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Coarse-grained simulations of an ionic liquid-based capacitor: II. Asymmetry in ion shape and charge localization
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 284114 (http://iopscience.iop.org/0953-8984/26/28/284114) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 147.188.128.74 This content was downloaded on 25/08/2014 at 22:14
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 284114 (8pp)
doi:10.1088/0953-8984/26/28/284114
Coarse-grained simulations of an ionic liquid-based capacitor: II. Asymmetry in ion shape and charge localization Konrad Breitsprecher1 , Peter Košovan1,2 and Christian Holm1 1
Institut für Computerphysik, Universität Stuttgart, Stuttgart, Germany Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University in Prague, Prague, Czech Republic 2
E-mail: [email protected] Received 23 December 2013, revised 24 January 2014 Accepted for publication 27 January 2014 Published 12 June 2014
Abstract
In this work, which is a continuation of part I, we introduce a primitive model for an ionic liquid (IL) that can account for the planar shape of cations typical for ILs like imidazolium. The model consists of a spherical anion and a triangular cation consisting of three spheres, where one or all three vertices of the triangle can carry electric charge. We use molecular dynamics simulations to study the differential capacitance Cd of an ionic liquid confined between two planar electrodes. Our goal is to elucidate the complex dependence of Cd on the electrode potential U in terms of simple entities such as the shape and charge distribution of the ions. For this purpose, we compare the results from the current model to the results based on the models with spherical cations that possess asymmetry in ion valence and shape that were analyzed in detail in part I of this work. We show that the various possible stackings of the triangles near the cathode lead to noticeable new features in Cd (U ) as compared to the spherical models. Different distributions of charges on the triangle lead to different preferred orientations of the cations near the cathode that are moreover potential dependent. Keywords: ionic liquids, capacitor, differential capacitance, ESPResSo (Some figures may appear in colour only in the online journal)
1. Introduction
between the oversimplified theoretical assumptions and overly complex real IL systems. In part I of this paper series [2], we introduced a coarse-grained model of an IL that acts as the dielectric of a capacitor. The model studied there consisted of spherical particles with a point charge located at their center. Model A featured asymmetry in ion size, characterized by the diameter ratio ϑ = dcation /danion ∈ {1, 1.5, 2}, and a valency of qcat = −qan = +1 e. The asymmetry was achieved by variation of the cation properties while properties of the anion were kept constant. The results for the symmetric case with ϑ = 1 could be compared to the mean-field theory of Kornyshev [7] and yielded a qualitative agreement in terms of a transition from a camel-shaped to the bell-shaped Cd (U ) with increasing volume fraction of the IL. The asymmetric case ϑ ∈ {1.5, 2} yielded asymmetric plots of Cd (U ), which
It is common in ionic liquids (ILs) for the cation and anion to differ in shape, size and valency. Various features of the differential capacitance (Cd (U )) of IL-based capacitors have been attributed to different kinds of asymmetries of the ions, both in experimental and in simulation studies [3–5, 8]. However, systematic knowledge of the influence of asymmetry on Cd (U ) seems to be lacking. It is close to impossible to gain such understanding from experiments or fully atomistic simulations, since exchanging one or several atoms in the chemical structure of real ions inevitably influences several different properties of the molecule at the same time. In an attempt to systematically study the influences of ion size, shape, and charge asymmetry, we provide a hierarchy of generic models of ILs, which bridge the gap 0953-8984/14/284114+08$33.00
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exhibited qualitatively similar features to Model B with a fixed asymmetry in ion size ϑ = 31/3 ≈ 1.44 and valency qcat ∈ {+3, +2, +1} going beyond the predictions of the mean-field theory. They revealed that Cd (U ) for an IL-based capacitor at a given electrode is predominantly determined by the packing ability of the species attracted to the respective electrode. Molecular-level insights into the layering of the model IL at the electrode surface could be obtained and changes in the layering behavior were correlated with specific features of the Cd (U ) dependence. In the present paper, called part II, we consider a model of IL with a cation that features an asymmetry in shape. This is achieved by binding three spherical particles to form an equilateral triangle, called Model C. In addition, we consider three different kinds of charge localization on the triangle and compare the results to those for Model B which has the same valency and excluded volume of the cation.
