Model of Water Sorption and Swelling in Polymer Electrolyte

Article pubs.acs.org/JPCB

Model of Water Sorption and Swelling in Polymer Electrolyte Membranes: Diagnostic Applications Motahareh Safiollah,† Pierre-Eric Alix Melchy,† Peter Berg,‡ and Michael Eikerling*,† †

Department of Chemistry, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada Department of Physics, NTNU, 7491 Trondheim, Norway

‡

ABSTRACT: This work expands on the poroelectroelastic model of water sorption in polymer electrolyte membranes. It links membrane water sorption and the evolution of the pore size distribution with the microscopic charge density distribution at pore surfaces and the microscopic shear modulus of polymeric pore walls. We evaluate different deformation modes of polymeric pore walls to derive stress−strain relationships that determine the law of swelling. Thereafter, the model is applied to different sets of water sorption data for membranes that were submitted to either hygrothermal aging or chemical degradation. The model-based analysis relates the macroscopic state of swelling to the evolution of the statistical distribution of the pore radius as well as of the microscopic fluid and elastic pressures. Changes in the surface charge density at pore surfaces and elastic pressure of pore walls, induced by degradation, can be rationalized. These insights are useful in view of understanding degradation mechanisms and their structural effects.

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INTRODUCTION A polymer electrolyte membrane (PEM) fulfills vital functions as a separator medium, proton conductor, and electronic insulator in polymer electrolyte fuel cells (PEFCs). The efficiency of the chemical-to-electrical energy conversion in PEFCs depends, among other factors, on the ability of the PEM to transport protons from anode to cathode. The proton conductivity of the PEM is a critical parameter for achieving high power density and performance in the fuel cell. Perfluorosulfonic acid (PFSA) ionomer membranes are the most widely employed PEMs in fuel cells, among which Nafion serves as a benchmark.1,2 The water content is the key variable to characterize the state of a PEM and determine its transport properties. In particular, the proton conductivity is highly sensitive to the water content. In a well-hydrated state, the unique molecular structure of PFSA-type ionomers leads to the formation of well-connected proton-conducting pathways.1,3−6 The large difference in solubility of highly hydrophobic perfluorinated backbones and the strongly hydrophilic acid head groups at ionomer side chains induces a strong phase separation between these components. As the relative humidity of the external gas phase is increased, water-filled ionic domains swell and coalesce. The growth of pores increases the fluid pressure exerted onto pore walls. The walls are formed by aggregates of hydrophobic polymer backbones with the dissociated anionic moieties exposed at the interface with the aqueous phase. With increasing water uptake, a percolating network of water-filled pores first forms and then undergoes continuous reorganization. The well-connected network of pores provides conductive © XXXX American Chemical Society

pathways for protons, while the perfluorinated backbone maintains the structural integrity of the film.7,8 A consistent description of water sorption and swelling under conditions relevant for PEFC operation lies at the heart of understanding transport properties, performance, and degradation phenomena in PEMs. Models for water uptake by vapor-equilibrated PEMs have been suggested by various groups.9−13 The models account for interfacial energies, elastic energies of polymer walls, as well as entropic effects. The shortcoming of the majority of water sorption models is that they employ an insufficient number of local equilibrium conditions, affecting their ability to describe the physical state of water as a function of external conditions. In some cases, explanations of water sorption data have postulated the existence of water vapor inside of the membrane, which was justified by assuming that a fraction of pores possess hydrophobic surface properties.11 These assumptions are, however, uncorroborated. There is evidence neither for a significant hydrophobic gas porosity nor for the existence of water vapor inside of PEMs. Pronounced gas-tightness of PFSA membranes and the collapse of the pore structure upon complete dehydration are clear arguments against these hypotheses. Herein, we build upon a previously introduced physical model of water sorption and swelling by Eikerling and Berg.14 The model employs the conditions of thermal, chemical, and Received: January 20, 2015 Revised: April 17, 2015

