Ion mixing, hydration, and transport in aqueous ionic systems

The study of how the dynamical properties of an ionic solution are determined from the structure of water is one of the classic problems in electrochemistry.^1–4 A less-studied problem is how the dynamical properties of one type of ions are affected by the presence of another type. Recently, we have studied how the self-diffusion constant of a fluoride ion is enhanced by the introduction of chloride ions, which we now call the “co-ion effects,” in an anion exchange membrane.⁵ Such co-ion effects on the transport properties may play an important role in the conductivity of a fuel cell when different ions that are constantly produced due to side reactions interact with the main charge carrier that produces the current. One such example is an anion-exchange membrane fuel cell^6,7 in which hydroxide, the main charge carrier, reacts with dissolved carbon dioxide to form carbonate species as byproducts.⁸ 

Our hypothesis to explain the co-ion effects is that the chloride has a lower affinity to hold onto its water solvation shell, and as the chloride approaches a cation, water is liberated to enhance ion diffusion, which is a monotonically increasing function of hydration. The reasoning in the hypothesis led us to believe that such co-ion effects should not be specific only to a membrane environment. One of the goals in this paper is to show that the co-ion effects also exist in simple aqueous ionic solutions. This problem is therefore important both for practical purposes because of the implications in fuel cells and for theoretical reasons because of its generality. For the range of ion concentrations that we study, it is an exceedingly difficult task to have an analytical theory that predicts the transport properties quantitatively.⁹ Through the use of molecular dynamics (MD) simulations with empirical force fields, we aim to provide ideas to further elucidate the mechanism of the co-ion effects in this study.

This study focuses on two different sets of simulations of cesium and benzyltrimethyl ammonium (BTMA) (Figure 1(a)) solutions that are paired with various ratios of fluoride to chloride and hydration levels λ, which is defined to be the number of water molecules per cation. We often use the symbols “F” and “Cl” to refer to fluoride and chloride anions, whose charges are fixed at −1e. For each cation, data are collected for %F at 100%, 90%, 60%, 40%, and 10% with %Cl = 100% − %F. For each F/Cl ratio, λ at 9, 10, 11, 12, 13, and 14 is simulated at temperature 300 K. Therefore, there are 25 setups with various F/Cl ratios and λ for each cation. Though not the focus of the article, data are also collected for the polymeric form of BTMA (poly(vinyl benzyltrimethylammonium) (PVBTMA), see Figure 1(b)).

In each simulation, there were 80 cations and anions in a simple cubic simulation box with periodic boundary conditions. The General AMBER Force Field (GAFF)¹⁰ was used for BTMA ion, the water model was SPC/Fw,¹¹ and the Lennard-Jones parameters for the simple ions (Cs⁺, F⁻, Cl⁻) were obtained from the work of Jensen and Jorgensen.¹² Coulombic interactions were treated by the Particle-Particle Particle-Mesh (PPPM) solver¹³ with an accuracy of 10⁻⁵ relative error in forces. The Lennard-Jones interactions and the real part of the Coulombic interactions had a cutoff distance of 14 Å. The partial charges were based on the RESP (Restrained ElectroStatic Potential) charges of three connected monomers which were calculated by Gaussian 09 and R.E.D. server.^14–17 The actual partial charges used for BTMA and its polymeric form (PVBTMA) are given in the supplementary material.¹⁸ A 1-fs time step was used for these simulations. For each setup, the structural and dynamical properties are calculated from six different constant NVE (microcanonical) trajectories of at least 10 ns whose initial conditions are sampled from the same constant NVT (canonical) ensemble at 300 K. For brevity, the equilibration steps are provided in the supplementary material.¹⁸ These simulations were carried out using LAMMPS Molecular Dynamics simulator.^19,20

Because the Lorentz-Berthelot combining rules were used and because the simple ions were originally parameterized for a TIP4P water model²¹ and geometric combining rules, we performed additional simulations to ensure the radial distribution functions (RDFs) between the simple ions and water are reasonable in a box of 472 SPC/Fw water molecules and each simple ion. These results are available in the supplementary material.¹⁸ To further test our observations, we also performed some simulations using the TIP4P-Ew water model²² for the Cs⁺ solutions at λ = 14. The simulation details and results using TIP4P-Ew are provided in the supplementary material.¹⁸ 