Figure 1. Sketch of an IL between two electrodes with an applied
voltage. A layer of cations accumulates at the cathode, followed by a less dense layer of anions. The layer structure gradually disappears towards the bulk.
2. Models and simulation methods
We consider a capacitor consisting of an ionic liquid positioned between two planar electrodes, as sketched in figure 1. The separation between the electrodes is sufficiently large that all density oscillations vanish in the middle region, and hence the IL density assumes the bulk values. We vary the surface charge density on the electrodes, and follow the IL response by performing equilibrium molecular dynamics simulations. Details of the simulation method, calculations of the electrode potential U from the ion density profiles, ρ(z), and the prescribed surface charge density at the electrode, σ , are provided in part I [2] and we omit them for brevity here. In figure 2, we provide sketches of Models A and B, which were studied in part I, and provide Model C studied in the current paper (part II). The cation in Model C consists of three spheres of diameter dcation = 1 nm connected by finite extensible nonlinear elastic (FENE) bonds [6] to form an equilateral triangle. One, two or three of the spheres carry an elementary charge +1 e at their center. An appropriate number of monovalent anions of size danion = 1 nm is present, to make the system overall electrically neutral. This is a minimalistic model which captures the flat shape and non-central localization of charge in pyridinium- or imidazolium-based cations in the IL. In terms of shape asymmetry, Model C can be compared to Model B studied in part I of the series, which has the same overall valency and excluded volume as Model C, but uses spherical cations with a centrally located charge.
Figure 2. Sketches and parameters of the generic IL models.
Model A: monovalent spheres with different ion size ratios ϑ = dcation /danion ∈ {1, 1.5, 2}; Model B: multivalent cations of volume Vcation = 3Vanion with charges q ∈ {+3, +2, +1}. Model C: spherical anions and triangular cations with different charge distributions on the triangle.
that the structure of the densely packed first layer of cations of Model C will very closely resemble the structure of Model A. The asymmetry of the Cd (U ) curves for Model C increases with decreasing charge on the cations and the maximum shifts towards positive potentials. The Cd (U ) on the cathode side decays faster with decreasing charge on the triangle. All these features have been observed in part I of our work [2], when the cation size was increased in Model A, or when the charge on the spherical cation was decreased in Model B. The Cd (U ) of Model C with q = +2 and q = +1 exhibits a clear shoulder on the anode side, in the range −1.1 V . U . −0.5 V. The Cd (U ) in the case of q = +2 in this range of potentials even becomes greater than the one for q = +3. No such behavior has been observed in the case of spherical particles with either asymmetry in size or asymmetry in valency. This
3. Results and discussion
In figure 3, we compare the capacitances Cd (U ) of all three triangular models. The plot shows that Cd (U ) for Model C, where all triangle constituents are charged (red curve in figure 3), almost coincides with the symmetric case of Model A studied in part I. This is not surprising, as the transition between these models is achieved by simply cutting the bonds in the triangle. Also, the close packing of equilateral triangles composed of spheres in a single layer is very similar to the close packing of non-bonded spherical particles. This implies 2
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Figure 3. Differential capacitance C d (U ) of Model C, with/without neutral constituents in the triangle, characterized by qtri . The similarities of the qtri = 3 case to the symmetric case of Model A and lower Cd (U ) (in the range U < 0) for decreasing qtri can be
explained with the charge per volume in the interfacial region.