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DOI: 10.1021/acs.jpcb.5b00486 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Figure 1. Structure formation from bundles of ionomers40 to different super structures confining pores: cylindrical aggregates,37 lamellar structures, and spherical cages.40

uptake in the membrane. Capillary condensation fails however for low water content.42 For λ > λS, equilibrium water uptake by pores in the PEM is determined by temperature, external relative humidity, and pressure. Moreover, the extent of pore swelling depends on elastic properties of pore walls and the density of anionic surface charges at pore surfaces. The osmotic pressure exerted by solvated protons plays a pivotal role in determining water sorption at the single pore level. In the second step, the model employs a convolution of single pore swelling with a statistical distribution of the surface charge density to describe macroscopic swelling in a random ensemble of pores. Figure 2 shows a unit cell that accommodates a single pore, surrounded by polymeric walls. Water in the pore is in

mechanical equilibrium of water at the pore level. Local capillary equilibrium of water in pores requires pressures to be balanced at all internal interfaces. Water uptake and swelling of a single pore is expressed as a function of external conditions and intrinsic material properties of PEMs. The former are temperature, relative humidity, and gas pressure; the latter include ion exchange capacity (IEC), shear modulus of polymer walls, and dielectric constant of water. In the following, we first discuss the phenomenology of swelling by briefly reviewing the poroelectroelastic model in the section “Model of Water Sorption and Swelling”. In the section “Stress−Strain Relationships of Microscopic Pore Walls”, we derive different stress−strain relationships of polymer walls based on different microscopic deformation scenarios. In the section “Pore Ensemble Model”, we explore relations between macroscopic effective water sorption, swelling, and statistical distributions of microscopic pore size and pressure. Diagnostic capabilities of the model are demonstrated in the section “Membrane Degradation Analysis”. From the macroscopic sorption data, we extract microscopic structural information and, based thereon, propose microscopic interpretations for the degradation studies we analyzed.

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MODEL OF WATER SORPTION AND SWELLING Various experimental techniques have been employed to unravel water sorption phenomena in PEMs and correlate them with membrane structure and transport properties.15−32 Although these techniques give valuable information, they cannot conclusively determine the microscopic and mesoscopic structure of the membrane. There is theoretical as well as experimental evidence that cylindrical fibers or bundles form the prevailing structural motif at the microscopic scale.33−35 These fibers define electrostatic and elastic properties of the PEM. They can assemble further into pore-forming superstructures. Structural models differ in terms of the shape of pores.36 The main types consider cylindrical pore structures,34,37−39 cage-like structures with spherical water inclusions,40 and lamellar-like flat pore structures.41 Figure 1 illustrates the main steps in the spontaneous structure formation from ionomer molecule to bundle and to different superstructures. The model of water sorption and swelling of Eikerling and Berg14 treats the water-filled PEM as a poroelectroelastic medium. In the first step, the model describes water sorption and swelling at the single pore level. It is observed that at sufficiently high water content, exceeding the amount of surface water, λ > λS, capillary condensation controls equilibrium water

Figure 2. Schematic view of a single pore.14 Region I, external vapor or liquid water phase; region II, pore water void of protons; region III, pore water including protons; region IV, elastic polymer.

equilibrium with a surrounding phase at controlled temperature, T, vapor pressure, PV, and gas pressure, Pg. Mechanical equilibrium of water in the pore can be expressed as a balance of pressures across interfaces in the system, involving gas pressure, Pg, capillary pressure, Pc, liquid water pressure in the pore, Pl, osmotic pressure due to solvated ions, Posm, and elastic pressure exerted by the polymer matrix, Pel. For mathematical convenience, pores are assumed to be cylindrical with radius R. The surface of pores is assumed to exhibit ideal wetting with cos θ = 1, where θ is the wetting angle. The dielectric permittivity of water in the pore, ϵr, is assumed to be constant. The surface charge density and the shear modulus G of the polymer phase are assumed to be uniform along the pore walls. The theory relates the charge density, σ0, at pore walls, defined in a reference state with radius R0, to a microscopic swelling parameter η, which is defined as the ratio of the B


