Chloride Enhances Fluoride Mobility in Anion Exchange Membrane

Dec 20, 2013 - John Savage , Ying-Lung Steve Tse , and Gregory A. Voth. The Journal of Physical Chemistry C 2014 118 (31), 17436-17445. Abstract | Ful...
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Chloride Enhances Fluoride Mobility in Anion Exchange Membrane/ Polycationic Systems Ying-Lung Steve Tse,†,§, ∥ Himanshu N. Sarode,¶ Gerrick E. Lindberg,†,§, ∥ Thomas A. Witten,‡,§ Yuan Yang,# Andrew M. Herring,¶ and Gregory A. Voth*,†,§, ∥ †

Department of Chemistry, ‡Department of Physics, §James Franck Institute, and ∥Computation Institute, University of Chicago, Chicago, Illinois 60637, United States # Department of Chemistry and Geochemistry and ¶Department of Chemical and Biological Engineering, Colorado School of Mines, Golden, Colorado 80401, United States S Supporting Information *

ABSTRACT: We have studied anion exchange membrane/polycationic systems with different percentages of fluoride and chloride as counterions by molecular dynamics simulations. We also experimentally measured the self-diffusion constant of fluoride in a diblock copolymer that has the same hydrophilic block and found satisfactory agreement with simulations within a factor of 2. At 300 K, our simulations showed that the self-diffusion constant of fluoride increases by about 70% when fluoride content decreases from 100% to 40% (and 60% Cl), and it increases by about 140% when fluoride content decreases from 100% to 10%. Increasing % Cl also slightly decreases the attraction between fluoride and the cations. We hypothesize that the root cause of the enhancement in fluoride mobility is due to the larger size of the chloride ion, which more readily loses its water solvation shells because of a lower charge/radius squared (surface electric field). This in turn frees up more water for ion transport. We believe this effect is likely more general than just the fluoride/chloride case reported here.

1. INTRODUCTION Anion exchange membrane (AEM) fuel cells have been gathering attention in recent years since they were proposed as a solution to the problems of the liquid electrolyte alkaline fuel cell.1,2 Because the oxygen reduction reaction is more facile in an alkaline environment,1 one of the most important advantages of AEM fuel cells is that they can utilize non-noble metal catalysts such as iron, cobalt, or nickel. Moreover, they could theoretically operate on a variety of more complex fuels such as the heavier alcohols (ethanol, propanol, ethylene glycol, etc.) and generate high power densities at moderate operating temperatures.3 Because of these advantages, AEM fuel cells show promise to replace proton exchange membrane (PEM) fuel cells built with perfluorosulfonic acid (PFSA) membranes that have been the most widely used and studied but suffer from high cost and limited fuel versatility. The AEM, at the heart of these fuel cells, has its own challenges. One major challenge is the chemical durability of the polycationic groups and their polymeric backbones under alkaline conditions.4 The self-diffusion constant of hydroxide is lower than that of an excess proton under most conditions, so high ion exchange capacities (IECs) or very wet conditions are needed for hydroxide transport to achieve conductivities that can compete with those of PEMs. Accordingly, maximizing conductivity, especially under conditions of lower relative humidity, has become a focus of AEM research. © 2013 American Chemical Society

Much of the research focus of AEMs has been on random copolymers made from hydrophilic, cationic moieties, but the study of block copolymers has gained traction.5−10 Recently, Tsai et al.11 synthesized and studied polystyrene-b-poly(vinyl benzyltrimethylammonium hydroxide), abbreviated as PS-b[PVBTMA][OH]. When both hydrophobic (PS) and hydrophilic (PVBMTA) domains are present, microphase separation makes it possible for a wide range of highly ordered morphologies, such as cylinders, lamellae, or gyroids,12,13 to form. Moreover, the presence of a hydrophobic block enhanced the mechanical stability of the membrane, which permits a higher density of cationic moieties. In the case of random copolymers, too many cationic groups could lead to the disintegration of the membrane at high hydration levels. Benzyltrimethylammonium (BTMA) cations are stable in an alkaline environment and do not suffer from Hofmann elimination because there are no β-hydrogen atoms. In addition, the polycationic form of BTMA (PVBTMA) has promising anionic conductivity.4,14 Because of these advantages, we have chosen to focus on ion transport in PVBTMA. Understanding anion transport in a polycationic system swollen with water is challenging. The motion of ions is Received: September 30, 2013 Revised: December 18, 2013 Published: December 20, 2013 845

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Figure 1. Simulated structure of the poly(vinyl benzyltrimethylammonium), PVBTMA, that is made of 20 monomers.

