J. Phys. Chem. B 2004, 108, 11953-11963
11953
Fine Morphology of Proton-Conducting Ionomers A. S. Ioselevich,† A. A. Kornyshev,*,‡ and J. H. G. Steinke‡ L. D. Landau Institute of Theoretical Physics of the Russian Academy of Science, 117940 Moscow, Russia, and Department of Chemistry, Faculty of Physical Sciences, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K. ReceiVed: January 21, 2004; In Final Form: May 20, 2004
The key factors that control the performance of perfluorinated sulfonic acid polymer electrolyte membranes cannot be deeply understood without a structural model of the material. Models of different complexity have been discussed in the literature. In this paper, we suggest a more detailed structural model of Nafion-type membranes, which results from a combined analysis of the ionomer molecular structure, data on swelling, small-angle diffraction, and conductivity as a function of water content. The analysis focuses on geometrical constraints on the self-organization of the polymer and possible patterns of phase segregation within it. The model identifies the percolation bottlenecks for proton transport and resolves controversies about the watercontent dependence of the activation energy of proton mobility. It also suggests a new framework for molecular dynamics simulations of proton and water transport in such media.
1. Introduction “High-Tech” Materials: Perfluorinated Ionomers. Perfluorosulfonated ionomers are widely used as membranes1 in liquid and gas separation,2-7 water electrolysis,8-12 electroorganic synthesis,13-17 catalysis,18-20 electrochemical synthesis of chlorine and alkali,21-23 nanoparticle synthesis,24,25 and protective clothing.26,27 In the past decade, these ionomers became the proton-conducting membrane materials for lowtemperature fuel cells (polymer electrolyte fuel cells (PEFC)).28 The latter are seen as perhaps the most promising, highly efficient renewable electrical power generators for emissionfree vehicles, portable power cartridges for laptops and cellular phones, small-scale residential applications for power back-up, or remote homes.29,30 Almost all leading car manufacturers are involved in research and development of PEFC, and some, like General Motors/Opel, DaimlerChrysler, Ford, and Toyota, are close to the production of sample parties of PEFC-driven vehicles. At the same time, there is rapid development of compact systems for portable and residential applications. As a key component of PEFC, polymer electrolyte membranes might soon reach every family on the planet, unless alternative betterperforming proton-conducting materials will be identified. The perfluorosulfonated ionomers are composed of perfluorinated polymer backbones with side chains containing acid groups, most commonly sulfonic acid end groups (-SO3H). In contact with water, the sulfonic acid groups dissociate and thus introduce protons as charge carriers; at the same time, the remaining SO3- side-chain end groups become hydrated. The presence of water is essential for the material to become proton conducting. In a fuel cell, protons generated at the anode by hydrogen oxidation in hydrogen PEFC or by methanol oxidation in direct methanol fuel cells (DMFC) (where the fuel is CH3OH instead of H2) travel through a film of such material toward the cathode. * To whom correspondence may
[email protected]. † Landau Institute of Theoretical Physics. ‡ Imperial College London.
be
addressed.
E-mail:
At the cathode, every two protons are consumed in a recombination reaction with one atom of oxygen, supplied to the catalyst surface from the air, and two electrons from the circuit that are generated through the hydrogen oxidation at the anode, to produce one water molecule. The film is typically 0.1 mm thick and is a membrane since it is almost impermeable to gases, which is crucial for gas-feed fuel cells, where hydrogen (or methanol) and oxygen reactions take place at two separated electrodes. However, water can diffuse through it as well as methanol. Water diffusion itself is not a negative factor, but electro-osmotic drag is. The reported electro-osmotic coefficient lies between 1 and 2, i.e., protons moving from the anode to the cathode drag, on average, one or two water molecules.31 Although the latter will return back via diffusion or hydraulic permeation, at the passage of a large current, this cannot happen too fast. Under stationary conditions, a gradient of water concentration will establish itself in the membrane.32-36 Lower concentration regions close to the anode will not function as efficient as water-saturated regions at the cathode side thus creating a limiting current in the cell.37 Whereas this problem could be solved in a fuel cell stack by smart water management, methanol crossover from the anode to the cathode on the other hand leads to considerable voltage losses. The requirements on polymer electrolyte membranes for the fuel cells are thus rather strict. An “ideal” membrane must (i) be chemically and mechanically robust and exhibit a stable performance in the temperature range of 80-150 °C with a conductivity not lower than 0.1 S cm-2 under currents of no less than 1 A/cm2, (ii) be impermeable to gases (and to methanol in the case of DMFC), and (iii) have an affordable price. The best-known polymer electrolyte membranes are various modifications of Nafion (DuPont de Nemours & Co), Dow (Dow Chemicals), Flemion (Asahi Glass), and Aciplex, which differ in their chemical structure, equivalent weight (EW, measured in g of dry polymer per mol of ion-exchange functionalities (SO3-)), and macroscopic thickness. Apart from their current exorbitant price, they meet the demands of hydrogen-oxygen fuel cells. However, Nafion and Flemion
10.1021/jp049687q CCC: $27.50 © 2004 American Chemical Society Published on Web 07/16/2004
11954 J. Phys. Chem. B, Vol. 108, No. 32, 2004
Figure 1. Popular cartoon of the microphase segregation in perfluorinated membranes (courtesy of A. Vishnyakov).