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where γ is the surface tension and ηc is the swelling parameter of a pore that is at capillary equilibrium. Posm(R) is the osmotic pressure at the pore wall given by

volume of the swollen cylindrical pore to the volume of the unit cell. The change of the interfacial charge density with swelling is described by an empirical scaling law ⎛ R ⎞α σ(R ) = σ0⎜ 0 ⎟ ⎝R⎠

P osm = R gTc H+(R ) (1 − α)/(1 + α)⎤ ⎡ ⎛ ⎞2α /(1 + α) R gT ⎛ ξ ⎞ σ ⎥⎜ ξ ⎟ ⎜⎜ ⎟⎟ = 2σ0⎢⎢ 0 − ⎥⎜⎝ η ⎟⎠ FR 0 ⎝ ηc ⎠ 4ε ⎣ ⎦ c

(1)

where α depends on the pore geometry and the extent of surface group reorganization upon swelling. A value of α close to zero indicates weak surface charge reorganization with swelling. A value of α close to 1 indicates strong reorganization of surface charges upon swelling. A value of α = 0.5 was considered in ref 14 to represent the case of cylindrical pores. Using the conservation of charge in a single pore and the empirical scaling law, the swelling parameter is given by ⎛ R ⎞1 + α πR2L = ξ⎜ ⎟ η= ν0 ⎝ R0 ⎠

In order to solve for ηc, we have to derive the relation between Pel and ηc, which will be discussed in the following section.

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STRESS−STRAIN RELATIONSHIPS OF MICROSCOPIC PORE WALLS Most models of swelling phenomena in polymer materials utilize the statistical theory of rubber elasticity to establish a relation between swelling pressure and ηc. One of the earliest models, developed by Flory,43 assumes random mixing of solvent and monomers; it uses the Bragg−Williams approximation to account for interactions between different components.44 Moreover, it considers a mechanism of affine swelling, in which there is no structural reorganization at the molecular scale and junction points between polymer chains are shifted proportionally to the macroscopic swelling deformation. Following this assumption, the change in the free energy due to the elastic deformation of the swollen polymer, Fel, can be calculated for a given representation of the polymer chain length distribution. Several models relate Pel to ηc based on the assumption of affine swelling.45−48 The Flory−Rehner theory considers an affinely deformed network of N walls. For such a system, the following expression for the free energy of deformation per unit volume can be derived

(2)

with ξ=−

R 0ρSO − 3

2σ0̅

(6)

(3)

where ν0 is the volume of the unit cell in the unswollen state, L is the length of the unit cell, ρSO3− is the volumetric charge density of anionic charges, and σ̅0 is the average wall charge density of pores. The main premise of the model is that local equilibrium of water in pores is established by a balance of pressures across all

F el =

1 G ∑ (c ln βij + βij 2 − 1) 2 ij

(7)

where the summation is over the walls and the microscopic axes. G = NKT/V0 is the shear modulus, and V0 is the volume of the dry polymer. The value of the parameter c should be in the range from 0 to 1.49 However, this parameter is of no relevance in our free energy calculations because the logarithmic terms cancel out. βij are components of the deformation tensor, β̿, which is defined as ⎛β β β ⎞ ⎜ x ,x x ,y x ,z ⎟ β ̿ = ⎜ βy , x βy , y βy , z ⎟ ⎜ ⎟ ⎜β ⎟ ⎝ z , x βz , y βz , z ⎠

Figure 3. Pressure equilibrium between fluid pressure, Pfl, and elastic pressure, Pel, at the pore wall for a certain σ0 value. As the swelling parameter increases, Pfl decreases and Pel increases. At the crossing point, a stable mechanical equilibrium is established, corresponding to the swelling parameter, ηc.

