Partition and Structure of Aqueous NaCl and CaCl2 Electrolytes in

Article pubs.acs.org/JPCC

Partition and Structure of Aqueous NaCl and CaCl2 Electrolytes in Carbon-Slit Electrodes R. K. Kalluri,† T. A. Ho,† J. Biener,‡ M. M. Biener,‡ and A. Striolo*,† †

School of Chemical, Biological and Materials Engineering, The University of Oklahoma, Norman, Oklahoma 73019, United States Nanoscale Synthesis and Characterization Laboratory, Lawrence Livermore National Laboratory, Livermore, California 94550, United States

‡

S Supporting Information *

ABSTRACT: We report molecular dynamics simulation results obtained for aqueous NaCl and CaCl2 solutions used as electrolytes in model electric double layer capacitors. The electrodes are carbon-slit pores of widths 0.65, 0.7, 0.79, 0.9, 1.2, and 1.6 nm. The applied voltage is represented as a uniform surface charge density on the pore surfaces. Toward replicating experimentally relevant conditions, the surface charge densities span between 0 (neutral pore) and 15 μC/cm2 (both positive and negative). Charge localization on pore entrances is not considered. As the neutral pores are charged, we monitor the accumulation of the ions from the bulk (at ∼1.8 M ionic strength) to the pores. Our results show that the ionic concentration inside the pores increases as the surface charge density increases, as expected. More interestingly, the surface charge density at which the ions begin to penetrate the pores increases as the pore width decreases and as the ion size and the ion hydration strength increase. The pore width at which the maximum partition coefficient obtained at the largest surface charge density considered varies with the ion type (0.65 nm pores for Na+, 0.9 nm pores for Ca2+, and 0.79 nm pores for Cl− ions). The density distribution of electrolytes within the charged pores depends on the water structure and on the hydration structure of the ions under confinement, which is ion-specific.

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electrolytes,16,17 and molecular simulations to quantify the energy barriers encountered by the ions as they enter narrow carbon-based pores from bulk solutions,18−20 as well as ion−ion and ion−water correlations under confinement.21−25 Ionic liquids, as used in EDLCs, have also been studied by molecular simulations.26−36 The simulation results provide needed understanding of the structure and dynamics of electrolytes under confinement,36 which enhances experiments. In this work we explore, via molecular dynamics simulations, the structure of aqueous NaCl and CaCl2 electrolytes as they partition between a bulk solution and two pores, one positively and one negatively charged, as a function of pore surface charge density and width. In a separate contribution, we discussed how the results from our simulations compare to experimental data obtained on similar systems.36 We provide extensive simulation details, including the charge distributions under confinement as a function of pore width, electrolyte composition, and surface charge density.

INTRODUCTION The deployment of alternative energy sources, including solar and wind, which are naturally intermittent, requires adequate energy-storage devices. Besides batteries, over the past decade promising research has been conducted in the field of electric double layer capacitors (EDLCs), often referred to as supercapacitors or ultracapacitors. EDLCs outperform conventional capacitors by utilizing high surface area electrode materials and thin electrolytic dielectrics.1−6 In EDLCs, electrolytes accumulate near charged surfaces.3,7 Because the amount of energy stored increases with the accessible surface area, nanoporous materials are used as electrodes. As the electrodes are required to conduct electricity, porous carbons are often the material of choice.8,9 The electrolytes can be composed of different aqueous and organic electrolytes, including ionic liquids.10 Aqueous electrolytes typically offer higher power densities because of their high ionic mobility, while organic electrolytes, in which the ions move more slowly than in water (EDLCs’ power density depends on the ion mobility), offer higher energy densities because the organic solvent decomposes at higher applied potentials than water does.7,9 Chmiola et al.11 showed that the EDLCs capacitance increases ∼3-fold when the electrolytes lose their solvation layer upon adsorption within subnanometer pores. These observations stimulated a number of experimental studies with the focus of producing electrode materials to improve the EDLCs’ performance,12−15 theoretical− experimental investigations to explain the behavior of confined © 2013 American Chemical Society

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METHODS AND ALGORITHMS We conducted equilibrium molecular dynamics simulations for NaCl and CaCl2 electrolytes dissolved in explicit water. All the results presented in the main text are obtained for electrolyte Received: January 7, 2013 Revised: May 9, 2013 Published: June 6, 2013 13609

dx.doi.org/10.1021/jp4002127 | J. Phys. Chem. C 2013, 117, 13609−13619

The Journal of Physical Chemistry C

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charged graphene−water interface.47 Local accumulation of charges at pore entrances was also not considered because our goal is to study the equilibrium partition of ions between bulk and confined spaces, not to quantify the free energy barriers encountered by the ions as they enter the charged pores. The 12−6 LJ parameters for unlike interactions were determined using Lorentz−Berthelot mixing rules.48 The cutoff distance for all interactions was set to 9 Å, and the long-range electrostatic interactions were treated using the particle mesh Ewald summation method.49 Each system was initially simulated in the NPT ensemble at 1 atm and 300 K. After the system attained constant volume, the simulations were conducted in the NVT ensemble at 300 K with a time step of 1 fs. All calculations were conducted using the simulation package LAMMPS.50,51 Equilibration was considered achieved when the concentrations of ions inside the pores did not change significantly within an interval of 8 ns. Only the last 5 ns of simulation, after equilibration, were used to obtain the average quantities discussed below. All simulations, each of at least 15 ns, were replicated four times to ensure reliability. For CaCl2 systems, because Ca2+ ions were found to enter the narrow pores very rarely, the achievement of equilibrium conditions was confirmed by initiating the simulations with a few ions confined within the narrow charged pores and by ensuring that the final concentration inside the pores corresponded, within statistical uncertainty, to the final concentration obtained when the simulations were initiated with all ions within the bulk aqueous solutions. A schematic of the simulation box is reported in Figure 1. The pores are arranged along the Z direction. When periodic boundary conditions are considered, one can identify two slitshaped pores separated by a graphitic slab composed by five graphene layers. As discussed in the Supporting Information, we confirmed that increasing the number of graphene layers between the slit pores does not significantly affect the results discussed herein. Within one simulation the width of both pores is the same (“symmetric” EDLC). The pore width is defined as the center-to-center distance between the carbon atoms of the bottommost layer of the top slab and the topmost layer of the bottom slab. Pore widths of 0.65, 0.7, 0.79, 0.9, 1.2, and 1.6 nm are considered. Smaller pores are not simulated because Chialvo and Cummings recently showed that water might not fill such pores, when neutral.22 It is also possible that wider carbon-based pores dewet,52 but typical active carbons used in experiments contain some oxygenated groups that secure water filling. The dimensions of the graphene sheets along the X and Y directions are 2.091 and 2.13 nm, respectively. Additional simulations, discussed in the Supporting Information, were conducted to test whether the lateral size of the pores might affect the simulation results. It was concluded that such size effects are limited, suggesting that in our simulations pore-entrance effects are also probably not very important in determining the equilibrium distribution of water and ions between bulk and pore volume. Both surfaces facing one pore bear charges of the same magnitude and sign. The two pores within the system bear the same surface charge density but different sign, which ensures that the entire system is electrically neutral. The main difference between the electrodes simulated here and those considered in our prior work21 is that in the current systems both carbon surfaces facing one pore volume bear charges of the same sign, while in our prior simulations a positive surface faced a negative one (“frontal” EDLC). The present protocol better reproduces experimental carbon-based EDLCs.

