An analytical model for the conductivity of polymeric sulfonated membranes

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An analytical model for the conductivity of polymeric sulfonated membranes

L. Pisani ⁎, M. Valentini, D.H. Hofmann, L.N. Kuleshova, B. D'Aguanno

CRS4, Parco Scientifico e Tecnologico, Polaris, 09010, Pula (CA), Italy

Received 25 October 2007; received in revised form 22 February 2008; accepted 19 March 2008

Abstract

In this work, we present a new approach to model the conductivity of polymeric sulfonated membranes. In the first part, we develop a modelfor the proton conductivity in a bulk acid solution. The model describes the vehicular and structural diffusion mechanisms at different protonconcentrations as a function of four parameters with well defined physical meanings. These parameters are adjusted by fitting the experimentalproton conductivity in a hydrochloric acid solution. In the second part, the effects of the presence of the porous membrane on the protonconductivity are added to the model. As a result, the model predicts the membrane conductivity as a function of its hydration level and porousstructure. In the absence of experimental data on the membrane porous structure, the model can be used to predict the membrane porous structureonce membrane conductivities are experimentally known. In this scope, in the third part, we provide a simple analytical description of the porousmembrane based on the percolation theory depending on two structure parameters. A side result of the conductivity model is to provide analyticalexpressions for the drag coefficient, pore diameters and water permeability. All obtained results compare very well with experimental data.© 2008 Elsevier B.V. All rights reserved.

Keywords: Proton conductivity; PEMFC; Modeling

1. Introduction

One of the most important properties of electrolytic mem-branes is the proton conductivity. Molecular dynamics (MD)techniques have been largely used in the study of polymericmembranes, since they provide insight into the role of atom–atom interactions and molecular structures on the protonconductivity [1–4]. However, the MD description of membraneconductivity has few drawbacks. In particular, classicalmolecular dynamics is not capable of describing the bondbreaking underlying the structure diffusion transport mechan-ism, which, when protons are the charge carriers, accounts forthe majority of the charge transport. Although considerableadvances have been made to overcome this limitation [5,6], theresults are still qualitative. The correct description of the struc-ture diffusion phenomena has been obtained in bulk water [7],but the computational complexity is too high to describe signi-ficant portions of membranes. This problem subsists also for

classical MD, as the space and time scales needed to accuratelydescribe the “macroscopic” membrane conductivity parameterare too large, compared to the space and time scales of atomsdynamics. The natural solution of this scale problem is todescribe the membranes at a higher level, either by using unitedatom MD techniques (multi-atom level) [4], or at a mesoscopiclevel [8]. Although these higher level techniques can be suc-cessfully used to predict membranes structures, they are of littleuse for conductivity prediction, as the conduction mechanismsstill remains at an atomic and subatomic level (the structurediffusion mechanism). As a consequence, from the moleculardynamics analysis of the ionic conductivity of complexmembranes we can expect a better understanding of the protonconduction mechanism, while the possibility of comparing withreal conductivity data is still outside of the MD realm.

A way to circumvent these limitations is to describe theconductivity by adopting a “phenomenological” point of view inspite of the “first principle” point of view of the MD basedapproaches. Some very good results have been obtained byphenomenological models for what concerns the description ofstructural aspects, as membrane swelling and Schroeder paradox

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⁎ Corresponding author.E-mail address: [email protected] (L. Pisani).

0167-2738/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.ssi.2008.03.029


[9–11], while the description of the conduction mechanisms stillremains questionable. In this trend are the works of Fimrite et al.[12] and Choi et al. [13], which describe the proton transport at amacroscopic level. While both works obtain good agreementwith the experimental conductivity in membranes, some doubtscan be cast on their prediction capability; in fact, in both cases,the proton–water interaction is considered to be independent onthe proton concentration. In this way, these models cannotpredict the correct proton conductivity in a simple bulk acidsolution so that the agreement obtained in the complex hydratedmembranes seems to be artificial. The same considerationapplies for the membrane conductivity models of Hsu et al. [14]and Weber and Newman [10], which are based on the per-colation theory. Both of these models predict a constant con-ductivity in the bulk solution limit.

In this work, we put the basis of an alternative approach tomodel the membrane conductivity. The basic idea is to startfrom the experimental proton conductivity of an acid solution(for example hydrochloric acid), and to proceed by modelingthe effects of the presence of the porous membrane on such aconductivity. In this way, the basic microscopic transport me-chanisms, including the structure diffusion mechanism, areincluded in the experimental data. Therefore, the level of de-scription of the system can be fitted on the complexity of theporous media. Here we will give an analytical (macroscopic)description of the porous membrane, and suggest how to refinethe structure model down to a mesoscopic level.

However and in order to avoid the use of experimental dataon an otherwise analytical expression, we also developed amodel for the proton conductivity in bulk acid solutions. It isonly after the testing of the accuracy of this model that wedecided to derive a fully theoretical expression for the protonconductivity in membrane.

In Section 2.1 we review the proton transport mechanisms ina bulk acid solution and derive a simple model for the protonconductivity at a varying proton concentration degree. Thesection also contains the model application and validation to thecase of the bulk hydrochloric acid solutions. Both the resultingconductivity and the considerations learned by this model areused as a starting point, in Section 2.2, for a mechanistic model

of proton conductivity in membranes. In Section 2.3, the para-meters characterizing the membrane structure are derived andthe final analytical expression for the proton conductivity isgiven. In Section 3 we present and discuss the results, obtainedfrom the application of the developed models.