The stress is applied normal to the surface of the wall, and we assume that wall deformation involves neither twisting nor torsion. Therefore, the deformation tensor can be written as

interfaces, as indicated in Figure 3. The governing equation for equilibrium of water in a single pore is el

fl

l

P =P =P +P

osm

⎛β 0 0 ⎞ ⎜ x ,x ⎟ ⎜ ⎟ β 0 0 β̿ = y,y ⎜ ⎟ ⎜0 ⎟ β 0 z ,z ⎠ ⎝

(4)

At capillary equilibrium with an external vapor phase, the liquid pressure can be written as 1/(1 + α) 2γ ⎛ ξ ⎞ ⎜⎜ ⎟⎟ P =P − R 0 ⎝ ηc ⎠ l

(8)

However, the assumption of affine swelling is not valid when macroscopically homogeneous swelling is accompanied by phase separation at the microscale. It has been observed that

g

(5) C


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The Journal of Physical Chemistry B β = ϕp−1/3

PFSA ionomer membranes exhibit nonaffine swelling; i.e., there is structural reorganization, in which the strain experienced by microscopic pore walls is significantly higher than the overall macroscopic swelling strain.50 Models that are based on the classical theory of rubber elasticity cannot be used to determine the swelling pressure in structures with nonaffine swelling behavior. A study that accounted for the nonaffine swelling of PFSA was conducted by Freger.51 Freger developed an expression for the free energy of a polymer swollen in water by assuming a volume-conserving deformation of each wall in the Flory− Rehner model.43 The model of Freger represents the swollen polymer matrix as a Voronoi tessellation; each polyhedral cell encapsulates a water droplet, as shown in Figure 4. Upon

(10)

where ϕp is the polymer volume fraction, defined as the ratio of the volume of the nonswollen network, V0, to the total volume. By summation over the principal axes of the film, we derive the free energy of deformation, using eq 7, for three different scenarios, following the formalism proposed in ref 47 1 F Ael = G(2β 2 + β −4 ) (11) 2 FBel =

1 G (β 2 + β − 2 ) 2

(12)

and 1 G(β 4 + 2β −2) (13) 2 Using eq 10, free energies of deformation can be written as 1 F Ael = G(2ϕp−2/3 + ϕp 4/3) (14) 2 FCel =

FBel =

1 G(ϕp−2/3 + ϕp2/3) 2

(15)

and 1 G(ϕp−4/3 + 2ϕp2/3) (16) 2 The change of the free energy at constant temperature and number of molecules is FCel =

Figure 4. The top panel shows a schematic cross section of the membrane represented by a Voronoi tessellation, adopted from ref 47. The zoomed in picture shows a single cell with pore walls (gray) and a swollen core (middle) as well as a single wall element (right). The bottom panel shows the three swelling scenarios considered in this work, corresponding to (a) in-plane isotropic deformation, (b) fixedlength deformation, and (c) elongation of wall-forming elements.

(17)

dF = − P dV The elastic pressure is obtained from ϕp2 ∂F el ∂F el =− P =− ∂V V0 ∂ϕp el

swelling, walls of each cell undergo compression in the thickness direction. We evaluate three scenarios for the deformation of polymer walls illustrated in Figure 4a, b, and c, which correspond to (A) isotropic in-plane deformation, (B) fixed-length deformation, and (C) elongation of wall-forming elements. For isotropic in-plane deformation, walls expand equally in both transverse directions. For fixed-length deformation, the length of the pore is conserved and polymer walls expand in the width direction only. In the third scenario, we consider elongation of polymer elements that form pore walls (bundles, fibrils, ribbons). Since the volume of the polymer phase is conserved during swelling, the product of the three deformation ratios in the deformation tensor has to be 1. The deformation tensors can thus be defined as ⎛ β −2 0 0 ⎞ ⎜ ⎟ βA̿ = ⎜ 0 β 0 ⎟, ⎜ ⎟ 0 β⎠ ⎝0

