Hydration of Ionomers and Schroeder's Paradox in Nafion - The

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J. Phys. Chem. B 2009, 113, 24–36

Hydration of Ionomers and Schroeder’s Paradox in Nafion Viatcheslav Freger* Zuckerberg Institute for Water Research and Department of Biotechnology and EnVironmental Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa 84105, Israel ReceiVed: February 15, 2008; ReVised Manuscript ReceiVed: October 9, 2008

Schroeder’s paradox, i.e., different uptake of a liquid solvent and its vapor, is analyzed for ionomers in general and Nafion in particular within a proposed general model of hydrated ionomers. The model considers four types of energy involved in a microphase-separated hydrated ionomer: hydration (solvation) of ionic groups, interfacial energy, and two distinct types of elastic energy associated, respectively, with inflation of the matrix upon hydration and varying stretching of the corona regions adjacent to the microscopic interface. By analyzing equilibrium in the bulk and surface regions of the polymer using approximate phenomenological relations for each contribution, it is shown that Schroeder’s paradox is a consequence of a composite interfacial-elastic “Laplace” pressure that is exerted on the aqueous microphase in equilibrium with a solvent vapor but exactly cancels out in equilibrium with a liquid as a result of structural rearrangements starting from the surface. The crucial difference with the previous models is association of this pressure with the microscopic polymer-liquid interface. The model conforms to the general picture of structural evolution of Nafion in the full range of hydrations and to the available scattering data, indicating a 2D morphology in a wide hydration range. It also allows for analysis of realistic nonequilibrium hydrated states stabilized by a transient rigidity of the matrix, apparently characteristic of Nafion at ambient conditions. Available data on hydrated Nafion strongly suggests that transient rigidity may be responsible for many of its unusual properties, such as unique morphologies, high conductivity, and its loss at high temperatures. Slow relaxation in the bulk and at the surface may also explain the controversies regarding observation of Schroeder’s paradox and the importance of thermal and hydration history. 1. Introduction It was observed that some polymeric materials differently take up solvent, i.e., swell in the liquid solvent and its vapor, despite the fact that all phases involved (polymer, liquid, and vapor) are in thermodynamic equilibrium. This phenomenon, referred to as Schroeder’s paradox,1 has been known for over a century and might be interpreted as a violation of thermodynamics, as it seems to suggest that there may be two different equilibrium states of a swollen polymer for the same chemical potential of the solvent. Nevertheless, for a number of polymers, this effect was shown to be reversible and reproducible without hysteresis, which strongly suggests it has a true thermodynamic origin.2-4 The polymers, for which this paradox was observed, with a few exceptions contain polar groups covalently attached to a hydrophobic backbone, and thus swell in water and some other polar solvents. A common feature of these systems is the propensity of the solvated (hydrated) polar groups to phaseseparate from the hydrophobic matrix and form microscopic aggregates. This paradox attracted much attention recently in view of the general interest in solid polymer electrolytes, particularly, Nafion (a perfluorinated ionomer with a Teflon-like backbone bearing side chains with sulfonate endgroups), as membrane material for solid polymer membrane fuel cells (SPMFCs).5-7 Such materials may be roughly divided into two groups: ionexchange resins8 and ionomers.9,10 In the resins, dissolution is prevented by chemical cross-linking, whereas, in ionomers, it is achieved by the hydrophobicity of an un-cross-linked matrix. Their performance, in particular, ionic conductivity, critically * E-mail: [email protected].

depends on the degree of hydration (water uptake).11 For this reason, maintaining appropriate hydration of the membrane (“water management”) is key to successful SPMFC design.12 As a part of this effort, modeling of water transport across a membrane requires knowledge of the thermodynamic activity of water as a function of water content in the membrane, i.e., water sorption isotherm. Such isotherms are usually measured by equilibration with a vapor in a range of activities.11,13-15 However, the ambiguity related to the occurrence of Schroeder’s paradox in Nafion and similar materials leaves one with the question of whether the isotherm correctly predicts the water activity as a function of water content, e.g., when the membrane directly contacts liquid solution at least at one side. A similar ambiguity arises in relation to other membrane processes utilizing polyelectrolyte membranes for water transport such as pervaporation.4 Recently, a few explanations of this phenomenon have been proposed. Vallieres et al.3 suggested this effect could result from multiple minima that may exist in the free energy expression given by the Frenkel-Flory-Rehner theory of swollen polymeric gels16,17 for certain combinations of parameters. Their hypothesis however requires certain relations between parameters and should lead to a swelling hysteresis that was not experimentally observed. A different approach was presented by Choi and Datta.18,19 Its central idea is that the microscopic interface between aqueous ion aggregates and the vapor phase presumably existing at the surface of vapor-equilibrated Nafion is curved and brings about an extra Laplace pressure on the aqueous phase inside the whole sample. This pressure increases the chemical potential of water, yet it should be absent for liquid-equilibrated samples, which

10.1021/jp806326a CCC: $40.75  2009 American Chemical Society Published on Web 12/15/2008

Hydration of Ionomers and Schroeder’s Paradox in Nafion may explain the difference between the two modes of equilibration. To calculate the extra Laplace pressure, the authors assumed that the matrix forms permanent hydrophobic pores filled with a water-like liquid that are open to the environment at the surface. The radius of pores was identified with the radius of aggregates in the bulk. Elfring and Struchtrup20 used a similar view as a basis for analysis of stability of water in Nafion but further assumed the possibility of a wide distribution of pore sizes. However, as pointed out by Vallieres et al.,3 the basic assumption of these studies ignores the fact that the open pore configuration at the surface would be unfavorable in terms of interfacial energy and therefore unstable. Finally, Weber and Newman21 proposed a phenomenological (“physical”) model that successfully explains nearly all existing data on structure, equilibrium, and transport of water in Nafion including Schroeder’s paradox; however, it contains a nonthermodynamic assumption of a constant water activity (equal to 1) for water contents beyond equilibrium with saturated vapor. This was rationalized by postulating that the phase state of the external medium drives changes in the microstructure that propagate from surface to inner bulk. In a follow-up publication considering transport of water,22 capillary pressure was added as part of the thermodynamic driving force for water transport, which is essentially equivalent to Choi and Datta’s assumption. While the above models seem to produce sensible results, they all rely on fairly questionable artificial assumptions. In this paper, we propose a more physically sound picture of swollen ionomers that avoids such assumptions. The main purpose is to demonstrate that that Schroeder’s paradox may indeed have a thermodynamic origin using a realistic picture of hydrated polymer. Our main premise is that Schroeder’s paradox is a direct consequence of the microscopic two-phase structure of these materials, well-established both experimentally and theoretically.9,10,23,24 It is to be stressed that the present model is not intended to replace rigorous approaches, such as self-consistent mean-field (SCMF) calculations,25,26 molecular dynamics,27,28 dissipative particle dynamics,29 or atomic structure calculations,30-32 nor does it aim at full analysis of the morphology of ionomers or the phase diagram. Rather, it attempts to offer the simplest possible phenomenological treatment, still capable of capturing the most essential physical features that help to understand its hydration and, to some extent, structural evolution. In the current situation, where even the most advanced models require significant computational efforts, yet still unable to address some important effects, such as matrix crystallinity or chain mobility constraints, the present model incorporates such effects in an approximate way suitable for semiquantitative analysis and practical engineering calculations. We start from construction of an expression for the total free energy and the chemical potential of water in a bulk polymer. This treatment is then extended to include surface equilibrium and is shown to result in Schroeder’s paradox. Finally, we compare the model with other available models and recent experiments and discuss practical implications including analysis of hydration, microstructure, and water transport in Nafion. 2. Theory 2.1. Free Energy in Ionomer Bulk. In the following, the solvent is water and the polymer is a hydrophobic backbone with attached ionic groups. Nevertheless, the model should apply to any strongly interacting solvent and strongly interacting and strongly solvated groups attached to a weakly interacting polymer backbone. The behavior of such systems closely