(which is very similar to Nafion but is processed under different conditions) are not suitable for DMFC, because they permeate methanol. Unfortunately both membranes cannot sustain the high temperatures at which the catalysis would have consumed all methanol supplied at the anode, which could have made methanol permeation no longer the issue. To a similar extent, this also refers to Dow and Aciplex. Melting (glass) transition of Nafion is near 140 °C, but the standard regime for Nafion operation lies close to 90° as water is difficult to keep in a membrane above 100 °C, even under pressurization. Creation of new membranes that could meet the listed demands, particularly sustaining higher temperatures, is a subject of intensive research38-48 and development49-52 in many polymer laboratories worldwide. The focus on Nafion in this paper is because it is still widely used and also is so far the best experimentally studied material. Chemical Structure and the Nature of Phase Segregation. The unique properties of polymer electrolyte membranes are related to their complex nanostructures. When exposed to water or other hydrophilic solvents, a polymer electrolyte undergoes nanoscale segregation into two subphases, often shown schematically as depicted in Figure 1. The hydrophobic subphase is formed by the perfluorinated polymer backbones and by the side chains except for their terminal SO3- groups, which will seek contact with water. The hydrophilic subphase is formed by water, mobile countercations, and SO3- groups. In the acidic form of the polymer, the countercations are protons and the hydrophilic subphase is the conducting medium when it spans (percolates) through the whole polymer. The proton mobility in a membrane was never observed to be better than that of pure water, but the density of mobile protons that can be reached there, depending on equivalent weight, approaches that of concentrated acids. This provides respectable levels of proton conductivity within the membrane provided the level of hydration is sufficiently high. While these basic principles are commonly accepted, there is no unified opinion on the “details” of the morphology of microphase segregation in polymer electrolytes despite a large number of investigations in this area.53,54 Nafion, best studied but still not fully understood, remains to be the “fruit fly” for the study of the interplay between patterns of the phase segregation and protonics. Nafion is formed from aqueous solution and has the general chemical structure
Ioselevich et al. The fluorine atoms confer on the -SO3H very high acidity, similar to that of sulfonic acid. The products of dissociation of CF3SO3H upon contact with water behave differently: SO3groups stay attached to the side chains, whereas the protons can freely move. There are three forms of this ionomer, sometimes labeled as shrunk (S), normal (N), and expanded (E) forms. The S form is acquired after a treatment at high temperatures; it corresponds to an ultra dry state, when all the residual water molecules are expelled from the polymer matrix. Since water works as a “plasticizer” for this polymer, the S form is rugged and essentially ruined for membrane applications. The E form is made from the polymer solution kept at high temperature and pressure. This form can have more than 50% water uptake, which is a swollen, physically cross-linked gel rather than a solid plastic material. The N form, even in its dry state (at zero extra water uptake), keeps “residual”, “chemisorbed” water: about two molecules per SO3- headgroup (the exact number, to our knowledge, has not been precisely measured). The dry state is a kind of a short-range ordered crystal hydrate. The wet state is a result of swelling through up to 40% of water uptake. In this paper, we will consider exclusively the N form. Experimental Studies and First Models. Information about the membrane structure comes from various sources, including small- and wide-angle X-ray and neutron scattering (SAXS and SANS),55-65 IR66 and Raman spectra,67 time-dependent FTIR,68 NMR,69,70,71 electron microscopy,72,73 positron annihilation spectroscopy,74,75 scanning probe microscopy,76,77 and scanning electrochemical microscopy (SECM).78,79 The group at DuPont55 was first to come to a conclusion about the inverted micellar structure of the aqueous subphase. The so-called Gierke model80 considers this subphase as built of approximately spherical water droplets of nanoscopic dimension, confined by anionic headgroups of the side-chains. In a dry state, the diameter of these droplets is of the order of D ≈ 2 nm and the droplets are disconnected from each other. With water uptake, the droplets grow up to D ≈ 4 nm, and aqueous necks emerge between them at certain intermediate water content. After emergence of a critical number of necks, a continuous pathway through the water subphase spans through the sample, and it becomes a proton conductor. The Gierke model,80 developed later by Mauritz and Rogers,81 implies that water droplets/hydrophilic ionic clusters and hydrophobic perfluorocarbon backbone (e.g., polytetra-fluoroethylene) constitute two subphases separated by high-area interface, but little is known about the molecular structure of this interface. The basic conclusions of the Gierke model were supported by the results of the DuPont,55 Kyoto,56 and Grenobles58,60 groups and by many further studies (see, e.g., refs 5865). Claims on the cylindrical micelles82 or flat lamellar structures83-85 have also been made, but the occurrence of such structures is more typical for E membranes.63 For N membranes, the concept of spherical or quasispherical64 micelles continues to be the most common conjecture about the phase segregation in the membrane, particularly in view of clear reports on the absence of elongated objects in the patterns of MaxEnt reconstruction of SAXS data.64 However, it has been difficult with SAXS and SANS techniques to determine more than just the position of a Bragg peak (called, sometimes, ionomer peak) characterizing the short range periodicity motif in the arrangement of micelles. Thus the average size of the micelles was, in fact, not known, but only the distance between the centers of the micelles. To determine the size of the micelles, one should have carried out the Gunier
Proton-Conducting Ionomers
J. Phys. Chem. B, Vol. 108, No. 32, 2004 11955 Whereas the conductivity gives information both on the global characteristics of the aqueous network inside the polymer and on the inherent proton mobility, the Arrhenius plots for different water content95 shed light, almost exclusively, on the elementary act of proton transfer. Simultaneous analysis of both features in fact helps to narrow down the choice of the structural model of the ionomer. Percolation properties of the membrane are also relevant. Conductivity is very small below the threshold value, φc, of the volume portion of excess water in the membrane, φ. Involving a high degree of simplification, it is sometimes asserted that the conductivity above the threshold grows roughly according to the empirical “Bruggeman-like” formula93
σ ) σr + (σ j - σ r) Figure 2. High-temperature data of Cappadonia et al.95 on activation free energy of similarly prepared samples of Nafion 117 with different water content, plotted as a function of excess water molecules per sulfonic acid group. To the accuracy of the point at φ ≈ 11, the activation free energy monotonically decreases with water content.97
