Persistent Subdiffusive Proton Transport in Perfluorosulfonic Acid

Letter pubs.acs.org/JPCL

Persistent Subdiffusive Proton Transport in Perfluorosulfonic Acid Membranes John Savage and Gregory A. Voth* Department of Chemistry, James Franck Institute, and Computation Institute, University of Chicago, Chicago, Illinois 60637, United States S Supporting Information *

ABSTRACT: Proton transport (PT) in solutions of small amphiphiles in water has previously been shown to be subdiffusive for long times. The present study analyzes simulations of hydrated perfluorosulfonic acid (PFSA) membranes in order to determine whether PT is also subdiffusive in these important amphiphilic systems. We show that PT is indeed subdiffusive for several hundred picoseconds for all hydration levels examined, and the subdiffusive behavior is highly dependent on water concentration. We also investigate the caging of the excess proton using a recently developed technique and show that the excess proton exhibits caging effects up to at least 1 ns in PFSA systems. In order to fully characterize the long-time behavior of PT in PFSAs, these results demonstrate that multiple nanosecond trajectories are needed, well beyond the current capabilities of ab initio molecular dynamics. SECTION: Energy Conversion and Storage; Energy and Charge Transport

P

In this Letter, we reanalyze self-consistent iterative MS-EVB (SCI-MS-EVB) simulations from a previous study32 of the 3M membrane at hydration levels (λ) of 5, 9, and 12. These hydration levels span the range from of the onset of percolation in the membrane at λ5 to a large amount of unbound water at λ12.33 The 3M membrane was chosen because of its low equivalent weight, which will increase the water concentration compared to a high equivalent weight polymer.34 Because the subdiffusive behavior of protons and water in amphiphiles was shown to be strongly dependent on water concentration,30 we show in Table 1 the water concentration used in all of the systems used in this study. We expect to see significant subdiffusive behavior because, even with a low equivalent weight polymer at high hydration level, the water concentration is still lower than that in the amphiphilic simulations. We first look at the dynamics of water in these systems as the vehicular motion of the excess proton should obviously be correlated with water dynamics. Moreover, it has been proposed that rearrangements of the hydrogen-bond network as a whole play a key role in long-range structural diffusion.19,31 Examining the mean-squared displacement (MSD) of water as a function of time in Figure 1a, we see at least two regions of distinct behavior. In the region below 3 ps, we see a very flat slope. This is attributed to the caging effect, that is, the first solvation shell waters trapping a central water molecule.30,35 After this point, one would expect to see an increase in slope as the molecules escape from these cages until we obtain a

roton transport (PT) in water is important across many areas of biology and chemistry.1−10 While the mechanism of PT in bulk water has been studied intensively,11−21 the effect of solutes22−25 and water confinement26−28 on PT is arguably more important to understand for most applications. As an example, understanding PT in perfluorosulfonic acid (PFSA) membranes is essential for improving the efficiency of protonexchange membrane fuel cells.29 Molecular dynamics (MD) simulations using the multistate empirical valence bond (MS-EVB) method, which includes structural diffusion of the excess proton, have shown that the PT in solutions of the amphiphiles urea or tetramethylurea (TMU) shows a marked slowdown compared to that in bulk water.30 Given that water is required for the hydrated proton to move, the reduction in PT in these systems is mostly caused by a reduction in water concentration. However, the chemical differences between urea and TMU also have a contribution, with solutions of the more hydrophobic TMU displaying slower PT rates. Excess protons in these amphiphilic systems show subdiffusive behavior for hundreds of picoseconds as the excess protons feel both the effect of the slowdown in water rotation caused by the interaction with the solutes, which is necessary for proton hopping,19,26,31 as well as a disruption of the water networks by the amphiphiles. Given the importance of understanding PT in PFSAs, we were motivated to investigate whether similar long-time subdiffusive behavior occurs in these systems and whether the charged hydrophilic groups or the more defined aggregation of the hydrophobic moieties because of the chemical bonding of the polymer would increase or decrease subdiffusive behavior compared to the solutions of amphiphiles. © 2014 American Chemical Society