⎛ β −1 0 0 ⎞ ⎜ ⎟ βB̿ = ⎜ 0 β 0 ⎟, ⎜ ⎟ ⎝0 0 1⎠

⎛ β −1 0 0 ⎞ ⎜ ⎟ βC̿ = ⎜ 0 β −1 0 ⎟ ⎜ ⎟ ⎜ 2⎟ β ⎠ 0 ⎝0

(18)

Differentiating eqs 14, 15, and 16 hence gives 2 PAel = G(ϕp1/3 − ϕp7/3) 3 1/3 ⎛ ⎛ 1 ⎞7/3⎞ 2 ⎛ 1 ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ ⎟ = G⎜⎜⎜ 3 ⎜⎝⎝ ηc + 1 ⎠ η + 1 ⎝ c ⎠ ⎟⎠ 1 G(ϕp1/3 − ϕp5/3) 3 1/3 ⎛ 1 ⎛ 1 ⎞ ⎟⎟ − = G⎜⎜⎜ 3 ⎜⎝⎝ ηc + 1 ⎠

(19)

PBel =

⎛ 1 ⎞5/3⎞ ⎜⎜ ⎟⎟ ⎟ η + 1 ⎝ c ⎠ ⎟⎠

(20)

and

and

2 G(ϕp−1/3 − ϕp5/3) 3 −1/3 ⎛ ⎛ 1 ⎞5/3⎞ 2 ⎜⎛ 1 ⎞ ⎟⎟ ⎟⎟ ⎟ = G ⎜⎜ − ⎜⎜ 3 ⎜⎝⎝ ηc + 1 ⎠ ⎝ ηc + 1 ⎠ ⎟⎠

PCel =

(9)

(21)

where the relation ϕp = 1/(ηc + 1) was used to relate the elastic pressure to the microscopic swelling parameter defined in eq 2. The elastic pressure in eqs 19, 20, and 21 approaches zero under dry conditions, as expected, and it rises linearly for a low degree of swelling. At a higher degree of swelling, Pel levels off

for the cases of isotropic, fixed-length, and elongation deformation of polymer walls, respectively. We use an empirical relation to express β as a function of the macroscopic deformation of the polymer fraction D


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parameter with the statistical pore density distribution, given as a function of wall charge density in the reference state, n(σ0). The volumetric expansion upon swelling is

in the case of isotropic deformation. Figure 5 illustrates the elastic pressure as a function of ηc for the three different

1 ΔV = V0 Nuc

∫0

σ0 max

n(σ0)η(σ0 , σ0,c) dσ0

(23)

with the function η(σ0, σ0,c) obtained from σ0,c =

2εR gT ⎛ ξ ⎞(1 − α)/(1 + α) ⎜ ⎟ FR 0 ⎝ η ⎠

⎧ ⎪ × ⎨1 − ⎪ ⎩

⎫ ⎛ F ⎞2 R 2 ⎡ ⎛ ⎞1/(1 + α)⎤⎛ ξ ⎞−2/(1 + α) ⎪ ⎥⎜ ⎟ ⎟⎟ 0 ⎢P el − P g + 2γ ⎜ ξ ⎟ ⎬ 1 + ⎜⎜ ⎥⎝ η ⎠ R0 ⎝ η ⎠ ⎪ ⎝ R gT ⎠ 2ε ⎢⎣ ⎦ ⎭

(24)

The parameter Nuc in eq 23 is a normalization factor that is defined as

Figure 5. Elastic pressure for the different deformation scenarios illustrated in Figure 4 as a function of the swelling parameter η.