analysis,63 but this part of the small-angle spectrum lies very close to the ionomer peak, and the accuracy of such determination is pretty low. The Porod part of the spectrum that helps to evaluate the overall surface of the micelles can give their diameter63 under an assumption of their spherical shape the ideality of which was recently debated. Additional information about the size distribution of micelles comes from the capillarypressure isotherms86 and IR spectroscopy,67 whereas electron microscopy64 discriminates the shape of the micelles. As for the necks, it was never clear how to obtain direct experimental information about them... Beginning with a seminal paper by Eisenberg87 a number of theoretical investigations focused on the elucidation of the patterns from micro-phase segregation in ionomers (for a review, see refs 54, 88, and 89). On the basis of phenomenological models, this research has solidified the concept of cluster formation and growth with water uptake. One of the models was focused on the arrangement between the polymer backbone and the aqueous aggregates. Assuming that each ionic aggregate is coated with a layer of polymer backbone, Yarusso and Cooper90 developed a modified hard-sphere model (the so-called “interparticle model”) that helped to rationalize SAXS data. Orfino and Holdcroft91 utilized this model for an estimate of the length of the channels joining aqueous clusters in Nafion. However, in view of the lack of information on the detailed structure of the hydrophobic polymer backbone, confused furthermore by observed batch dependence,92 it became almost common place to consider the details of these patterns too confusing to be taken into consideration in a theoretical model. There are however additional sources of information associated with the membrane performance, which could help discriminating different “fine-structure” features. On a qualitative level, first steps in this direction have been made in a conceptual review by Pourcelly and Gavach.93 Proton Conductivity. This was measured as a function of water content for various membranes (for reviews, see refs 28 and 93). For Nafion, the temperature dependence of conductivity has been studied, giving the free energy of activation of proton conductance (Figure 2) through Arrhenius plots.94,95 These data have been rationalized with a semiphenomenological model of membrane swelling and effective medium percolation theory.96
( )
φ - φc τ θ(φ - φc) 1 - φc
(1)
with the Heaviside function θ(x) ) 1 at x > 0 and 0 at x < 0. The “residual” conductivity at low water content, σr, is at least 3 orders of magnitude lower than the conductivity of the saturated sample, σ j , close to the threshold τ ≈ 1.5. Far above the threshold, there is a crossover to a regime roughly following the same law but with the exponent smaller than unity. The activation energy varies dramatically between the value of 0.1 eV for a water-saturated sample and 0.35 eV for a dry sample. Theoretical models98,99 and molecular dynamics,100 including the basic modes of proton states and proton transfer in water,101,102 have been recently applied for a study of the proton transport in an environment of a “representative channel”.103 These studies, not yet coupled with the overall membrane architecture, were focused on the key parameters of the interaction of proton complexes and water with ionized charged groups localized near the surface of a hydrophobic skeleton. A number of factors were revealed determining the activation energy of proton mobility, including the width of the channel, distance between the side chains, and their lability.100,103 Need for a Detailed Morphological Model: The Goal of this Paper. But how representative were the structures? To answer this question, one needs a detailed morphological model. This model must “digest” all the available information about the membrane structure and performance. We will try to develop such a model in this paper. In particular, we will refer to the data on the correlation between the macroscopic swelling of the membrane and the distance between micelles.56,63 We will give a possible explanation why in Nafion a different law is obeyed than initially expected.104 Although not very accurately measured, this correlation is important for verification of the structural model. We will also discuss aspects of the neck formation, which will result in estimates of their size parameters, which have a strong effect on membrane performance. 2. Mesoscopic Architecture of Nafion Existing Models. The standard model proposed to describe the properties of Nafion and similar substances views the system as more or less a solid matrix constituted of polymer backbones interdispersed with roughly spherical pores filled with water. The side chains are distributed along the perimeter of these pores so that all the sulfonic headgroups are located on the surface of a nearest water cluster, forming inverted micelles. With the water uptake, the size of each pore grows, as well as the distance between the pores. As a result, the system swells: its volume increases with the water uptake. It is also expected that, while being isolated at low humidity, the water-filled pores gradually
11956 J. Phys. Chem. B, Vol. 108, No. 32, 2004 become connected by narrow channels when the humidity increases. At a certain humidity level, the percolation network is established, and from that point, the proton conductivity will monotonically grow with water content. There have not been any attempts, however, to build and explore the class of geometric constructions underlying such picture. Basically, the only experimentally available geometric characteristic of the intramembrane organization is the dependence of the average distance d between the micelles related to the level of humidity, found in SAXS and SANS experiments. In some cases also, the dependence of the sample volume on the humidity has been obtained. All the other characteristics, such as mean cluster diameter D, the number of headgroups per water cluster, etc., cited in the literature, are usually extracted from the above experimental data, based on certain theoretical assumptions. The abrupt onset of the proton conductivity suggests that necks joining the clusters should exist at high and moderate water content while ceasing to exist at low water content. It should be noted that the existence of the micelle channel structure postulated in the standard model of Gierke et al.,55 though very plausible and widely accepted, is still under debate. In particular, in recent papers,105-108 it has been argued that the micelles may be considerably elongated. Besides the popular model of Gierke, there exist a number of alternative models. These alternative models do not invoke a phase-transition-like formation of channels connecting the inverted micelles. The model due to Vishnyakov and Neimark109 introduces a fluctuational formation of transient water bridges between the micelles without establishing stationary percolation through the water subsystem. Certainly such processes should exist, but it remains hard to explain the abrupt conductivity crisis in terms of these processes. One can also speculate about a complex interplay of different modes of protonic transport in wet membranes: the transport via “bulk” water, the transport along the water-polymer interface, and the transport along narrow one-dimensional water channels (which was shown to be especially effective in carbon nanotubes110 and other systems).111 Molecular dynamics simulations of protonic transport100,103 in channels of fixed cross sections showed that the activation energy is considerably affected by the surface only for extremely narrow compartments, which cannot represent the main part of the aqueous domains in the membrane. In the present paper, we dwell on the micelle-channel model, not undermining the value of other models, especially the one with the temporary water bridges, which, in our opinion, may describe the conductivity in the low water content phase. In the latter phase, the protonic transport seems to be more sensitive to the details of the membrane structure, since in the absence of channels, the protons are forced to come into closer contact with the polymer. Indeed, the results of Eikerling et al.112 show that the conductivities of two slightly chemically different membranes considerably differ from each other in the low water content regime, while at high water contents, they are practically identical. Housing Micelles in the Polymer Skeleton: Pressing Questions. The bottleneck for proton conductance in Nafion and related substances is associated with the process of proton transfer between adjacent water clusters so that, to work out a theory of proton transport, it is necessary to understand the structure and properties of the channels. It should be noted that the very existence of channels is already a puzzle. Indeed, one could have guessed that these channels have the “kinetic” origin,
Ioselevich et al.