J. Phys. Chem. B, Vol. 113, No. 1, 2009 25 corresponds to block copolymers in the so-called strong segregation regime far above order-disorder transition.33,34 Due to the very large difference in interactions, the interface between the microphases is sharp and therefore the fixed ionic groups, counterions, and water molecules are assumed to form a perfectly separated microscopic phase in the hydrophobic matrix. Notably, the ionic groups, e.g., sulfonic groups in Nafion, are part of the polar phase, which therefore exists in the dry polymer as well. It is usually assumed and indeed confirmed by molecular models30 and scattering analysis6,35 that in a weakly to moderately hydrated solid ionomer the ionic species and water aggregate form spherical or rod-like “inverted micelles”. The fixed ionic groups are located at the surface of the aggregates, whose interior is filled with water and free counterions. As hydration increases, the morphology may undergo transition to a bicontinuous phase,36,37 similar to diblock34,38 and linear alternating multiblock copolymers.33,39 At still larger hydrations, the structure may eventually be inverted, i.e., “normal micelles” formed.37,40-42 We write the total change of the free energy of such swollen polymer per unit dry volume as follows:

∆F ) ∆Fo + ∆Fs + ∆Fe

(1)

where ∆Fo is the osmotic (hydration) term that accounts for polar group dilution and all interactions within the aggregates, i.e., between ionic groups, counterions, and water, ∆Fs is the interfacial energy associated with the aggregate-matrix interface, and ∆Fe is the elastic energy of the matrix polymer. The second term may be approximately viewed as the residual free energy associated with incomplete phase separation and the presence of a microscopic interface. Equation 1 neglects the residual translational entropy of aggregates, which is small since it is roughly proportional to 1/Q, where Q is the usually large aggregation number. Given the number of polar groups per unit dry volume n or, alternatively, the polymer volume per group Ve ) 1/n, the macroscopic state of the system in equilibrium is uniquely determined by the average number of water molecules per polar group λ. The osmotic term in eq 1 may be quite complex, particularly when the aggregates are not very dilute, and includes many interrelated types of interactions, such as ion hydration, hydrogen bonding, ion association, etc., as well as the entropy of ion dilution. An approximation used here is that ∆Fo depends only on λ. This assumption essentially separates all interactions dependent on the size of aggregates, i.e., related to the presence of an interface and attachment of groups to the backbone, and lumps them into the second term. The hydration energy ∆Fo per ionic group is then written as

fo ) ∆Fo/n ) go(λ)

(2)

where go(λ) is some decreasing function of λ, whose form is inessential for the following analysis, yet a possible approximate expression is discussed below. The second interfacial term in eq 1 is customarily written as43,44

∆Fs ) γA or

26 J. Phys. Chem. B, Vol. 113, No. 1, 2009

Freger

fs ) ∆Fs /n ) γσ

(3)

where γ is the interfacial tension and A is the total interfacial area per unit dry polymer volume. Similar to eq 2, eq 3 gives this energy per ionic group, where σ is the area per group. The value of γ will be in general dependent on λ but should weakly change for the more important range of moderate and large λ, when the aggregates are mostly composed of water. Note that both ∆Fo and ∆Fs monotonously change with λ and σ, respectfully. To ensure existence of a finitely hydrated microscopic phase at equilibrium, i.e., a minimum of the free energy both at a finite volume (λ) and a finite surface area (σ), the elastic term must contain contributions depending on both λ and σ, i.e.,

fe ) ∆Fe/n ) fe(λ,σ) Surprisingly, the previous models of swollen ionomers usually introduced in various ways only the λ-dependence of the elastic energy. However, the σ-dependence was introduced for dry ionomers (λ ) 0).43,44 The present study essentially removes this inconsistency for swollen ionomers as well. Schroeder’s paradox naturally arises as part of this analysis as a thermodynamic phenomenon. 2.2. Bulk Characteristics of an Ionomer: General Consideration. The chemical potential of water and structural characteristics of an ionomer are found by minimizing the free energy per group

f ) fo+fs+fe)go(λ) + γσ+fe(λ,σ)

(4)

with respect to λ and σ. A Lagrange multiplier µsthe chemical potential of solventshas to be introduced to account for equilibration with an external reservoir of solvent; therefore, it is the Gibbs energy g ) f - µλ that is actually minimized. It is convenient to split up fe into two terms, one independent of σ and the other containing all σ-dependence, i.e.,

fe)fd(λ) + fc(λ,σ)

(5)

in which only the second term may actually counterbalance fs, whereas the first may only oppose the volume expansion of the polymer together with fs. Minimizing first with respect to σ at a given λ yields

γˆ )

∂[f + f ] ( ∂σ∂g ) ) ( ∂σ ) s

λ

e

λ

)γ+

( ) ∂fc ∂σ

λ

)0

(6)

Since all groups are assumed to be located at the interface and their total number is fixed, the first equality is in fact the thermodynamic definition of surface tension; therefore, γˆ may be viewed as the composite interfacial-elastic tension that must be zero to ensure thermodynamic stability of the microscopic interface. (Recall that a microscopic fluid phase with γ > 0 is always unstable.) Equation 6 may be solved for σ to yield the dependence σ(λ). Consequently, the chemical potential of water is obtained by minimization with respect to λ at constant σ

µ(λ) )

∂f ∂λ

|

) σ)σ(λ)

|

dgo dfd ∂fc ) µo(λ) + + + dλ dλ ∂λ σ Vπd(λ) + Vπs(λ, σ) (7)

The three terms in eq 7 have a clear physical meaning: µo(λ) is the osmotic (hydration) part of the chemical potential of water in the polymer. The second and third terms both counterbalance the osmotic term, and are represented as a product of V, the molecular volume of water, and some pressure. The elastic pressure πd(λ) is associated with elastic expansion of the matrix and is directly calculated from the expression for fd(λ). The pressure πs, obtained by substituting σ(λ) into the last term in eq 7 after partial differentiation, depends on interfacial tension γ through σ(λ) and on hydration through both λ and σ(λ). This pressure is intimately related to the microscopic interfacial equilibrium and may be interpreted as a modified Laplace pressure involving both interfacial and elastic contributions. Note the crucial difference between the two pressures: πd is the average extra pressure in the whole polymer phase including both matrix and aggregates in excess of the pressure in the external phase, whereas πs, similar to the regular Laplace pressure, is the extra pressure within the aggregates in excess of the pressure in the matrix phase. 2.3. Bulk Characteristics of an Ionomer: Specific Relations. Realistic approximate equations of the above general form may be constructed based on previously considered limiting cases. One limiting case is when fs + fc negligibly varies compared to fd associated with elastic expansion of the matrix. In the absence of microphase separation, the volume-dependent elastic energy for swollen covalently cross-linked gels is usually calculated using the Flory-Rehner theory or its later modifications. This classic model assumes that the whole gel phase expands affinely; i.e., polymer strands are homogeneously stretched.16,45 However, in the present case of microphaseseparated polymer, the matrix surrounding expanding aggregates is incompressible and free of solvent and therefore it maintains a constant volume and is deformed in a nonaffine manner resembling inflation of a balloon.46 The inflation will be restricted for a covalently cross-linked matrix. However, for an un-cross-linked transiently elastic (i.e., viscoelastic) matrix, characteristic of ionomers, increasing hydration will gradually transform the matrix from an inflated network of spherical or cylindrical partitions separating isolated aggregates to a web of stretched connected rods embedded in a continuous aqueous phase and, ultimately, to a liquid dispersion at very high hydrations. Such transformation will however be possible only at time scales exceeding the relaxation time of the matrix polymer. Within a narrow hydration range for times much shorter than relaxation times and negligible interfacial energy, this elastic energy may be calculated by assuming that partitions are elastically uniaxially compressed or rods are uniaxially stretched. This yields a linearized expression46,47

Fe)Fd ≈ Gef(1 - φ0)(φw - φ0)2 or per group

fe)fd ≈ GefVe(1 - φ0)(φw - φ0)2

(8)

Here, φw ) Vλ/(Ve + Vλ) is the water volume fraction in the polymer, φ0 is the value of φw corresponding to the relaxed state of the matrix, and Gef is the effective elastic modulus of the matrix depending on its morphology and φw. Thus, for a