Received: July 11, 2014 Accepted: August 20, 2014 Published: August 20, 2014 3037

dx.doi.org/10.1021/jz5014467 | J. Phys. Chem. Lett. 2014, 5, 3037−3042

The Journal of Physical Chemistry Letters

Letter

Table 1. Water Concentration Used for Each Simulated System water molecules simulation box length (Å) water concentration (M)

urea 1

TMU 1

urea 2

TMU 2

3M λ12

3M λ9

3M λ5

230 20.57 43.9

230 20.67 43.3

230 25.31 23.6

230 25.38 23.4

480 35.57 18.65

360 33.96 15.31

200 32.62 9.57

Figure 1. (a) Water MSD as a function of time plotted on a log−log scale for the systems of interest. (b) Water MSD/time as a function of time plotted on a log−log scale. Dashed lines in both plots show the fit to eq 2 with parameters in Table 2. (c) Proton center of excess charge (CEC) MSD as a function of time plotted on a log−log scale for the systems of interest. (d) Proton CEC MSD/time as a function of time plotted on a log− log scale. Dashed lines in both plots show the fit to eq 2 with parameters in Table 3

Table 2. Parameters for the Fits of Equations 2 and 3 to the Water MSD for Each System α Dα (Å2/ps) Dw (Å2/ps) C (Å2)

urea 1

TMU 1

urea 2

TMU 2

3M λ12

3M λ9

3M λ5

1 0.18 0.18 0.13

0.97 0.15 0.12 8.69

0.95 0.14 0.10 11.95

0.90 0.13 0.06 22.80

0.80 0.208 0.04 65.38

0.78 0.17 0.03 43.35

0.66 0.07 0.005 12.26

Table 3. Parameters for the Fits of Equations 2 and 3 to the Proton CEC MSD for Each System α Dα (Å2/ps) Dp (Å2/ps) C (Å2)

urea 1

TMU 1

urea 2

TMU 2

3M λ12

3M λ9

3M λ5

1 0.18 0.207 −80.73

0.97 0.18 0.195 −146.81

0.86 0.20 0.0597 69.12

0.87 0.14 0.052 36.22

0.74 0.14 0.02 21.49

0.77 0.08 0.01 17.46

0.34 0.12 0.0005 4.53

constant slope corresponding to “normal”, long-time diffusion, which should occur on the time scale of tens of picoseconds.36 We can use the Einstein relation, which relates the MSD to the diffusion coefficient as ⟨r 2⟩t →∞ = 6Dw t

where 0 < α < 1 because this is subdiffusive behavior and Dα is a parameter with the same units as the diffusion coefficient. As has been discussed previously,30 only urea 1 shows a switch to normal diffusion in the expected 10 ps time scale. For the other systems, we fit the MSD to eq 2 in the region that we determine to be subdiffusive and therefore obtain the characteristic parameters, which are shown in Table 2. The exponent α clearly follows a decreasing trend with respect to water concentration. Dα is noticeably higher for the PFSA systems, which seems to stem from the very different short-time behavior of these systems compared to the amphiphilic solutions. As well as this subdiffusive behavior, it is interesting to examine the long-time water diffusion behavior. In previous work,30,32,37,38 the behavior of MSD/t has been used to

(1)

where ⟨r2⟩ is the MSD and Dw is the water diffusion coefficient. In most of these systems where the water dynamics have been slowed, we do not see a constant slope at intermediate time scales and instead see power law behavior, which we can fit using the relation ⟨r 2⟩ = 6Dα t α