Figure 3. A sketch of a bundle of three Nafion polymer chains. The number of side groups per unit length for such a bundle is three times larger than for a single chain.
as rudimentary passes, via which extra water has been delivered to micelles from outside in the course of the sample-preparation process. The samples, however, are claimed to be carefully equilibrated, and the proton conductivity does not seem to show any tendency to vanish in equilibrium. It forces one to admit that the channels are equilibrium (or, at least, quasiequilibrium) features of the system. Why could it be energetically favorable to form narrow channels? To answer this question, one has to examine the entire structure of Nafion. The first problem, which one encounters, trying to give a rationale to the model of inverse micelles, is how is it possible to arrange curved (presumably, spherical) housings for them out of real molecules? These molecules are known to have quite stiff backbones, with a persistence length Lpers g d. Indeed, measurements suggest Lpers > 10 nm. It means that solid “cages” that confine the micelles, whose characteristic size ranges between 1 and 4 nm, should be constructed out of long strings which are almost straight on the micelle size scale. Since, as we know, the cages should in fact be partially permeable for protons, the walls cannot be continuous, but we assume that they are rather cellular in structure, with windows wide enough to provide proton permeation. Such cages can be viewed as being made of stiff strings, constituted by either single-polymer molecules or “bundles” of those, which will compose the “skeleton” of the polymer. The tendency to form bundles is caused by physical attraction between the perfluorinated backbones of polymer molecules. Such an attraction is the reason for structural stability of Teflon, whose perfluoro backbone is the same as Nafion but without the side-chain comonomer incorporation. The presence of sidechains reduces the ability to form backbone bundles, but one can easily imagine small bundles made up of at least a few polymer chains to exist, (see Figure 3). Formation of bundles would further increase the relevant persistence length. Thus, if the bundles were there, our assumption about the suppression of string bending on the length scale of the order of intermicellar distance d would acquire additional footing. The strings stick to each other at their crossings due to interbackbone forces that are not large for each individual crossing point as the contact surface area is relatively small. Potential barriers, which the system has to overcome, when two crossing backbones slide with respect to each other should not be significant. Still, since there are many such crossings, the polymer material keeps its integrity, but it reconstructs easily with the water uptake so that linear dimensions of cages increase.
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J. Phys. Chem. B, Vol. 108, No. 32, 2004 11957
Figure 4. Three possible constructions of a four-coordinated cage: (a) a truncated tetrahedral cage, an open structure with shared hexagonal facets; (b) the same as part a but enforced by three principle diagonals on each facet; (c) the same as part a but enforced by 4 short (side) diagonals on each facet.
One can envisage this process as the sliding of strings with respect to each other. It is certainly hindered by the side chains, but since the distance between the subsequent side chains is relatively large, their effect can be reduced to production of rare defects in the structure, which can be removed in the process of equilibration. The “glassy” system will inevitably have a certain degree of disorder. First, as we will see in the next section, there are several kinds of admissible geometrical arrangements for the backbones, and they may coexist in the neighboring “crystallites”. We will also see that the geometry of a crystalline domain is correlated with the number of strings in bundles forming the domains. Accommodating different domains is possible due to a finite persistence length of the backbone; this will require binding and unbinding of the bundles. How exactly it will occur is difficult to say. But in any case, we should be prepared to admit a distribution of crystalline structures at a limited degree of crystallinity. The latter may be extending to the scales considerably larger than persistence length due to a stabilizing effect of the micelles formation. Admissible Superlattices. Since the strings are almost straight, the system is expected to have crystalline order at large distances compared to the size of an individual cage. Thus, as a first approximation, we ignore the disorder and consider an ideal superlattice of the cages, constructed according to the following rules: • Construct a cage, made of straight linear segments, and then organize a periodic superstructure out of identical cages. • The result should be such that the collection of finite segments constituting all the cages is equivalent to a system of infinite straight lines. The second requirement (we will call it the string-continuity condition) poses a severe limitation on a variety of possible forms of an individual cage and makes the search for an optimal form feasible. The simplest example of a cage that fits the string continuity condition would be a cube with a micelle placed in its center. It would mean that the system of micelles forms a simple cubic lattice (we are reminded that the long-range order in this system is presumably destroyed by disorder so that only the short-range one is left). A cubic lattice, however, seems to be unlikely due to mechanical stability considerations. A cage with triangular faces would be much better in this respect. Generally, for the system with isotropic interactions, it would have been natural to expect ordering in one of densely packed lattices (e.g., face-centered cubic). However, we do not see any way for constructing a cage that could be a basis for a densely packed (12 coordinated) lattice and would satisfy the