effect is presumably related to orientational restructuring of the triangular cations. It is corroborated by the fact that it occurs before the shoulder at |U | ≈ 2 V, which was attributed in part I to first-layer saturation. In the following, we analyze in detail the density profiles and orientations of the triangles. 3.1. The anatomy of layers and triangle orientations
Like in part I, we examine now the density profiles of ions at the electrodes and try to link them to the features of Cd (U ). The number density profiles of particles which constitute the ions at various electrode surface charges do not exhibit any specific features in any of the variations of Model C. Therefore we omit them in the following figures and merely state that they exhibit peaks which reveal layering at each electrode and this layering diminishes towards the bulk. The height of the peaks increases while their width shrinks with increasing electrode surface charge, but their position does not change significantly. This is in contrast to the case for spherical ions with asymmetry in size or charge studied in the preceding work [2], which exhibited a significant shift, disappearance or reappearance of the total density peaks at different locations as the electrode surface charge was varied. We begin the discussion with the case qcat = 3, since its Cd (U ) is very close to that of the symmetric Model A from our preceding paper. Profiles of various quantities for this system are shown in figure 4. The integrated charge density in figure 4(a) exhibits a remarkable symmetry between the anode and cathode. This once again resembles the behavior of the symmetric Model A and is in line with the similarity of the capacitances Cd (U ) of Models A and C. The oscillating pattern in the charge density indicates a typical overcharging effect, which decays within a few layers. At higher surface charges the cathode exhibits a slightly stronger overcharging than the anode. At the cathode, the anions are expelled from the first and third layers and accumulate in the second layer (figure 4(b)), like for the symmetric Model A. The profile of the
Figure 4. Profiles of various quantities for qcat = 3: the integrated
charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes. 3
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Figure 5. Snapshot of the submodel with qtri = +3 at U = −0.32 V. For illustration, it is not the individual spheres of the triangles that are drawn but only a frame connecting the centers of the constituents. Also, only the first layer of cations is shown.
center of mass of the triangular cations in figure 4(c) exhibits three peaks at the cathode. For the triangular cations we use the center of mass profile instead of accounting separately for the constituent particles in order to facilitate comparison between different triangular models and also to enable comparison of the density profiles with the order parameter profiles discussed below. The three peaks in figure 4(c) correspond to triangles lying flat on the electrode at z = 1 nm and triangles perpendicular√ to the electrode with two segments at its surface (z = 1 + 3/6 √ ≈ 1.3 nm) and with one segment at the surface (z = 1 + 3/3 ≈ 1.6 nm). At zero electrode surface charge, the two more distant peaks dominate, which is a consequence of the perpendicular orientations being entropically more favorable. With increasing surface charge, the portion of the flat-lying triangles grows and the third peak gradually disappears. The second peak starts to diminish at σ = 0.72 e nm−2 and eventually disappears at even higher σ (outside the range of σ in figure 4(c)). Further information about the orientations of the triangles can be conveniently expressed using an order parameter. For the angle α between the normal vector of the plane defined by a triangle and the z-axis, we define the order parameter as the second Legendre polynomial of cos(α): D E S = hP2 (cos α)i = 32 cos2 (α) − 21 . (1)
Figure 6. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +3 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange). There is a
strong trend towards flat alignment for increasing electrode charge σ with a small share of perpendicular orientation.
present, which is rather misleading. Therefore we propose that the overall tendency of cations for a particular orientational order at the cathode can be well characterized by the product of the order parameter and the number density of the center of mass S(z)ρcom (z) shown in figure 6. This has the advantage that it suppresses the peaks in S(z) in regions of low density, and enhances those in high density regions. Since the density is strictly positive, S(z)ρcom (z) preserves the information regarding the preference for perpendicular or flat alignment contained in the sign of S(z) and not in ρcom (z). With its scale zoomed to the close vicinity of the cathode, figure 6 reveals the increasing significance of the flat-lying cations with increasing σ in comparison with the almost σ -independent values for the perpendicular ones. Finally, at high surface charges, dense packing can be achieved only with flat-lying triangles. A further increase in electrode surface charge beyond the first-layer saturation might lead again to triangles perpendicular to the electrode, entirely filling up the first layer and protruding into the second layer. We observed signs of such extreme behavior at the highest electrode charges studied, but this is presumably of little practical significance since layer saturation is hard to exceed in experiments on real ILs [1]. The behavior at the anode as revealed from figure 4 is similar to but less complex than that at the cathode. At zero surface charge, the density profiles are mirror images of those at the cathode. Again, S(z) shows strong orientational ordering at 31 nm with very few cations present in this region. The double peak in cation density, which corresponds to the two possible perpendicular cation orientations, is transferred with increasing electrode charge to another double peak which corresponds to flat (≈30.3 nm) and perpendicular (≈29.8 nm)