solutions at the ionic strength of 1.8 M. In the Supporting Information, we discuss one representative simulation conducted to study the effect of salt concentration on the observed results. Water was modeled using the simple point charge extended (SPC/E) model,26 with which we have ample experience.23,37−41 Sodium (Na+) and chloride (Cl−) ions were modeled as charged Lennard-Jones (LJ) spheres by employing the parameters proposed by Dang and collaborators.42 Calcium (Ca2+) ions were modeled as charged LJ spheres by using the parameters employed by Predota et al.43 Polarization effects were not considered, although such effects have been found important in predicting the correct distribution of ions at the water−air interface,44 particularly for those ions that are easily polarized (large negatively charged ions such as Br− and I−). Ignoring polarization effects should not introduce significant artifacts when the pores are charged, the condition that is relevant for EDLCs applications. We are currently assessing the validity of this assumption. In Table 1 we report the composition of the Table 1. Composition of the Simulated Systemsa system

pore width (nm)

water molecules

Na+ ions

1 2 3 4 5 6 7 8 9 10 11 12

1.6 1.2 0.9 0.79 0.7 0.65 1.6 1.2 0.9 0.79 0.7 0.65

1670 1670 1670 1670 1670 1670 5000 5000 5000 5000 5000 5000

54 54 54 54 54 54 0 0 0 0 0 0

a

Ca+2 ions Cl‑ ions 0 0 0 0 0 0 54 54 54 54 54 54

54 54 54 54 54 54 108 108 108 108 108 108

Ionic strength = 1.80 M.

Table 2. Atomic Charges (q) and Lennard-Jones Parameters (σ and ε) for Carbon Atoms, SPC/E Water, and Ions Considered in Our Simulations atom C O H Na+ Ca2+ Cl−

q (e) −0.8476 +0.4238 +1 +2 −1

σion (Å)

εion (kcal/mol)

3.40 3.166

0.0557 0.1554

2.254 2.895 4.401

0.1 0.1 0.1

systems simulated. In Table 2 we report LJ simulation parameters for carbon atoms in the pores, water molecules (the LJ site is centered at the oxygen atom of the SPC/E water), and the electrolytes considered here. The carbon atoms within the carbon-slit electrodes were modeled as LJ spheres with parameters taken from Cheng and Steele.45 To mimic applied potentials, partial charges were placed only on those carbon atoms facing the pore volume. The charges were uniform throughout one pore, and the charge densities considered ranged from 0 (neutral pores) to 15 μC/cm2. Neither image charges nor surface polarization was considered.46 In a recent report we concluded that the polarization of graphene insignificantly affects the structure and dynamics of water at the 13610



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Figure 1. Left: Schematic of the simulation box organized in two graphite slabs. The arrangement of the graphitic slabs yields two carbon-slit pores. Gray spheres represent carbon atoms. Ions and water are not shown for clarity. Right: Representative snapshot of a simulated system with charges placed on carbon atoms to mimic applied potentials. The pore width is 1.6 nm. Gray spheres represent neutral carbon atoms, blue spheres represent carbon atoms that bear partial positive charge, purple spheres represent carbon atoms that bear partial negative charge, red and white spheres represent oxygen and hydrogen atoms of water, respectively, yellow spheres represent sodium ions, and green spheres represent chloride ions.

(both co- and counterions) within the pores in terms of a partition coefficient, Γ. This coefficient is the ratio between the molar concentration of the ions inside the pores and that in the bulk. The concentrations in the pore and in the bulk are calculated by counting the numbers of ions and water molecules within the slit pore and in the region outside the pore, respectively. In Figure 2 we report Γ in the six pores considered (different symbols) as a function of the surface charge density for sodium, calcium, and chloride. To ease comparison, we define the reduced pore size (R*) as the ratio between the effective pore width (center-tocenter pore width minus the LJ diameter of one carbon atom) and the LJ diameter of the ion of interest (Na+, Ca2+, or Cl−). The results are shown for the accumulation of Na+ ions within the negatively charged pores (top left panel), for Cl− ions within positively charged pores (top right panel) for the NaCl system, for Ca2+ ions within the negatively charged pores (bottom left), and for Cl− ions within positively charged pores (bottom right) for the CaCl2 system. The data in Figure 2 show that, for Na+, Ca2+, and Cl− ions, when the pores are neutral, Γ is different than zero only when the pore size is either 1.6 or 1.2 nm. This is because the effective pore width is larger than the size of the hydrated ions only for these pores (these pores have effective pore width of 1.265 and 0.865 nm, respectively, while hydrated Cl−, Na+, and Ca2+ ions are of size ∼0.72, ∼0.66, and ∼0.7 nm, respectively).60 The hydrated ions can enter these neutral pores. To enter smaller pores, the ions must lose part of their hydration shell, which is energetically unfavorable, unless the pores are charged (see below). We point out that, for wide neutral pores, Γ for Na+ is larger than that for Ca2+ (∼0.66 and ∼0.45 for Na+ as opposed to ∼0.35 and ∼0.24 for Ca2+ in 1.6 and 1.2 nm pores, respectively). Although these differences might be due, in part, to the different bulk concentrations considered (to maintain constant ionic strength, the Na+ concentration differs from the Ca2+), they are also due to the different hydrated ions size and to different hydration strength,57 consistent with observations reported in several prior reports.16,18,19,21 Note that the partition coefficient for Cl− from the NaCl solution is smaller than that from the CaCl2 solution (compare top right and bottom right panels). This is a consequence of the lower molar concentration of Cl− ions in the CaCl2 solution compared to the NaCl solution (in our simulations we maintained constant the ionic strength). The effect of

