membrane core. These two models will be compared inection 4.

Fig. 7 shows the water desorption from Nafion 1110 as aunction of temperature. A comparison of the time for desorption

ig. 5. Water sorption into Nafion 112, 115, 1110, and 1123 at 60 ◦C. The massptake is normalized by the final mass gain of the sample.

Fig. 6. (A) Normalized water sorption as a function of time normalized by thesquare of the sample thickness for Nafion 112, 115, 1110, and 1123 at 60 ◦C.(B) Normalized water sorption as a function of time normalized by the samplethickness for Nafion 112, 115, 1110, and 1123 at 60 ◦C.

98 P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106

fion 1

sssc1

6nFttcfwmosoii

Fbt

3

iathwfiifitowt

Fig. 7. Water desorption from a saturated Na

hown in Fig. 7 and the time for adsorption shown in Fig. 4hows that desorption is an order of magnitude faster than waterorption at the same temperature. At 30 ◦C water desorption isompleted within 1000 s, whereas water sorption into Nafion110 took >10,000 s.

Water desorption from membranes of different thicknesses at0 ◦C is presented in Fig. 8. Water desorption is faster for thin-er membranes. The water desorption data has been replotted inig. 9 as the rate of mass loss versus the normalized water con-

ent in the membrane. The rate of mass loss was obtained fromhe slope of the mass versus time graph. The normalized waterontent is proportional to the number of waters absorbed per sul-onic acid group, λ. The mass loss appears to only depend on theaters absorbed per sulfonic acid group and is independent ofembrane thickness. This result suggests that the absolute rate

f water loss does not scale with total mass of the membrane, but

cales with external surface area of the membrane. Water des-rption, like water sorption, appears to be controlled by eithernterfacial mass transport or water liberation at a shrinking-corenterface.

ig. 8. Water desorption from saturated Nafion 112, 115, 1110 and 1123 mem-ranes at 60 ◦C. The mass loss is normalized to the total mass loss after longime.

asd

F1

110 membrane as a function of temperature.

.2. Water permeation measurements

Water permeation was measured between a high water activ-ty (either liquid water or humidified nitrogen) and a low waterctivity in nitrogen. The relative humidity in the effluent of bothhe wet side and dry side of the membrane was measured so weave a direct measure of the thermodynamic driving force forater permeation across the membrane. A typical set of data

or water permeation through a Nafion 115 membrane is shownn Fig. 10. The relative humidity in the effluent of the dry sides recorded as a function of time for fixed nitrogen flow rate,xed water activity at the wet side and fixed temperature. Once

he relative humidity at the dry side reached steady-state onef the fixed parameters was changed and the relative humidityas recorded until a new steady-state was obtained. Typically the

ime constant for responding to parameter changes was ∼10 min

t 21 ◦C to ∼1 min at 90 ◦C. We would wait at least 10 time con-tants after making a parameter change to record steady-stateata.

ig. 9. Rate of mass loss from water desorption from saturated Nafion 112, 115,110 and 1123 at 60 ◦C as a function of the water content in the membrane.

P.W. Majsztrik et al. / Journal of Memb

Fig. 10. Steady-state water permeation data for Nafion 115 as a function ofnlt

ipihiirtf

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3ast1

NIitmtm

bFpoatrtf

fivfmapwotio

ttfWms

fptm1

TP

T

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itrogen flow rate at the gas side at different temperatures. The liquid side wasiquid water (aw = 1). Two different membrane samples were tested at each ofhe four different temperatures.

Permeation rates were calculated based on the relative humid-ty, temperature and nitrogen flow rate using Eq. (3). Theermeation rate was plotted as a function of relative humid-ty at the dry side of the cell and extrapolated to zero relativeumidity, corresponding to infinite nitrogen flow rate, where thenterfacial mass transfer resistance at the membrane–gas sides minimized. Table 2 summarizes the extrapolated permeationate to zero RH at the dry side. The permeability increases withemperature and the permeation rate increases with temperatureaster for the thinner membrane.