Hydration of Ionomers and Schroeder’s Paradox in Nafion

J. Phys. Chem. B, Vol. 113, No. 1, 2009 27

hypothetic incompressible, isotropic, and linearly elastic matrix of shear modulus G, Gef may be shown to assume the value 2 /3G for dilute spherical aggregates46 that eventually decreases down to 1/6G for a network of connected rods47 and vanishes when the matrix breaks up to a colloidal dispersion of normal micelles. The limiting value Gef ) 2/3G for the relaxed dry state (φw , 1 and φ0 ) 0) agrees with other models proposed for this type of energy for isolated spherical aggregates.46,48-50 Admittedly, accounting for non-uniformity of inflation strain may lead to a larger Gef than this estimate, yet eq 8 will remain valid. For an un-cross-linked transiently rigid matrix, it is more expedient to view Gef and φ0 as phenomenological parameters depending on pretreatment, i.e., thermal and/or hydration history. This yields the elastic pressure

∂fd πd ≡ ≈ 2Gef(φw - φ0) ∂(Vλ)

(9)

Notably, for φ0 ) 0 (dry matrix is relaxed), πd changes linearly for small φw and vanishes at φw ) 0, whereas in the classic Flory-Rehner or related models of homogeneous gels πd is always finite, since the elastic energy is roughly proportional to Gφw rather than to Gφw2.16,45 The other limiting case is when the (un-cross-linked) matrix completely lacks rigidity, i.e., Gef hence fd and πd are zero. For strongly microscopically segregated polymers such as ionomers and block copolymers, a σ-dependent elastic energy emerges in the so-called “corona” region adjacent to the microscopic interface, in which backbone chains have to be stretched to allow aggregation of ionic groups.43,44,51 The energy of stretching rapidly increases with the size of aggregates, which ensures that the phase separation remains microscopic. The existence of such highly stretched areas around aggregates in dry ionomers is well established.52 Dreyfus introduced this energy in an approximate way.40,51 A rigorous approach was first developed by Semenov34 for diblock copolymers with strongly segregated blocks and was later extended to dry ionomers by Nyrkova et al.43,44 The latter model in an approximate form is applied here to hydrated ionomers. This eleastic energy per ionic group, hereafter referred to as the “corona” energy, increases with the local radius of curvature of the micelle-matrix interface (R) and decreases with the interfacial area per chain (σ). For smaller σ, the chains in the corona are more crowded and thereby more strongly stretched, while R determines the distance from the interface, over which this stretching extends. For ideal chains in the so-called strong segregation regime, this energy may be approximate expressed as (see the Appendix)34,43,44

fc ) BR/σ

(10)

where B is an elastic constant specific for the polymer and micelle geometry. Remarkably, with only somewhat varying B, the same expression applies both to inverted and normal micelles (i.e., concave and convex interfaces) and, up to the leading terms, to spherical, rod-like, and lamellar geometries (in the latter case, R is the lamellae thickness), as explained in the Appendix. This result, originally derived for diblock copolymers,34 also applies to telechelic ionomers with ion groups at both ends or long-chain ionomers bearing many such groups spaced along a backbone chain or connected to it through side chains, as in Nafion and other practically important ionomers. In such cases, B is only expected to increase compared to

Figure 1. Schematic 2D representation of the structure of a real solid ionomer at low hydrations: (1) liquid aqueous phase inside micelles; (2) hydrophobic matrix; (3) stretched matrix domains (corona) adjacent to the micelle surface; (4) matrix crystallites serving as cross-links; (5) middle planes of partitions separating micelles and subject to elastic stretching upon inflation.

telechelic ionomer by a constant factor. To avoid inessential complications, in the following approximate treatment, B is viewed as a polymer-specific phenomenological parameter weakly dependent on the micelle geometry. Apparently, even in the absence of chemical cross-links, real long-chain ionomers, such as Nafion, may have a finite Gef at experimentally accessible time scales and thus experience both types of elastic strain. Partial crystallinity, glassiness of the matrix, or entanglements may slow down relaxation and lead to finite rigidity over realistic times. Moreover, the ionic aggregates themselves may serve as transient cross-links, and therefore, long backbone chains sharing many aggregates may from “bridges” when the matrix is under stress, similar to linear multiblock copolymer.53 Such a transient modulus should then strongly depend on the thermal, hydration, and mechanical history of the sample. The regions experiencing the two types of strain, inflation and corona, must be somewhat separated in space and the character of local strain is different, as illustrated in Figure 1. Inflation moves farther apart the crystalline or glassy domains serving as physical cross-links, and thus, nearly affine local deformation develops in the areas adjacent to each dashed line and bounded by nearby micelles (Figure 1). On the other hand, groups pulled out of the surrounding corona create a permanent localized strain that is superimposed on the inflation strain and, as will be shown, actually decreases with hydration. Most of the corona energy is then contained in the vicinity of the curved interface,43 whereas the inflation energy is more or less evenly spread over the whole matrix. The inflation energy fd may be enough to preserve the macroscopic shape of the sample and prevent dissolution upon hydration but may still be unable to significantly distort the microscopic interface. The local microscopic configuration of micelles may be insensitive to the macroscopic rigidity of the matrix and only depend on the level of hydration. This assumption is obviously questionable, and its analysis is beyond the scope of this paper. However, it seems to hold true for Nafion, since solid hydrated Nafion (Gef > 0) and its solutions (Gef ) 0) show similarly positioned “ionomer” peaks in SAXS spectra at the same hydrations.36 The recent analysis by SchmidtRohr and Chen also indicates that the crystallites serving as physical cross-links in Nafion are significantly more widely spaced than water channels35 and are then expected to weakly affect the immediate vicinity of the micelle surface. It might


Freger

R ) b(Ve - Vg)φ/σ

(13)

then be reasonable to accept this assumption as a crude model. This amounts to assuming that πs may be calculated as for a nonrigid matrix. The inflation and corona energies both act to compress the aggregates and thus oppose the hydration; therefore, the pressures due to the two effects (i.e., normal stress at the interface) are added up. Since the pressures, πd and πs, are derivatives of the respective energies (cf. eq 7), in the following, it is taken as an approximation that

Equations 4, 8, 10, 11, and 13 result in an expression for the total free energy with two independent variables σ and λ. The minimization procedure outlined in section 2.2 then yields the equilibrium values of σ and R at given λ (or φ)

σ(λ) ) {2b(Ve - Vg)B/γ}1/3φ1/3 ) σ∞φ1/3

(14)

fe ≈ fd + fc

R(λ) ) {b2(Ve - Vg)2γ/2B}1/3φ2/3)R∞φ2/3

(15)

(11)

where fd and fc are given by eqs 8 and 10, respectively. Equation 11 is then formally identical to eq 5. Note that the possibly related elastic constants Gef and B involved in fd and fc are viewed here as independent phenomenological parameters. The rigorous condition γˆ ) 0 (eq 6) is independent of the presence of fd in the total free energy; otherwise, the microscopic interface will be unstable. This means the corona chains may remain stretched for an infinitely long time, perfectly counterbalanced by the interfacial tension. This emphasizes another principal difference between fd and fc and hence πd and πs. The pressure πs will arise even in a perfectly rubbery un-cross-linked matrix and at given hydration does not tend to decrease with time. On the other hand, the pressure πd is crucially dependent on the presence of some kind of cross-links. Without permanent cross-linking, the matrix will gradually yield to the inflation stress; thus, πd will decrease and hydration will increase with time, slowly approaching that of an ideal nonrigid ionomer. This means that experimentally observed hydration of a solid ionomer may be far from the true equilibrium state, subject to the initial state of relaxation of the matrix and time scale of equilibration experiments. This may be accounted for through historydependent phenomenological parameters Gef and φ0. Minimization of the resulting total free energy has to account for the fact that the variables λ, σ, and R in eq 10 are not independent. Since the total volume per group is Ve/(1 - φw) ) (Ve - Vg)/(1 - φ), where Vg is the volume of an ionic group with counterion and φ ) (Vg + Vλ)/(Ve + Vλ) is the total volume fraction of the polar microphase including the ionic groups, the interfacial area per unit total volume is σ(1 - φ)/(Ve - Vg). By viewing the polymer as a random symmetric two-phase system, in the full range of hydrations, the specific area may be approximately calculated as follows54,55