string continuity condition! This can be done, however, for a four-coordinated diamond lattice. The idea of a tetrahedrically coordinated system of micelles with a local order of diamond lattice was initially put forward by Gebel,63 supported by some indirect indications to that. We are not aware of any direct experimental measurements of micelle coordination number. So far, this conjecture should be considered as a plausible hypothesis. In the lattice model, the ionomer is built of cages and an equivalent number of voids. The three simplest candidates for such four-coordinated cages are shown in Figure 4. Each cage of these “short-listed” structures is a tetrahedron with edges 3a, with four small tetrahedral voids (with edges a) being cut from all four vertices of the large one. In the “minimal model” (Figure 4a), each cage consists of 4 triangles and 4 empty hexagons, and the adjacent cages share a hexagonal facet via which the channel can be easily formed; perhaps, too easily?! Indeed, the hexagonal facet is so wide that the channel would form already at the lowest humidity level, which would contradict the fact of absence of proton conductance at low water content. This does not apply to the models in parts b and c of Figure 4. In Figure 4b, the basic construction of the cage is the same as in Figure 4a but with three principal diagonals on each facet of the hexagon. In Figure 4c, the principal diagonals are replaced by side ones, the facets looking as if covered by starlike lids. The windows in these cages are considerably smaller, and the channels will not be that easily formed. Geometric Estimates and Constraints. Let us denote the length of the cage edge by a. What is then the total length of edges per cage? Structure a. Any cage has 16 edges in total. Each of them is shared by three adjacent cages, so that there are 16/3 edges per cage. The corresponding perimeter is
Pcell ) (16/3)a
(2)
Structure b. In addition to the above 16 edges, there are 6 × 4 edges, constituting the lids. They are shared by two cages each so that
Pcell ) [(6 × 4)/2 + 16/3]a ) (52/3)a
(3)
Structure c. Here the additional length due to the lids is (4 × 6x3/2)a ) (12x3)a so that
Pcell ) (12x3 + 16/3)a
(4)
11958 J. Phys. Chem. B, Vol. 108, No. 32, 2004
Ioselevich et al.
In all the three models, the distance between the centers of nearest micelles d is related to a through
a)
x23d ) 0.82d
(5)
The distance between the center of the cage and the center of hexagonal facet is
Rmin ) x(3/8)a ) 0.61a
(6)
The distance between the center and any vertex is
Rmax ) x(11/8)a ) 1.17a
(7)
A volume per cage (that is, the volume of cage itself plus volume of one adjacent void) is
(3a)3x2 1 Vcell ) 1 - ) 2x2a3 12 9
(
)
(8)
Part of this volume (1/24) is taken by a tetrahedral void and the rest (23/24) by the cage itself. A Relationship between Macroscopic and Microscopic Swelling. The expectation104 that the volume of ionomer V should scale with intermicellar distance d as V ∝ d 3 is generally wrong. There should be no universal scaling law, as it depends on the assumptions about the changes of morphology with swelling. We derive, however, the relation between V and d for the tetrahedral cage structures under an assumption that each cage is proportionally scaled with d, while the infrastructure of the strings (the number m of backbones in the bundle) is not altered. The latter means that the total amount of “construction material”, the total length L of all strings together, is conserved. On the other hand, one can write
L)
V V P ∝ Vcell cell d2
(9)
(where the weight fraction of water can be neglected). Then we obtain
FVcell ) Pcellµ
(10)
should be observed, provided that the assumption of affine swelling is valid. Note that the applicability of this law does not depend on a particular form of the cage and the number of facet-enforcing diagonals in it, but the proportionality coefficient does. Thus if the structure changes abruptly, there will be a change in the slope of V-d2 dependence. If the structure variation proceeds continuously, there will be no straight line in these coordinates. Also, since there is no true long-range order, possible rearrangement of the differently structured domains with swelling may cause deviations from this law that will be small, however, if the domains are large enough. Experimental data shown in Figure 5 exhibit roughly a straight line that approves the scaling law (10) and the affine model of swelling. The unknown accuracy of the data would not justify at this point speculations based on an apparent slight change of the slope at intermediate humidity values. The Number of Strands in the Bundle. Strands of the polymer seemingly aggregate into bundles, making the bundles as a whole more “hairy” than each individual string. It is important to estimate the number, n, of strands in the backbone strings. Let us consider a specific weight F of a dry Nafion
(11)
where
µ)n
w δ
is the weight of a polymer per unit length. Here w is the weight of a dry polymer per one sulfonic-group side chain (related to EW), and δ is an average distance between the side chains (w ) 1.82 × 10-21 g and δ ) 17 A for Nafion 117). As a result, we obtain following estimates for n
n)β
Thus, since L is constant, we conclude that
V ) constant‚d2
Figure 5. Macrosopic vs microscopic swelling of Nafion. The data of Fujimura et al.56 for the ionomer peak position vs volume expansion of HSO3-1100-EW-Nafion for different water content: Fujimura’s λB3 ∝ V and λd ∝ d. Replotted in λB3 T λd2 coordinates, these data roughly approve the scaling law (10). Solid circles correspond to the data obtained by variation of relative humidity of the membrane. Open circles mark the data obtained by soaking the membrane in aqueous solutions.