Because of computing power limitations, in our simulations only one pore is considered for each electrode, both pores in the same system are considered of the same width (“symmetric” EDLCs), and the two pores are carved out of the same graphitic monolith. The graphene layers are continuous in the Y direction but not along the X direction, which allows us to maintain a “bulk” region in contact with the charged pores. After NPT equilibration, the water density in the bulk region corresponds to that of bulk liquid water. The salt concentration in the simulated systems was always large enough to prevent depletion of the ions in the bulk, which could lead to saturation effects.53,54 Periodic boundary conditions are implemented along the three directions. Additional simulation details are available elsewhere.21 Our simulation results are complementary to others, obtained using Monte Carlo methods. For example, Yang et al.55 showed that, within carbon-slit nanopores, predictions of the formation of the electric double layer based on the continuum Guoy− Chapman model fail to reproduce simulation results obtained treating water both implicitly (i.e., in the primitive model) and explicitly. Instead of molecular dynamics, to investigate the equilibrium partition of ions within charged pores it would be preferable to implement the grand canonical Monte Carlo algorithm. Jamnik and Vlachy56 already in 1995 employed such a technique, coupled with the Poisson−Boltzmann equation, to predict the partition of electrolytes between bulk and cylindrical pores as a function of surface charge density and mixture composition. It would be desirable to extend such studies by implementing an atomistic description of water. However, as pointed out by Yang et al.55 and also by Hou et al.,57 the acceptance rate would be extremely low should an explicit model be used to describe water molecules. Other alternative approaches include solving the extended Nernst−Planck equation, following for example Bowen and Mukhtar,58 which would require estimates for hindrance factors and effective hydration ion sizes, or classic density functional theory, which has recently been employed by Jiang et al.59 to investigate solvent effects on the performance of organic supercapacitors.

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RESULTS AND DISCUSSION As a first approximation, the EDLCs’ energy density increases as the concentration of counterions within the charged carbon pores increases. We quantify the electrolyte concentration 13611



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Figure 2. Partition coefficients for sodium (top left) and chloride (top right) ions in NaCl systems; calcium (bottom left) and chloride (bottom right) in CaCl2 systems as a function of surface charge density. Results obtained for pores of width 1.6, 1.2, 0.9, 0.79, 0.7, and 0.65 nm are represented by empty diamonds, full squares, empty triangles, full circles, empty squares, and empty circles, respectively. Both the reduced pore size (R*) and center-to-center pore width (H) are reported for completeness. Lines are guides to the eye.

above a pore-dependent and ion-dependent threshold, Γ obtained for the narrow pores increases with the surface charge density faster than it does in the wider pores considered. The threshold seems to shift to higher surface charge densities when ions of similar size but higher hydration strengths are compared (e.g., Na+ vs Ca2+), in good qualitative agreement with experimental observations.11,36 As the pore width decreases further, it becomes more and more difficult for the ions to enter the charged pores. This is clear when the results for Cl− ions are considered (Cl− ions are bigger than Na+ ones, making this observation more evident). At 15 μC/cm2 surface charge density, Γ decreases as R* decreases from 1.03 to 0.72. Qualitatively similar results are expected for Na+ ions should the pore widths be decreased below 0.65 nm. Although these results suggest a practical limit on the energy density that can be achieved with EDLCs, they also suggest that it might be possible to maximize the EDLC energy density by employing electrodes with different pore size distributions, depending on the electrolytes used. The results for partition coefficients of sodium and chloride ions implicate that, for attaining maximum energy density, the ions should be of similar size to those of the pore width considered. The results obtained for the partition coefficient of calcium ions are somewhat different. For these ions the highest

salt concentration on the partition coefficients is further discussed in the Supporting Information. For all the ion−pore combinations considered, within the thermodynamic conditions simulated here, Γ increases as the surface charge density increases. However, the results in Figure 2 show a strong dependence on pore width but also on the ion type. In general, the wider the pores are (see, for example, the 1.6 nm pores), the more slowly Γ increases as the surface charge density increases for both NaCl and CaCl2 systems. Because the EDLCs capacitance (charge stored per applied voltage) is related to the slope of the curves in Figure 2, our results suggest that narrow pores are better suited to enhance the EDLCs’ energy density than wide ones, in qualitative agreement with experimental observations.11 Our results show that at low applied surface charge densities the narrower pores yield small Γ for all the ions considered. See, for example, the results obtained for Cl− in the 0.7 nm or narrower pores (top right panel of Figure 2): Γ remains close to zero even when the surface charge density is 5 μC/cm2, at which conditions Γ ∼ 0.82 for the 1.6 nm pore. It is likely that when the surface charge density is low in the narrow pores, attractive pore−ion electrostatic interactions are not sufficiently strong to compensate for the loss of hydration water experienced by the ions as they enter the pores. Once the surface charge density is 13612



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Figure 3. Density profiles for oxygen of water (dashed black) and sodium (continuous red) and chloride ions (continuous blue) within positively charged slit-shaped pores at surface charge density 12.5 μC/cm2. Atomic oxygen densities are shown on the left Y axes and atomic densities for ions are shown on the right Y axes. The top panels are for results obtained for pores of width 1.6, 1.2, and 0.9 nm and the bottom panels for results obtained for pores of width 0.79, 0.7, and 0.65 nm, respectively. The vertical distance from the pore surface, H, is measured from the center of the carbon atoms.