σ(1 - φ)/(Ve - Vg) ) bφ(1 - φ)/R

(12)

where b is a numerical coefficient depending on the aggregate dimensionality and geometry. The factor φ(1 - φ) describes the area in a random symmetric two-phase system per total volume, hence the factor 1 - φ on the left-hand side. Viewing the system as a 2D or 3D Voronoi tessellation, in which the fraction φ of Voronoi cells is occupied by the aqueous phase, the value b ) 1.455 may be proposed for spherical micelles. This value applies to random 3D Voronoi polyhedral cells of density (4R)-3,56 and well compares to b ) 1.5 for a dilute dispersion of spheres of mean curvature 1/R, i.e., diameter 4R. Similarly, for the 2D case, a natural suggestion is b ) 2, which is exact both for random Voronoi polygons of density (2R)-2 and for dilute circles (or aligned cylinders) of diameter 2R.56 Equation 12 then yields

Adelbert et al. derived the theoretical relation σ ∼ γ-1/3 for dilute micelles in solution along similar lines.40 They also demonstrated its semiquantitative agreement with experiment for solutions of Nafion in different polar solvents. Also, R ∼ Ve2/3 is equivalent to the well-known dependence of the microdomain size for block copolymers on the molecular weight of the block in the strong segregation regime.34,38 It is seen that both σ and R increase with hydration. The increase in σ also means that chain crowding and local corona stretching decrease with hydration, though the total corona energy per group (eq 10) increases. The parameters σ∞ and R∞ have the meaning of limiting values at very high hydrations, i.e., when Vλ . Ve, φ f 1, and σ and R become independent of λ. The aggregates in this situation assume the configuration of normal micelles, and the ionomer breaks up to a dispersion or forms a very loosely connected network of micelles. The existence of normal-type micelles in solution was confirmed for Nafion and similar ionomers.9,37,40 The effective values of σ∞ and R∞ ) b(Ve - Vg)/σ∞ are likely to somewhat vary with hydration, if γ and B vary. In such a case, subject to relevant hydration range, effective σ∞ and R∞ accessible from scattering experiments57 may conveniently replace “molecular” phenomenological parameters γ and B. In the other limiting case of small hydrations Vλ , Ve or φg < φ , 1, where φg ) Vg/Ve is the volume fraction of dry aggregates (usually small), the aggregates are expected to assume the inverted micelle configuration with the aqueous microphase inside. The transition from convex to concave interface (looking from the matrix side), i.e., from inverted to normal micelles, around φ ∼ 0.5 is actually embedded in eq 12 and subsequent relations. After substituting the equilibrium values of σ and R, the free energy becomes

3 3 f ) g0(λ) + fd(λ) + γσ(λ) ) g0(λ) + fd(λ) + γσ∞φ1/3 2 2 (16) The factor 3/2 results from scaling fs ∼ σ and fc ∼ σ-2 (eqs 10 and 13) and may change if the scaling of fc with R and σ is modified (eq 10). The chemical potential then has the form of eq 7, in which πd is given by eq 9 and the modified Laplace pressure πs is

πs )

1 γσ∞ -2/3 bγ φ (1 - φ)2 ) (1 - φ)2 2 Ve - Vg 2R

(17)

It is seen that πs decreases with φ and vanishes as φ f 1. When πs vanishes, the chemical potential of water is fully determined by the σ-independent terms


µ ≈ µo(λ) + Vπd(λ)

(18)

Solving this equation for λ gives a limiting σ-independent hydration designated λ*(µ) and used below. In the special case of an ionomer with a nonrigid matrix, i.e., with Gef ) 0, referred to as an “ideal” ionomer, the function λ*(µ) is simply reciprocal of the osmotic term µo(λ), meaning that at φ f 1 the attachment of ionic groups to the backbone has no effect on the chemical potential of water. 2.4. Superstrong Segregation vs Strong Segregation. It was postulated43,58 that in dry ionomers the interfacial area per group predicted by eq 14 might reach the limit σmin imposed by the densest packing of the groups or maximal stretching of the corona chains, e.g., when the strands connecting the groups are too short. This regime called the superstrong segregation regime43 is characterized by a constant σ ≈ σmin independent of λ. In this regime, the “Laplace” pressure πs exists as well, but it is entirely due to the corona energy. The modified Laplace pressure and aggregate size then become

πs )

bB (1 - φ)2 2 σmin

R ) b(Ve - Vg)φ/σmin


Figure 2. Schematic representation of micelles at the ionomer surface: (a) a hypothetical open micelle at the surface facing a vapor phase; (b) an equilibrium micelle “sunk” under the surface facing a vapor phase; (c) a hypothetical “flat” surface facing a liquid phase; (d) a transitional state at the surface facing a liquid; (e) equilibrium state at the surface facing a liquid (normal-type micelles)sthe micelles will stay connected for a rigid matrix. The round shapes are schematic and may represent cross sections of both spherical and cylindrical structures.

(19)

(20)

πs changes more slowly in this regime than in eq 17. However, as hydration increases, σ might eventually exceed σmin (cf. eq 14). In such a case, the system will cross over to the strong segregation regime and πs will decrease further according to eq 17. In either case, πs vanishes at φ ) 1. 2.5. Equilibrium at the Ionomer Surface and Three-Phase Line. As depicted in Figure 2a, at the surface of a swollen ionomer there may in general be three different phases: the hydrophobic matrix, vapor (if the polymer contacts water vapor), and a solution phase that may be internal (inside the aggregates) and external (if the polymer contacts an external solution). Correspondingly, there are three interfacial tensions involved: solution-vapor γ1, matrix-vapor γ2, and matrix-solution γ12 (Figure 2a). The equilibrium configuration at the surface should in general be found by functional minimization of the total energy including interfacial energies of all interfaces under appropriate constraints. The constraints should conserve the polymer volume (i.e., λ) and, at a given λ, the macroscopic surface of the sample. In general, the shape of all interfaces is found by solving a corresponding Euler-Lagrange equation.59 In the case of fluid phases under negligible gravity, the well-known solution is that each interface maintains its own constant mean curvature and a constant Laplace pressure in each phase. In addition, the three tensions at the three-phase contact line must satisfy the Weierstrass-Erdmann condition60 or Neumann construction,61 i.e., form a force triangle, otherwise the three-phase contact may not exist and one interface will disappear. The matrix elasticity complicates the situation, since if the microscopic stress in the matrix phase around an aggregate is nonuniform and anisotropic, then not all curvatures are necessarily constant. For instance, consider a hypothetical “open” inverted micelle at the surface of an ionomer facing a vapor phase (Figure 2a). In general, the tangential stress in the matrix around a spherical or a cylindrical micelle decays with the distance from the micelle (see the Appendix); therefore, the varying tangential stress will have to be balanced by a varying

curvature of the external matrix-vapor interface. This complication is however inessential, since it is easily seen that an “open” micelle will always be unstable for a viscoelastic matrix. Indeed, γ12 is identified with γˆ , which must be zero at any location on this interface (to ensure its stability), including the three-phase line. Since in general γ1 * γ2, the Neumann construction is impossible. As the surface tension of water is much higher than that of a hydrophobic matrix, γ1 > γ2, the matrix phase must “spread” over the unstable “open” micelle; i.e., the latter must “sink” under the surface to become a regular “closed” bulk micelle (Figure 2b). Disregarding fluctuations, the surface will then be entirely composed of the matrix-vapor interface, which indeed agrees with experimental observations.62,63 For G ) 0 or for a G value only sufficient to keep the macroscopic shape of the polymer sample but weakly affecting the local interfacial equilibrium (see section 2.3), the external surface of the sample will be nearly liquid-like and microscopically flat; therefore, the pressure in the matrix will be the same as in the external medium. Given µ, the chemical potential of water in the external vapor phase, the hydration, and structure of the ionomer will be described by the bulk equilibrium equations of section 2.3. 2.6. Surface Phase in a Liquid-Equilibrated Ionomer. In the case of liquid equilibration, the situation at the external surface will be different from exposure to vapor. In this case, if γ1 ) 0, then γ2 will be the largest interfacial tension. Like surfactant molecules in water, the groups will “rise” to or “adsorb” on the polymer surface that will become similar to the internal matrix-micelle interface. Unlike a surfactant monolayer on water, a flat external surface (R ) ∞) schematically shown in Figure 2c will be unstable. Instead, to minimize the free energy, the groups at the surface facing an infinite solution will tend to assume the limiting values σ ) σ∞ and R ) R∞ and the configuration of normal micelles. This may be explicitly seen by viewing the polymer surface as a separate surface “phase” and, starting from the hypothetical flat surface shown in Figure 2c, writing its free energy per group as