2. Model

It is well agreed that the most important parameter tocharacterize the state of a hydrated sulfonated membrane is itshydration level λ, defined as the number of water molecules foreach sulfonate group. Such a number λ can also be used in bulkacid solution which dissociates at very low water content,making free one proton for each acid molecule. In this case therelation between λ and the proton concentration cH+ is:

k ¼ cH2O

cHþ¼ q cHþð Þ

cHþMH2O� Macid

MH2Oð1Þ

Here, cH2O is the water concentration, M is the molecularweight of water and acid, and q is the density of the solutionexpressed in grams per liter. For a hydrochloric acid solution,the experimental solution densities at different acid concentra-tions can be found in [15].

By using Eq. (1), the experimental proton conductivity in abulk acid solution can be expressed in terms of λ, and compareddirectly with the proton conductivity in sulfonated membranes.In Fig. 1, the experimental proton conductivities in a bulk HClsolution and in Nafion and Dow membranes are reported as afunction of λ. The dashed line is obtained by considering non-interacting protons in a bulk solution, where the conductivity issimply proportional to the proton concentration:

rnon�interacting ¼ KcHþ : ð2Þ

It is seen that both bulk concentration effects and the pre-sence of the porous membrane should be taken into account inorder to have a reliable conductivity model.

2.1. Proton conductivity in the bulk acid solution

Protons in an acid solution form hydronium ions H3O+.

Proton transport through the solution occurs by two mainmechanisms:

1. migration of hydronium ions (vehicular diffusion)2. jumping of a proton from a hydronium ion to an adjacent

water molecule (structure diffusion).

The first transport mechanism is classic and it encounterstranslational friction resistance.

The second mechanism implies the breaking and formationof hydrogen–oxygen bonds. The activation energy for thismechanism is partially due to the hydrogen bond potentialbarriers and partially to the orientation of the hydronium ion andthe water molecule in the transition state.

Fig. 1. Proton conductivity as a function of λ for non-interacting protons(dashed line) in hydrochloric acid solutions [15,18], Nafion [19,24,25] and Dowmembranes [24,25].

466 L. Pisani et al. / Solid State Ionics 179 (2008) 465–476


For low ionic concentrations, the protons move independentlyfrom each other, and the solution conductivity is expressed byEq. (2). However, when the ionic concentration increases, someinteractions between the charge carriers are unavoidable.

In particular, the ions attract and polarize water moleculesaround them thus reducing both the translational and rotationaldegrees of freedom of the solvent water. In the following model,the reduction of the translational degrees of freedom is asso-ciated with the vehicle diffusion mechanism, while the reduc-tion of the rotational degrees of freedom is associated with thestructure diffusion mechanism.

2.1.1. Vehicle mechanismEach proton, forming a hydronium ion, subtracts one water

molecule from the bulk water solvent, thus reducing both thetranslational and rotational degrees of freedom of the solvent.The negative ions also tend to attract and polarize water mole-cules around them: for example, the chloride ion forms hydro-gen bonds with four water molecules. Such structure, althoughmore rigid than bulk water is not completely rigid and thehydronium ion can partially penetrate it.

Altogether, we can describe the concentration effect on thevehicular conduction mechanism as the subtraction of a numbernsub of water molecules for each dissociated acid molecule fromthe bulk water solvent volume. The bulk water fraction ε̣bulk,therefore, is rescaled by a factor that depends straightforwardfrom the hydration level λ:

ebulk ¼ k� nsubk

: ð3Þ

The conductivity originated from the vehicular diffusionmechanism σV, therefore, should be rescaled by the same factor.We can write:

rVcKVcHþebulk; ð4Þ

where cH+ is the proton concentration, while the vehicular dif-fusion coefficient KV doesn't depend on concentration.

2.1.2. Structure mechanismThe presence of an ion in the solution, besides subtracting a

water molecule from the solvent, tends to polarize the watermolecules around it. So, when a second hydronium ion comesclose to the first one, it finds stiffer water. Since the structurediffusion mechanism requires the rotation of the water mole-cules, around each ion there is a volume of polarized water(presumably the first solvation shell of the ion) where thestructure diffusion mechanism is inhibited to the other protons.To evaluate the effects of water polarization on the structurediffusion mechanism, we define:

• Vpol the volume of polarized water around a mole of disso-ciated acid molecules

• Vbulk the bulk water volume

• VS the portion of Vbulk still available to the structure diffu-sion, and assume that:

• only bulk water molecules belong to Vpol, since a hydroniumion cannot stay in the first solvation shell of another hydro-nium ion.

• The polarized water volumes around the ions have a randomspatial distribution, as they can overlap.

Now, suppose to add to the solution 1 mol of acid and nsub molof water in such a way to keep Vbulk unchanged. Following thetwo above assumptions, the probability for the polarized bulkwater around the new acid of being yet unpolarized is VS/Vbulk.Then, VS varies according to:

AVS

AnHþ¼ �Vpol

VS

Vbulk: ð5Þ

The differential equation Eq. (5) can be easily solved andexpressed in terms of the volume fractions:

VS ¼ Vbulke�

VpolVbulk

nHþ

eS ¼ ebulke�

Vpolebulk

cHþ ¼ 1� nsubk

� �e�

npolk�nsub

ð6Þ

where npol=cH2O Vpol is the number of polarized water mole-cules per dissociated acid molecule.