Water permeation through Nafion from a humidified gastream was compared to permeation from liquid water. Theow rate of humidified nitrogen through the wet side of theermeation cell was sufficiently high (∼2–3 L/min) such thathe change in humidity between the inlet and outlet was negli-ible. Measurements were taken from humidified streams at 30,0 and 80% RH at 30, 50, 70, 80 and 90 ◦C. Dry nitrogen wasowed through the dry side of the cell at different flow rates.he water flux was determined using Eq. (3). To determine theermeability, ko, the water flux was normalized by the overallhermodynamic driving force, aL

w − agw.

Permeation rates for water through Nafion 112 and 1110 at0 and 90 ◦C are compared in Fig. 11. The fluxes increased by

pproximately an order of magnitude from 30 to 90 ◦C (note thecale on the y-axes for the two graphs). At 30 ◦C the water fluxhrough Nafion 112 is 1.2× larger than the water flux through110 at the high nitrogen flow rate. At 90 ◦C the flux through

mtwt

able 2ermeation rates of water through Nafion 112 and 1110

emperature (◦C) Nafion 115 extrapolatedpermeation rate to 0% RH(mol/cm2 s × 105)

Nafion 1110 extrpermeation rate t(mol/cm2 s × 105

0 2.0 1.60 3.4 2.60 5.4 4.12 9.2 6.22 11.6 7.3

rane Science 301 (2007) 93–106 99

afion 112 is 2× larger than the water flux through Nafion 1110.f permeation was limited by diffusion the flux should scalenversely with membrane thickness, so the water flux throughhe 112 membrane should be 5× of the flux through the 1110

embrane. Since the water flux varies much less with membranehickness than expected for diffusion limited process permeation

ust be limited by interfacial mass transport and not diffusion.Water permeation through Nafion 112, 115 and 1110 mem-

ranes from 80% RH into dry nitrogen at 80 ◦C is shown inig. 12. In contrast to permeation with liquid on the wet side theermeation rate from water vapor on the wet side does not levelut to a constant value at high nitrogen flow rates. The perme-tion rate increases initially at low nitrogen flow rates, then goeshrough a maximum and decreases with higher nitrogen flowates. The water activity on the dry side of the membrane con-inually decreases with increasing nitrogen flow so the drivingorce for water permeation increases.

The difference in permeation from liquid water and humidi-ed gas is shown explicitly in Fig. 13. The water flux is plottedersus the nitrogen flow rate for Nafion 115 and 1110 at 80 ◦Cor 80% humidified N2 and liquid water on the water side of theembrane. The difference in water permeation between liquid

nd vapor at the wet side of the membrane is remarkable. Theermeability (normalized flux) is more than a factor of 2 largerhen there is liquid water present. Furthermore, within the rangef flow rates we were able to explore, there was no fall off inhe permeation with nitrogen flow rate when liquid was present,n contrast to the fall off in the permeation seen when there wasnly vapor present.

The difference between permeation from a liquid comparedo permeation from a humidified gas is accentuated at higheremperatures; at 30 ◦C there was no evidence that the water fluxrom a humidified gas goes to zero at high nitrogen flow rates.

e suggest that the decrease in water flux is a result of theembrane completely drying out and forming an impermeable

kin at the dry side.The relative importance of interfacial mass transport to dif-

usional mass transport can be assessed by fitting overall massermeation coefficients, ko, at different membrane thicknesseso determine values of kg and c0

wD. Both kg and c0wD were deter-

ined from the water permeation data through Nafion 112 and110 given in Table 2 and their values are listed. Both interfacial

ass transport and diffusion increase with increasing tempera-

ure, but the interfacial mass transport coefficient increases fasterith temperature than the diffusivity, which results in a shift in

he limiting transport resistance with increasing temperature.

apolatedo 0% RH)

k∞g (mol/cm2 s × 105) c0

wD (mol/cm s × 107)

2.1 ± 0.8 18 ± 123.6 ± 1.3 24 ± 205.8 ± 2.2 35 ± 25

10.5 ± 2.8 38 ± 2713.6 ± 4.0 39 ± 31

100 P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106

Fig. 11. Water flux through Nafion 1110 and Nafion 112 at 30 and 90 ◦C as a functmembrane was liquid water.