˜f(λ˜ ) ) go(λ˜ ) + fd(λ˜ ) + γσ˜ +

BVe σ˜ 2

(21)


Freger

where the tilde designates the values in the surface “phase” and the corona term is obtained by replacing R in eq 7 with the approximate thickness of the stretched domain near the surface Ve/σ˜ and neglecting the inessential numeric factor (equal to 1 for the flat interface in Figure 2c). To find the equilibrium values, we should minimize a Gibbs energy that conserves the total projected (macroscopic) surface area of the sample A0, the number of water molecules, and the number of groups in the surface phase. (Note that the number of groups in the surface phase is not fixed as in the bulk phase.) Correspondingly, this requires three Lagrange multipliers: the effective surface tension of the hydrated surface γ˜, chemical potential of water µ, and chemical potential of groups η. γ˜ may be omitted by minimizing the Gibbs energy per unit A0 area, i.e.,

(

)

BVe ˜ G 1 A˜ ) go(λ˜ ) + γσ˜ + 2 - µλ˜ - η A0 A0 σ˜ σ˜ ˜ BVe go(λ) - µλ˜ - η A˜ ) γ+ 3 + A0 σ˜ σ˜

(

)

(22)

where A˜ is the area of the (curved) interface between the matrix and external solution; therefore, A˜/σ˜ is the total number of groups in the surface phase. To minimize the energy, the surface will curve (Figure 2d); i.e., A˜/A0 will change. Minimization with respect to A˜/A0, σ˜, and λ˜ yields

σ˜ (λ˜ ) ) (3BVe /γ)1/3

(23)

µ˜ (λ˜ ) ) µo(λ˜ ) + Vπd(λ˜ ) ) µ

(24)

It is seen that σ˜ coincides with σ∞ from eq 14 within an inessential numerical factor and λ˜ is identical to λ*(µ), in which the effect of πs vanishes (cf. eq 18 and the following discussion). This indicates that the surface will tend to break up to separate “islands” that assume the configurations of dilute normal micelles (Figure 2e). If the matrix in the surface domain retains rigidity, the micelles will stay connected; therefore, inclusion of the πd term in µ˜(λ˜ ) is important. Since the surface phase is in equilibrium with the bulk and the ionic groups have freedom to exchange between the surface and bulk regions, it must always hold that

µ˜ (λ˜ ) ) µ(λ) ) µ

(25)

η˜ (λ˜ ) ) η(λ)

(26)

Here µ(λ) ) df/dλ and η(λ) ) f - λ(df/dλ), and therefore, µ(λ) and η(λ) are both fully determined by the same function f(λ) and identical relations hold between µ˜ , η˜ and ˜f. Substituting these relations into eqs 25 and 26 and differentiating with respect to λ, it is easily shown that surface and bulk groups must have identical free energies and identical hydrations:

λ ) λ˜ ) λ*(µ)

That means that the effect of extra Laplace pressure πs on ionomer hydration must Vanish for liquid equilibration. The vanishing of πs in the bulk results from the fact that the whole matrix phase, everywhere bounded by the matrix-solution interface, is under a pressure lower by πs than the solution phase. In the process of equilibration with a liquid solution (i.e., transition from Figure 1d to 1e), external and internal interfaces have to adopt the same configuration to equalize the pressure in the matrix. The excess pressure πs inside the aqueous aggregates in the bulk is then exactly compensated by the pressure in the matrix lower by ∆P ) πs than in the external solution; therefore, the external and internal liquid phases are under the same pressure. Note that this adds to the bulk free energy f(λ) a term of the form -∫V∆P dλ that ensures eq 27. The final equilibrium state in liquid will obviously be different from the hydration in vapor of the same µ, since then the external surface is approximately flat and the matrix is under the same pressure as the external medium (∆P ) 0), whereas the pressure inside the aggregates is larger by πs. This results in Schroeder’s paradox, which could then be no more paradox than the Kelvin-Laplace equation predicting that the chemical potential of liquid in a droplet differs from bulk liquid. It is instructive to consider the case of an ideal nonrigid ionomer placed in pure liquid water. In this case, πs and πd both approach 0 and the ionomer should form a dispersion of micelles. Since without covalent cross-links all ionomers are viscoelastic, i.e., have G ) 0 under infinitely lasting load, such ionomers are ultimately expected to break up to a dispersion in water. For some ionomers and conditions, however, this may never be observed, if the relaxation time exceeds any sensible time scale of experiments. 2.7. Construction of Hydration Isotherms and a Possible Form of µo(λ). In order to use eq 7 in practical calculations or model fitting, an expression for µo(λ) or, equivalently, λ(µο) should be explicitly given. A reasonable option, suitable for concentrated ion-water mixtures, could be the BET isotherm. This isotherm, first proposed for multilayer surface adsorption,64 describes well the activity of water in concentrated electrolyte solutions.65 This success may be explained by the parallels between the first adsorption layer in BET and the primary hydration of ions, as well as water-like interactions more weakly affected by the ions in the higher adsorption layers. The standard BET model poses no limitations on the number of layers, which becomes infinite at saturation, and has the following functional form

λBET(a) ) λh

Ca (1 - a)(1 + Ca - a)

(28a)

Here λh and C are parameters characterizing, respectively, the capacity of the first hydration layer and the strength of primary hydration, and a is the activity of water (relative humidity) defined as a ) exp[(µ - µ0)/kBT], with µ0 being the chemical potential of pure water. In the case of hydrated ionomers, the right-hand side of eq 34 is modified by simply replacing a with a product ka, i.e,

λo(µ) ) λBET(ka) ) λh f(λ) ) ˜f(λ˜ )

(28)

Cka (29) (1 - ka)(1 + Cka - ka)

(27) where k is an “activity coefficient” given by


(

k ) exp -

)

(

µ - µo(λ) V{πs + πd} ) exp kBT kBT

)

(30)

The sorption isotherm is constructed by solving eq 29 with k given by eq 30, in which λ appears at both sides through πd and πs. Equations other than eq 28 may also be used and treated similarly. Note that no artificial limitation, such as enforcement of a finite number of layers in the BET model,18,19 is necessary to ensure a finite λ for a solid material at a ) 1. Importantly, for liquid equilibration, the effect of πs vanishes. Therefore, the relation

( )

k ) exp -

Vπd kBT

(31)

has to be used instead of eq 30. This indicates that Schroeder’s paradox, originally defined for pure solvent and saturated vapor, may be generalized and lead to two different isotherms, depending on whether the external reservoir of solvent with varying activity is vapor or liquid. In regimes where k is nearly constant, e.g., in the superstrong segregation regime, the model becomes equivalent to the socalled modified BET or GAB isotherm that was found to successfully describe water uptake by many polar polymers66 including ionomers.67 3. Discussion. The Case of Nafion Due to the importance of Nafion, its structure, hydration, and other properties have been very extensively studied, both experimentally and theoretically,7,30,68 which allows direct comparison with the present model. The subsequent discussion therefore mostly focuses on this material. 3.1. Comparison with Previous Models. It is easy to see that the two basic conclusions of the present model, the presence of a modified Laplace pressure πs in vapor equilibration and its vanishing for liquid equilibration, qualitatively coincide with the basic premises of the Choi and Datta model.18,19 This seems to provide some basis for this model and explain its success in fitting the hydration data. However, a closer inspection shows one fundamental difference with the present model. While the former associates the Laplace pressure with the interface between the internal liquid and external medium, the present one shows it is actually associated with the interface between the internal liquid and polymer matrix. Even though πs varies as the reciprocal aggregate radius in the strong segregation regime, i.e., similar to the regular Laplace pressure of a liquid interface, the factor -γ1 cos θ, where θ is the water-matrix contact angle, relating πs to R-1 in Choi and Datta’s model is replaced here with a different parameter (1 - φ)2(bγ/2) (eq 17). The similarity between the two models is then superficial; it is actually nonexistent in the superstrong segregation regime. One may also recognize parallels between the conclusions of section 2.6 and Weber and Newman’s postulation that contact of vapor-equilibrated Nafion with liquid eventually changes the microstructure and chemical potential of water in the whole sample, starting from the surface.21 Indeed, the external surface, previously depleted of ionic groups, becomes unstable and undergoes a transition setting up a thermodynamic driving force (“capillary pressure gradient”) propagating into the sample, which ultimately leads to a new equilibrium state, i.e., Schroeder’s paradox. Again, the present analysis suggests that the capillary pressure, which Weber and Newman associated with