In Eq. (6), εs is the fraction of volume allowed to the structurediffusion mechanism. The corresponding conductivity, there-fore, should be rescaled by the same factor. We can write:

rScKScHþeS ¼ KScHþ 1� nsubk

� �e�

npolk�nsub : ð7Þ

2.1.3. Total conductivity and parameter guessAs the two conductivity mechanisms work in parallel, the

total proton conductivity of the bulk solution σbulk is the sum ofthe two partial conductivities:

rbulk ¼ rV þ rS ¼ cHþk� nsub

kKV þ KSe

� npolk�nsub

� �: ð8Þ

This expression contains four parameters: nsub, npol, KV andKS. Since these parameters have a well defined physical meaning,it is possible to associate guess values to them. By comparingthese values with the fitted ones, we can check if the physicalmeaning of the parameters is retained after the fitting procedure.

The parameter nsub represents the number of water moleculessubtracted from the bulk water volume for each dissociated acidmolecule, and should be worth about 2. At this regard, it isinteresting to note that conductivity experiments show that atλb2 Nafion membrane behaves as an insulator. This behaviorin general is interpreted as a percolation threshold in the Nafionpore structure [16]; we observe that the model presented here,when nsub=2, predicts the same behavior for the bulk solutionand we believe that the correct interpretation of the observedconduction threshold lies on the “freezing” of the water mole-cules around the ionic charges.

467L. Pisani et al. / Solid State Ionics 179 (2008) 465–476


The parameter npol is the number of polarized water mole-cules for each dissociated acid molecule and should be roughlyequal to the number of water molecules in the first solvationshell of the hydronium ion and its counter-anion, approximately10.

The sum of KV and KS is the proton conductivity at infinitedilution, which, at 25 °C, is worth about 0.35 S/cm [15], whiletheir ratio is a measure of the relative importance of the twoconduction mechanisms in the limit of low proton concentrationand is related to the so called amplification factor A [17]:

A ¼ rS þ rVrV

ð9Þ

A cHþ ¼ 0ð Þ ¼ KS þ KV

KV: ð10Þ

From the experimental data [17] we get a value of about 3.5for the ratio KS/KV.

2.1.4. Parameter fitting and validationTo validate the above conduction model and the parameters

estimates, we compare the results obtained from it with theexperimental proton conductivity in a hydrochloric acid solu-tion at 25 °C. The experimental values are obtained bymultiplying the hydrochloric acid conductivity given in [15]by the transference number [18], which is the fraction of thetotal current carried by the protons. Instead the theoreticalconductivity is obtained by a partial optimization of the para-meters around their guess values to fit the experimental data.The optimized parameters, together with the guess values, areshown in Table 1. It can be noted that the optimized values arequite close to their theoretical guesses.

In Fig. 2, the proton conductivity is shown as a function ofthe proton concentration. The model results are in excellentagreement with the experimental data in the whole concentra-tion range. In Fig. 2B, the equivalent conductivity (σ/cH+) isshown in order to emphasize the low concentration range. Theagreement is again excellent. Only at very low proton concen-trations (below 0.1 M) the experimental equivalent conductivityshows an increase that is not reproduced by the model results.

For further validation, in Fig. 3 we show the amplificationfactor A as estimated from Eq. (9). Once again, the agreement isexcellent for all the experimental points [17] beside the first oneat very low concentrations.

As the conductivity is the sum of the structure and vehiclediffusion mechanism, while the amplification factor is related totheir ratio, both mechanisms are well described by the model.Furthermore, as the fitted parameters are close to their guessvalues, we can conclude that the hypotheses underlying the

model are correct. In particular, the concentration effects on thebulk solution proton conductivity can be interpreted in terms of:

1. subtraction of one water molecule for each ion from the bulkwater volume

2. exclusion of the polarized water volume surrounding eachion from the structure diffusion mechanism.

While the KV and KS parameters are related to the con-ductivity at the infinitely diluted solution limit and, therefore,must be independent on the nature of the anions, the nsub andnpol parameters, in principle, should depend on the anionic

Table 1Guessed and optimized parameters for the bulk conductivity

Parameter Guess value Optimized value

nsub 2 2.07npol 10 9.2KV 0.078 S/cm 0.081 S/cmKS 0.272 S/cm 0.251 S/cm

Fig. 2. Comparison between model and experimental proton conductivitiestaken from [15] as a function of the proton molarity. A: conductivity in S/cm;B: equivalent conductivity in S/cm L/mol. In B the low concentration limit isemphasized.

Fig. 3. Comparison between model and experimental amplification factor takenfrom [17] as a function of the proton molarity.



species. In particular, npol depends on the extension of thehydration shell of the anion and can be retrieved from MD orexperimental structural data. On the other hand, nsub=2 in-dicates that one water molecule is subtracted for each of both theproton and the chloride ion. Thus, it seems that nsub is deter-mined mainly by the absolute value of the ionic charge, and thevalue of 2 obtained for the hydrochloric acid solution should beapproximately correct for each single charged counter-ion, in-cluding the sulfonic group of PFSA membranes.

2.2. Proton conductivity in the membrane

The main effects on the proton conductivity due to thepresence of the membrane are:

1. non-uniform concentration of charges in the acid solution;2. lowered diffusion due to themembrane–solution friction forces;3. an extended path for the particles moving through the tor-

tuous membrane;4. exclusion of the solid volume fraction to the conductive

phase.