Nuc =

scenarios. In the section “Pore Ensemble Model”, we evaluate the relevance of these deformation scenarios by comparison with experimental data. For a single pore that is in capillary equilibrium at given external conditions, the relation between ηc and σ0,c is given by 2εR gT ⎛ ξ ⎞ ⎜⎜ ⎟⎟ FR 0 ⎝ ηc ⎠

⎧ ⎪ × ⎨1 − ⎪ ⎩

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∫0

σ0 max

n(σ0) dσ0

(25)

Knowledge of the statistical pore density distribution as a function of surface charge density, n(σ0), is a requirement to predict the water sorption behavior of a PEM. However, we cannot determine it from existing experimental data. We will thus assume its form according to the following general requirements: (1) it is a continuous distribution, (2) its integral is consistent with the ion exchange capacity, which is defined as the number of moles of sulfonic acid groups per gram of dry polymer, and (3) it has a single peak. The maximum wall charge density σ0max can be determined from ab initio simulations of arrays of charged surface groups.52 Calculations suggest a maximum density of sulfonic head groups of σ0max = −0.52 C m−2. For σ0 < σ0max, water molecules and hydronium ions would be squeezed out from the space between anionic head groups and strong electrostatic interactions between anions would destabilize the polymer matrix. σ0max is thus the cutoff charge density of the distribution. To validate the model, it is essential to compare it with experimental water sorption data. For this purpose, we have to relate the volumetric expansion to the water content, λ. Water content is normally defined as the number of water molecules per anionic headgroup. The volumetric expansion of the PEM,

(1 − α)/(1 + α)

σ0,c =

V0 = ν0

⎫ ⎛ F ⎞2 R 2 ⎡ ⎛ ξ ⎞1/(1 + α)⎤⎛ ξ ⎞−2/(1 + α) ⎪ 2 γ ⎢ ⎥ 0 el g ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 1 + ⎜⎜ ⎢P − P + R ⎜⎝ η ⎟⎠ ⎥⎜⎝ η ⎟⎠ ⎪ ⎝ R gT ⎠ 2ε ⎣ 0 c ⎦ c ⎭

(22)

PORE ENSEMBLE MODEL SAXS, SANS, porosimetry, and water sorption studies of PEMs show a dispersion in pore sizes and an evolution of the pore size distribution during water uptake. Eikerling and Berg related the dispersion in pore sizes to fluctuations in the surface charge density in the reference state, σ0. The total water sorption of the membrane is obtained by summing up elementary contributions from each unit cell. Mathematically, this corresponds to a convolution of the single pore swelling

Figure 6. (a) Statistical pore density distribution vs surface charge density of an ensemble of pores. The red dashed line shows a Gaussian distribution function and the black solid line a log-normal distribution function. (b) Differential pore volume as a function of pore radius for the different pore density distributions shown in part a. E


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The Journal of Physical Chemistry B normalized to the dry PEM volume, is related to λ via the following relation ΔV = IECρp Vw̅ λ V0

was chosen deliberately as R0 = 1 nm. The surface tension of water, γ, is an experimentally well-measured quantity. For water in pores, we assume perfect wetting, thus cos θ = 1. We calculated the mean wall charge density in the reference state, σ̅0, from the structural characteristics of Nafion (grafting density and aggregation33); this approach yields σ̅0 = 0.08 C m−2 for membranes with an IEC of about 0.9 mmol g−1. For the dielectric permittivity, εr, modeling studies suggest a value in the range of 10−15 in the surface water layer at pore walls and 80 for bulk-like water in the pore center.53 We have used an effective dielectric permittivity of εr = 25 for water. With this information and comparing model isotherms with experimental data, we found that the standard deviation, dσ0, which will give us best agreement, is 0.034 C m−2 for fresh membranes. A value of 0.5 was used for α. Therefore, G was left over as the only free parameter that we determined from the analysis of water sorption data for a fresh membrane. We used R-square as a statistical measure to evaluate the goodness of our fits. A Gaussian distribution n(σ0) results in a Gaussian differential pore volume and a log-normal distribution n(σ0) in a log-normal differential pore volume. As is apparent in Figure 6b, the two curves mostly differ at large values of the pore radius. In general, the log-normal distribution of n(σ0) yields a better agreement with experimental pore size distribution functions for polymer electrolyte membranes.54 Figure 7a exhibits the differential pore volume as a function of pore radius at different RH values. Figure 7b shows the statistical pore density distribution vs elastic pressure. Upon increase of the RH, the elastic pressure at the pore level shifts to larger values. The statistical pore density distribution vs elastic pressure becomes wider. Therefore, increasing the water uptake results in a more heterogeneous pressure distribution in the PEM, which could induce larger stress fluctuations and an increased propensity to mechanical degradation.55,56 Experimental values of the shear modulus for Nafion range from 250 to 300 MPa.55 These values represent an effective macroscopic shear modulus. The input of the model is the microscopic shear modulus for polymer bundles or walls that confine pores. Exact experimental values for this microscopic property are not available. However, it is reasonable to expect that the microscopic shear modulus at the pore level is significantly larger than its macroscopic counterpart, which is a volume-averaged parameter.57 In agreement with this inference,