J. Phys. Chem. B, Vol. 113, No. 1, 2009 31 a hypothetical vapor-liquid interface,22 is actually associated with the external and internal matrix-liquid interfaces. 3.2. Hydration Experiments and Testing of Schroeder’s Paradox. Observation of Schroeder’s paradox has been nearly always conducted by equilibrating Nafion and other ionomers with water or its vapor at a ) 1, as in Schroeder’s original work. Such studies were reviewed recently and many indeed report a difference between vapor and liquid equilibration.3,69 However, recent careful experiments by Onishi et al.69 claimed complete absence of Schroeder’s paradox at a ) 1 and different temperatures when the thermal history was properly taken into account. A similar view was suggested earlier by Cornet et al.70 It may be noted that the case of a ) 1 is distinctly different from equilibration in unsaturated conditions, both experimentally and theoretically. First, an ideal nonrigid ionomer is infinitely hydrated at saturation (φ ) 1) in both regimes, since in this case πs ) 0 (eqs 17 or 19). This means that, if the polymer dissolves, Schroeder’s paradox disappears at saturation. However, if it remains solid (G > 0 and φ < 1), Schroeder’s paradox is expected for a ) 1 as well. Measurements at a ) 1 may also present an experimental problem, since a thin film of condensed vapor may readily form on the ionomer surface and, once formed, will be difficult to prevent or remove. In this case, the vapor equilibration conditions become indistinguishable from liquid equilibration and no paradox can be observed. A solid polymer pre-equilibrated in liquid may have its ionic groups “sticking out”, as in Figure 2d or e, for some time. If a film forms at the surface while it is still hydrophilic, it will be “pinned” by the surface groups and may neither dewet nor let the surface groups rearrange and “sink” below the surface, as in Figure 2b. Such surface film of water may be microscopically thin and visually undetectable. The thin liquid film may be further stabilized by “dissolving” Nafion. Initial condensation of water introduces large gradients of πs at the surface that will “pull” micelles out of the matrix (cf. Figure 2d and e), and thus promote formation of a stable Nafion dispersion in a thin surface layer. Since the gradient of πs develops first at the surface, the surface layer may reach this fully relaxed liquid-like state upon prolonged exposure to saturated vapor long before such relaxation is attained in the entire bulk. Once formed, this layer will then be infinitely stable, just as Nafion solutions, and impossible to remove without drying the surface for a time necessary to let the surface groups rearrange and “sink” below the surface, as in Figure 2b. The sample permanently kept at saturation may then eventually approach the hydration of liquid equilibration. This will occur despite the fact that most of the polymer will be in a nonrelaxed solid state, for which Schroeder’s paradox is expected. The ambiguity related to likely artifacts at saturation may be removed by using unsaturated conditions a < 1. While vapor isotherms have been routinely measured in this range, corresponding liquid-equilibrated isotherms were measured only recently using an osmotic stress technique, i.e., using solutions of a solute (stressor) strongly excluded from the polymer and only acting to reduce solvent activity.71 Availability of isotherms has an additional advantage that it allows for experimentally estimating the (modified) Laplace pressure πs, if the same hydration may be obtained in two modes at different water activities. The results indeed demonstrated the drastic difference in the levels of hydration between the two equilibration modes in contrast to the conclusions by Onishi et al.69 The Choi and Datta model18,19 was originally used for interpretation; however, the difference between isotherms greatly exceeded any sensible model predictions. The large discrepancy found could be

32 J. Phys. Chem. B, Vol. 113, No. 1, 2009 interpreted by assuming much smaller aggregate radii at the surface, as compared to the bulk. However, the present analysis leaves room for another interpretation, namely, that liquidequilibrated solid Nafion is not in true thermodynamic equilibrium; therefore, large differences in πd, as well as πs, may be involved. Indeed, many published reports14,15 and our own experiments indicate that Nafion reaches equilibrium with vapor within 1-3 days and within no more than a few hours in liquid. This relatively rapid (probably mass-transfer-limited) process is followed by a much slower relaxation, which is often ignored, since it is extremely slow in vapor at ambient temperatures but may accelerate in liquid or when the temperature increases. Expansion or shrinking of Nafion by preboiling or drying at high temperature69,72 may be viewed as a kind of such accelerated relaxation and dissolution at around 200 °C36 as its extreme case. The equilibration times in experiments of Onishi et al. were much shorter in liquid (1 h to 2 days) than in vapor (up to 2.5 months),69 suggesting that vapor equilibration may have involved a different, very slow relaxation process. Although the authors could not observe liquid on the sample surface and dried the expanded (preboiled) samples in ambient air for several minutes prior to vapor equilibration, formation of a stable microscopically thin liquid film in these experiments on the surface cannot be ruled out. The drying time of several minutes could completely remove surface water yet could be insufficient to let the groups rearrange. The saturation conditions and very long vapor equilibration times could eventually make the vapor and liquid equilibration indistinguishable. On the other hand, the equilibration times in ref 71 were much shorter for vapor equilibration (4 days) compared to the liquid (osmotic stress) experiments that lasted 7-31 days required to ensure that the osmotic stressor solution had attained the desired water activity. In this situation, the samples in solution were exposed to larger and longer internal stresses. As a result, πd could drop, which could increase the difference between the liquid and vapor isotherms much beyond πs. Notably, the results reported in the same study for permanently cross-linked Dowex resins, in which πd for given λ was unlikely to change with time, exhibited Schroeder’s paradox that could be reasonably interpreted in terms of πs alone. 3.3. Segregation Regime in Nafion. An interesting question is the actual segregation regime in Nafion. This question may be important for quantitative modeling of real systems. The most convenient fingerprint is the parameters σ and R deducible from scattering experiments (SAXS and SANS).57 Equation 13 predicts that at all hydrations and irrespective of micelle type and geometry σ should be proportional to γ-1/3 in the strong segregation regime, while in the superstrong segregation regime it will be independent of γ. Experiments of Aldebert et al. showed that in solutions of Nafion in different solvents σ is indeed proportional to γ-m, with m ≈ 1/4,40 only somewhat smaller than the ideal value m ) 1/3. This dependence suggests that the regime in Nafion solutions is closer to strong segregation. The small deviation could be explained by nonideality of the short and stiff Nafion chains and partial crystallinity of the matrix. On the other hand, σ in hydrated solid Nafion and concentrated solutions was found to be fairly independent of the equivalent volume Ve,37,73 which disagrees with eq 14 and could point to the superstrong segregation. However, it could also be related to the strong nonlinear rigidity of the matrix. Also, most studied systems were quite significantly hydrated; therefore, the

Freger factor φ1/3 in eq 14 could only slightly vary. Overall, the variation of σ with γ seems to indicate that the actual regime is closer to the strong segregation. 3.4. Stability and Evolution of Nafion Structure with Hydration. Gebel proposed a general scheme for the evolution of the Nafion structure with hydration, mostly based on scattering data.36 According to it, spherical aggregates (inverted micelles) are formed at low hydration. When hydration increases, a continuous “cluster network”74 is first formed and around φ ∼ 0.5 the continuous perfluorinated matrix breaks up to a network of rod-like micelles of normal type, which at still higher hydrations becomes a liquid dispersion of free micelles. Very recently, this scheme was modified by Schmidt-Rohr and Chen35 who successfully interpreted the thorough scattering and other data by the Grenoble group36,41,42,75 and other results, and concluded that the aqueous aggregates at low hydration have the form of elongated cylindrical channels, apparently stabilized by the stiffness and partial crystallinity of the matrix polymer. The present model readily conforms to this picture, only requiring “plugging in” a 2D morphology in the whole range of hydrations. Interestingly, apart from this recent evidence, the rod- or channel-like structure seems to agree with the long-known experimental value of the linear relation, obtained when the “microscopic swelling”; i.e., increase in the Bragg spacing d relative to the dry state (d0) is plotted versus the linear expansion ratio (“macroscopic swelling”) and the value of initial slope, equal to about 5.5-6.13,36 It has been already pointed out and shown by simulations25,32 that the nearly linear relation between the two may naturally follow from the evolution of the microstructure and does not necessarily indicate a lamellar (1D) morphology, as was previously assumed.76 Indeed, one can derive scaling of the Bragg spacing from the present model using the following arguments. The average volume of a unit cell enclosing one aggregate scales as dD, where D ) 2 for rods (cylinders) and D ) 3 for spherical aggregates. On the other hand, the aggregate surface area scales as RD-1. Using eq 12 for area per volume, we obtain