The first three effects can be considered mesoscopic, as theyinvolve the membrane–solution interface (pore structure level);the last effect is macroscopic.

In this section, we describe a simple strategy to account forall these effects.

2.2.1. Non-uniform concentrationAs the negative charges are fixed on the lateral polymer

chains, there is a spatial separation between the protons, whichmove inside the bulk of the liquid solution, and the sulfonic ionswhich are bounded to the pore surface. In turn, this separationwill produce electrostatic forces, which will cause a non-homogeneous arrangement of the protons inside the pores.

With regard to this, we can distinguish two limiting topologies:

1. at a low hydration level, the liquid proton solution issurrounded by the negatively charged membrane walls(channels)

2. at a very high hydration level, the negatively charged mem-brane dissolves in the solution and its fragments are sur-rounded by the liquid solution.

In the first topology, the attraction forces that the negativecharges have on the protons average to zero. On the other hand,the protons close to the channel center push the other protonstoward the channel walls thus generating a non uniform chargedistribution.

In the second topology, the protons are attracted toward thenegatively charged membrane fragments. The protons close tothe membrane partially shield the electrostatic attraction forces.Again, the proton charges will be more concentrated close to themembrane walls. Above a certain degree of hydration, theproton charges will occupy a fixed volume of solution aroundthe membrane fragments, leaving, elsewhere, a proton-freefraction of water.

In the literature there are several works describing the dis-tribution of charges inside a hydrated acidic polymer (see [19]for a review). The shape of charge density profiles appear to becomplicated and to depend strongly on the hydration degree.However, when the effect of the non-homogeneous water pola-rization is taken into account, the proton concentration profilebecomes quite smooth a few angstroms away form the channelsurface [20]. Moreover, by considering that a possible conduc-tivity gradient will generate a force pushing the protons towardthe higher conductivity regions, we expect a further smoothingof the charge profiles. In summary, we expect a relevant effectof the non-uniform charge distribution on the conductivity onlyi) at very high hydration levels, where the dimensions of thewater clusters are so large that non-homogeneity can't beavoided, and ii) in the region very close to the channel surface,where the water is highly polarized. Since, in the operatingconditions of sulfonated membranes, λ is never higher than afew tens, the first case can be neglected. The second effect canbe treated, as we made for the bulk acid conductivity, in terms ofexcluded polarized volumes. The main difference here is thatthe hypothesis of random spatial distribution of the polarizedvolumes does not hold anymore for the fixed negative ions. As aconsequence, by excluding overlaps between the polarizedvolumes Vpol− around two different sulfonic groups, while stillallowing random superposition of the polarized volumes Vpol+

around two protons or proton-sulfonic group pairs, Eq. (5)assumes a slightly different expression:

AVS

AnHþ¼ �Vpolþ

VS

Vbulk� Vpol�

VS

Vbulk � nHþVpol�: ð11Þ

The differential equation Eq. (11) can be solved and expres-sed in terms of the volume fractions:

VS ¼ Vbulk � nHþVpol��

e�VpolþVbulk

nHþ

eS ¼ ebulk � cHþVpol��

e�Vpolþebulk

cHþ ¼ 1� nsub þ npol�k

� �e�

npolk�nsub ;

ð12Þwhere npol−= cH2O Vpol− is the number of surface watermolecules per sulfonic group, while npol+=cH2O Vpol+ is thenumber of polarized water molecules around each hydroniumion.

The structure diffusion term of the conductivity becomes:

rS ¼ KScHþ 1� nsub þ npol�k

� �e�

npolþk�nsub : ð13Þ

By comparing Eq. (13) with Eq. (7), we observe that, as aconsequence of the different assumptions, the number ofpolarized water molecules around the anionic charges hasmoved from the exponential to the pre-exponential term.

2.2.2. Membrane–solution friction forcesWhen an electric field is applied to the system, the protons

are pushed by the electrostatic forces toward the negativeelectrode, thus generating an electric current.

Water molecules are dragged by the moving ions and be-cause the negative charges are fixed, this results in a global



motion of the water in the direction of the proton current. Onone hand, this movement, causes a power loss due to the water–membrane friction forces but, on the other hand, causes areduction of the energy dissipated by water–protons frictionforces. Overall, the velocity of water motion will be such as tominimize the global power loss.

To describe the dynamics of the solution inside the mem-brane, we use a simple macroscopic approach. We consider thehydrated membrane system to be comprised by three compo-nents: the solid medium incorporating the negatively chargedsulfonic groups (anions of the acid solution), the protons (in-cluding the hydronium ions) and the bulk water (excluding thehydronium ions). Each component has its own average velocity:the solid medium is motionless; the proton velocity is pro-portional to the current density, while the bulk water velocity vcan be obtained by minimizing the global power loss. Thepower dissipation comes from the frictions between: (i) bulkwater–solid medium, (ii) protons–bulk water and (iii) protons–solid medium and each term depends on the relative velocities.