We previously showed⁵ that the self-diffusion constants of fluoride increase as %F decreases for hydration level (λ) 14 in PVBTMA. In Figure 2(a), we can see that the enhancement effect is also present for lower hydration levels. The enhancement for the fluoride diffusion constant from 100% to 10% F at λ = 14 is about 2.3 times, and that at λ = 9 is about 4.6 times. A main message in this paper is to show that this effect does not only exist in a polymer system but it is also general in other more simple ionic solutions. Similar enhancement effects are observed in fluoride/chloride ionic solutions with more simple BTMA or cesium cations as shown in Figures 2(b) and 2(c). Since water is the medium of ion transport in these systems, it is not surprising that, for the same %F and λ values, the fluoride self-diffusion is highest with Cs⁺, then BTMA, and finally PVBTMA as the connectivity of water domains or the water volume fraction increases. The enhancement effect is not only exclusive to the fluoride. Since the self-diffusion constant of chloride experiences similar trends, its results are provided in the supplementary material.¹⁸ Some results for the same Cs⁺ setups using TIP4P-Ew water are provided in the supplementary material and they are found to be consistent with what we report here for SPC/Fw water.

The only two differences between the chloride and fluoride models are their mass and ionic radius (proportional to the Lennard-Jones σ parameter), and it is the latter that is the root cause of the enhancement effect. Chloride is 25% larger in radius and its surface electric field is weaker that of fluoride. Because of this, chloride does not hold on its solvation water molecules as tightly as fluoride and tends to bind more strongly with the cations. The statements are further illustrated by the RDFs (or g(r)) in Figures 3 and 4. The RDFs between the cation and the halide ions in Figures 3(a) and 3(b) indicate that there is a stronger binding between the cation and chloride in either the BTMA or the cesium solutions. Changing either λ or %F has an observable but minimal effect. Increasing λ or decreasing %F slightly decreases the interactions between the cations and both halide ions. On the other hand, Figures 4(a) and 4(b) clearly show that the strongest interactions with water come from fluoride as indicated by the first tall peak. Decreasing %F slightly increases the water coordination around chloride and fluoride, but decreases that for the cations.

Our previous study⁵ indicated that, relative to fluoride, chloride loses more waters of solvation as it coordinates with the cation. We have found that this behavior also holds for the cations studied here, as shown in Figure 5. This supports the mechanism we proposed for the observed enhancement of diffusion: replacement of fluoride by chloride has the effect of decreasing the number of bound waters in solvation shells in favor of unbound waters. As one chloride ion approaches its closest cation, it loses approximately one more water than fluoride and there is a net gain of one extra free water molecule for the medium in BTMA, and it is about a net gain of two extra free water molecules in the Cs⁺ solution. Given that the halide ions are not overly attracted to the cations, as in the cases of BTMA and cesium solutions, the ion motilities for fluoride and chloride should be enhanced. This hypothesis also suggests the equivalence of changing chloride content and hydration level in affecting the self-diffusion constants of the halide ions. From here onwards, we will call this the “water liberation hypothesis.”

Changing %F also has an interesting effect on the water structure. As shown in Figure 6, when %F is lowered, the second solvation shell is significantly changed and becomes more like bulk-water when compared to the bulk water simulation. These shifts suggest blockages of the chloride ions at about r = 3.3 Å as shown by the green line in the figures. The presence of chloride ions physically forces the water molecules in the second solvation to spread out. There has been substantial experimental debate about whether the presence of ions in a single ionic salt can cause changes to the water structure beyond the first solvation shell, as discussed by Bakker and co-workers^23–28 and Fayer and co-workers.²⁹ We have a different, but related, situation in the present work in which two different anions co-exist in solution. Our results suggest that the influence of fluoride or chloride on the water structure goes beyond the first solvation shell, even though the degree of change depends on the sizes of both anion(s) and cation(s). As we shall see next, the change in the second peak is correlated to the changes of water dynamical properties.