Fig. 12. Water permeation through a Nafion 112, 115 and 1110 membranes asa function of the nitrogen flow rate at the gas side. The wet side is 80% RHhumidified gas at 80 ◦C.

Fig. 13. Water permeability through Nafion 115 and 1110 at 80 ◦C from liquidwater and 80% RH humidified nitrogen.

amhsdasmflaft

fraw

Fwtww

ion of nitrogen flow rate at the dry side of the membrane. The wet side of the

The water permeability at 80 ◦C through Nafion 112, 115nd 1110 as a function of the average water activity across theembrane, (aL

w + agw)/2, is plotted in Fig. 14. Different RH

umidities were employed at the wet side of the membrane topan the range of average water activities from 0.15 to 0.75. Theata show that permeation at 80 ◦C increases linearly with aver-ge membrane water activity. Closer examination of the datahows that ratio of permeation rates does not scale directly withembrane thickness. The ratio of the slopes of the normalizedux versus average water activity is 1:1.8:2.3 for 1110:115:112,nd the expected ratio based on membrane thickness is 1:2:5;rom which we conclude that even at 80 ◦C interfacial massransport is a significant resistance.

Fig. 15 is a plot of the in plane membrane resistance as a

unction of the average water activity across the membrane. Theesistance decreases with increasing average membrane waterctivity, as expected. To see if the resistance data was consistentith a linear water gradient across the membrane we calculated

ig. 14. Summary of normalized water flux as a function of average membraneater activity at 80 ◦C for Nafion 112, 115 and 1110. The data sets correspond

o different water activity on the high activity side of the membrane. The linesere added to show that normalized water flux scales with the average membraneater activity.

P.W. Majsztrik et al. / Journal of Memb

Fig. 15. Membrane resistivity as a function of average water activity in a Nafion115 membrane at 70 ◦C during the permeation experiments. The lines show thepredicted resistance assuming a uniform gradient across the membrane where theNc

tNfibcTwm

4

p

1

2

3

4

5

67

ppi

rg

mmW“gWhttoe

iidwmvi

timtiSdflrwm

piimawbstppihc

fcbwff

afion resistivity is given by 107 exp(−14(RH/100)0.2) � cm. The three data setsorrespond to different fixed water activity on the “liquid” side of the membrane.

he membrane resistance assuming the proton conductivity ofafion is given by σ = 10−7 exp(14a0.2

w ) � cm [39]; the datat is shown as the solid lines in Fig. 15. The measured mem-rane resistance decreases more slowly with increasing waterontent than that predicted based on a linear water gradient.his suggests that the water content is more homogeneous, asould be expected if diffusion is fast compared to interfacialass transport.

. Discussion

The key results from the water sorption/desorption and waterermeation experiments may be summarized as follows:

. Water sorption into Nafion 1110 has a characteristic rate con-stant of ∼5 × 10−4 s−1 at 40 ◦C increasing to ∼4 × 10−3 s−1

at 90 ◦C.. The rate of water sorption scales inversely with membrane

thickness.. Water desorption has a characteristic rate constant approxi-

mately 10 times greater than water sorption.. The initial rate of water desorption scales with external sur-

face area.. Water permeation is limited by interfacial mass transport

from the membrane to the gas for thin Nafion membranes.. Water permeation increases with temperature.. Water permeation through Nafion is enhanced when driven

by liquid water compared to water vapor.