from the negatively charged surfaces (top left panel of Figure 4), yielding a density maximum in correspondence of a region depleted of water molecules. While Cl− ions appear to lose part of their hydration shell, Na+ ones seem to maintain it, which is in qualitative agreement with our prior results.21 The density profiles provide some evidence according to which ion−ion correlations exist within the wide charged pores (note the presence of some Cl− ions in the negatively charged pore and, vice versa, some Na+ ions within the positively charged pore). These results are consistent with simulation results obtained within the primitive model by Federov and Kornishev,23 although much weaker ion−ion correlations are observed in our simulations, probably because of several reasons, including that the aqueous electrolyte solutions of Figures 3 and 4 are in equilibrium with a bulk reservoir, water is explicitly considered in our calculations, and the surface charge densities considered here are lower than those simulated in ref 23. Results similar to those just discussed are obtained in the 1.2 nm wide pores. It is worthwhile to point out that when frontal capacitors were considered in our prior simulations, the 1.2 nm wide pore showed significant ion−ion correlations that yielded enhanced partition coefficients.21 In the present configuration (both surfaces across the pore volume bear a charge of the same sign), ion−ion correlations are not accentuated, and Γ for the 1.2 nm pores is larger than that observed for wider pores, but not as high as those observed in narrower ones (see Figure 2). More interesting are the results for the ionic structure within the 0.9 nm pores (top right panels in Figures 3 and 4). As expected from results in Figure 2, no pronounced Na+−Cl− correlations are observed within these pores, as the number of coions is negligible. Consistent with results obtained in wider pores, Cl− ions within positively charged pores are found to accumulate

partition coefficients are obtained for pores of width 0.9 nm (i.e., R* = 1.95). This is probably due to the much stronger hydration strength of Ca2+ compared to Na+, which increases the effective size of Ca2+ ions compared to their LJ size parameter. This suggests that the maximum achievable EDLCs energy density depends on the hydration properties of the ions employed in the electrolyte. The results discussed above can be further analyzed by quantifying the number and distribution of individual species within the various simulated pores. In Figures 3 and 4, we report axial density profiles for oxygen atoms of water and Na+ and Cl− ions within the positively (Figure 3) and negatively charged slit-shaped pores (Figure 4) at the surface charge density of 12.5 μC/cm2 for the NaCl system. In both figures the results are shown for 1.6, 1.2, and 0.9 nm pores in the top panels and for 0.79, 0.7, and 0.65 nm pores in the bottom panels. The results obtained for the density distribution of oxygen atoms of water molecules change when pores of the same width are either negatively or positively charged. This is a consequence of the different orientation of confined water, dictated by the surface charge. Density profiles of water oxygen and hydrogen atoms within the charged pores, in the absence of ions, are reported as Supporting Information. By comparing the density profiles in the Supporting Information to those in Figures 3 and 4, we conclude that the ions’ presence has little effect on the structure of confined water (only a small increase in the number of confined water molecules is observed). This is due to the rather small number of ions found within the charged pores. Within the 1.6 nm wide pores our results suggest that Cl− ions accumulate on contact with the positively charged surfaces (top left panel in Figure 3), yielding a density maximum just above the first hydration layer, while Na+ ions accumulate slightly further 13613



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Figure 4. Density profiles for oxygen of water (dashed black) and sodium (continuous red) and chloride ions (continuous blue) within negatively charged slit-shaped pores at surface charge density 12.5 μC/cm2. Atomic oxygen densities are shown in the left Y axes and atomic densities for ions are shown in the right Y axes. The top panels are for results obtained for pores of width 1.6, 1.2, and 0.9 nm and the bottom panels for results obtained for pores of width 0.79, 0.7, and 0.65 nm, respectively. The vertical distance from the pore surface, H, is measured from the center of the carbon atoms.

fact that the effective width of the 0.79 nm pores approximately corresponds to the LJ diameter of Cl− ions, which must accumulate at the pore center to minimize steric repulsions with the pore surfaces. Note that, out of all pores considered, the 0.79 nm ones yield the largest Γ for Cl− ions at high surface charge density, suggesting that stripping the hydration shell maximizes EDLCs energy density, corroborating the interpretation provided by Chmiola et al.11 However, note that not only the desolvation of ions is important but also the simultaneous attractive interaction with both charged pore surfaces. Within the 0.7 nm wide pores, positively and negatively charged, Cl− ions yield one dense layer while Na+ ions yield two near the pore center (bottom middle panels in Figures 3 and 4, respectively). Size effects can explain these differences. While the large Cl− ions are constrained to the pore center, the smaller Na+ ions lose only a portion of their hydration shell. It is possible that by preferentially occupying a position near one of the two charged surfaces these ions minimize the perturbation on their hydration shell while maximizing the attractive interaction with the charged surface. As the pore width further decreases, Na+ ions continue to accumulate near the two negatively charged graphene surfaces, and fewer and fewer Cl− ions can penetrate the pores, as quantified in Figures 3 and 4, yielding the partition coefficients shown in Figure 2. Over all, the results in Figures 3 and 4 suggest that the propensity of electrolytes to lose their hydration shell upon confinement within charged slit-shaped pores depends not only on pore width but also on ion type and both pore−water and ion−water correlations. At the surface charge densities considered in our simulations, the highest partition coefficients are obtained when (i) the ions lose part of their hydration shell

on contact with both charged surfaces, yielding two adsorbed layers (Figure 3, top right panel). However, the atomic density within these layers is larger than that found in wider pores, which leads to the higher Γ shown in Figure 2. The Na+ ions (Figure 4, top right panel), instead of yielding two adsorbed layers, one near each of the negatively charged surfaces, form one dense layer near the pore center. The position of this layer corresponds to a region in which the density of the oxygen atoms of water is near zero, suggesting that these adsorbed Na+ ions can maintain a relatively unperturbed hydration shell, they do not compete with water molecules for space, and they interact simultaneously with both oppositely charged surfaces. In the case of frontal capacitors the 0.9 nm wide pores showed partition coefficients smaller than 1.2 nm wide ones, because the pore size was too small to permit synergistic Na+−Cl− correlations.21 For the capacitor configuration considered here the 0.9 nm pores show larger Γ than the 1.2 nm ones, probably because of the synergistic interaction of the confined ions with both charged surfaces. The density profiles just discussed suggest that in the case of Na+ the enhanced Γ observed for 0.9 nm pores is not entirely due to the disruption of the ions hydration shell, as could have been expected on the basis of interpretation of experimental results.11 Within the 0.79 nm pore, our results show the formation of one layer of Na+ ions in positively charged pores (Figure 4, bottom left panel) that spans the entire pore width, probably because the two layers formed by the confined water molecules have merged, and the Na+ ions cannot maintain a full hydration shell in such narrow pores. The density distribution of Cl− ions within the positively charged 0.79 nm pore (Figure 3, bottom left panel) shows that instead of the two layers obtained in wider pores, one Cl− layer forms near the pore center, in between the layers formed by oxygen atoms of water. This result is due to the 13614