Fδd2 w
(12)
{ }
with the proportionality constant different for the indicated structures
x2 (a) ) 0.354 4 x2 (b) ) 0.109 β) 13 x2 ) 0.072 (c) 9x3 + 4
Given F ) 2.075 g/cm3 and d ) 35 Å, we find
{ }
8.4 (a) n ) 2.6 (b) 1.7 (c)
(13)
A bundle of three polymer chains is a favorable, stable configuration, and case b looks most reasonable. A bundle of three chains allows maximizing of the hydrophobic/fluorophilic interactions of the perfluorinated backbone and at the same time allows for maximum exposure of the sulfonic acid side chain to the micellar water pool (rich hair image!). Larger bundle sizes are conceivable, though would they inadvertently require
Proton-Conducting Ionomers segments of different polymer chains to aggregate more akin to the shape of a braid. Inverted Micelles. At low water content, nanosized water droplets form in each cage the center of inverted micelle. The side-chain headgroups of each cage lie in water close to the surface of the droplet as it costs energy to immerse the hydrophobic body of the side chain into water, at least to the level of the energy of hydrogen bonds lost by water molecules neighboring the side chain. Moving a headgroup out of the water costs more energy than thermal energy. The upper estimate is roughly half of the free energy of ion hydration in the bulk, the quantity >0.5 eV. Even 1/10 of it will be two times larger than kBT. This energy is likely to be larger than the attraction energy per junction of two crossing backbone strings, which means that the polymer will swell with the addition of more water. As in the case of charged lipid mesophases,113 the surface tension of the micelle will not be an important parameter of the problem because the surface tension of water is dramatically reduced by hydration of charged headgroups. The form of the droplet in each cage will fluctuate and adjust the capability of the headgroups to imbed into the droplet. Necks Emerging: Phase Transition? We want to understand how the necks between the inverted micelles emerge, because the emergence of these water channels gives rise to nonnegligible proton conductivity. Further development and the growth of the width of the channels with the increase of the water content allows the conductivity to reach its maximum value at which Nafion is tended to be used in fuel cell applications. Building an accurate theory of neck formation is a difficult task. Therefore, let us first consider qualitatively the main effects that should be present in such a theory and discuss a simple parametrization, presented in Appendix A, that demonstrates their interplay. The milestones are: (1) water sorbs into the membrane interior because it hydrates the headgroups; the hydration always asks for more water and (2) water does not dissolve the ionomer at ambient conditions. Its skeleton has an finite elasticity modulus because of the physical attraction of the fluoride fragments in the backbones, which suppresses the polymer expansion. The mechanism and sequence of sorption steps are seen to be as follows. (1) In a state close to a dry one, the membrane accommodates water in the kernel of inverted micelles. With each extra amount of water accommodated in the micelles, the skeleton expands, but water is still confined in each elementary cell of the skeleton without building water necks (bridges) between the cages.114 At the critical water content, which is the point of the birth of the necks, the membrane starts to conduct protons. To avoid misunderstanding, we stress here that in the “dry state” the proton conductivity of the membrane, though being very low, is, however, finite. The dry-state conductivity may be due to one of the fluctuational mechanisms of the transport, which are always present in the system at finite temperature. The question is: which of these mechanisms is the most effective one? Although we do not yet have a definite answer, we expect that the scenario with the temporary water bridges109 might be relevant. The same fluctuational mechanism might also be responsible for establishing the equilibrium water content in the “dry” state, in response to the change of external conditions. In the absence of stationary channels (which would be the most natural and efficient path for reaching the equilibrium humidity), the excess water can, nevertheless, slowly travel from the surface of the membrane to its interior via the fluctuational temporary channels. Therefore, the equilibration time in the dry state should
J. Phys. Chem. B, Vol. 108, No. 32, 2004 11959 be much longer than in the wet one. One should have in mind, however, that the water content equilibration is usually held at special conditions, such as boiling, which facilitate fluctuations responsible for transport. (2) There is no visible jump of conductivity at this point. If all windows in the cage were the same, e.g., as in model b, all necks emerge simultaneously and one would expect a nonsmeared phase transition. However the necks are so narrow that the conductivity through them will be very small. If even there is a small jump of conductivity at the critical point, due to high Ohmic resistance, the accuracy of impedance measurements will be too low to detect the jump. (3) If there are “windows” of different kinds, the wide ones will be filled first, and later, at larger water content, the narrower ones. Transition if any will be strongly smeared. (4) Why do the necks emerge at all? Increase of the water content by filling the interior cages meets resistance of the skeleton on one hand; on the other hand, it does not utilize all the possible voids in the skeleton. Since water molecules like to hydrate other water molecules (hydrogen bonding), water bridges emerge because this is the option for water aggregation in the hydrophobic environment of the windows. The growth of the water-neck cross sections is impeded by the contact of water with the hydrophobic framework of the window. (5) If water movement into the channels were followed by side-chain headgroups, this would relax the hydrophobic effect. However, hardly would it be entropically favored. (6) After the “percolation” point, conductivity grows monotonically and its activation energy decreases, and therefore the neck cross-sections must increase with water content. How can this happen? Only with the expansion of the whole skeleton which leads to the expansion of the windows. Thus first necks emerge, and then their width grows together with the further expansion of the skeleton. (7) How does the transition from the “neckless” to “neckrich” situation takes place? In each cage, the droplet may be in one of two states: (i) an isolated state or (ii) a state with branches growing into the “windows”. The balance of forces that decides which state has lower free energy is formed by the following factors: (a) hydration of the headgroups (favors cage expansion), (b) entropic confinement resistance of headgroups to shrink within the same cage with the increase of the droplet size (favors cage expansion), (c) skeleton elasticity (favors cage contraction), (d) hydrogen bonding of water molecules through the necks, and (e) entropic tendency of water deconfinement. Although the water molecule joining the droplet will possibly get more hydrogen bonds than when it joins the neck, necks will emerge utilizing the free void space without the costly expansion of the skeleton. Does surface tension of the droplet play a role? If the droplet is covered by the charged headgroups, the surface tension will be of minor importance. In the necks, however, this may be different. Theory. A common way to describe the transitions in amphiphilic systems, if they are of second order or weak firstorder type, is the Landau-Ginzburg approach.115 Before writing the free-energy functional,116 one should find appropriate variables, describing the skeleton, water droplets, necks, side chains. By its nature, such a theory is predominantly descriptive, as it gives almost directly what has been put in it. The predictive part will be the correlation functions of Gaussian fluctuations of the specified variables, calculated for the chosen free-energy functional. These correlation functions are interesting, if one can measure them using some scattering techniques.
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Ioselevich et al.