The essential result from these studies is that water trans-ort in and through Nafion membranes is not simply a diffusionrocess, but interfacial mass transport across the gas/membrane

nterface is major resistance to water transport.

Why should there be a substantial interfacial mass transportesistance for water to leave a Nafion membrane and go into theas phase? If Nafion phase separates as proposed in the Gierke

tmww

rane Science 301 (2007) 93–106 101

odel [40] (and other models as well [41]), then the phase whichinimizes the interfacial energy should segregate to the surface.hen Nafion is exposed to a gas phase the low surface energy

Teflon-like” phase is expected to segregate to the membraneas interface; this is even true for a saturated vapor atmosphere.ater entering or leaving the membrane must pass through this

ydrophobic layer which will inhibit mass transport betweenhe membrane and gas phases. This is analogous to having ahin layer of oil on a pool of water. The oil will inhibit the ratef water evaporation, even though it does not affect the overallquilibrium.

When in contact with liquid water Nafion’s hydrophilic phases expected to segregate to the membrane/liquid interface. Thenterfacial mass transfer resistance for water is expected toecrease if the surface of the membrane is hydrophilic. Theater permeation measurements reported here showed that per-eation from liquid water was much greater than that from water

apor, consistent with the hypothesis that there is a differencen interfacial mass transport resistance.

We believe there are two complementary effects that con-ribute to the difference in water transport when liquid waters present. First, is that liquid water will induce a hydrophilic

embrane surface reducing the interfacial mass transport resis-ance. In addition, there is an increase in the water contentn the membrane when exposed to liquid water, the so-calledchroeder’s paradox [31,34,36,42]. Even with no change iniffusivity the increased c0

w would result in a greater waterux. Both these phenomena will increase the permeationate through the membrane; from the transport measurementse cannot distinguish the relative contributions of these twoechanisms.Our results presented here and the interpretation contradicts

revious studies that have attempted to model water diffusionn Nafion. These studies have suggested that water diffusionncreases with water content at low water content, rising to a

aximum and then decreases with water content. Fig. 14 showsclear increase in normalized water permeation with averageater activity across the membrane. The only way this coulde consistent with the previous studies modeling water diffu-ion is if the water permeation rate is limited by interfacialransport such that diffusion plays little role in determining theermeation rate. Both Rivin et al. [17] and Ge et al. [3] haveointed out that interfacial mass transport should be importantn water permeation and sorption into Nafion; the data presentedere shows the importance of interfacial mass transport muchlearer.

Fig. 14 also shows that water permeation from water vaporell to zero when the water activity at the dry side went to zero. Inontrast Fig. 13 indicates that Nafion 112, 115 and 1110 mem-ranes did not dry out when liquid water was present on theet side of the membrane. We suggest that this is due to the

ormation of a thin impermeable skin at the membrane/gas inter-ace. This impermeable skin appeared to form more readily on

hicker membranes; we only saw evidence with 112 (50 �m)

embranes when the liquid side had a relative humidity <30%,hereas with 1110 (250 �m) membranes the impermeable layeras detected when the liquid side was at 80% RH. Since our

102 P.W. Majsztrik et al. / Journal of Membrane Science 301 (2007) 93–106

F orptiog he mes g reac

rmsstubd

fTbmmfpi

1

2

3

1

2

mt

Nw

tfaPmsoafcerTta

mbtfisolved the Fickian diffusion equation with a boundary conditionof a known concentration at the membrane/gas interface (forexample see Morris and Sun [12]). The proper boundary con-dition is not the concentration at the membrane/gas interface,

Table 3Comparison of initial desorption rate and permeation rate

ig. 16. Schematic of the steps involved in water permeation through and water sas stream or liquid water to dry gas. They differ because of the composition at torption into a membrane, showing the shrinking-core model where the swellin

esults indicated that water flux was only weakly dependent onembrane thickness we suggest that the thicker membranes can

ustain larger swelling stresses across the membrane from waterorption. When liquid water is present the greater water sorp-ion will result in greater membrane swelling. The larger waterptake from liquid may create sufficient stress to keep the mem-rane from shrinking on the dry side so the impermeable skinoes not form.