RD-1/dD = bφ(1 - φ)/R

(32)

d ) d∞φn-1/D(1 - φ)-1/D

(33)

and, using eq 15,

where d∞ = b-1/DR∞ and n should vary between 2/3 (strong segregation) and 1 (superstrong segregation). Since φ includes the group volume, φ ) φw + φg(1 - φw), where φg ) Vg/Ve is the volume fraction of the ionic groups in the dry polymer. The Bragg spacing of dry polymer is then d0 ) d∞φgn-1/D(1 - φg)-1/D, which yields the microscopic swelling for the full range of φ

( ) ( ( )

d - d0 φ ) d0 φg

n-1/D

φ φg

1-φ 1 - φg

)

-1/D

-1)

n-1/D

(1 - φw)-1/D - 1 ≈

[(n - D1 )

]

1 - φg 1 + φ (34) φg D w

where the last approximate equality is the linearized expression for small φw.



Figure 3. (a) Microscopic vs macroscopic swelling for different micelle geometries (eq 34); (b) Bragg spacing of aggregates vs polymer volume fraction φp for D ) 2 (eq 33). For all curves, d∞ was adjusted to fit the value d ) 2.7 nm for dry polymer. For solid curves, a constant d∞ was used, while, for dashed curves, d∞ was allowed to vary linearly with φp from 1.8 to 5.5 nm at very large hydrations.

On the other hand, for isotropic swelling, the macroscopic linear expansion ratio is (1 - φw)-1/3 - 1 ≈ φw/3; therefore,

slope )

(d - d0)/d0 -1/3

(1 - φw)

-1

[(

≈3 n-

]

1 1 - φg 1 + D φg D (35)

)

where the last approximate expression is the initial slope for small hydrations. The calculated microscopic versus macroscopic swelling and the Bragg spacing for D ) 2 and D ) 3 in a wide range of hydrations are shown as solid curves in Figure 3 in a form that allows direct comparison with the scattering data presented by Gebel for solid Nafion and its solutions.36 The plots were built using eqs 34 and 33, respectively, and the volume of the sulfonic group Vg ) 68 Å3 and the dry volume per group of Nafion 117 Ve ) 870 Å3.24,37 The value of d∞ was adjusted to fit the Bragg spacing d ) 2.7 nm for dry Nafion.36 The curves for D ) 2 show a much better agreement with scattering data. First, there is indeed an almost linear relation between the macroscopic and microscopic swelling for D ) 2 in a wide hydration range, whereas it is visually different for D ) 3 (Figure 3a). Second, the predicted scaling exponent -1/D between d and φp ) 1 - φw approached at low φp in Figure 3b is in agreement with Gebel’s value -0.5. This result is obviously not new, since the rod-like geometry of dilute Nafion micelles has long been known. Third, the theoretical value of the initial slope (eq 35) is 7.4 for D ) 2 in the strong segregation regime (n ) 2/3), which is only somewhat larger than the experimental value, whereas for spherical aggregates in the strong segregation regime the slope is nearly twice as large and still larger for superstrong segregation. The only notable deviation is the values of d at large hydrations (low φp) that are about 40% off Gebel’s values. This discrepancy is most probably explained by the variation of the effective parameters γ and B over a wide hydration range, following large compositional changes in the polar microphase. To account for this effect, the dotted lines in Figure 3 are calculated for D ) 2 using the same Vg and Ve but allowing d∞ to linearly vary with φp from 1.8 (to fit d ) 2.7 nm for dry Nafion) to 5.5 nm (to fit the whole dilute range). These curves are in a good agreement with Gebel’s data. It may then be concluded that scattering data may be reasonably explained by the present model provided a 2D morphology of Nafion is adopted, in agreement with the analysis by Schmidt-Rohr and Chen.35

The elongated water channels well explain the exceptionally high ionic conductivity of Nafion or similar perflourinated ionomers, which largely exceeds the conductivity of many other solid polyelectrolytes at similar ionic contents and hydrations.77 This behavior is analogous to the large conductivity and reduced percolation threshold of composites containing elongated conductive particles dispersed in an insulting matrix, e.g., in carbonnanotube-filled polymer composites.78 However, elongated structures, at least at low hydrations, seem to contradict the general phase stability considerations.33,39,58 Not surprisingly, they were not obtained in computer simulations, which predict connected spherical aggregates,25,27,29,32 as in the earlier picture by Hsu and Gierke.74 A likely explanation is that once again one may deal with a frozen nonequilibrium structure that originates from chain alignment and partial crystallinity inside micelles in solution that survives the solution-casting or extrusion process used to prepare solid Nafion membranes. Partial crystallinity or high stiffness of Nafion chains may serve as a kinetic and, to a certain extent, thermodynamic stabilizing factor far below the temperature of crystallization. The strongest evidence in favor of this assumption is the dependence of the properties on thermal and hydration history.69,72,79 Another support comes from the fact that stable Nafion films may be prepared without any crystallinity,80,81 which indicates that crystallites in solid Nafion are not necessarily in an equilibrium state. Due to the very long relaxation in ambient conditions, at experimental time scales, differently pretreated samples may be differently but reproducibly swollen and reswollen,36 similar to permanently cross-linked gels. The relaxation can however be accelerated and stability lost at elevated temperature or under a very large stress. Indeed, at elevated temperatures (>90 °C) and high humidity (i.e., high swelling pressure πd), solid Nafion starts losing its ionic conductivity, which is usually accompanied by some loss of water.7,82 It is likely that the thermodynamically unfavorable long water channels simply break up to more stable yet poorly connected spherical aggregates. The loss of water may then be explained by an increase in the intra-aggregate pressure πs, more weakly opposed by the rigidity of the matrix. For instance, Park et al. showed that samples of Nafion 117 equilibrated at different temperatures at a ) 0.98 first show an increase of conductivity for short equilibration times (0.5 h); however, after prolonged exposure (48 h), apparently, sufficient for structural changes, the conductivity and hydration drop.82 Notably, the drop in hydration after 48 h monotonously depends on temperature in the range 30-90 °C without any sign of sharp transition, which


Freger

suggests that one deals with a relaxation effect steadily accelerated with temperature. 3.5. Modeling of Water Transport. In many applications, such as fuel cells or pervaporation, an ionomer membrane may simultaneously contact both liquid and vapor. The existence of the two types of boundaries should be taken into account in boundary conditions in the diffusion equation for transport of water