“F” and “Cl” to refer to fluoride and chloride, respectively. In the simulation setups, various ratios were studied: % Cl = 0%, 10%, 40%, 60%, and 90%, and all setups satisfied % F + % Cl = 100%. The general AMBER force field (GAFF)17 was used for PVBTMA; the water model was SPC/Fw;18 and the LennardJones parameters for the halides were obtained from the work of Jensen and Jorgensen.19 The Lorentz−Berthelot combining rules were used. Since the halide models were originally parametrized for the TIP4P water model20 and the geometric combining rules, we performed additional tests and confirmed the radial distribution functions and the coordination numbers for the halides with water in our setups are consistent with the original results (see Figure S6b in the SI). Coulombic interactions were treated by the Particle−Particle Particle−Mesh (PPPM) solver21 with an accuracy of 10−5 relative error in forces. The Lennard-Jones interactions and the short-range part of the Coulombic interactions had a cutoff distance of 14 Å. The partial charges were based on the RESP charges of three connected monomers which were calculated by Gaussian09 and the R.E.D. Server.22−25 The actual partial charges used are given in Figure S2 in the SI. A 1 fs time step was used for all simulations. For equilibration, the systems were first relaxed using constant number, pressure, temperature (NPT) dynamics at 400 K and 1 atm for 12 ns. The temperature was then brought down to the desired temperature at the same pressure for another 6 ns with constant NPT dynamics. The systems were finally equilibrated with constant number, volume, temperature (NVT) dynamics at the desired temperature for at least 12 ns more (see Table S1 in the SI for the final density values). After equilibration, six configurations were chosen from the last 2.5 ns of the same trajectory to become the initial states of six separate trajectories in the subsequent constant number, volume, energy (NVE) calculations, which used the same control parameters as those of constant NVT calculations but without a thermostat. Data presented in this paper were collected from these separate constant NVE trajectories for at least 10 ns. The energy drift was found to be on the order of 1 kcal/mol/ns. Two snapshots of a simulation run are shown in Figure 2 to illustrate what the simulated environment is like.

was made of 20 monomers. The simulation box was a simple cube with periodic boundary conditions. There were four polymer chains in the simulation box, and they were built simultaneously with a random morphology by a Monte Carlo algorithm developed by Knox and Voth.15 To ascertain that the local properties (local transport, RDFs, etc.) reported in this paper have already reached their asymptotic limits, a smaller system with four chains of 10 monomers was also studied. We found that a factor-of-two change in chain length has no significant effect, and thus the 10-monomer chains (and certainly the 20-monomer chains) have reached the asymptotic behavior of the reported calculations. We have provided some of these tests in Figure S1 in the Supporting Information (SI). The simulations were carried out using LAMMPS.16 Three hydration levels (λ, defined as the number of water molecules per cation) at 5, 9, and 14 and three temperatures at 300, 328, and 353 K were studied. The counteranions consisted of both fluoride and chloride ions. Since fluorine and chlorine in our simulations are always in their ionic form, we use the symbols

Figure 2. Two snapshots of a simulation run that are separated by about 3 ns in time. A specific fluoride ion is shown as red, and its trajectory between the frames is shown as orange in (b). The other colors are yellow (other fluoride ions), dark blue (nitrogen), transparent gray (rest of PVBTMA atoms), and transparent light blue (water oxygen and hydrogen atoms). The hydration level was 14, and the temperature was 300 K. The sizes of the atoms are altered to aid visibility. The snapshots were prepared in Visual Molecular Dynamics (VMD).26

thwarted by steric blockades and Coulombic barriers. Simulations can shed light on the nature of these barriers and suggest synthetic means to mitigate them. Moreover, modern simulations have sufficient atomic-scale realism to anticipate quantitative agreement with experiment. To maximize this potential agreement, we examine diffusivity, directly measurable with pulsed-field gradient spin−echo (PFGSE) nuclear magnetic resonance (NMR) experiments. To provide the simplest comparisons, we chose to study fluoride ions as the diffusing species. The fluoride ion (19F−) is readily addressable by the NMR experiments, and it can also be reasonably modeled by classical force fields in molecular dynamics (MD) simulations. However, it turns out that the binding between BTMA cations and chloride ions, the counterions remaining from the syntheses, is so strong that complete exchange to fluoride cannot be easily accomplished experimentally. Interestingly, the remaining presence of chloride strongly influences fluoride transport in these AEM/polycationic systems. We find that when there is more chloride the fluoride mobility is enhanced. Since the mechanism that leads to the enhancement provides insight into how ion mobility is affected by the local environment, this paper also focuses on the interplay between the coexisting anions and the cationic groups. This effect should be general and not specific to just the fluoride/chloride case reported here.