By averaging over the available configurations and spatially resolving S along the z-axis, one can identify preferred orientations: triangles lying flat on the electrode are characterized by S → 1, while triangles perpendicular to the electrode (α = 0) yield S → 0.5. The order parameter profile as a function of the triangle center of mass position is shown in figure 4(d). The location of the peaks at the cathode correlates very well with the density profile in figure 4(c). Strong orientational ordering at z = 1 nm can be observed right from σ = 0, even though at low σ cations are scarce in this region. With increasing σ , the orientational ordering in the first layer reaches the limiting value of S = 1. In the second peak, a decrease in S with increasing σ shows that cations increasingly prefer perpendicular orientation, when one of their constituents is not directly at the electrode. This picture is further supported by a simulation snapshot in figure 5. The order parameter profile reveals strong orientational preference also in regions where almost no particles are 4
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orientation of triangles in the second layer. In this region, the anions are depleted and they accumulate around 29 nm. Let us now compare the behavior of Model C with qcat = 3 to those for the other two cases, qcat = 2 and qcat = 1. The profiles for the case qcat = 2 are shown in figure 7. The integrated charge density profile in figure 7(a) reveals quite some symmetry between the cathode and anode sides, like for qcat = 3. The profiles on the anode side are very similar to those for qcat = 3, so we will not discuss them in detail. We rather focus on the cathode, which reveals interesting information about the reorientation of cations. The anion density profile at the cathode (figure 7(b)) shows expulsion of anions from the first layer and their accumulation in the second, like for qcat = 3. In the cation center of mass density profile (figure 7(c)) on the cathode side, the third peak around 1.6 nm diminishes with increasing σ , but in contrast with the case for qcat = 3, the second peak at 1.3 nm is dominant over the first one at 1.0 nm. This suggests a different reorientation mechanism of the triangles with qcat = 2. Interestingly, the cation order parameter profile at the cathode varies very little with σ . Just the peak at 1.3 nm becomes enhanced with increasing σ , while the other peaks remain unaffected. At higher σ , a third peak appears around 3 nm in the cation density as well. The product S(z)ρcom (z) in figure 8 reveals a feature which is barely distinguishable in the density profile of figure 7(c): with increasing σ , the peak in S(z)ρcom (z) at 1.3 nm constantly increases, while that at 1.0 nm first grows but then diminishes again for U . −0.4 V. We recall that the shoulder in Cd (U ) in figure 3 occurs at the same potential. At low U both the flat alignment of the triangles and the perpendicular alignment, with two vertices at the electrode, are equivalently preferred. As long as void space is available at the electrode, both of these orientations provide two elementary charges in the first layer and hence equal interaction energy. When the first layer is filled up with cations, it can still accommodate more charge when the flat triangles change their orientation to perpendicular, moving the neutral constituent to the second layer and providing extra void space in the first layer to accommodate the charged vertices of other triangles. Although the triangles with qcat = 2 contain 2/3 of the charge of those with qcat = 3, they can achieve the same charge density in the first layer at the cost of extra orientational entropy. Therefore, at low U the Cd (U ) curve for qcat = 2 falls faster than that for qcat = 3. At higher U , when the extra entropy loss can be overcome, Cd (U ) for the divalent triangles exceeds that for the trivalent ones, so finally they end up with a comparable charge density in the first layer. The limiting situation then features all triangular cations standing perpendicular to the electrode, with the two charged constituents in the first layer and the neutral one in the second layer. This change in the polarization mechanism from simple accumulation to reorientation shows up as a shoulder around U = −0.4 V in Cd (U ) in figure 3. The two situations are illustrated by simulation snapshots in figure 9: at low electrode charge (top picture), both flat triangles and those perpendicular to the electrode with two vertices in the first layer are present. At higher electrode charge (bottom picture), the higher density close to the electrode due to the increased attraction
Figure 7. Profiles of various quantities for qcat = 2: the integrated
charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes. 5