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Figure 5. Density profiles for oxygen of water (dashed black) and calcium (continuous red) and chloride ions (continuous blue) within positively charged slit-shaped pores at surface charge density 12.5 μC/cm2. Atomic oxygen densities are shown on the left Y axes, and atomic densities for ions are shown on the right Y axes. The top panels are for results obtained for pores of width 1.6, 1.2, and 0.9 nm and the bottom panels for results obtained for pores of width 0.79, 0.7, and 0.65 nm, respectively. The vertical distance from the pore surface, H, is measured from the center of the carbon atoms.

molecules, these yield only one layer within the positively charged 0.7 nm pores, while they form two layers (albeit not well pronounced) within the negatively charged pores of the same width, in agreement with our density profiles. The Cl− ions form two layers within positively charged pores of width 0.9 nm but, because of steric effects, only one layer, near the pore center, within pores of width 0.79 and 0.7 nm. Na+ ions form one dense layer near the pore center, in a region somewhat depleted of water molecules, within the negatively charged pores of width 0.9 nm (top middle panel), while, in an effort to maintain part of their hydration shell, they distribute rather uniformly across the pore width in the 0.79 nm pores and yield two layers within the 0.7 nm pores (bottom middle panel). On the contrary, Ca2+ ions accumulate near the pore center in all cases considered in Figure 7. The snapshots shown in Figure 7, combined with the density profiles described above, suggest that complex phenomena including synergistic, and sometimes antagonistic, pore−ion, ion−water, and ion−ion correlations need to be accounted for to fully understand the mechanism of energy accumulation in EDLCs with aqueous electrolytes. The structural results discussed above show that when the applied surface charge density changes, a number of structural details vary inside the various pores. In particular, water molecules change their orientation, different amounts of ions enter the pores, and the ions distribute differently within the pores. All these changes can be quantified simultaneously using charge density profiles. Once the charge density distribution is known, it is possible to integrate the Poisson equation to obtain electrostatic potential profiles.1,61 Others implemented similar procedures to calculate the potential drop between negatively and positively charged pores62 or that between a free-standing charged surface and bulk electrolyte solutions.27 Further, the capacitance can be computed from the surface charge density and

on entering the pores, (ii) they can simultaneously interact favorably with both charged surfaces, (iii) the surface charge density is sufficiently large to overcome the loss of hydration water, and (iv) the ions do not experience steric hindrance upon entering the pores. These results are consistent with experimental observations.36 In Figure 5, we report axial density profiles for oxygen atoms of water and Ca2+ and Cl− ions within the positively charged slitshaped pores at the surface charge density of 12.5 μC/cm2 for the CaCl2 system. In Figure 6 we report the correspondent density profiles within the negatively charged pores. Qualitatively, the results are similar to those discussed for NaCl electrolytes (Figures 3 and 4, respectively). The main difference is that when Ca2+ ions are confined within narrow negatively charged pores (bottom panels in Figure 6) they accumulate near the pore center, while Na+ ions show a preference for accumulating near the pore surface (bottom panels in Figure 4). This is probably due to the stronger hydration strength for Ca2+ ions than for Na+ ones. To visualize the structure of confined electrolytes just discussed, we report in Figure 7 representative simulation snapshots for aqueous ions confined within 0.9 nm (top), 0.79 nm (middle), and 0.7 nm wide pores (bottom). We consider positively and negatively charged pores in contact with NaCl electrolytes (left and middle, respectively) and negatively charged pores in contact with CaCl2 electrolytes (right panels). These snapshots provide visual evidence for the different orientation of confined water molecules discussed in the Supporting Information (i.e., note that the hydrogen atoms are near the negatively charged surfaces in the middle and right panels and far from the positively charged ones in the left panels), as well as of the changes in structure of confined electrolytes in response to changes in pore width. As a consequence of the different orientation of confined water 13615



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Figure 6. Density profiles for oxygen of water (dashed black) and calcium (continuous red) and chloride ions (continuous blue) within negatively charged slit-shaped pores at surface charge density 12.5 μC/cm2. Atomic oxygen densities are shown on the left Y axes, and atomic densities for ions are shown on the right Y axes. The top panels are for results obtained for pores of width 1.6, 1.2, and 0.9 nm and the bottom panels for results obtained for pores of width 0.79, 0.7, and 0.65 nm, respectively. The vertical distance from the pore surface, H, is measured from the center of the carbon atoms.

Figure 7. Representative simulation snapshots of ions confined within the 0.9 nm (top), 0.79 nm (middle), and 0.7 nm (bottom panels) carbon slit pores at a surface charge density of 12.5 μC/cm2. Results are shown for the positively and negatively charged pores for the NaCl system (left and middle, respectively) and for the negatively charged pores for the CaCl2 system (right). Gray spheres represent charged carbon atoms, red and white spheres represent oxygen and hydrogen atoms of water, respectively, yellow spheres represent sodium ions, blue spheres represent calcium ions, and green spheres represent chloride ions.

(top and bottom panels, respectively). The results obtained for other pores are presented as Supporting Information. The charge density profiles shown are obtained by considering the contribution due to each atomic species within the various pores (ions, but also oxygen and hydrogen atoms of water). In fact, the anisotropic orientation of water within charged pores yield significant changes in the charge density profile, as pointed out by others.61 In the wider pores considered, as the surface charge

potential drop. For electrodes such as those simulated here, the capacitance could be calculated when the potential drop from the center of the pores to the bulk solutions is known, following Feng and Cummings.62 Such extensive calculations were not attempted here but will soon be reported for other systems. In Figure 8 we show selected charge density profiles obtained for NaCl solutions within pores of width 1.6 and 0.7 nm (left and right panels, respectively), positively and negatively charged 13616



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Figure 8. Charge density profiles within pores of width 1.6 and 0.7 nm (left and right panels, respectively). The pores considered are positively (top) and negatively charged (bottom). Different lines represent results obtained at different surface charge densities, as explained in the legends. Vertical dashed− dotted lines indicate the pore centers. Vertical dashed and vertical dotted lines indicate the position of the first layer formed by Cl− ions near positively charged pore surfaces, and that of the layer formed by Na+ ions near negatively charged pore surfaces, respectively. As the peak position changes with surface charge density, the peak positions observed at 10 μC/cm2 were used for these lines. Note that within the 0.7 nm pores (top right panel) the Cl− ions accumulate at the pore center.