2. SIMULATION DETAILS The structure of the PVBTMA that we simulated is shown in Figure 1. Each monomer had a net charge of +1, and each chain

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3. EXPERIMENTAL DETAILS The polymer studied experimentally was the diblock copolymer polyethylene-b-poly(vinyl benzyltrimethylammonium fluoride/ chloride), PE-b-[PVBTMA][F/Cl]. The PVBTMA moiety was the simulated cation described in the previous section. The counteranions consisted of both fluoride and chloride (more details below). The mass ratio of PE to PVBTMA was 2 to 3, and the ion exchange capacity (IEC) was determined to be 1.7 meq/g. The membrane was exchanged from the chloride form to the fluoride form by incubating it in a 4 M solution of benzyltrimethylammonium fluoride at room temperature in a pressure bomb at 500 PSI for 7 days. The exchanged membrane was thoroughly washed with deionized water for 24 h and dried before using. A small sample of the exchanged membrane was sent to Galbraith Laboratories (Knoxville, Tennessee, USA) to determine the final % F, and it was found to be only 44%. Water uptake was measured using a Dynamic Vapor Sorption (DVS) apparatus, DVS Advantage from Surface Measurements Systems Ltd. The hydration levels (λ) were calculated gravimetrically by passing a humidity controlled nitrogen gas and noting the change in weight of the membrane. Humidity was cycled twice from 0% to 95% in steps of 20% intervals at 26 °C. The membrane was allowed to equilibrate for two hours at each humidity level before recording the final weight of the membrane. λ at 95% relative humidity was found to be 13.5 ± 1.1 (this led us to focus on λ = 14 in simulations). The self-diffusion constants of fluoride (19F) at temperatures of 30, 40, 50, and 55 °C were determined by pulsed-field gradient spin−echo (PFGSE) nuclear magnetic resonance (NMR) measurements. The experiments were carried out using a Bruker AVANCE III NMR spectrometer and 400 MHz (1H frequency) wide bore Magnex magnet. 19F (376.02 MHz) diffusion measurements were made using a 5 mm Bruker singleaxis DIFF60L Z-diffusion probe. The 90° pulse length was on the order of 5 μs. The range of gradient strength g was 0−500 G/cm, which was incremented in 16 steps. The maximum value of the gradient was chosen such that the signal decayed completely. The duration of the gradient pulse δ was 1 ms, and the time between pulses Δ was 50 ms (results using Δ = 20 or 100 ms are within 15% of 50 ms results reported here). The NMR tube was maintained at saturated conditions of humidity by putting deionized water at the bottom of the sample tube (see Figure S3 in SI for a schematic view of the NMR tube used). The self-diffusion constants D were determined by fitting the measured data to the Stejskal−Tanner equation27 S /S0 = exp[−γ 2g 2δ 2(Δ − δ /3)D]

Figure 3. Measured dependence of S/S0 is plotted as a function of the gradient strength g at 55 °C. The best fit to eq 1 is shown as the red line. The self-diffusion constant of fluoride D was found to be 5.50 × 10−6 cm2/s in this example. The other experimental results are shown in Figure 6b.