Water sorption/desorption experiments also suggest that dif-usion is faster than other processes governing water transport.he process of water sorption and permeation in a Nafion mem-rane is summarized schematically in Fig. 16. More complicatedodels for water sorption/desorption may be proposed, but theechanism presented here is the simplest one that can account

or the data. Neglecting mass transport limitations in the gashase there are at least three processes that must be consideredn water sorption from water vapor:

. Interfacial transport across the membrane/gas interface intothe membrane.

. Diffusion through the membrane from the membrane/gasinterface to the interior of the membrane.

. Swelling of the membrane to accommodate water at the sul-fonic acid site.

Water desorption must consider two steps:

. Diffusion from the interior of the membrane to the mem-brane/gas interface.

. Interfacial transport from membrane/gas interface into thegas phase.

There is an asymmetry between sorption and desorption. Theembrane must swell to accommodate the volume change when

he water molecule coordinates with a sulfonic acid site in the

T

58

n into Nafion membranes. At the left is the permeation from either a humidifiedmbrane–gas or membrane–liquid interface. At the right is a schematic of watertion at the shell–core interface is rate limiting.

afion. However, the membrane does not need to shrink for theater to leave the swollen membrane.The early stages of desorption are similar to water permeation

hrough a membrane. In desorption water moves by diffusionrom the bulk of the membrane to the membrane/gas interfacend then must be transported across the membrane/gas interface.ermeation across the membrane adds the additional interfacialass transfer resistance at the liquid/membrane interface. As

hown by Fig. 9 the initial rate of desorption was independentf the thickness of the membrane. This initial rate should bepproximately equal to the permeation rate across the membranerom liquid water (a small difference is expected due to theoncentration gradient across the membrane in the permeationxperiment). Table 3 is a comparison of the initial desorptionates from Nafion 1110 with the steady-state permeation rates.he two rates are remarkably similar supporting the conjecture

hat the rate-limiting step for both is the interfacial transportcross the membrane/gas interface.

As desorption proceeds, and the water content of theembrane decreases, diffusion to the membrane/gas interface

ecomes slower and more significant. To properly analyze theransient water desorption from a Nafion membrane both inter-acial mass transport and diffusion in the membrane must bencluded. Previous investigators that analyzed water sorption

emperature Initial desorption rate (g/cm2 s) Permeation rate (g/cm2 s)

0 3.3 × 10−4 4.7 × 10−4

0 8.3 × 10−4 11.2 × 10−4

emb

bEfli

ptsp

w

β

ESbBavettbptnciat

acawhFWsawoesii

Itsirduwsgfmtbesaw[

fsmWacait

{

Th

−

T(t

−

P.W. Majsztrik et al. / Journal of M

ut rather the flux at the membrane/gas boundary, as shown inq. (7) where kg is the interfacial mass transfer coefficient. Theux in the membrane must match the mass transport across the

nterface:

∂c

∂t= D

∂2c

∂x2 , c = c0 for all x at t = 0,

D∂c

∂x= kgc at x = ±Lm

2(7)

Rivin and co-workers suggested that interfacial mass trans-ort should be considered in the analysis of water sorption, buthey did not have sufficient data to distinguish between diffu-ion and interfacial transport [17]. The solution to (7) has beenresented by Crank [43] and is given in Eq. (8):

�M(t)

�M∞= 1 −

∞∑n=1

8 sin2(βn/2)

β2n + βn sin βn

exp

(−β2

n

(D

L2m

)t

)(8)

here βn are the roots to the equation:

n tanβn

2= Bi

q. (8) is functionally the same as that presented by Morris andun [12], but the characteristic time is scaled by a Biot num-er, which is the ratio of external transport to internal transport,i = (kg)(Lm)/D. For thin membranes the Biot number is small;t long times the leading term in the series solution has an eigen-alue β1 = √