J)-

DwC ∇µ RT

(36)

where J is the diffusion flux, C ) Vφw is the concentration of water in the polymer, and Dw is the diffusion coefficient of water that strongly depends on φw.77 If other mechanisms of water transport occur (e.g., electro-osmosis or convection), the diffusion flux will be a part of the full transport equations. Since water transport takes place inside the polymer, the use of eq 36 requires that the chemical potential be defined in such a way that a state with a given hydration will be assigned the same chemical potential irrespective of whether it is equilibrated with a liquid or a vapor. This means that a certain equilibration mode must be chosen as a reference. If liquid equilibration is chosen, the term Vπs has to be subtracted from the chemical potential of water in the vapor phase at any boundary contacting a vapor phase. For a given µ in vapor, this term should be calculated using eq 17 (or 19) after solving for λ in eq 7 in conjunction with eqs 6 and 17 (or 19). A curious situation may arise when an ionomer film is inserted between a liquid and a saturated vapor. In a seemingly paradoxical way, this will set up a gradient of chemical potential in the film that will make water to diffuse to the vapor side, even if liquid and vapor are in equilibrium. However, eventually, the vapor side will be covered with a molecularly thin film of liquid and the gradient will vanish. The amount of water transferred through the ionomer-liquid boundary will only be the one necessary to fully hydrate the film to the level of liquid equilibration. On the other hand, if the vapor is not saturated and the rate of evaporation is such that the vapor side remains dry, a liquid film will not form and the Laplace pressure πs will arise and act to enhance the chemical potential gradient and water transport. This situation often occurs in fuel cells and pervaporation membranes. The picture of water transport will not be much different from the one proposed by Weber and Newman,22 in which the gradient of capillary pressure is added as part of the thermodynamic driving force. The important difference is however that the present model calculates the “capillary” pressure based on the state of the internal matrix-liquid interface rather than a hypothetic and, most likely, nonexistent intrapolymer vapor-liquid interface. Conclusions A fairly general physical picture of a swollen ionomer is proposed that involves four types of energy: hydration (solvation) of ionic groups, interfacial energy, and two distinct types of elastic energy associated, respectively, with inflation of the matrix and stretching of the corona regions adjacent to the microscopic interface. Using simple phenomenological relations for each type of energy, an approximate model is constructed suitable for analysis of hydration and structural evolution of swollen non-cross-linked ionomers in the ideal nonrigid and realistic transiently rigid states. This model also seems to

conform to the scattering data, which point to a 2D morphology of Nafion in a wide hydration range. Using this picture, it is shown that Schroeder’s paradox is a natural consequence of the microphase-separated structure arising from the balance between the interfacial and corona energies, together forming the interfacial area-dependent part of the total free energy. The microphase separation brings about a large modified Laplace pressure increasing the chemical potential of solvent (water) in equilibrium with a solvent vapor. In contrast to the previous models, this pressure appears to be associated with the microscopic aggregate-matrix interface; however, it should similarly vanish when the polymer directly contacts a liquid solvent. This leads to differences in equilibrium solvent uptake from vapor and liquid (Schroeder’s paradox) and must be accounted in water transport in a material simultaneously contacting vapor and liquid phases. The transition between different swelling modes is driven by a misbalance of interfacial tensions involved and subsequent structural rearrangements starting from the polymer surface. The experimental observation of this effect may however have been obscured in some cases, particularly in saturated vapor, due to possible formation of a thin surface layer of liquid solvent or Nafion solution. Analysis of available data on hydration of Nafion strongly suggests that the transient rigidity of the very slowly relaxing perflourinated matrix may affect its observed behavior. The propensity of the Nafion matrix to forming elongated crystallites may stabilize the unusual structure of Nafion, either kinetically, through large transient rigidity of crystalline regions serving as physical cross-links, or thermodynamically, through forcing elongated cylindrical structure upon aqueous ionic channels or micelles. This may explain seemingly contradictory results on equilibrium hydration and Schroeder’s paradox by different authors, as well as the exceptionally high conductivity and its loss at high temperatures. This also rationalizes the need to account for the thermal, hydration, mechanical, and processing history of Nafion while analyzing its properties and behavior. Acknowledgment. The author is grateful to Klaus-Dieter Kreuer and Martin Ise for introducing him to the subject of Nafion and to Yoav Tsori and Simcha Srebnik for discussions. Financial support from the Israeli Science Foundation (grant 340/04) is gratefully acknowledged. Appendix The Corona Energy and Parameter B. The elastic energy of two ideal corona chains connected to a group on the surface of an inverted micelle is34,43

∫RRφ Rφ ) 3kBT(la)-1 ∫R

fc ) 3kBT(la)-1

-1/D

(dr/dn)2 dn

-1/D

(dr/dn) dr

(A1)

where l and a are the Kuhn and monomer lengths and r(n) is the radial location of monomer n counting along the chain starting from the group. The integral may be viewed as an analogue of mechanical (or electrostatic34) energy and (dr/dn)2 as a local mechanical (or electrostatic) energy density, i.e., pressure or (radial) stress. It is a reasonable approximation34,43 that the corona region of a micelle is entirely filled by the chains (two per group) emanating from this micelle. For a spherical

Hydration of Ionomers and Schroeder’s Paradox in Nafion (3D) micelle, this means r3(n) ) R3 + (3/2π)QVmn, where Vm is the monomer volume, which gives

2Vm R2 QVm dr ) ) 2 dn σ r2 2πr

(A2)

where σ ) 4πR2/Q is the interface area per group. Integration of eq A1 then yields

fc )

6kBTVm R (1 - φ1/3) la σ

(A3)

which means B ) 1 - φ1/3 in units of 6kBTVm/la. The factor 1 - φ1/3 is close to 1 for small hydrations, when 3D inverted micelles may exist. For inverted rod-like micelles (2D), a similar analysis shows that dr/dn ) (2Vm/σ)R/r and B ) ln φ-1/2, i.e., again B ∼ 1 in the relevant φ range, and for lamellae dr/dn ) 2Vm/σ and B ) 1. For large φ, the micelles will tend to have a normal configuration. It may be shown that for 3D normal micelles B ) 3π2/40 ) 0.740 and for 2D B ) π2/12 ) 0.822.34 It is seen that for a corona composed of ideal chains B ∼ 1 in units of 6kBTVm/la for all above configurations. As discussed by Nyrkova et al.,43 different architectures, e.g., a random linear ionomer vs telechelic or an ionomer with side chains, may be accounted for through constant factors multiplying the above values of B, which may then be approximately viewed as a polymer-specific constant. The above relations show that the radial stress decays with distance from the micelle as

.

τr ∼

( dndr )

2

∼ r-2(D-1)

(A4)

This means the tangential stress, τθ ) (1/2r)[∂(r2τr)/∂r] for D ) 3 or τθ ) (1/r)[∂(rτr)/∂r] for D ) 2, decays in the same manner but has the opposite sign; i.e., the matrix is compressed in the tangential direction. References and Notes (1) Schroeder, P. V. Z. Phys. Chem. 1903, 45, 75. (2) Tager, A. A.; Tsilipotkina, M. V. Usp. Khim. 1978, 47, 152. (3) Vallieres, C.; Winkelmann, D.; Roizard, D.; Favre, E.; Scharfer, P.; Kind, M. J. Membr. Sci. 2006, 278, 357. (4) Freger, V.; Korin, E.; Wisniak, J.; Korngold, E. J. Membr. Sci. 2000, 164, 251. (5) Curtin, D. E.; Lousenberg, R. D.; Henry, T. J.; Tangeman, P. C.; Tisack, M. E. J. Power Sources 2004, 131, 41. (6) Diat, O.; Gebel, G. Nat. Mater. 2008, 7, 13. (7) Kreuer, K. D.; Paddison, S. J.; Spohr, E.; Schuster, M. Chem. ReV. 2004, 104, 4637. (8) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. (9) Ionomers: Characterization, Theory and Applications; Schlick, S., Ed.; CRC Press: Boca Raton, FL, 1996. (10) Ionomers: Synthesis, Structure, Properties and Elasticity; Tant, M. R., Mauritz, K. A., Wilkes, G. L., Eds.; Chapman & Hall: 1997. (11) Zawodzinski, T. A.; Derouin, C.; Radzinski, S.; Sherman, R. J.; Smith, V. T.; Springer, T. E.; Gottesfeld, S. J. Electrochem. Soc. 1993, 140, 1041. (12) Weber, A. Z.; Newman, J. Chem. ReV. 2004, 104, 4679. (13) Fujimura, M.; Hashimoto, T.; Kawai, H. Macromolecules 1982, 15, 136. (14) Jalani, N. H.; Choi, P.; Datta, R. J. Membr. Sci. 2005, 254, 31. (15) Morris, D. R.; Sun, X. D. J. Appl. Polym. Sci. 1993, 50, 1445. (16) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (17) Flory, P. J.; Rehner, J., Jr. J. Chem. Phys. 1943, 11, 521.

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Hydration of Ionomers and Schroeder's Paradox in Nafion - The

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