2.2.2.1. Bulk water–solid medium. The solid medium iscomposed by a polymeric matrix with the charged sulfonicanions attached to its lateral chains. The presence of the polymermatrix, by constraining the anionic charges into a rigid structureand the water to flow inside narrow channels, causes powerlosses. In particular, the water velocity inside the channels is notuniform (the water around the channels center is faster) thusgenerating water–water frictions. This “indirect” water–solidfriction term can be estimated through thewater viscosity and theDarcy law. The average velocity v of a fluid (water) of viscosityμ subject to a force per unit volume f inside a porous media ofpermeability kp can be expressed by the Darcy law:

f ¼ Akp

v: ð14Þ

The power per unit volume, dissipated by the water solidfriction forces, is given by:

WH2O ¼ vf ; ð15Þand, by using Eq. (14), we obtain:

WH2O ¼ Akp

v2: ð16Þ

2.2.2.2. Protons–bulk water. The power dissipated by theprotons–bulk water friction forces can be written by using theJoule law:

WHþ ¼ 1

rbulkI � cH2OF

kv

� �2

; ð17Þ

where the term in parentheses is the proton current density in thereference frame of the moving water.

2.2.2.3. Protons–solid medium. The protons–anions frictionsare already included in the σbulk term. The presence of the solidmatrix increases the polarized water volume and changes thestructure conductivity as described by Eq. (13).

2.2.2.4. Total power losses. The global power loss can beobtained by summing Eqs. (16) and (17). We obtain:

W ¼ Akp

þ c2H2OF2

k2rbulk

� �v2 � 2IcH2OF

rbulkkvþ I2

rbulk: ð18Þ

The velocity of water can be obtained by minimizing theglobal power loss. We get:

v ¼ IcH2OFrbulkk

Akp

þ c2H2OF2

k2rbulk

� ��1

: ð19Þ

By substituting Eq. (19) in Eq. (18), we obtain:

W ¼ I2

rbulk þ rdrag; ð20Þ

where we have defined:

rdrag ¼c2H2OF

2kp

Ak2: ð21Þ

By comparing Eq. (20) with the Joule law, we find anexpression for the effective proton conductivity, which alsotakes into account for the water motion:

ref f0 ¼ rbulk þ rdrag: ð22Þ

2.2.2.5. Drag coefficient. Our approach also allows derivingan expression for the drag coefficient, i.e. the number of watermolecules following each proton. The drag coefficient is linkedto the water velocity through the following expression:

kD ¼ cH2OFI

vtot ð23Þ

In Eq. (23), we have used the total water velocity (vtot),including the hydronium ions:

vtot ¼k� 1ð Þvþ vH3O

þ

k: ð24Þ

By using Eqs. (19) and (23), we can derive an expression forthe drag coefficient:

kD ¼ IH3Oþ

Iþ k� 1ð Þ 1þ rbulk

rdrag

� ��1

; ð25Þ

where the first term on the right hand side of Eq. (25) is relatedto the amplification factor A defined by Eq. (9):

IH3Oþ

I¼ 1

Aþ A� 1

A1þ rbulk

rdrag

� ��1

: ð26Þ

Finally, we get:

kD ¼ 1Aþ k� 1

A

� �1þ rbulk

rdrag

� ��1

: ð27Þ



In the results section, this expression will be compared withthe experimental results obtained for Nafion as a function of thehydration level.

2.2.3. Porosity and tortuosityThe presence of the solid media precludes a fraction of the

volume to the conductive liquid phase and extends the path thatthe moving particles should walk to cross the tortuous media.These two effects can be described at a macroscopic level byusing the porosity, ε, and the tortuosity, τ, and by rescaling theconductivity and permeability coefficients. They become:

ref f ¼ esref f0 ; ð28Þ

kp ¼ esk0p : ð29Þ

This last expression can be further modified by using theclassical Poiseuille expression [21] for the permeability ofcylindrical pores of diameter d, kp

0 =d2/32:

kp ¼ esd2

32: ð30Þ

Because the membrane pores are not cylindrical, the para-meter d should be intended as an average value.

By inserting Eq. (28) into Eq. (22), we get:

ref f ¼ es

rbulk þ rdrag� �

: ð31Þ

By inserting Eqs. (21), (8), (30) and (13) into Eq. (31), weobtain:

ref f ¼ es ½cHþ Kv 1� nsub

k

� �þ KS 1� nsub þ npol�

k

� �e�

npolþk�nsub

� �

þ c2H2OF2d2

32Ak2 �: ð32Þ

2.3. Modeling of the porous media

The conductivity expression Eq. (32) contains four para-meters which depend on the structure of the liquid phase inside

the membrane: the volume fraction of the liquid phase ε, itstortuosity τ, the average pore diameter d and the number ofsurface water molecules per sulfonic group npol−. In this section,we model the porous structure in order to express these para-meters in terms of the hydration level λ.

2.3.1. PorosityThe porosity parameter ε̣ represents the ratio between the

volume occupied by the liquid solution and the total volume. Itcan be easily expressed in terms of the hydration level λ,equivalent membrane weight EW, molecular water weightMH2O, and membrane and water densities, ρmem and ρH2O, as:

e ¼ kEWMH2O

qH2Oqmem

þ k: ð33Þ

If we consider that the density of water is about 1 g/cm3 [10],its molecular weight MH2O is 18 and by defining the equivalentmembrane volume EV=EW/ρmem as the membrane volume persulfonic group, we obtain:

eck

EV18 þ k

: ð34Þ

2.3.2. TortuosityThe tortuosity of porous media in general depends on the

pore volume fraction, shape and connectivity. However, forsome class of materials, theoretical (or phenomenological)relations exist expressing the tortuosity as a function of theporosity only. The best known among these expressions is theBruggeman relation [22]:

s ¼ e�0:5; ð35Þ

which has been proven to be a good approximation for a largeclass of porous materials.