Bi, which results in desorption being controlled byxternal mass transport with a characteristic time Lm/kg. For veryhick membranes the Biot number is large and β1 = π, so at longimes desorption is controlled by internal diffusion in the mem-rane with a characteristic time L2

m/D. The permeation studiesrovided values of the interfacial mass transfer coefficient andhe diffusivity given in Table 2. Based on those values the Biotumber ranged from ∼1 to 5, so the desorption experimentsorrespond to conditions where the interfacial mass transport ismportant and the effective rate constant for water sorption is anveraged value between those for diffusion and interfacial massransport.

The analysis of water sorption into Nafion membranes mustlso be modified to replace the fixed concentration boundaryondition with a flux boundary condition. However, if diffusionnd interfacial mass transport were the only factors controllingater sorption then water uptake and water desorption shouldave similar rate constants. It is clear from the data presented inigs. 4 and 7 that these rates differ by an order of magnitude.ater sorption differs from desorption because the polymer must

well to accommodate water molecules coordinating to sulfoniccid sites. It is not necessary for the polymer to shrink for theater to desorb. Conceptually, we can use the two-phase modelf Gierke and Hsu [44] to understand the swelling. The water

nters the hydrophilic sulfonic acid domains and causes them towell, stretching out the hydrophobic Teflon-like domains. Thiss analogous to blowing up a balloon, except the Nafion is notdeally elastic but has a finite rate of visco-elastic deformation.

Uswm

rane Science 301 (2007) 93–106 103

Water sorption into Nafion is similar to the problem of CaseI diffusion in hydrogels that has been discussed extensively inhe literature [45–47]. The visco-elastic response of the gel towelling is generally slow compared to water diffusion. Depend-ng on how water diffusivity depends on water content can giveise to two interpretations of water uptake. If water diffusivityoes not vary much with concentration then water will distributeniformly through the membrane and swell uniformly. Whenater diffusion decreases with water content the water a swollen

hell forms. The water diffuses rapidly through the swollen shelliving a “nearly uniform” concentration in the shell. A front isormed between the swollen shell and the “dry” core of poly-er. The rate-limiting step for water uptake is the swelling of

he polymer at the shell–core interface. Case II diffusion cane represented by adding a visco-elastic term to the diffusionquation [47]. Alternatively, if the time scales for diffusion andwelling are disparate one can treat the front motion by assumingsteady-state flux through the shell to the shell–core interfacehere a “swelling” reaction occurs (a “shrinking-core” model

48]).The sorption data in Fig. 6A and B indicate that the time

or water uptake scales with length which is consistent with thehrinking-core model for water uptake. A simple shrinking-coreodel for water uptake is schematically illustrated in Fig. 16.ater is sorbed at the gas/membrane interface, diffuses throughshell of swollen membrane and then reaction occurs at the

ore where water causes the membrane to swell. A simplifyingssumption is that the rate of swelling is slow compared to thenterfacial mass transport and the diffusion, so we treat thosewo steps as being at pseudo-steady-state:

Swelling reaction} = {Rate of diffusion to core interface}= {Rate of external mass transport}

(λeq − λ)ρp

Ewc0

wkraw=Dc0w

amgw − aw

(Lm/2) − x=kgc

0w(1 − amg

w ) (9)

he mass uptake scales with the growth of the swollen shell (weave assumed λ ≈ 0 in the core):

dNw

dt= Am

λeqρp

Ew

dx

dt= Am

λeqρp

Ewc0

wkraw (10)

he concentration at the shell–core interface is found from Eq.9) to give the overall mass uptake in terms of the three resis-ances to transport:

dNw

dt= Am

λeqρp

Ew

×krc0

w

1/[(λeqρp/Ew)kr]+1/{D/[(Lm/2) − x]}+1/kg

(11)

nder the shrinking-core approximation the thickness of thehell is directly proportional to the fractional water uptake,hich can be substituted into Eq. (11) (In the case of the planarembrane the shell is the thickness of the swollen slab on either