(26)

where ρp is the polymer density and V̅ w is the molar volume of the water. The volume change due to water uptake can be obtained from 1 ΔV = V0 Nuc

∫0

ηc

dηn(σ0)η

dσ0 dη

(27)

The differential pore volume as a function of pore radius is obtained from 1 + 2α ⎛ ⎞ dσ0 ⎟ (1 + α)ξ 2 ⎛ R ⎞ d ⎛ ΔV ⎞ 1 ⎜ σ n ( ) ⎜ ⎟= ⎜ ⎟ 0 dR ⎝ V0 ⎠ Nuc ⎜⎝ R0 dη ⎟⎠ ⎝ R0 ⎠

(28)

Figure 6b shows the differential pore volume as a function of pore radius obtained, when Gaussian and log-normal distribution functions are used for n(σ0), as shown in Figure 6a. Parameters employed in this calculation are given in Table 1. In the list of parameters, we have a good understanding of Table 1. Reference Set of Parameters parameter temperature gas pressure ion exchange capacity density of dry polymer reference radius surface tension of water contact angle mean wall charge density dielectric permittivity standard deviation of wall charge density distribution reorganization parameter shear modulus

symbol

value

unit

T Pg IEC ρp R0 γ θ σ̅0 εr dσ

298 0.1 0.91 2000 1 0.072 0 −0.08 25 0.034

K MPa mol kg−1 kg m−3 nm N m−1

α G

0.5 0.38

C m−2 C m−2

GPa

the first seven parameters. Temperature and gas pressure are externally controlled. The IEC and ρp have been measured in the experiments that we have analyzed. The reference radius

Figure 7. (a) Differential pore volume as a function of pore radius of the ensemble upon increasing relative humidity (25, 50, 75, and 90%). (b) Statistical pore density distribution vs elastic pressure upon increasing relative humidity (25, 50, 75, and 90%). F


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Figure 8. (a) Sorption isotherm calculated using the fixed-length deformation (black line), isotropic in-plane deformation (red line), and elongation (blue dashed line) scenarios, defined in Figure 4, of polymer walls in comparison with two sets of experimental data.59,60 (b) Volumetric swelling of water, methanol, and ethanol sorption isotherms of Nafion 1100. The circles, triangles, and squares represent experimental data for water, methanol, and ethanol sorption.59−61