each of the negatively charged surfaces, instead of accumulating near the pore center. The charge density profiles shown in Figure 8 reflect this distribution. As the charge density increases, the layers of positive charge become more intense and shift closer to the pore surfaces. Simultaneously, a positive peak in the charge density profile appears near the pore center. This peak is due to hydrogen atoms of water molecules rather than to Na+ ions.

density increases, our results show that an oppositely charged layer forms close to the surface. The density of this charged layer increases as the surface charge density increases. In the positively charged pores two negatively charged layers can be detected near each charged surface. The layer closer to the surface is more intense than the further one, as expected. One unexpected feature within the positively charged 1.6 nm wide pores is the appearance of a peak of positive charge closer to the surface than any other charge layer. This layer is due to hydrogen atoms of water molecules. It is known that water molecules near a neutral graphite substrate project some of their hydrogen atoms toward the surface, in an effort to minimize the number of water−water hydrogen bonds lost due to the presence of the solid substrate.37 At low surface charge densities these hydrogen atoms give rise to significant charge densities. As the positive surface charge density increases, the orientation of water molecules at the carbon− water interface changes and the contribution of such hydrogen atoms to the charge density near the solid substrate decreases. In the negatively charged pores of width 1.6 nm, more and more hydrogen atoms are found near the charged surfaces as the charge density increases, leading to a pronounced contribution to the charge density profiles. In the pores of width 0.7 nm, the effects of water orientation can still be observed, but the pore width is so narrow that the layer of water molecules formed on one surface interacts with the one formed on the opposite pore surface. The Cl− ions accumulate near the center of the positively charged pores because of steric effects, as discussed above. The charge density profiles reflect this arrangement of ions within the pore. As pointed out above, Na+ ions tend to form two layers, one near

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CONCLUSIONS In summary, we have reported molecular dynamic simulation results obtained for aqueous NaCl and CaCl2 electrolyte solutions as they partition from a bulk solution to slit-shaped carbon pores at different surface charge densities. The pore widths considered are 0.65, 0.7, 0.79, 0.9, 1.2, and 1.6 nm. In a simplistic representation of EDLCs, each electrode was simulated as one carbon-slit pore. The width of both pores within one simulation was the same (symmetric EDLCs). Mimicking typical experimental conditions for aqueous electrolytes, the surface charge densities simulated range from 0 (neutral pores) to 15 μC/cm2. The results are quantified in terms of partition coefficients (concentration of ions within the pores divided by that in the bulk) and explained on the basis of the structure of the confined solutions as observed from the number of confined water molecules, counterions and co-ions, density profiles across the pore width, representative simulation snapshots, and charge density profiles. At low surface charge densities (small applied potentials) the ions prefer to remain in the bulk. As the surface charge density increases, the ions enter the pores, but higher surface charge densities are required for the ions to enter the smaller pores, 13617



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in part because of the energy penalty encountered by the ions as they lose part of their hydration shells. At the highest surface charge densities considered, the narrower pores yield higher partition coefficients, which is in qualitative agreement with experimental observations available in the literature, notably those of Chmiola et al.,11 although these were obtained for organic electrolytes. In addition to this qualitative agreement, our results show evidence of ion-specific effects that are due to different ion sizes and also to differences in the ions hydration strengths. Specifically, pores of width 0.79 nm, whose effective pore width corresponds to the Lennard-Jones diameter of Cl− ions, yield the maximum partition coefficient for Cl−, while the narrowest pores considered, 0.65 nm, yield the maximum partition coefficient for Na+. In the case of Ca2+, the maximum partition coefficients are obtained when the pore width is 0.9 nm, almost twice as large as the Lennard-Jones size parameter for these ions, probably because Ca2+ ions are found to be strongly hydrated in bulk aqueous solutions. For these ions the hydration shells effectively increase the ions diameter. These results suggest that carbon-based materials of different pore size distributions could be used as EDLCs electrodes to maximize performance. Detailed analysis of the structure of confined electrolyte solutions shows that ion−ion correlations are only important for wide pores and at relatively small surface charge densities. However, water−ion correlations are important, in particular within the narrower pores simulated, and need to be taken into consideration, together with ion−pore interactions, when the correct distribution of ions within the charged pores is required, e.g., in predicting the EDLCs capacitance. For example, Cl− ions are always found in part desolvated within the charged pores, even when the pore width far exceeds the size of the hydrated ions, and the distribution of water within 0.9 nm wide pores seems responsible, in part, for the accumulation of Na+ ions near the pore center. Although limited simulations were conducted that vary the bulk salt concentration, our results suggest that even this parameter might affect the electric double layer capacitors performance. Despite limitations in the procedures implemented herein (e.g., polarization effects are not described, the electrodes simulated are of limited size, and realistic descriptions of the carbonaceous pores typically used for electrodes are not implemented), the detailed description presented is important for achieving a better understanding of electric double layer capacitors and it could not be attained using simulation or theoretical models that overlook the presence of water molecules. Applications that benefit from our results include energy storage and water desalination (capacitive water desalination).

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Article

AUTHOR INFORMATION

Corresponding Author

*Phone: 405-325-5716. Fax: 405-325-5813. E-mail: astriolo@ ou.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Work at the University of Oklahoma was supported, in part, by the U.S. Department of Energy, under contract number DESC0001902. Generous allocations of computing time were provided by the Oklahoma Supercomputer Center for Education and Research (OSCER) and by the National Energy Resources Supercomputer Center (NERSC). Work at LLNL was performed under the auspices of the US DOE by LLNL under Contract DE-AC52-07NA27344. Project 12-ERD-035 was funded by the LDRD Program at LLNL. The Authors wish to thank Deepthi Konatham, Dr. Naga Rajesh Tummala, Dr. Matthew D. Merrill, and Dr. Michael Stadermann for helpful discussions.