1 emb

s

x

Tiu

(

stuc

N

rstmrbmfib

rpNlnp1csaar

r

dcmcactmesmsbr

5

Npmmvp

1

2

3

4

5

6

r

04 P.W. Majsztrik et al. / Journal of M

ide of the “dry” core):

= Lm

2

Nw(t)

Nmaxw

(12)

he mass uptake as a function of time can be determined byntegrating Eq. (11). There are two limiting cases for the massptake predicted by the shrinking-core model:

(i) Reaction controlled regime,λeqρkr/2EwD[(Lm/2) − x] 1.

In the reaction controlled regime water uptake is limitedby the rate that the polymer swells. The mass uptake scaleslinearly as a function of time. The mass uptake as a functionof time in the reaction controlled regime is given by Eq. (13).The reaction controlled regime applies to all membranes atshort times and extends to completion of water uptake forthin membranes:

Nw(t) = 2Amλeqρ

Ew(krc

0wt) (13)

ii) Diffusion controlled regime,λρkr/2EwD[(Lm/2) − x] 1.

In the diffusion controlled regime water diffusion through thehell is slow compared to the swelling reaction. This occurs forhick membranes. In the diffusion controlled regime the massptake scales at t1/2 and is given by Eq. (14). The diffusionontrolled regime applies to long times:

w(t) = 2Am

(2λeqρDc0

w

Ewt

)1/2

(14)

The functional forms for diffusion and reaction controlledegimes are useful in explaining how the mass change shouldcale as a function of time. In the reaction controlled regime,he normalized mass uptake should scale as time divided by

embrane thickness (t/Lm), whereas in the diffusion controlledegime the normalized mass uptake should scale as time dividedy membrane thickness squared (t/L2

m). Fig. 6 shows the nor-alized mass uptake plotted as functions of t/Lm and t/L2

m. Thet is much better for the former, consistent with water sorptioneing primarily limited by the swelling reaction.

The rate constant for the water sorption into Nafion 1110anges from 0.0008 to 0.01 s with increasing temperature. Waterermeation studies provide lower limits for water diffusivity inafion. We can neglect interfacial mass transport and get a lower

imit to the diffusivity, D ≈ (permeation rate)(membrane thick-ess)/(water concentration). The diffusivities obtained from theermeation studies vary with temperature from 3 × 10−6 to× 10−4 cm2/s. Based on those values the characteristic rateonstant for water sorption limited by diffusion in Nafion 1110hould vary from 0.005 to 0.1 s−1. These rate constants aren order of magnitude larger than the observed rate constants

dding further support to the hypothesis that water sorption iseaction limited.

We choose to represent the swelling reaction as a simpleeaction with mass action kinetics. The rate depended on the

mcsi

rane Science 301 (2007) 93–106

egree of unsaturation of the sulfonic acid sites and the con-entration of water in the membrane. The swelling process isuch more complex, involving chemical and mechanical pro-

esses including the solvation of sulfonic acid groups by waternd stretching the hydrophobic teflonic domains. The chemi-al reactions are probably faster than the mechanical relaxationimes associated with water sorption. Stress relaxation and creep

easurements of Nafion both show a range of relaxation timesxtending from 10 to 10,000 s [33]. The effective time con-tant for water sorption is ∼1000 s, which is within the range ofechanical relaxation times for Nafion. However, based on the

orption and transport measurements reported here it is impossi-le to identify any specific rate-limiting process for the swellingeaction.

. Conclusions

Water sorption, desorption and permeation in and throughafion 112, 115, 1110 have been determined as functions of tem-erature. Sorption and permeation are controlled by interfacialass transport, diffusion and polymer swelling. Temperature,embrane thickness and the presence of liquid water or water

apor all affect the rates of the different transport and reactionrocesses in Nafion. The key results are:

. Water permeation is limited by interfacial mass transportacross the membrane/gas interface for thin membranes andat low temperature. At higher temperatures and with thickermembranes the diffusional resistance across the membranebecomes important for water permeation.