is greatly reduced when the simulation is done with chloride and fluoride in the proportions used experimentally. Thus, both simulation and experiment suggest that the presence of chloride enhances the mobility of fluoride. We propose that this effect originates from the difference in size between fluoride and chloride. We provide evidence for it by analyzing the water coordination around fluoride/chloride, the free energy differences of ion binding, and the mean residence times. Justification of the Simulation Approximations. As noted above, the experimental system is a microphase-separated diblock material, while our simulation treats only a homogeneous solution of the ion-containing species. Here we discuss why such a simulation is appropriate for representing the ion transport properties of interest. Small-angle X-ray scattering data show that PS-b-PVBTMA in a lamellar phase has the total width of a hydrophilic domain and a hydrophobic domain together that is between 35 and 50 nm.28 An all-atom simulation that includes both domains would contain millions of atoms, but as we argue below, such large-scale atomistic simulations would not be needed to study these systems. Since it is evident that the ions stay in the hydrophilic domains almost all of the time, studying ion transport by simulating only the hydrophilic block, as in our simulations, should capture the most important features of local ion transport on the nanometer length scale and nanosecond time scale. However, the measured diffusivity by NMR diffusion experiments is on a micrometer length scale and millisecond time scale. Even though diffusivity depends on the heterogeneities in the local environment, it also potentially depends on the constraints on the hydrophilic domains and further on the confinement of ions into the hydrophilic channels, which may be blocked or tortuous due to the microphase separation with the hydrophobic domains. Therefore, it is important to question how the hydrophobic domains can potentially affect ion transport and whether we can compare local diffusion information from MD simulations with NMR diffusion data. When both hydrophobic and hydrophilic domains are present, mircophase separation (see Figure 4) has three obvious effects: (1) well-defined phase boundaries (Figure 4a) are formed so that the intersection point of the hydrophobic block and the hydrophilic block on a chain is effectively tethered in place (see also polymer brush

(1)

where S is the signal amplitude; S0 is the signal amplitude at g = 0; and γ is the gyromagnetic ratio. A sample plot is shown in Figure 3.

4. RESULTS AND DISCUSSION In this section, we first justify why we can compare local transport data from simulations with NMR diffusion data when the length and time scales involved are different by several orders of magnitude. Then we report our simulation results and compare them with experimental data where available. The main result is that the observed diffusion is markedly faster than the simulated diffusion with 100% fluoride in PVBTMA. However, the discrepancy between simulation and experiment 847

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tethering constraint has a negligible effect on ion motion on nanometer and nanosecond scales. The second effect caused by the alignment of the lamellae cannot be easily captured with our all-atom simulations because of the limitation on system sizes, but we can gain insights into this effect from experimental data on ionic conductivity. The hydrophobic domains potentially limit conductivity just as they limit diffusivity. However, a study by Park and Balsara32 found that the conductivity that is normal to the plane of the membrane, σ⊥, increases only by 30% when comparing random to optimally aligned lamellae. Moreover, geometrical reasoning was used to argue that the alignment for lamellar phases alone cannot yield more than a 1.5-fold conductivity gain in lamellar phases.33−35 Our setups with only hydrophilic blocks and periodic boundary conditions would be similar to a continuous conductive band in perfectly aligned lamellae. Such a continuous band should be well described by the replicas of a central simulation cell with periodic boundary conditions. The local self-diffusion constant from simulation will be a good estimate of the experimental value for such a system, if we are able to obtain a convergent estimate of the self-diffusion constant before the particles of interest move in a length scale that is not much larger than the simulation box width (we see diffusive behavior on a length scale of about 1 nm in Figure 5), whereas the box length is about 4 nm. Typically in our simulations, the MSD reaches the linear regime on the time scale of a few nanoseconds, and this implies that the system is roughly homogeneous on the length scale traveled by a fluoride ion during that time. Furthermore, the insight above suggests that the conductivity in our setups would not be different by more than 50% if we were able to simulate different alignments of lamellae by incorporating multiple (maybe more than a few) hydrophilic and hydrophobic blocks. In light of all these, local diffusion data collected in simulations in which only hydrophilic blocks are present should be meaningful, and this provides an opportunity for direct comparison between simulation and experiment diffusion data for diblock copolymers. The last effect is concerned with how the water uptake of the membrane changes because of the hydrophobic blocks. Our simulations can directly study how this effect can influence ion transport by simulating at different hydration levels. Even though we have collected data for different hydration levels (5, 9, and 14), the focus of this paper is on the trends in ion transport at constant hydration level 14. For better clarity of the discussion, we have put the diffusion data for lower hydration levels (5 and 9) in Figure S4 in the SI. Comparison of Fluoride Self-Diffusion Constant between Simulation and Experiment. Self-diffusion constant data from both simulations and experiments are compared in this subsection. In Figure 6a, the fluoride MSDs are plotted as a function of time and % F (and % Cl). The selfdiffusion constants were extracted from 1/6 of the long-time slopes of the MSD curves. Figure 6a clearly shows fluoride mobility is enhanced by the presence of chloride. The effect is significant; the self-diffusion constant increases by about 140% from 100% F to 10% F. This is surprising, considering that the major difference comes from only the sizes of the halide ions. A similar effect can also be seen in our simulations of simple ionic sodium/chloride/ fluoride solutions. These results are reported in Figure S7 in the SI.