Such a relation is valid only when the pore phase is wellconnected. Actually, pore closures phenomena occur, which, atporosities lower than a characteristic percolation threshold(εbε0) can interrupt mass (and charge) passing through the

Fig. 4. Graphical representation of Eq. (37) for n0s =12.

Fig. 5.MDsimulation [23] of the radial distribution function (RDF) and coordinationnumber of the sulfonic sulfur–water oxygen pair at two different λ values.



pores. Percolation theory provides a conductivity expressionthat includes the percolation threshold:

ref f ¼ e� e0ð Þtreff0 : ð36ÞIt is easy to see that Eq. (35) is a particular case of Eq. (36)

for ε0=0 and t=1.5.In the following, we will use the expression given by Eq. (36)

with ε0=0 and t=1.5 as initial guess values for the fittingprocedure.

2.3.3. Number of surface water moleculeTo evaluate npol− as a function of λ, we need to model the

porous structure.The backbone of the polymers is strongly hydrophobic while

the sulfonated sites are hydrophilic, and, as soon as the acidsolution permeates the membrane, the sulfonated sites-solutioncontact surface will be maximized while the backbone-solutioncontact surface will be minimized and will become approxi-mately zero. As a consequence, we expect that at low hydrationlevels all the water molecules in the solution belong to itssurface, npol−∼λ, while above a certain value of λ, the polymer-solution surface remains approximately constant, and npol− willreach a constant value n0

s when λNn0s. This behavior can be

represented by the smooth function:

npol� ¼ 1

ns4

0

þ 1

k4

!�14

: ð37Þ

In Fig. 4 Eq. (37) is represented for n0s=12.

In Fig. 5, MD simulation results [23] for the radial dis-tribution function (RDF) of the sulfonic sulfur–water oxygenpair are shown. It is seen that there is structured water at adistance between 3 and 5.5 Å from the sulfur atom. The amountof water contained in this shell is represented by the coor-dination numbers (dashed lines) and is about 10 molecules forλ=12 and 12 molecules for λ=24. These results are coherentwith the theoretical scheme presented above when n0

s∼12.

2.3.4. Pore diameterTo evaluate the water–membrane surface, we shall assign

areas to the npol− water molecules on the surface. When thewater is less then what is needed to fully hydrate the sulfonicgroups, it is reasonable to assume that the water molecules aresurrounded by the oxygen atoms of the sulfonic groups and allof the water molecule surface, about 4πrwater

2 , belongs to thepolymer-water interface. When there is water in excess, thesurface water molecules are in contact with the sulfonic groupson one side only, while elsewhere are surrounded by other watermolecules. In these conditions and by assuming a cubic latticeof water molecules, we estimate that only a fraction of the watermolecule surface, about 4rwater

2 , belongs to the polymer-waterinterface. Overall, an expression of the water surface per sul-fonic group following this behavior can be obtained by modi-fying Eq. (37) as follows:

Swaterc4r 2water1

ns4

0

þ 1

pkð Þ4 !�1

4

: ð38Þ

The water volume per sulfonic group can be simply obtainedby

Vwater ¼ kMH2O

NAqH2Oc3k� 10�29m3: ð39Þ

The water molecule diameter can be derived either by thewater oxygen–water oxygen radial distribution function or bytaking the cubic root of the molecular water volume. In bothcases we obtain rwater∼1.5 Å. The volume to surface ratio ofwater is:

Vwater=Swaterck4

ns40

þ 1p4

!14

3 AB: ð40Þ

The volume to surface ratio can be related to the pore diameterby assuming simple pore shapes. It is worth d/4 for cylindrical

Fig. 6. Pore diameters of Nafion and S-PEK membranes as a function of theliquid phase saturation [19].

Table 2Water properties and Faraday constant

μ Water viscosity 0.89×10−3 kg m−1s−1

cH2O Water concentration 0.555×105 mol m−3

F Faraday constant 0.96485×105 C mol −1

Table 3Specific model parameters

nsub 2.07KV 0.81 S/cmKS 0.251 S/cmnpol+ 5n0s 12d0 1.2×10−7 cm



pores and d/6 for spherical ones. By comparing such relationswith Eq. (40) we can derive the average pore diameter:

dcd0k4

ns4

0

þ 1p4

!14

; ð41Þ

where the factor d0 is worth 1.2 nm for the cylindrical poregeometry and 1.8 nm for the spherical geometry.

In Fig. 6, Eq. (41), for both cylindrical and spherical poregeometries, is compared with the experimental pore diametersof the Nafion 1100 and S-PEK 700 membranes obtained bysmall angle X-ray scattering experiments [19]. It is seen that theexperimental data for both membranes are fully comprisedbetween the cylindrical and spherical theoretical curves. For thefollowing, we will use the values obtained by supposing cylin-drical pores, because they are closer to the experimental data.

By substituting Eqs. (34), (36)–(37) and (41) in Eq. (32),we get our final expression for the proton conductivity in themembranes:

ref f ¼ kEV18 þ k

� e0

!t½cHþ KV 1� nsubk

� ��

þKS 1� nsubk

� k4

ns4

0

þ 1

!�14

0@

1Ae�

npolþk�nsub

1Aþ

132 c

2H2O

F2d20ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2

ns40

þ A2

pkð Þ4r �:

ð42Þ

3. Results

In this section, after a short review on the model parameters,the expressions of the conductivity Eq. (42), permeabilityEq. (30) and of the drag coefficient Eq. (27) will be used andcompared with experimental data.