An alternative to the Landau-Ginzburg approach is the general mean field theory, which does not invoke any expansion in the small order parameter. The principles of this theory are discussed in Appendix A. It suggests that the transition is likely to occur as first order. Shrinking at the Transition? If the transistion is first order, the volume of the membrane may slightly shrink at the transition as water will be more homogeneously distributed over the membrane. One may also expect hysteresis in water sorption, observed in the capillary pressure isotherms obtained for Nafion membranes.117 Of course, there could be a number of other reasons for hysteresis in such systems, associated with filling the complex network space86 rather than the effects based on what occurs in a unit cell. Without a more sophisticated theory, it would not be wise to dwell on this point any further. But the absence of visible hysteresis in the dependence of conductivity on water content does not contradict our conjecture. The jump of conductivity at the “neck transition” is small, beyond the experimental accuracy, so that the hysteresis in conductivity could hardly be detected. In our model, the first-order transition will be accompanied by the drop of both R and d. Consequently, at the transition (cf. Figure 5), the thickness of membrane will drop down, as d2/3, to increase continuously again with the further increase of water content. Again, will this be experimentally detectable? Somehow, this effect will be drastically diminished by inhomogeneous broadening. 3. Proton Conductance Controlled by the Necks Our picture and the earlier-developed concepts of the proton transport along the interior surfaces of membrane99,100,103 can explain experimental data for conductivity dependence on water content assuming that the necks, when they emerge, are quite narrow. This is in line with expectations following from the geometrical structure of the cages. Thus with the growth of the necks with water sorption, the activation energy of the proton mobility may easily grow several times. This phenomenon was sought in the MD simulations103 but not found because these simulations have dealt with much wider channels, essentially closer in size to the characteristic size of water droplets in the inverted micelles. In other words, in refs 100 and 103, wide and narrow channels have been studied, whereas the key point is in the difference between narrow and Very narrow channels. If we literally take our structural model, we may not need any Bruggemann-like formulas but may assume that necks, when emerging, will always provide connectivity and what matters is how large would be the conductivity of each neck. Just to minimize the number of free parameters, let us assume that the main effect of the water sorption will be on the activation energy. In general, this must not be so. Indeed, there is a tradeoff between the concentration profile of protons near the wall and their mobility,99 and the simulations in relatively wide channels103 have demonstrated that the pre-exponential factor also grows with the water content; furthermore, not all necks appear simultaneously! But if we still take such a crude approximation, results will be quite close to those of Cappadonia et al., at least for their not “hot-pressed” samples, where Arrhenius plots for different water content merged more or less to the same value of the pre-exponential factor. Let us take then the data of Figure 2 and approximate them by an exponential function of excess water content, φ
Ew(φ) ) E h + (Er - E h )e-φ/∆φ
(Er > E h)
(14)
with E h and Er standing for activation energy of proton mobility
Figure 6. Typical plot of conductivity as a function of excess water content (in water molecules per sulfonate groups) resulting from the variation of the activation free energy of proton transport through the micelle-connecting channels following the growth of the “radii” of the channels. Calculation rests on approximating the data of Figure 2, as described in the text.
in water-saturated and dry states, respectively. We then plot conductivity to the accuracy of the pre-exponential factor as σ ∝ exp{-Ew(φ)/kBT} (Figure 6). This graph reminds us of the published plots for conductivities of proton-conducting ionomers as a function of water content,28 although this construction ignores all random network percolation aspects (for discussion, see, e.g., ref 96). However, not withstanding the possible importance of such aspects, we want to stress that the key point may lie in the properties of individual channels. Of course the true statistical theory of conductance in these systems is more complicated, but mapping the problem on bond percolation, we must not forget that with the increase of the water content the conductivities of the bonds must change.118 And this is the main message from this analysis. 4. Concluding Remarks We presented a detailed structural model of Nafion, which responds to the constraints imposed by structural, conductivity, and swelling data, geometrical “common sense”, and does not seem to contradict the wisdom of polymer chemistry. Not all assumptions of the model are directly experimentally proved, and not all the parameters of the model have been measured; so far the model remains speculative and one of the plausible models, among others. These are the assumptions of large persistence length, quasicrystallinity, and four-coordination motif (the latter is less critical but essential for the specific construction that we have studied). The model is lattice based. Obviously, ionomers are rather disordered, glassy materials, and the same unit cell will not be repeated over the whole sample. This will lead to inhomogeneous broadening of all characteristics calculated on the basis of a cell model. In particular it will smear the first-order transitions. With proper modification, this model could probably be extended to other similar perfluoro-sulfonates and perhaps even to stiff, nonfluorinated polymer backbones with sulfonic acid side chains. The picture of proton conductance, including the effects of water content and equivalent weight, comes out as a natural consequence of the suggested membrane mesoscopic structure. This establishes the bridge between the architecture and performance of the membrane. As we have discussed in the