Figure 4. (a) Schematic diagram showing the interfaces (dashed gray) that are formed by the hydrophobic blocks (black) and the hydrophilic blocks (blue). (b) Schematic diagram of a diblock copolymer with lamellar morphology that has well-defined orientations.

models),29,30 (2) well-defined orientations of the lamellae (Figure 4b) are also formed, which affect the tortuosity of ion transport pathways, and (3) water uptake as a function of relative humidity is changed. The first effect can be investigated by fixing one end31 of each polymer chain as if they were grafted on the hydrophobic−hydrophilic interface and comparing the difference in the self-diffusion constants. In an MD simulation, one of the most common ways to extract the self-diffusion constant of a particle is by studying the long-time behavior of its meansquared displacement (MSD) which is defined as N

MSD(t ) =

1 ⟨∑ |ri(t ) − ri(0)|2 ⟩ N i=1

(2)

where the angular brackets denote an ensemble average of systems whose initial states are sampled from a canonical (constant NVT) distribution and propagate in time with constant NVE dynamics. For a diffusive particle, its long-time MSD grows linearly in t, and the self-diffusion D is related to the long-time MSD by the Einstein formula D = lim MSD(t )/6t t →∞

(3)

In Figure 5, the MSDs of fluoride are shown for two sets of simulations with free polymer chains and with polymer chains

Figure 5. Fluoride MSDs with free polymer chains (red) and with chains with one end tethered (black) are plotted as a function of time. The green asymptotic line has a fixed slope of 1. The conditions of the setups were 100% F, 300 K, and hydration level 14.

with their one end fixed. These results show there are no statistical differences for the two sets of simulations and suggest that the first effect is not significant for ion transport. The green line in the figure has a slope of 1, and it passes through both the red and black data points beyond 3 ns. This shows that the 848

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Figure 6. (a) Fluoride MSD as a function of time and % F content at 300 K and λ = 14. (b) Fluoride self-diffusion constants as a function of % F and reciprocal temperature at λ = 14. The lines are exponential fits that were used to obtain the nominal activation energy barriers ΔEa for the diffusion processes. The ΔEa values are reported in the table. a “NMR Exp” refers to the NMR experiments described in the section “Experimental Details”. The experimental ΔEa was found to be 6.9 kcal/mol by fitting all four points (dotted orange line), but a better fit (solid orange line) was obtained without fitting the data point at 303 K (ΔEa = 5.0 kcal/mol).

lower than those of fluoride (within 15%), even though the selfdiffusion constant for the chloride model in water at 298 K is considerably larger than that of the fluoride model (1.55 × 10−5 vs 1.08 × 10−5 cm2/s, see Figure S6a in the SI). The setups with 100% F were also studied at λ = 5 and 9. Increasing λ increases fluoride mobility dramatically. At 300 K, the self-diffusion fluoride increases from 2.1 × 10−9 to 1.1 × 10−7 to 7.2 × 10−7 cm2/s from λ = 5 to 9 to 14; it increases by 2 orders of magnitude. See Figure S4 in the SI for other data at 328 and 353 K. Strong, but relatively smaller, effects of hydration were also seen in hydroxide conductivity.28 Potentials of Mean-Force and Residence Times. To understand the interplay between the halides (fluoride or chloride) and the ammonium cations, we compare the strength of ion binding between the ammonium cation and the anion, which can be understood in terms of the potential of mean forces (PMFs) in Figure 7. The PMFs, obtained from the radial distribution functions by the reversible work theorem,36,37 tell us how much free energy the system needs to move an anion away from an ammonium cation center as a function of the distance between the two centers. At 100% F, the free energy difference ΔF to move a fluoride ion out of the energy well at 4.75 to 7.25 Å is about +0.46kBT (0.27 kcal/mol at 300 K), whereas the ΔF to separate a chloride ion from the cation is about +1.8kBT. Thus, the N−Cl binding is significantly stronger than the N−F binding. We can understand more about the ion binding by calculating the average time that an anion spends in the ammonium cation energy well. One metric for such an average time is the mean of residence times. A residence time between the anion and the cation is defined to be the time that starts from the moment when the two centers gets closer than r0 and ends at the moment when they get farther apart than r0 for longer than or equal to t*.38 We choose t* to be 2 ps and r0 to be the boundary of the first energy well in the PMF, namely, 7.25 Å for N−F and 6.5 Å for N−Cl (marked by the green and orange vertical lines in Figure 7). The boundary of the first energy well for N−F is not entirely clear, but we confirmed that