3.1. Parameters

Eq. (42) contains a large number of parameters. In thissection we give their values and briefly discuss them.

In Table 2, the values of water viscosity, water concentrationand Faraday constant are listed. The variations of viscosity withtemperature should be taken into account when operating attemperatures different from 25 °C.

In Table 3, the specific parameters of the model are listed.The first three parameters are obtained by fitting the experi-mental proton conductivity in the hydrochloric acid solution asexplained in Section 2.1 and reported in Table 1. Since theparameters KS and KV are directly linked to the limiting protonconductivity in electrolyte solutions, when operating at tem-peratures different from 25 °C, their value should be rescaledaccording to the formula [26]:

K Tð Þc 1þ 0:0139 T � 25ð Þ½ �K 25Bð Þ: ð43ÞThe value of the parameter npol+ corresponds to the number

of polarized water molecules around a hydronium ion as dis-cussed in Sections 2.1.4 and 2.2.1. The number of water mo-lecules on the surface of a sulfonic group n0

s has been estimatedfrom MD simulation results presented in Section 2.3.3, while

Table 4Membrane properties and hydration condition

EV Equivalent membrane volumeλ Hydration levelcH+ Proton concentration

Table 5Membrane structure parameters

ε0 Percolation thresholdt Tortuosity parameter

Fig. 7. Comparison between model and experimental conductivity for theNafion 1100, Dow 800 and S-PEEKK 700 membranes as a function of thehydration level. The experimental data are taken from [19,24,25].



the value of d0 comes from the assumption of cylindrical poreshape (Section 2.3.4).

In Tables 4 and 5, the parameters characterizing themembrane, its hydration condition and pore structure arereported. These parameters are the variables of the model. Theequivalent volume EV is a fundamental membrane character-istic and is related to the equivalent weight EW, commonlyspecified after the membrane name (e.g. Nafion 1100, Dow 800etc.), and to the membrane density (about 2 g/cm3 for Nafionand Dow membranes [10]) while the hydration level λ and theproton concentration cH+ represent its hydration condition; λand cH+ are linked by the first equality in Eq. (1). The structureparameters ε0 and t are quite difficult to obtain a priori. In thiswork we use them as fitting parameters.

3.2. Conductivity

3.2.1. Conductivity as a function of λIn Fig. 7, we compare the experimental conductivity

measured for the Nafion 1100, Dow 800 and S-PEEKK 700membranes at a varying hydration degree, with the theoreticalconductivity results from Eq. (42). The experimental data aretaken from Ref. [19,24,25]. Since the experiments are per-formed at slightly different temperatures ranging from 22 to30 °C, in order to uniform the data to a common temperature of25 °C we have rescaled the conductivities according to Eq. (43).

The model results are shown for two values of the tortuosityparameter: t=1.5 (corresponding to the Bruggeman relation)and t=1.75. The percolation threshold has been left equal tozero for the Nafion and Dow membranes, while a percolationthreshold different from 0 (ε0=0.22) has been chosen to fit theconductivity of the S-PEEKK membrane. We observe a quitegood agreement between theory and experiment. In particular,all the qualitative behaviors are well reproduced: the shape ofthe curves, the conductivity thresholds and the differencesbetween Nafion 1100, Dow 800 and S-PEEKK 700 all matchthe experimental data. Moreover, a “quantitative” agreementcan be observed between the data form Ref. [19,25] and themodel results obtained by using t=1.75 while the data fromRef. [24] agrees better with the model result obtained by usingt=1.5, probably indicating that the specific membrane treatmentmay influence its tortuosity.

In Fig. 8A, we show the experimental conductivity of anumber of membranes: together with the Nafion and Dowmembranes, the conductivity of a series of sulfonated poly-styrene/polyvinyl chloride (SPS/PVC) composite membranes[27] and of sulfonated polyimides (SPI) membranes [28] isshown. Differently from the Nafion and Dow conductivity data,each point of the SPS/PVC and SPI series corresponds to aparticular membrane with its own equivalent volume. It can be

Fig. 8. Conductivity of a series of membranes as a function of the hydrationlevel. In (B) the conductivity values have been normalized according to Eq. (44)and compared with theory.

Fig. 9. Comparison between model and experimental conductivity of Nafion1100 as a function of temperature.

Fig. 10. Comparison between model and experimental hydraulic permeability ofNafion 1100 taken from [30], as a function of the hydration level.



seen that, while the data have common qualitative behaviors(increase of conductivity with λ, no apparent percolation thres-hold) the conductivity values are quite spread out and eachmembrane follows a different trend. In Fig. 8B, the same dataare normalized as follows:

rnorm ¼ r=EV18

þ k

� �t

ð44Þ

with t=1.75. By looking at Eq. (42), it appears that, when thepercolation threshold is zero, such normalization takes intoaccount for the volumetric differences between different mem-branes. We see that after the normalization the experimentalpoints are much closer to each other and follow closely thetheoretical curve.

3.2.2. Conductivity as a function of temperatureAs discussed in Section 3.1, the model conductivity depends

on temperature through the parameters KS, KV and μ. In Fig. 9,we show the Nafion conductivity as a function of temperaturefor two hydration values: λ=5 and λ=16. We observe that thetheoretical curves have the same behaviour as the experimentalpoints [19].