Decreasing %F not only enhances the ion mobilities in these systems but also enhances water self-diffusion significantly in both BTMA and cesium ionic solutions (see Figures 7(a) and 7(b)). For instance, at λ = 14 and 10% F, the water self-diffusion constant recovers almost 90% of its value in bulk water. This is quite remarkable, considering a solution that has a rather high concentration at about 4 mol/L (molar). The trend reported here is consistent with the experimental findings in which the self-diffusion of water in a CsCl solution is faster than that of a CsF solution at the same concentration.^30,31 These changes to water transport and its structural properties should be not a coincidence.

The concept that relates the presence of ions to strengthening or weakening the hydrogen-bond structure of liquid water was first introduced by Cox and Wolfenden.³² The subsequent work of Frank and Gurney^2,33,34 has influenced our understanding of how the structure of liquid water is related to the water dynamics. Small ions such as fluoride ion or lithium ion are usually identified as “structure makers” that increase the strength of (intermolecular) water hydrogen bond (H-bond) network, whereas large singly charged monoatomic ions such as the iodide ion or the cesium ion are often identified as “structure breakers” that loosen up the water H-bond structure and promote faster water dynamics. Even though this broad classification of structure breakers or makers may be somewhat over-simplified, it is worthwhile to study how the H-bonds in our systems are affected by the presence of chloride.

We have adopted the definition of H-bonding by Luzar and Chandler.³⁵ After an H-bond is formed based on the specific geometric criteria, as the waters move, any such hydrogen bond eventually fails to satisfy the geometric criterion, and at that point, the H-bond is said to be broken. By keeping track of the formation and breakage of each H-bond in the system, the lifetimes of H-bonds can be collected and the arithmetic average of these lifetimes can be computed. Note that even though any H-bonding definition contains some arbitrariness, it has been previously shown³⁶ that many different common choices for the relevant geometric variables, including the ones used here, all lead to quite similar static and dynamical results.

In Figure 7(b), we show the average lifetime of a H-bond for the BTMA and cesium solutions. It is clear that the lifetime decreases as either the hydration level or chloride content increases. [Xu, Stern, and Berne reported that hydrogen bond kinetics is slowed down by the inclusion of electronic polarizability in the simulation.³⁷ Since our force fields are not polarizable, the absolute values reported are probably underestimated, but the trends should not be affected as a longer residence time of water within the first solvation shell of fluoride than chloride is also observed when polarizability is included.³⁸] This enhanced water dynamics due to an increase in chloride content can again be linked to the fact that chloride, being a bigger ion, has a weaker surface electric field and does not hold onto its water neighbors as tightly as fluoride. This point is more clearly demonstrated in terms of the potential of mean force, calculated from the radial distribution functions from Figure 4, by the reversible work theorem.³⁹ For the cesium solutions, the free energy barrier for a water to move away from the first solvation shell of a fluoride ion to its second shell is 2.2 kcal/mol, whereas the free energy barrier is only 1.2 kcal/mol in the case of chloride (a similar difference in the barrier is seen for the BTMA solutions). This stronger interaction between fluoride and water than between chloride and water is also observed in vibrational spectroscopy experiments.^27,40 The influence of chloride on the potential of mean force is consistent with a more disordered water structure and faster water dynamics around it. We further hypothesize that the more disordered water structure and faster water dynamics, caused by chloride, facilitates faster ion transport for all ions.

Since either increasing λ or decreasing %F increases the self-diffusion constants of the halide ions in the systems, contour lines of the diffusion constants with positive slopes can be drawn as shown in Figure 8. The lines for chloride and fluoride are roughly parallel in both cases, and this suggests that there is a single variable that controls the halide self-diffusion constants. We hypothesize that this unknown variable is strongly related to the water liberation hypothesis, more disordered water structure, and faster water dynamics. The reasoning that we have offered so far should not be only limited to the anions. Similar effects to the self-diffusion constant for the cesium ion are observed as shown by the green lines in Figure 8(b). This further supports the idea of a single universal transport mechanism for all simple ions in these systems.