The self-diffusion constant of fluoride as a function of % F and reciprocal temperature at constant hydration level 14 were plotted in Figure 6b. As expected, the self-diffusion constant increases as the temperature increases. The experimental data (44% F) are shown as orange in the figure. When the experimental data are compared with 40% F simulation data, we see that the calculated self-diffusion constant is 45% and 48% lower than the experimental value at 303 K (300 K in simulation) and 323 K, respectively. The agreement is reasonable, and this underestimation by the simulations is not unexpected because the simulated fluoride model also diffuses too slowly by about 25% in the case of pure water (see Figure S6a in the SI). Moreover, the relative humidity inside the sealed NMR tube, which had a deionized water reservoir, was probably close to 100% instead of 95% (λ = 14). The values of the activation energy ΔEa of the fluoride diffusion process at different % F are obtained from the slopes of the curves reported in the table in Figure 6b. The barrier decreases monotonically as a function of % Cl. Moreover, the enhancement of fluoride mobility becomes smaller when the temperature increases. Fluoride diffusion generally increases monotonically when % Cl increases, but because the enhancement becomes less significant as temperature increases, the 100% F data point (black) is higher than the 90% F data point (red) at the highest temperature studied (353 K or 0.00283 K−1). The experimental value of ΔEa for the diblock membrane with 44% F was found to be 6.9 kcal/mol. If the last experimental data point, which is not fitted well by an exponential curve, is not included, the estimated ΔEa is 5.0 kcal/mol. The calculated ΔEa for 40% F from simulations, 5.2 kcal/mol, is in good agreement with the experimental value. Since this paper focuses on fluoride diffusion, the discussion on chloride transport is kept to a minimum; the chloride MSDs and self-diffusion constants are given in Figure S5 in the SI. However, many of the arguments provided below for fluoride are also applicable to chloride. Chloride is affected by the presence of fluoride and diffuses more slowly when % F increases (or diffuses faster when % Cl increases). The selfdiffusion constants of chloride were usually found to be slightly 849

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relationship works well for the temperature range studied. However, as the diffusion constant goes to zero, the residence should become infinitely long-lived (and 1/τ should go to zero exactly). The nonzero x-intercept means that this simple linear relationship has to eventually break down for larger residence times. As we see from the free energy differences and the mean residence times, replacing fluoride with chloride decreases the effective attraction between the remaining fluoride and the cations. It is tempting to attribute this to be the source of the enhancement effect. However, from 100% to 10% F, the decrease in the free energy difference is no more than 0.4kBT, and the decrease in the mean residence time is only about 20%. The decrease in the strength of N−F ion binding alone cannot fully explain an increase of 140% in the fluoride self-diffusion constant. Surface Electric Field and Water Coordination Number. The only two differences between the fluoride model and the chloride model in the simulations are the mass and the size (σ in the Lennard-Jones potential). The latter causes the surface electric field of fluoride to be about 1.7 times higher than that of chloride, so fluoride should have tighter water solvation shells. This is supported by simulation data shown in Figure 10a.41 If we plot the number of water neighbors in the first solvation shell as a function of the distance r from the halide to its closest nitrogen neighbor (Figure 10b), we see that chloride loses its water more easily than fluoride as it approaches the closest cation neighbor.42,43 Therefore, for a given hydration level, there is more free water (and higher water chemical potential) when fluoride is replaced by chloride. As we reported earlier, the increase in hydration has a large effect on fluoride mobility. We believe this is mainly responsible for the enhancement effect. Because the medium is more hydrated, this notion also explains why chloride also diffuses faster when % Cl is higher. However, this may mean that the enhancement would disappear or become less significant if we worked at constant humidity (constant water chemical potential) instead of constant hydration level (constant number of waters per cation).