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layers shifted in the x y-plane with respect to each other. Additionally, the tilted configurations are more degenerate and hence entropically more favored than the perpendicular ones. In agreement with that, the order parameter in figure 7(d) at z = 1.3 nm saturates around S ≈ −0.35 and does not increase with increasing σ . Indeed, if the first layer is almost filled with upright oriented triangles, they also occupy almost 50% of the available space in the second layer. The remaining space is left for the anions, which seems to be sufficient to avoid affecting the overcharging too much. Finally, we address the case with qcat = 1. Again, we omit a detailed discussion of the anode profiles which possess features very similar to those of the previous two cases, and focus only on the vicinity of the cathode. The integrated charge density profile of this system (figure 10(a)) is quite different from those for the previous two cases. It exhibits a pronounced asymmetry between the anode and cathode. On the cathode side, the first-layer saturation occurs already at σ ≈ 0.58 e nm−2 . In addition, the plateau in the integrated charge density, which indicated layer saturation in the spherical IL models studied in part I [2], now rises with further increase in σ . After the first-layer saturation, overcharging in further layers is strongly suppressed. Furthermore, anions are expelled from the first and second layers at the cathode and accumulate around 3 nm. Finally, in the cation density profile the first peak at 1.0 nm disappears at rather low σ , while the second peak at 1.3 nm increases and later the third peak at 1.6 nm reaches the height of the second one. Similarly, the first peak in the order parameter decreases with increasing σ while the latter two increase. This behavior can be understood by considering the packing of triangles with a single charged vertex, which maximizes the surface charge at the electrode. The maximum charge density is attained in a 1:1 mixture of triangles with one (charged) and two (one charged, one neutral) vertices at the electrode surface. The notion that this is the limiting structure at high σ is supported by the equal heights of the second and third peaks in figure 10(c), by the simulation snapshot in figure 11 and also by the product S(z)ρcom (z) in figure 12. Especially the latter reveals that the flat-lying triangles soon disappear, despite the strong peak in S(z), and the proportion of the other two orientations increases simultaneously. Finally, we note that the cation structure which maximizes the charge density at the cathode also entirely fills up the second layer with neutral particles. Since the plateau in the integrated charge density of figure 10(a) grows with increasing σ , we anticipate that at σ ≈ 0.58 e nm−2 all anions are expelled from the first two layers but still more charge density at the electrode can be gained by reorientation and more efficient packing of the cations. Because at this state the second layer is entirely filled with neutral particles and counterions can only occupy the third layer, overscreening effects are strongly suppressed, as can also be seen from figure 10(a).
Figure 8. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +2 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange): the triangle
gets tilted and the neutral constituent acts as a latent void for additional charges.
Figure 9. Snapshot of the submodel with qtri = +2 at U = −0.3 V
(top) and U = −1.5 V (bottom). Neutral parts of the triangles are colored in gray; charged ones are blue.
causes a change in structure: the previously flat aligned triangles rise, leaving the two charged vertices in the first layer and moving the neutral vertex away from the electrode. The final preferred orientation is not perpendicular (S → −0.5), but tilted with S(1.3 nm) ≈ −0.35, as can be observed in both snapshots of figure 9. With this tilt, more efficient close packing of the first and second layers can be achieved, which resembles the hexagonal close packed structure: two
4. Conclusions
In the present paper (part II) we extended the investigation of part I [2] on the behavior of a generic ionic liquid between two electrodes. While in part I we focused on spherical ions 6
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Figure 11. Snapshot of the submodel with qtri = +1 at U = −1.83 V.
Figure 12. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +1 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange): triangles
start with flat alignment and get tilted with increasing surface charge. An efficient packing in the first layer has to include neutral constituents and is achieved by a mixture of two perpendicular orientations (sketched bottom right).
asymmetric in size (Model A) and valency (Model B), we investigated in part II the asymmetry in shape which we introduced by connecting three spherical particles to form a single triangular cation. One, two or all three particles of the triangle can carry an elementary charge and hence can produce an asymmetry in charge localization. The fully charged triangular cation showed similar behavior of Cd (U ) to the symmetric spheres of Model A. This was due to the like behavior of the first electrode layers produced by the same dense packing in these two models, supported by the fact that a triangle close to the electrodes will most likely be oriented with its normal vector perpendicular to the electrode plane. The similarity in layer filling leads to the observed similarity of the Cd (U ) curves. A general trend that
Figure 10. Profiles of various quantities for qcat = 1: the integrated charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes.