From the investigated temperature range, no drop in con-ductivity is observed. However, it is known that, whenoperating at a fixed relative humidity, membrane hydration λincreases with temperature and when the temperature raisesabove a certain value a conductivity drop is observed [29]. Alsothis behaviour is qualitatively reproduced by the model pre-sented here, as it forecast a conductivity drop for very largevalues of λ, where, however, the effect of non-uniform protonconcentration becomes too important to allow a quantitativeestimate of the phenomenon (see Section 2.2.1).

3.3. Permeability

The hydraulic permeability of electrolytic membranes, besideshaving influence on the conductivity, as shown in Eq. (21), is animportant property in itself since it strongly affects all the phe-nomena involving the transport of liquids through the electrolyte,including membrane dehydration and methanol crossover. InFig. 10, we compare the results from the theoretical expression,

Eq. (30), with the experimental permeability of Nafion 1100 [30]for different water contents. The simulation has been performedby using the same model parameters as used for the conductivitycalculation. The agreement with the experimental values isexcellent for values of the hydration level up to λ=21, while athigher λ values, the experimental points show a hump, probablyindicating a structural change.

3.4. Drag coefficient

In Fig. 11, we compare the theoretical drag coefficient resultsfrom Eq. (27) with the experimental data for both Nafion 1100and S-PEK 730 membranes. Since the experimental data arequite scattered, the experimental points for Nafion have beenobtained by fitting the data reproduced in figure 15 of Ref. [19].The comparison is overall very good.

By looking at Eq. (27), it is interesting to note that at lowhydration, the drag coefficient depends strongly on the am-plification factor. In the absence of a collective water motion,indeed, only the hydronium motion contributes to the watertransport and the drag coefficient becomes approximately equalto the inverse of A. The good agreement between model andexperimental drag coefficient at low λ, therefore, indicates thatthe model correctly predicts the relative importance of vehicularand structure conductivity mechanisms in the membrane. Inparticular, it confirms the further reduction of the structureconductivity in membranes caused by water polarizationaround the sulfonic groups, as described by Eq. (13) (in spiteof Eq. (7)).

4. Conclusions

One of the main results of this work is the separation of themembrane resistance to protons transport in two nearlyindependent contributions: the resistance of the homogeneousacid solution and the additional resistance of the solid polymermatrix. The first term is very complicated to describe as itinvolves molecular, atomic and sub-atomic level dynamics, butit is nearly independent on the specific acid composition andcan be easily retrieved from experimental data. The secondcontribution depends on the membrane porous structure and canbe described at a mesoscopic level. The only parameter used inthe two sub-models with different values is the volume ofpolarized water around the anionic species that in the bulk acidsolution are randomly distributed in space while in the sulfonicmembrane are fixed on the lateral polymer chains.

The results from Eqs. (8) and (42), which are the maintheoretical contribution of this work, are in a qualitative andquantitative agreement with experimental data as shown inFigs. 2 and 7. A number of other physical properties, whichhave been derived as intermediate results (average pore dia-meter, hydraulic permeability) or side results (amplificationfactor, drag coefficient) have been compared with experimentaldata as well. All the comparisons show good agreements, thusdemonstrating the robustness of the theoretical castle.

From the results shown in Fig. 7 it also appears that Nafionand Dow membrane have a nearly zero percolation threshold

Fig. 11. Comparison between model and experimental drag coefficient takenfrom [19] (see text) as a function of ε.



ε0̣. It means that the (liquid phase) porous structure remains wellconnected also at very low hydration levels. This is probablythe main reason for their superior conductivity with respect tothe S-PEEK membrane whose percolation threshold is ε̣0=0.22(a typical value for stochastic media). In the absence of apercolation threshold, the differences in conductivity are mainlydue to differences in the membrane equivalent volume.

A natural extension of this approach is the study of theproton conductivity in complex membranes containing inor-ganic inclusions. The presence of hydrophilic inclusions en-hances the membrane hydration. On the other hand, the watersurrounding the inclusions, following their hydrophilic nature,can be polarized or even immobilized. These effects can beeasily taken into account by this model as the parameters n0

s andnsub are tailored to describe them.

SymbolsA Amplification factorci Concentration of species id Pore diameterEV Equivalent volumeEW Equivalent weightf Force per unit volumeF FARADAY constantI Current densitykD Drag coefficientkp PermeabilityK ConstantMi Molecular weight of species in NumberNA Avogadro numberr RadiusS Surfacet Tortuosity parameterT Temperaturev Water velocityV VolumeW Power per unit volume

Greekε Volume fractionλ Hydration levelμ Water viscosityρ Densityσ Conductivityτ Tortuosity

Subscripts and superscripts0 Reference valuebulk Bulk watereff Effectivemem Membranepol Polarized waterpol− Water polarized by negative chargespol+ Water polarized by positive charges

S Structure mechanisms Surfacesub Subtracted from bulktot TotalV Vehicular mechanism

Acknowledgments

This work has been carried out with the financial contribu-tion of the Sardinia Regional Authorities and of the ItalianMinistry for University and Research, MIUR, Progetto FISRNuovi Materiali elettrolitici e sistemi elettrodici innovativi percelle a combustibile polimeriche, NUME www.nume.it.

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An analytical model for the conductivity of polymeric sulfonated membranes

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