Figure 7. Free energy of ion binding between N and halide (solid lines for F and dashed lines for Cl) as a function of N−halide distance and % F (and % Cl). The inset shows the energy minima for the N−Cl curves. The temperature is 300 K and λ = 14. The green (orange) vertical line at 6.5 Å (7.25 Å) is our choice for the boundary of the first energy well for N−F (N−Cl). The blue vertical line at 8.8 Å is the average distance between two N atoms, assuming the distribution of their positions is uniform in space.

changing r0 from 6.5 to 7.25 or 8.3 Å does not change the trends reported here. The mean residence time τ between the two ions is defined to be the average of all the residence times collected for the pair.39,40 Figure 8 shows the mean residence times for (a) N−F and (b) N−Cl, respectively. The results are consistent with the free energy differences given in Figure 7. Because chloride binds more strongly than fluoride to the cation, it should take longer on average for chloride to escape from the attractive well by thermal fluctuations. If the mean residence time τ is the characteristic time scale for the fluoride diffusion process, then a simple random walk model with time step τ and step size C·rN, where C is a constant and rN is the average distance between two ammonium cations, should describe the self-diffusion constants reasonably well. The value of rN was chosen to be 8.8 Å (see the blue vertical line in Figure 7). The diffusion constant predicted by this random walk is D = (C·rN)2/τ, which suggests that we plot D vs 1/τ. Figure 9a shows 1/τ as a function of 1/T, and the trends largely resemble those in Figure 6b. Figure 9b shows a plot of fluoride self-diffusion constant DFluoride as a function of 1/τ. A linear relationship between the two is clearly shown. The linear

5. CONCLUSIONS Counterions naturally left in the synthesis of an ionomer are often exchanged by a different kind of ion. In this study, we attempted to replace chloride with fluoride experimentally to understand ion transport by PFGSE NMR experiments.

Figure 8. (a) Mean residence time between N and fluoride as a function of reciprocal temperature and % F. (b) Same as (a), but it is between N and chloride. 850

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Figure 9. (a) Reciprocal of the mean residence time 1/τ is plotted as a function of inverse temperature and % F. (b) The self-diffusion constant of fluoride DFluoride vs 1/τ at different temperatures studied. Each color represents a different temperature. Each temperature has five data points for five different % F setups.

Figure 10. (a) Number of water neighbors of fluoride (black) or chloride (red) as a function of distance away from the anion. The dotted lines indicate the radii of the first solvation shells and the corresponding water coordination numbers. The temperature is 300 K and λ = 14. (The curves are insensitive to % F or % Cl). (b) The number of water molecules in the first solvation shell as a function of the distance r between a halide ion and its closest nitrogen neighbor. See also refs 42 and 43.

However, the binding between chloride and the BTMA cation is strong enough that 100% exchange with fluoride was found to be experimentally difficult to achieve. The fluoride content for the diblock copolymer studied experimentally was 44%. The incomplete exchange led us to study how chloride affects fluoride diffusion by simulations that revealed some unexpected, but potentially important, consequences. In this study, we simulated only the hydrophilic part (PVBTMA) of a diblock copolymer and provided arguments why such a setup would be reasonable for studying ion transport. By comparing the experiment with a similar % F at hydration level λ = 14, the diffusion constants from simulations were about 45% lower, but the agreement is reasonable. The simulations underestimated the experimental values probably because the simulated fluoride model diffuses more slowly by about 25% also in the case of water, and the relative humidity inside the sealed NMR tube was probably close to 100% instead of 95% (λ = 14). The activation energy barrier for fluoride diffusion for 40% F was found to be 5.2 kcal/mol from simulations. This is in good agreement with the experimental value that was found to be 5.0 kcal/mol if the outlier data point at 303 K is not fitted. The free energy difference in ion binding shows that chloride binds more strongly to the ammonium cation than fluoride. Because the N−F binding is weaker, the mean residence time, τ, is lower than that for N−Cl binding. From the insight of a

random walk model, we found a linear relationship between the fluoride self-diffusion constant and 1/τ (mean residence time reciprocal) at the temperatures studied. The self-diffusion constant of fluoride was found to increase as % Cl increases (and % F decreases) in our simulations. At 300 K, the self-diffusion constant of fluoride increased by about 70% when fluoride content decreased from 100% to 40%, and it increased by about 140% when fluoride content decreased from 100% to 10%. When % Cl increases, the attraction between fluoride and the cation is slightly weakened, but this factor alone cannot explain such significant increases in fluoride mobility. On the other hand, as chloride binds with the cation, water given up by chloride effectively increases the hydration in the medium. This would explain why a higher % Cl increases the self-diffusion constants of both fluoride and chloride. It also would imply that the enhancement in ion mobility might disappear or lessen if we worked at constant humidity (constant water chemical potential) instead of constant hydration level (constant number of waters per cation). Experimentally, because we could not easily control % F in the membrane, this enhancement has not been confirmed by experiment. However, we found a much improved agreement between simulation and experiment when they were compared at a similar fluoride content at about 40% (instead of comparing with the 100% F simulation data and assuming chloride does 851