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Acknowledgments
came along with the replacement of charged components of the triangular molecule with neutral ones was the overall reduction of Cd (U ) on the cathode side of the curves—similar to the results from Model B, where the valency was varied. As the potential increased, the triangles were restructured such that the charged constituents became located closest to the electrodes. With a single neutral particle in the triangle, the preferred alignment changed from flat to perpendicular orientation for increasing potential. Such potential-dependent reorientations could result in a local enhancement of the Cd (U ) curves, if the rearrangement led to an increase of charge close to the electrode. Finally, the case of two neutral components and a single charge left on the triangles resulted in a mixture of two perpendicular alignments that most efficiently occupied the available space at the electrode. Our analysis showed that the charging mechanism of ionic liquid-based capacitors included (re)orientation effects, which were affected by the polarity of the underlying molecules. At low voltages, displacement of cations by anions at the cathode and the opposite behavior at the anode were the dominant mechanisms of the capacitor charging. In a simplified argument the orientational rearrangement brought a charged particle closer to the electrode and moved a neutral particle further away, which is just half of the energy gain when charges of opposite signs are exchanged. The orientational rearrangement occurred at higher voltages and appeared as a shoulder in the Cd (U ) dependence. This behavior could not be observed in a generic model with spherical particles but should be present in most real ILs, since they typically contain bulky cations with the charge localized in a smaller region. In line with our earlier investigation in part I, we observed also here that the general shape of Cd (U ) was mainly controlled by the charge per occupied volume in the very first layer.
The authors acknowledge financial support by the DFG through the SimTech cluster of excellence and SFB 716. PK thanks the MSMT of the Czech Republic for financial support under grant LK 21302. References [1] Bazant M Z, Kilic M S, Storey B D and Ajdari A 2009 Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions Adv. Colloid Interface Sci. 152 48–88 [2] Breitsprecher K, Koˇsovan P and Holm C 2014 Coarse grained simulations of an ionic liquid-based capacitor I: density, ion size, and valency effects J. Phys.: Condens. Matter 26 284108 [3] Fedorov M V, Georgi N and Kornyshev A A 2010 Double layer in ionic liquids: the nature of the camel shape of capacitance Electrochem. Commun. 12 296–9 [4] Fedorov M V and Kornyshev A A 2008 Ionic liquid near a charged wall: structure and capacitance of electrical double layer J. Phys. Chem. B 112 11868–72 [5] Georgi N, Kornyshev A A and Fedorov M V 2010 The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: electrostriction versus lattice saturation J. Electroanal. Chem. 649 261–667 [6] Grest G S and Kremer K 1986 Molecular dynamics simulation for polymers in the presence of a heat bath Phys. Rev. A 33 3628–31 [7] Kornyshev A A 2007 Double-layer in ionic liquids: paradigm change? J. Phys. Chem. B 111 5545–57 [8] Lauw Y, Horne M D, Rodopoulos T, Nelson A and Leermakers F A M 2010 Electrical double-layer capacitance in room temperature ionic liquids: ion-size and specific adsorption effects J. Phys. Chem. B 114 11149–54
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Coarse-grained simulations of an ionic liquid-based capacitor: II. Asymmetry in ion shape and charge localization
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 284114 (http://iopscience.iop.org/0953-8984/26/28/284114) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 147.188.128.74 This content was downloaded on 25/08/2014 at 22:14
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 284114 (8pp)
doi:10.1088/0953-8984/26/28/284114
Coarse-grained simulations of an ionic liquid-based capacitor: II. Asymmetry in ion shape and charge localization Konrad Breitsprecher1 , Peter Košovan1,2 and Christian Holm1 1
Institut für Computerphysik, Universität Stuttgart, Stuttgart, Germany Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University in Prague, Prague, Czech Republic 2
E-mail: [email protected] Received 23 December 2013, revised 24 January 2014 Accepted for publication 27 January 2014 Published 12 June 2014
Abstract