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REFERENCES

(1) Conway, B. E. Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications; Kluwer Academic/Plenum Publishers: New York, 1999. (2) Burke, A. Ultracapacitors: Why, how, and where is the technology. J. Power Sources 2000, 91, 37−50. (3) Kotz, R.; Carlen, M. Principles and applications of electrochemical capacitors. Electrochim. Acta 2000, 45, 2483−2498. (4) Arico, A. S.; Bruce, P.; Scrosati, B.; Tarascon, J.-M.; van Schalkwijk, W. Nanostructured materials for advanced energy conversion and storage devices. Nat. Mater. 2005, 4, 366−377. (5) Chu, A.; Braatz, P. Comparison of commercial supercapacitors and high-power lithium-ion batteries for power-assist applications in hybrid electric vehicles I. Initial characterization. J. Power Sources 2002, 112, 236−246. (6) Halper, M. S.; Ellenbogen, J. C. SupercapacitorsA brief overview. MITRE report number MP 05W0000272; MITRE Corp.: McLean, VA, 2006. (7) Winter, M.; Brodd, R. J. What are batteries, fuel cells, and supercapacitors? ChemInform 2004, 35, no−no. (8) Zhang, L. L.; Zhao, X. S. Carbon-based materials as supercapacitor electrodes. Chem. Soc. Rev. 2009, 38, 2520−2531. (9) Simon, P.; Gogotsi, Y. Materials for electrochemical capacitors. Nat. Mater. 2008, 7, 845−854. (10) Lewandowski, A.; Olejniczak, A.; Galinski, M.; Stepniak, I. Performance of carbon−carbon supercapacitors based on organic, aqueous and ionic liquid electrolytes. J. Power Sources 2010, 195, 5814− 5819. (11) Chmiola, J.; Yushin, G.; Gogotsi, Y.; Portet, C.; Simon, P.; Taberna, P. L. Anomalous increase in carbon capacitance at pore sizes less than 1 nanometer. Science 2006, 313, 1760−1763. (12) Kim, B.; Chung, H.; Kim, W. Supergrowth of aligned carbon nanotubes directly on carbon papers and their properties as supercapacitors. J. Phys. Chem. C 2010, 114, 15223−15227. (13) Yu, D.; Dai, L. Self-assembled graphene/carbon nanotube hybrid films for supercapacitors. J. Phys. Chem. Lett. 2009, 1, 467−470. (14) Largeot, C.; Portet, C.; Chmiola, J.; Taberna, P.-L.; Gogotsi, Y.; Simon, P. Relation between the ion size and pore size for an electric double-layer capacitor. J. Am. Chem. Soc. 2008, 130, 2730−2731. (15) Izadi-Najafabadi, A.; Futaba, D. N.; Iijima, S.; Hata, K. Ion diffusion and electrochemical capacitance in aligned and packed singlewalled carbon nanotubes. J. Am. Chem. Soc. 2010, 132, 18017−18019. (16) Chmiola, J.; Largeot, C.; Taberna, P.-L.; Simon, P.; Gogotsi, Y. Desolvation of ions in subnanometer pores and its effect on capacitance and double-layer theory. Angew. Chem., Int. Ed. 2008, 47, 3392−3395.

ASSOCIATED CONTENT

S Supporting Information *

Additional simulation results obtained when more than five graphene layers are used to separate the two charged pores, additional simulations conducted to assess size effects on our results, additional simulations conducted for bulk electrolyte ionic strength different than 1.8 M, water density profiles within charged pores in the absence of electrolytes, and charge density profiles within the various pores considered in the main text. This material is available free of charge via the Internet at http://pubs.acs.org. 13618



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(38) Striolo, A. Water self-diffusion through narrow oxygenated carbon nanotubes. Nanotechnology 2007, 18, 475704. (39) Striolo, A. The mechanism of water diffusion in narrow carbon nanotubes. Nano Lett. 2006, 6, 633−639. (40) Argyris, D.; Ashby, P. D.; Striolo, A. Structure and orientation of interfacial water determine atomic force microscopy results: Insights from molecular dynamics simulations. ACS Nano 2011, 5, 2215−2223. (41) Argyris, D.; Cole, D. R.; Striolo, A. Dynamic behavior of interfacial water at the silica surface. J. Phys. Chem. C 2009, 113, 19591−19600. (42) Dang, L. X. Mechanism and thermodynamics of ion selectivity in aqueous solutions of 18-crown-6 ether: A molecular dynamics study. J. Am. Chem. Soc. 1995, 117, 6954−6960. (43) Pøedota, M.; Zhang, Z.; Fenter, P.; Wesolowski, D. J.; Cummings, P. T. Electric double layer at the rutile (110) surface. 2. Adsorption of ions from molecular dynamics and X-ray experiments. J. Phys. Chem. B 2004, 108, 12061−12072. (44) Jungwirth, P.; Tobias, D. J. Specific ion effects at the air/water interface. Chem. Rev. 2005, 106, 1259−1281. (45) Cheng, A.; Steele, W. A. Computer simulation of ammonia on graphite. I. Low temperature structure of monolayer and bilayer films. J. Chem. Phys. 1990, 92, 3858−3866. (46) Kjellander, R.; Greberg, H. Mechanisms behind concentration profiles illustrated by charge and concentration distributions around ions in double layers. J. Electroanal. Chem. 1998, 450, 233−251. (47) Ho, T. A.; Striolo, A. Polarizability effects in molecular dynamics simulations of the graphene−water interface. J. Chem. Phys. 2013, 138. (48) Allen, M. P.;Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 2004. (49) Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. A smooth particle mesh Ewald method. J. Chem. Phys. 1995, 103, 8577−8593. (50) Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 1995, 117, 1−19. (51) http://lammps.sandia.gov/. (52) Leung, K.; Luzar, A.; Bratko, D. Dynamics of capillary drying in water. Phys. Rev. Lett. 2003, 90, 065502. (53) Pell, W. G.; Conway, B. E.; Marincic, N. Analysis of non-uniform charge/discharge and rate effects in porous carbon capacitors containing sub-optimal electrolyte concentrations. J. Electroanal. Chem. 2000, 491, 9−21. (54) Zheng, J. P.; Huang, J.; Jow, T. R. The limitations of energy density of electrochemical capacitors. J. Electrochem. Soc. 1997, 144, 2026−2031. (55) Yang, K. L.; Yiacoumi, S.; Tsouris, C. Electrosorption capacitance of nanostructured carbon aerogel obtained by cyclic voltammetry. J. Electroanal. Chem. 2003, 540, 159−167. (56) Jamnik, B.; Vlachy, V. Ion partitioning between charged micropores and bulk electrolyte solutionMixtures of monovalent and divalent counterions and monovalent co-ions. J. Am. Chem. Soc. 1995, 117, 8010−8016. (57) Hou, C.-H.; Taboada-Serrano, P.; Yiacoumi, S.; Tsouris, C. Monte Carlo simulation of electrical double-layer formation from mixtures of electrolytes inside nanopores. J. Chem. Phys. 2008, 128, 044705−044708. (58) Bowen, W. R.; Mukhtar, H. Characterisation and prediction of separation performance of nanofiltration membranes. J. Membr. Sci. 1996, 112, 263−274. (59) Jiang, D. E.; Jin, Z. H.; Henderson, D.; Wu, J. Z. Solvent effect on the pore-size dependence of an organic electrolyte supercapacitor. J. Phys. Chem. Lett. 2012, 3, 1727−1731. (60) Smith, D. E.; Dang, L. X. Computer simulations of NaCl association in polarizable water. J. Chem. Phys. 1994, 100, 3757−3766. (61) Wander, M. C. F.; Shuford, K. L. Alkali halide interfacial behavior in a sequence of charged slit pores. J. Phys. Chem. C 2011, 115, 23610− 23619. (62) Feng, G.; Cummings, P. T. Supercapacitor capacitance exhibits oscillatory behavior as a function of nanopore size. J. Phys. Chem. Lett. 2011, 2, 2859−2864.