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not affect fluoride transport). Even though this study focused on the interplay between chloride, fluoride, and the cation, these results should be general to other systems in which there is more than one kind of mobile ion.

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ASSOCIATED CONTENT

* Supporting Information S

Test data for system sizes, partial charges, densities, NMR tube setup, fluoride diffusion data at hydration levels 5, 9, and 14, chloride transport data in both membrane and bulk water, and diffusion data for sodium/fluoride/chloride solutions are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: 773-702-7250. Fax: 773-702-0805. E-mail: gavoth@ uchicago.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Army Research Office for funding this Multidisciplinary University Research Initiative (MURI) under contract W911NF-10-1-0520. S.T. acknowledges the Croucher Foundation for a postdoctoral research fellowship. H.S. thanks Yifan Li and Prof. Dan Knauss for supplying the membranes for the NMR experiments. We are grateful to the members of the Anionic Transport in Organic Media MURI team for numerous fruitful discussions and support.

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REFERENCES

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The Journal of Physical Chemistry C

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(31) The velocities of these atoms are set to zero, and their masses are increased to 109 amu; they are allowed to respond to the forces exerted on them and move very slightly so that the linear momenta are conserved. (32) Park, M. J.; Balsara, N. P. Anisotropic Proton Conduction in Aligned Block Copolymer Electrolyte Membranes at Equilibrium with Humid Air. Macromolecules 2010, 43, 292−298. (33) Kinning, D. J.; Thomas, E. L.; Ottino, J. M. Effect of Morphology on the Transport of Gases in Block Copolymers. Macromolecules 1987, 20, 1129−1133. (34) Singh, M.; Odusanya, O.; Wilmes, G. M.; Eitouni, H. B.; Gomez, E. D.; Patel, A. J.; Chen, V. L.; Park, M. J.; Fragouli, P.; Iatrou, H.; et al. Effect of Molecular Weight on the Mechanical and Electrical Properties of Block Copolymer Electrolytes. Macromolecules 2007, 40, 4578−4585. (35) Panday, A.; Mullin, S.; Gomez, E. D.; Wanakule, N.; Chen, V. L.; Hexemer, A.; Pople, J.; Balsara, N. P. Effect of Molecular Weight and Salt Concentration on Conductivity of Block Copolymer Electrolytes. Macromolecules 2009, 42, 4632−4637. (36) Kirkwood, J. G. Statistical Mechanics of Liquid Solutions. Chem. Rev. 1936, 19, 275−307. (37) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (38) After a residence is confirmed to have ended, we also look for other residences for the same pair that occur afterwards in the same trajectory. (39) This definition of “mean residence time” is related to, but not the same as, the definition of “residence time” given by Impey, Madden, and Mcdonald.40 Our definition does not rely on the decay of the correlation function. (40) Impey, R. W.; Madden, P. A.; McDonald, I. R. Hydration and Mobility of Ions in Solution. J. Phys. Chem. 1983, 87, 5071−5083. (41) Note, in bulk water, the water coordination number in the first solvation shell for chloride is higher than fluoride. The trend here in membrane is opposite. See the supplemental text and Figure S6b in the SI. (42) This is calculated by first finding the closest nitrogen neighbor of a halide ion, then finding the number of water molecules in the first solvation shell (whose radius is defined in Figure 10a) around the halide ion. (43) In Figure 10a, the average water coordination number in the first solvation shell for fluoride is 6.1, and that of chloride is 5.6. However, in Figure 10b, it seems that the average for chloride would be greater than 5.6. The explanation is that chloride tends to be closer to its closest nitrogen neighbor (see the PMF), so the weight for smaller distances r is higher, and the average is closer to the coordination numbers associated with smaller r. Also note that the coordination numbers when the halides are far from the closest nitrogen are more similar to the values in bulk water (see also Figure S6b in SI).

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