(2) 3038



Letter

Figure 2. (a) Water MSD as a function of time plotted on a linear−linear scale for the systems of interest. Dashed lines show the fit to eq 3 with parameters in Table 2. (b) Proton CEC MSD as a function of time plotted on a linear−linear scale for the systems of interest. Dashed lines show the fit to eq 3 with parameters in Table 3

respect to water concentration, as has been seen previously for the amphiphiles. Combined with the water results from above, this shows that excess proton subdiffusion can be well correlated to water concentration alone, and chemically very different species behave similarly at a given water concentration. Again, similar to the water dynamics, the MSD/t plot for the CEC is misleading with regards to the switch from subdiffusive to diffusive behavior, as seen in Figure 1d. We can fit the CEC MSD from 500 to 1000 ps to eq 3 (with Dw replaced with Dp) and obtain the diffusion coefficients much earlier that the MSD/t plot alone would suggest. Using Dp obtained using eq 3 (with Dw replaced with Dp), we can see in Figure 3 the effect that water concentration has on

determine the long-time diffusion coefficient because this will converge to 6Dw at long times according to eq 1. However, it can be possible to obtain the diffusion coefficient at earlier times by noting that the MSD becomes linear earlier than the MSD/t plot would suggest. This can be seen in the nonlogarithmic plot of the MSD in Figure 2. In fact, at medium times, eq 1 can be better written as ⟨r 2⟩ = 6Dw t + C

(3)

which would result in MSD/t behaving according to ⟨r 2⟩ C = 6Dw + t t

(4)

From eq 4, we can clearly see that if C is large compared to 6Dw, the curve MSD/t will take a long time to converge toward 6Dw, giving the impression that subdiffusive behavior is lasting much longer than that in reality, as seen in Figure 1b. Using eq 3, we have fit from 500 to 1000 ps to determine Dw and C for all systems. We can see that Dw follows a trend similar to the subdiffusive behavior, with the systems showing most subdiffusivity having the lowest diffusion coefficient. The diffusion coefficients obtained for the amphiphilic systems from this analysis also match that obtained from the MSD/t analysis earlier. When we look at the values of C that we have obtained, we can see how the MSD/t analysis matches this analysis well because for the amphiphilic systems, the intercept is small relative to 6Dw, and therefore, MSD/t will quickly converge to give the diffusion coefficient. In contrast, the intercepts in the PFSAs not only are much larger in absolute terms but, more importantly, are also much larger relative to 6Dw. For example, C is 400 times larger than 6Dw for λ5, and eq 4 shows us that MSD/t will still have a 10% contribution from the intercept even at 4 ns, while the MSD has started to become linear at 400 ps. This highlights the danger of using MSD/t to determine the subdiffusive behavior of highly confined systems such as these. In reactive SCI-MS-EVB simulations, it is necessary to concretely describe the position of the hydrated excess proton charge defect; therefore, we define the center of excess charge (CEC) as a sum over the positions of waters participating in a MS-EVB complex, weighted by their relative MS-EVB state probabilities.39 Examining the dynamics of the CEC in the same manner as the water dynamics, we again see that the only system that shows normal diffusion at short times is the urea 1 system, while all others exhibit subdiffusive behavior. Fitting the MSD to eq 2 in the subdiffusive regime, we can see that both α and Dα for PT follow an overall monotonic decrease with

Figure 3. Excess proton CEC diffusion coefficients of each system plotted as a function of their water concentration. The value for pure water is obtained from the original MS-EVB3 paper.39

the long-time proton diffusion coefficient. Even with the differences between each of these systems, there is a consistent dependence on water concentration. Over a range of water concentrations between 10 and 56 M, we see a 2 orders of magnitude difference in PT, yet PT for different chemical species is well within the same order of magnitude for a given water concentration. It is interesting that the plot in Figure 3 is nonmonotonic in that the behavior of the curve for water concentration 10−22 M is rather different from the behavior for >22 M, which is linear. This appears to reflect a rather different proton diffusion mechanism for the low water concentration PFSA system, likely due to the influence of the negatively charged sulfonate groups. The MSD intercepts, C, obtained from eq 3 for the CEC show different behavior compared to the intercepts obtained 3039



Letter

Figure 4. Probability density distributions for the relative angle θ between the displacement vectors of a proton CEC for successive steps. The definition of the angle is shown in the lower insets for example time steps of 1 or 2. The upper inset shows a trajectory for an individual proton, with positions shown every 100 fs (gray), 1 ps (blue), and 10 ps (red). Any probability values above 3 are plotted as red in order to show the most detail in the rest of the plot. (For quick reference, cos(180°) = −1, cos(90°) = 0, cos(0°) = 1.)