The sensitivity of the ion diffusion constants as a function of %F and λ is different in the two systems, and therefore, the slopes (Δ%F/Δλ) shown in Figures 8(a) and 8(b) are not the same. The more positive slopes for the BTMA system are due to the fact that the ion diffusion constants are increased more significantly when λ is increased as shown in Figures 2(a) and 2(b). Therefore, a larger increase in the fluoride content is needed to compensate for an increase in λ for the BTMA system. The slopes for the BTMA contour lines are about 30% (this means, to maintain the same self-diffusion constant, 30% of additional fluoride is needed when λ is increased by 1), whereas that of the Cs⁺ lines is about 10%. From Figure 5, for the case of BTMA, one chloride nets about 1 extra water molecule, but the slope in Figure 8 implies one chloride should have the same effect on the diffusion constant as changing λ by about 3. What this means is that the water liberation hypothesis alone does not fully account for the change in the diffusion constant. Some, if not all, of the rest of the contribution will come from the changes in the water equilibrium and dynamical properties around chloride.

This study of the co-effects was first motivated by our work to understand ion transport in anion-exchange membranes. In this work, we have further shown that such co-ion effects are quite general and do not only exist in polyionic systems. In the simple ionic solutions of BTMA and cesium ions, we have similarly shown that the self-diffusion constant of fluoride is enhanced as the chloride content increases. One reason for the enhancement, as shown in our previous study in polyionic systems, is that water surrounding the chloride ion is liberated as it approaches a cation.

The self-diffusion constant of water, which we did not previously investigate, is also enhanced by the presence of chloride. This increase in water transport is correlated with a clear shift of the second solvation water shell around a water molecule as shown in the O–O radial distribution function. This spreading in the water solvation structure is accompanied by a decrease in the average lifetime of a hydrogen bond (H-bond). The presence of a relatively bulky chloride ion and changes in the water structure have reminded us of the classic picture of structure makers and breakers, brought forth by Cox, Wolfenden, Frank, and Gurney, in which a bulky ion disrupts the water structure and each water can then more easily orient or move. These observations in the changes of water structure have also led us to a related discussion of whether the ion influence on the water structure goes beyond the first solvation shell. In our cases of fluoride and chloride, the results suggest these effects do affect the second solvation shell. The correlated faster water dynamics should play an important role in the enhanced ion mobilities in addition to our water liberation hypothesis.

There is no reason to believe these co-ion effects should be restricted to only fluoride and chloride. It is plausible that ion dynamics could be slowed down by introducing a different co-ion that behaves as a structure maker that would strengthen the orderliness of the water structure. To make matters more interesting (and complicated), it is also plausible proton-transferring species such as the hydronium, and the hydroxide may thrive on a stronger H-bond network as protons are passed through the H-bonds. It is well-known that proton transfer is expedited in a stable water wire.^41–43 

When the contour lines of ion self-diffusion constants are plotted as a function of hydration level λ and fluoride content, parallel lines of positive slopes are observed. While the sign of the slopes is not at all surprising because decreasing the fluoride content has the similar effect of increasing λ on the diffusion constants, it is not at all obvious that the lines should be parallel. This parallelism implies the ion motilities are controlled by a single universal transport mechanism. Since all ions share the same transport medium, water, it is reasonable to assume that the single transport mechanism is caused by changes in water. We hypothesize that there is effectively more water in the medium due to the water liberation hypothesis and considerably faster water dynamics around a chloride ion due to the weaker attractions. By studying the slope of the contour lines, even though we found that the water liberation hypothesis by itself does not account for all the change in the diffusion constants, the other hypothesis related to the more disordered water structure and faster dynamics can co-exist, work collectively with the water liberation hypothesis, and manifest as a single ion transport mechanism.

This research was supported by an Army Research Office Multidisciplinary University Research Initiative (MURI) under Contract No. W911NF-10-1-0520. S.T. acknowledges the Croucher Foundation for a postdoctoral research fellowship. The computational resources in this work were provided in part by a grant of computer time from the U.S. Department of Defense (DoD) High Performance Computing Modernization Program at the Engineer Research and Development Center (ERDC) and Navy DoD Supercomputing Resource Centers and, in part, by the University of Chicago Research Computing Center (RCC).