(17) Huang, J.; Sumpter, B. G.; Meunier, V. Theoretical model for nanoporous carbon supercapacitors. Angew. Chem., Int. Ed. 2008, 47, 520−524. (18) Feng, G.; Qiao, R.; Huang, J.; Sumpter, B. G.; Meunier, V. Atomistic insight on the charging energetics in subnanometer pore supercapacitors. J. Phys. Chem. C 2010, 114, 18012−18016. (19) Yang, L.; Garde, S. Modeling the selective partitioning of cations into negatively charged nanopores in water. J. Chem. Phys. 2007, 126, 084706−084708. (20) Shim, Y.; Kim, H. J. Nanoporous carbon supercapacitors in an ionic liquid: A computer simulation study. ACS Nano 2010, 4, 2345− 2355. (21) Kalluri, R. K.; Konatham, D.; Striolo, A. Aqueous NaCl solutions within charged carbon-slit pores: Partition coefficients and density distributions from molecular dynamics simulations. J. Phys. Chem. C 2011, 115, 13786−13795. (22) Chialvo, A. A.; Cummings, P. T. Aqua ions−graphene interfacial and confinement behavior: Insights from isobaric−isothermal molecular dynamics. J. Phys. Chem. A 2011, 115, 5918−5927. (23) Fedorov, M. V.; Kornyshev, A. A. Towards understanding the structure and capacitance of electrical double layer in ionic liquids. Electrochim. Acta 2008, 53, 6835−6840. (24) Lynden-Bell, R. M.; Rasaiah, J. C. Mobility and solvation of ions in channels. J. Chem. Phys. 1996, 105, 9266−9280. (25) Tang, Y. W.; Chan, K.-Y.; Szalai, I. Structural and transport properties of an SPC/E electrolyte in a nanopore. J. Phys. Chem. B 2004, 108, 18204−18213. (26) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. The missing term in effective pair potentials. J. Phys. Chem. 1987, 91, 6269−6271. (27) Yang, L.; Fishbine, B. H.; Migliori, A.; Pratt, L. R. Molecular simulation of electric double-layer capacitors based on carbon nanotube forests. J. Am. Chem. Soc. 2009, 131, 12373−12376. (28) Vatamanu, J.; Borodin, O.; Smith, G. D. Molecular insights into the potential and temperature dependences of the differential capacitance of a room-temperature ionic liquid at graphite electrodes. J. Am. Chem. Soc. 2010, 132, 14825−14833. (29) Feng, G.; Qiao, R.; Huang, J.; Dai, S.; Sumpter, B. G.; Meunier, V. The importance of ion size and electrode curvature on electrical double layers in ionic liquids. Phys. Chem. Chem. Phys. 2011, 13, 1152−1161. (30) Feng, G.; Cummings, P. T. Supercapacitor capacitance exhibits oscillatory behavior as a function of nanopore size. J. Phys. Chem. Lett. 2011, 2, 2859−2864. (31) Rajput, N. N.; Monk, J.; Singh, R.; Hung, F. R. On the influence of pore size and pore loading on structural and dynamical heterogeneities of an ionic liquid confined in a slit nanopore. J. Phys. Chem. C 2012, 116, 5169−5181. (32) Wang, S.; Li, S.; Cao, Z.; Yan, T. Molecular dynamic simulations of ionic liquids at graphite surface. J. Phys. Chem. C 2009, 114, 990−995. (33) Chen, S.; Wu, G.; Sha, M.; Huang, S. Transition of ionic liquid [bmim][PF6] from liquid to high-melting-point crystal when confined in multiwalled carbon nanotubes. J. Am. Chem. Soc. 2007, 129, 2416− 2417. (34) Chen, S.; Kobayashi, K.; Miyata, Y.; Imazu, N.; Saito, T.; Kitaura, R.; Shinohara, H. Morphology and melting behavior of ionic liquids inside single-walled carbon nanotubes. J. Am. Chem. Soc. 2009, 131, 14850−14856. (35) Kondrat, S.; Georgi, N.; Fedorov, M. V.; Kornyshev, A. A. A superionic state in nano-porous double-layer capacitors: Insights from Monte Carlo simulations. Phys. Chem. Chem. Phys. 2011, 13, 11359− 11366. (36) Kalluri, R. K.; Biener, M. M.; Suss, M. E.; Merrill, M. D.; Stadermann, M.; Santiago, J. G.; Baumann, T. F.; Biener, J.; Striolo, A. Unraveling the potential and pore-size dependent capacitance of slitshaped graphitic carbon pores in aqueous electrolytes. Phys. Chem. Chem. Phys. 2013, 15, 2309−2320. (37) Striolo, A.; Gubbins, K. E.; Gruszkiewicz, M. S.; Cole, D. R.; Simonson, J. M.; Chialvo, A. A.; Cummings, P. T.; Burchell, T. D.; More, K. L. Effect of temperature on the adsorption of water in porous carbons. Langmuir 2005, 21, 9457−9467. 13619


Partition and Structure of Aqueous NaCl and CaCl2 Electrolytes in

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