At time intervals below ∼10 ps, a clear preference for low values of cos(θ) is shown for all systems. This corresponds to the proton turning back in the direction that it came and matches well with the picture of proton caging by the surrounding waters. For the amphiphilic systems, both decreasing the water concentration and changing from urea to TMU extends the amount of time the caging effect is seen. This correlates well with the rotational slowdown of water molecules seen in these systems. However, caging cannot account for the slowdown in the 10−500 ps regime as this region shows the signs of a random walk in the amphiphilic systems. The PFSA systems show qualitatively different behavior, with the caging effect extending to all values of Δ sampled. This change is much greater than the difference in water concentration or subdiffusive properties would suggest and suggests different PT behavior between the amphiphilic solutions and PFSA membranes. The results in the present study reveal the presence of substantial subdiffusive PT behavior in the 3M PFSA membrane, which is similar to what has been seen before in amphiphilic solutions. This subdiffusive behavior continues up to as much as 400 ps of simulation time, and furthermore, caging of the excess proton is observed up to 1 ns and possibly much longer. Therefore, in order to fully explore the long-time diffusive behavior with statistical certainty, multinanosecond simulation times are needed. It is important to also emphasize that the true nature of PT in water requires inclusion of the Grotthuss hopping

from the water MSDs, especially in the amphiphilic systems. The high water concentration amphiphiles show a negative intercept, which, as can be seen more clearly in the Supporting Information, means that there is a switch to faster diffusive behavior after an initial slower diffusive regime between 10 and 500 ps. The low water concentration amphiphilic systems show a positive intercept, which is significant compared to 6Dp, meaning that the MSD/t will take a long time to converge. The PFSA systems, which already showed a large intercept for water diffusion, show an even greater intercept for proton diffusion, with C 1500 times greater than 6Dp for λ5, ruling out any chance of seeing convergence of the MSD/t plot in reasonable simulation time. Given that the CEC shows slower PT between 10 and 500 ps in all systems, even in the high water concentration amphiphiles, it is interesting to investigate whether this is an effect of water cages being slowed by the presence of amphiphiles. In order to investigate the caging effect of the water and the hydrophobic moieties on the motion of the excess proton CEC, we can use a recently developed analysis method.40 In this method, the relative angle θ between the vectors of motion for two successive time intervals is used as a probe of the directional changes of the CEC. By scanning different time intervals (Δ), we can gain insight into the evolution of the caging of the CEC. A uniform distribution of cos(θ) is expected for a random walk in three-dimensions, and therefore, we have plotted the distribution of cos(θ) versus Δ in Figure 4 to highlight deviations from normal diffusion. 3040



Letter

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mechanism and hence a simulation methodology able to capture bond breaking and forming, such as the MS-EVB method used in this study. Although ab initio molecular dynamics (AIMD) simulations can capture this mechanism, the feasible simulation times for this method are on the order of 100 ps41−43 (in addition to the limited simulation size accessible to AIMD). Figures 1c and 4 show that this time scale is likely inadequate to describe long-time PT in PFSAs. However, by using recent algorithmic developments in the SCIMS-EVB algorithm, the long-time PT behavior in PFSAs will be explored in the future.

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ASSOCIATED CONTENT

* Supporting Information S

Plots of the excess hydrated proton CEC MSD of each system individually with the linear and power law fits. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Chemical Sciences, Geosciences, and Biosciences, through Grant No. DE-FG02-10ER16171. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-06CH11357.

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REFERENCES

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Persistent Subdiffusive Proton Transport in Perfluorosulfonic Acid

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