Proton-Conducting Ionomers paper, it is harder to explain these dependencies on the basis of models of long cylindrical or flat pores, as these pores have to be very narrow to exert an effect on proton conductance activation energy. There are a number of directions where the model could be developed. This is, first of all, the theory of neck formation. Phenomenological field-theoretical models similar to those used in the theory of lipid mesophases could be used here as well as rheological models of the fusion of inverted micelles. Theory of forces that cause swelling should be developed to account for electrical double-layer effects and “entropic forces”. The model also suggests an essentially new environment for molecular dynamic computer simulations of proton transport in the membrane: instead of a motion inside a fixed flat or cylindrical pore, one should describe the transport of protons between the cages, very strongly assisted by the side chains. In the flatpore simulations, the lability of the side chains strongly accelerates the proton transport, in the very narrow neck, the effect of the side chains will be even stronger. It is hard to see the necks experimentally, and we do not know whether any new methods will soon emerge that could detect them or exclude their existence. We only know that the molecular structure of the ionomer, hypothesis of quasicrystallinity, and the periodical motif of water droplets, interpreted in terms of the inverted micelle model of the ionomer, will require necks in the proton-conducting state. Our model is in a way analogous to cubic phases of inverted micelles in lipids, where the fatty tails are, however not free but attached to the polymer backbones, which also have to aggregate subject to the interaction between the backbones, side chains, and water. Different models can be imagined based on elongated aqueous aggregates similar to cylindrical phases of inverted micelles that could probably be rationalized in the spirit of the present work, as well. But they must not contradict the conditions of the membrane stability and homogeneous swelling typical for the best membranes. Appendix A: On the Theory of “Neck” Transition The state of the system in a continuum approximation may be described in terms of the following variables: the distance between the centers of neighboring micelles, d, the form of the micelle, defined by the function R(n), which determines the distance R between the center of a micelle and its surface, seen in the direction n, and the distribution σ(n) of headgroups over the surface of the micelle. The free energy of the system consists of the following parts: (i) the elastic energy of the backbone Fbackbone(d), it depends only on the size of cages and reaches its minimum at certain d ) d0, (ii) the hydrophobic part of the side-chain energy Fchain[d,{R(n)},{σ(n)}], which is a functional of the form of the micelle and of the distribution of headgroups over its surface, it contains both elastic and entropic contributions. Its minimization with respect to R(n) and σ(n) at fixed d would give certain solution R0(d|n) and σ0(d|n); (iii) the “double-layer repulsionfree energy”, which is the free energy of squeezing the proton clouds in the micelle, Fdouble-layer{R(n)}, {σ(n)}, it takes into account the Coulomb interaction of the charged species, protons and anionic headgroups, in water droplets; and (iv) the hydration free energy Fhydration{R(n)}, {σ(n)}, of headgroups in a confined water droplet. The total free energy should be minimized with respect to variables R(n), σ(n), and d, subject to the following constraints: the total amount of polymer (i.e, the summary length of all bundles in all cages) is conserved and the total amount
J. Phys. Chem. B, Vol. 108, No. 32, 2004 11961 of water (i.e., the volume of a micelle times number of cages) is conserved. We expect that there will be two types of extrema for this variational problem: the solutions, corresponding to closed micelles, they should exist at relatively low water content, and the solutions, corresponding to the system of micelles, connected by channels, existing at larger water content. In the intermediate range of water content, the two extrema coexist, and the stable equilibrium state corresponds to that one, which gives lower free energy. At certain critical water content, the first-order phase transition occurs, which interchanges the stable and the metastable solutions. The reason for these expectations is as follows: while at small d, the form of the surface R0(d|n) preferred by the Fchain term is the closed one, at large d, it apparently should have “windows”. Indeed, for very large d, this surface would look like a system of cylinders drawn near each bundle which constitute the cages. Such a structure, obviously, describes a connected water space. On the other hand, with increasing water content, the optimal d(φ) value determined mainly by the competition between the elastic and the double-layer/hydration terms shifts to larger values. Somehow, there should be a transition between these two limiting cases! In the present article, we do not solve this problem explicitly neither do we discuss the explicit form of the free energy contributions. This will be done elsewhere. Here we mention only one basic difficulty: no matter how much simplified form one would choose for each of the contributions to the free energy, it would inevitably involve some “force constants”. Where to get the estimate for these constants? Unfortunately, most of them cannot be extracted from available experimental data. Still, choosing some generically reasonable estimates, one can draw some qualitative conclusions. Note also that a proper consideration should start with selfconsistent theory of absorption isotherm, where water is distributed between the inverted micelles and necks of adjustable size subject to all the pertinent interactions in the model and taking into account the main entropic effects (see, e.g., a seminal paper by Mauritz and Rogers).81 The “birth” of the necks should be manifested in the sorption isotherm. Acknowledgment. The authors are thankful to Michael Eikerling (SFU), Eckhard Spohr (FZJ), and Ulrich Stimming (TUM) for useful discussions. A.A.K. acknowledges the support of the Royal Society Wolfson Merit Research Award that has made this work possible References and Notes (1) Heitner, W. C. J. Membr. Sci. 1996, 120, 1. (2) Sportsman, K. S.; Way, J. D.; Chen, W. J.; Pez, G. P.; Laciak, D. V. J. Membr. Sci. 2002, 203, 155. (3) Hu, W.; Tanioka, A. J. Colloid Interface Sci. 1999, 212, 135. (4) Hu, W.; Adachi, K.; Matsumoto, H.; Tanioka, A. J. Chem. Soc., Faraday Trans. 1998, 94, 665. (5) Jiang, J. S.; Greenberg, D. B.; Fried, J. R. J. Membr. Sci. 1997, 132, 255. (6) Yamaguchi, T.; Koval, C. A.; Noble, R. D.; Bowman, C. N. Chem. Eng. Sci. 1996, 51, 4781. (7) VanZyl, A. J.; Linkov, V. M.; Bobrova, L. P.; Timofeev, S. V. J. Mater. Sci. Lett. 1996, 15, 1454. (8) Laurence, R. J.; Wood, L. D. U.S. Patent 4,272,353, 1981. (9) Takenaka, H.; Toriaki, E.; Kawami, Y.; Wakabayashi, N. Int. J. Hydrogen Energy 1982, 7, 397. (10) Ota, K.; Minoshima, H. Electrochemistry 2003, 71, 274. (11) Okada, T.; Satou, H.; Yuasa, M. Langmuir 2003, 19, 2325 (12) Linkous, C. A.; Anderson, H. R.; Kopitzke, R. W.; Nelson, G. L. Int. J Hydrogen Energy 1998, 23, 525. (13) Ogumi, Z.; Inaba, M.; Ohashi, S. I.; Ushida, M.; Takehara, Z. I. Electrochim. Acta 1988, 33, 365.
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