. Water permeation through Nafion from humidified gasstreams decreases with increasing driving force across themembrane at low water activity on the dry side of the mem-brane. This wrong way behavior appears to be due to themembrane becoming less permeable at lower water content.

. Water permeation through Nafion from liquid water is muchgreater than permeation from humidified gas streams.

. Water desorption from saturated membranes is limited bythe interfacial mass transport resistance at the membrane/gasinterface. It is suggested that the Nafion surface is enrichedwith Teflon at the membrane/gas interface, and it is transportacross this layer that is the limiting mass transport resistance.

. Water sorption from humidified gas is limited by the rateof swelling of the polymer membrane to accommodate thewater.

. Water sorption into Nafion can be described by a shrinking-core model, where there is rapid diffusion of water acrossa swollen shell to the interface of the dry Nafion corefollowed by a slow reaction as water sorption swells thecore.

The large discrepancies in diffusivities for water in Nafioneported in the literature the result of different experiments

easuring different rate processes. Water sorption measures a

ombination of the rate of polymer swelling and water diffu-ion, water desorption measures a combination of the rate ofnterfacial mass transport at the Nafion/gas interface and water

embrane Science 301 (2007) 93–106 105

dcpat

A

0at

Greek symbolsβn eigenvalues to the diffusion equationλ number of water molecules sorbed per SO3λeq equilibrium number of water molecules sorbed

per SO3 at water activity aLw

ρ density of dry polymer (∼2 g/cm3)ρ membrane resistivity (� cm)

R

[

[

[

[

[

[

[

[

[

P.W. Majsztrik et al. / Journal of M

iffusion, water permeation measures a combination of interfa-ial mass transport and diffusion. Only by careful analysis ofermeation and sorption data for different membrane thicknessnd different temperatures it is possible to identify the individualransport rates.

cknowledgement

The authors thank the National Science Foundation (CTS-354279 and DMR-0213707 through the Materials Researchnd Science Engineering Center at Princeton) for its support ofhis work.

Nomenclature

aiw activity of water in phase i

Am area of membraneAmi

w activity of water at the membrane phase i interfacecw concentration of water in the membranec0

w equilibrium concentration of water in the mem-brane

D water diffusivity ([=] cm2/s)Dsorption water diffusivity based water uptake experimentDdesorption water diffusivity based water loss experimentD0 water diffusivity of a dry membrane ([=] cm2/s)D1 water diffusivity of fully hydrated membrane ([=]

cm2/s)Ew equivalent weight of polymer

(mol SO3/g polymer)F Faraday’s constant (96,468 coulomb/mol)kdesorption effective rate constant for water loss (1/s)kg effective mass transfer coefficient at

gas/membrane interfacekL effective mass transfer coefficient at liq-

uid/membrane interfaceko overall permeation coefficient (g/cm2 s)kr effective rate constant of polymer swelling ([=]

cm4/mol s)ksorption effective rate constant for water uptake (1/s)Lm thickness of dry membrane (cm)Lw thickness of water swollen layer in membrane

(cm)M(t) mass of a membrane at time t (g)M0 initial mass of membrane at time zero (g)M∞ final mass of membrane (g)Nmax mass of water in a saturated membrane (g/cm3)Nw mass of water in the membrane (g/cm3)Pw partial pressure of water (bar)P0

w(T ) vapor pressure of water at temperature T (bar)R gas constant (82.05 cm3 bar/mol K)RH relative humidity (RH = 100 × Pw/P0

w)Q inlet nitrogen flow rate at standard conditions

(cm3/min)T temperature of measurementTref reference temperature (298 K)

[

mρw density of water (1 cm3/g)

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