an increase in shear modulus has been observed in ionomer thin-film systems, studied experimentally in ref 58. Sorption isotherms have been calculated for the three deformation scenarios discussed in the section “Stress−Strain Relationships of Microscopic Pore Walls”, namely, the isotropic deformation, fixed-length deformation, and axial stretching scenario. The results are compared with two sets of experimental data59,60 in Figure 8a. Experimental data have been corrected by subtracting the amount of surface water from the total water content reported in experimental studies. The fixed-length deformation scenario gives the best agreement with water sorption data, if a realistic value of the microscopic shear modulus, G, is used. A rationalization of this finding at the microstructure level is beyond the scope of our work. It will require detailed molecular level simulations, which we are currently performing in our group. A sensitivity analysis with respect to the key parameters, viz., εr, dσ, α, and G, was conducted; for a meaningful analysis, we evaluated the parameter variation necessary to bring about a change in water sorption by ±5% at 90% RH. The analysis revealed a high sensitivity to the εr (8% variation required) and G (10% variation required) and a lower sensitivity to dσ (30% variation required) and α (25% variation required). It is worth noting that experimental studies also exhibit significant uncertainty because of the dependency of water sorption data to pretreatment procedures, as can be seen in Figure 8a. We also calculated the volumetric expansion of Nafion in methanol and ethanol and compared it with available experimental data.61 Figure 8b shows good agreement of the model with these data. Sets of parameters used for calculations are listed in Table 2. The volumetric expansion of the PEM is largest in ethanol followed by methanol and water. These trends can be rationalized with the identified changes in εr and G. As εr decreases, the osmotic pressure that swells pores is bound to increase. Moreover, the less polar solvents result in weaker phase separation, softening the polymer matrix and leading to a reduction of G. The net effect is a larger volumetric expansion. For the purpose of displaying this effect, experimental data for λ have been converted to ΔV/V0, using the linear relation in eq 26. To achieve good agreement with experimental data, we had to use a lower degree of surface charge reorganization, represented by α, and a larger σ̅0 for swelling in ethanol and methanol. These findings suggest that

Table 2. Microscopic Shear Modulus, Dielectric Constant, and Molecular Weight Corresponding to the Sorption Isotherms in Figure 8b water shear modulus (G) (MPa) dielectric constant (ϵr) molecular weight (g mol−1) reorganization parameter (α) average wall charge density (σ̅0) (C m−2) surface tension (γ) (N m−1) molar volume (V̅ ) (m3)

methanol

ethanol

380

230

205

25 18.01

13 32.04

10 46.07

0.5

0.2

0.2

0.08

0.15

0.21

0.072

0.022

0.022

18.06 × 10−6

40.50 × 10−6

58.30 × 10−6

more pronounced structural changes occur when water is exchanged for ethanol and methanol. The parametric effect of α is shown in Figure 9. For α = 1, the PEM swells more because of the stronger reorganization of surface charges.

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MEMBRANE DEGRADATION ANALYSIS In this section, we use the model to analyze water sorption data of PEMs that have undergone structural changes. On the basis of this analysis, we infer a structural rationale for the observed changes in water sorption. We evaluate data sets from different

Figure 9. Effect of the parameter α. G


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Figure 10. Hygrothermal degradation: (a) Water sorption isotherms recorded in aging study after 0, 80, and 280 days.62 Dots represent experimental data, and lines correspond to curves calculated in the model. (b) Statistical pore density distribution vs surface charge density corresponding to the water sorption isotherms in part a. (c) Differential pore volume as a function of pore radius corresponding to the water sorption isotherms in part a. (d) Statistical pore density distribution vs elastic pressure, corresponding to the water sorption isotherms in part a.

the corresponding statistical pore density distribution vs surface charge density, as shown in Figure 10b. Good agreement of the model with experimental data is achieved if in the course of aging the average surface charge density and the shear modulus of polymer walls increase. The total number of pores decreases drastically, as indicated by the area under the curve in Figure 10b. Moreover, reorganization of surface groups during swelling becomes more pronounced, as indicated by the value of α that increases upon aging. The increasing trend for the shear modulus is in agreement with experimental findings for the macroscopic shear modulus, shown in Table 3. This behavior is

experimental protocols, exploring the impact of hygrothermal aging and chemical degradation. We use again the reference set of parameters in Table 1 for fresh membranes. We keep σ̅0 and dσ0 constant for chemical degradation studies because the change in IEC is small (

Model of Water Sorption and Swelling in Polymer Electrolyte

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