In this work, which is a continuation of part I, we introduce a primitive model for an ionic liquid (IL) that can account for the planar shape of cations typical for ILs like imidazolium. The model consists of a spherical anion and a triangular cation consisting of three spheres, where one or all three vertices of the triangle can carry electric charge. We use molecular dynamics simulations to study the differential capacitance Cd of an ionic liquid confined between two planar electrodes. Our goal is to elucidate the complex dependence of Cd on the electrode potential U in terms of simple entities such as the shape and charge distribution of the ions. For this purpose, we compare the results from the current model to the results based on the models with spherical cations that possess asymmetry in ion valence and shape that were analyzed in detail in part I of this work. We show that the various possible stackings of the triangles near the cathode lead to noticeable new features in Cd (U ) as compared to the spherical models. Different distributions of charges on the triangle lead to different preferred orientations of the cations near the cathode that are moreover potential dependent. Keywords: ionic liquids, capacitor, differential capacitance, ESPResSo (Some figures may appear in colour only in the online journal)
1. Introduction
between the oversimplified theoretical assumptions and overly complex real IL systems. In part I of this paper series [2], we introduced a coarse-grained model of an IL that acts as the dielectric of a capacitor. The model studied there consisted of spherical particles with a point charge located at their center. Model A featured asymmetry in ion size, characterized by the diameter ratio ϑ = dcation /danion ∈ {1, 1.5, 2}, and a valency of qcat = −qan = +1 e. The asymmetry was achieved by variation of the cation properties while properties of the anion were kept constant. The results for the symmetric case with ϑ = 1 could be compared to the mean-field theory of Kornyshev [7] and yielded a qualitative agreement in terms of a transition from a camel-shaped to the bell-shaped Cd (U ) with increasing volume fraction of the IL. The asymmetric case ϑ ∈ {1.5, 2} yielded asymmetric plots of Cd (U ), which
It is common in ionic liquids (ILs) for the cation and anion to differ in shape, size and valency. Various features of the differential capacitance (Cd (U )) of IL-based capacitors have been attributed to different kinds of asymmetries of the ions, both in experimental and in simulation studies [3–5, 8]. However, systematic knowledge of the influence of asymmetry on Cd (U ) seems to be lacking. It is close to impossible to gain such understanding from experiments or fully atomistic simulations, since exchanging one or several atoms in the chemical structure of real ions inevitably influences several different properties of the molecule at the same time. In an attempt to systematically study the influences of ion size, shape, and charge asymmetry, we provide a hierarchy of generic models of ILs, which bridge the gap 0953-8984/14/284114+08$33.00
1
c 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 284114
K Breitsprecher et al
exhibited qualitatively similar features to Model B with a fixed asymmetry in ion size ϑ = 31/3 ≈ 1.44 and valency qcat ∈ {+3, +2, +1} going beyond the predictions of the mean-field theory. They revealed that Cd (U ) for an IL-based capacitor at a given electrode is predominantly determined by the packing ability of the species attracted to the respective electrode. Molecular-level insights into the layering of the model IL at the electrode surface could be obtained and changes in the layering behavior were correlated with specific features of the Cd (U ) dependence. In the present paper, called part II, we consider a model of IL with a cation that features an asymmetry in shape. This is achieved by binding three spherical particles to form an equilateral triangle, called Model C. In addition, we consider three different kinds of charge localization on the triangle and compare the results to those for Model B which has the same valency and excluded volume of the cation.
Figure 1. Sketch of an IL between two electrodes with an applied
voltage. A layer of cations accumulates at the cathode, followed by a less dense layer of anions. The layer structure gradually disappears towards the bulk.
2. Models and simulation methods
We consider a capacitor consisting of an ionic liquid positioned between two planar electrodes, as sketched in figure 1. The separation between the electrodes is sufficiently large that all density oscillations vanish in the middle region, and hence the IL density assumes the bulk values. We vary the surface charge density on the electrodes, and follow the IL response by performing equilibrium molecular dynamics simulations. Details of the simulation method, calculations of the electrode potential U from the ion density profiles, ρ(z), and the prescribed surface charge density at the electrode, σ , are provided in part I [2] and we omit them for brevity here. In figure 2, we provide sketches of Models A and B, which were studied in part I, and provide Model C studied in the current paper (part II). The cation in Model C consists of three spheres of diameter dcation = 1 nm connected by finite extensible nonlinear elastic (FENE) bonds [6] to form an equilateral triangle. One, two or three of the spheres carry an elementary charge +1 e at their center. An appropriate number of monovalent anions of size danion = 1 nm is present, to make the system overall electrically neutral. This is a minimalistic model which captures the flat shape and non-central localization of charge in pyridinium- or imidazolium-based cations in the IL. In terms of shape asymmetry, Model C can be compared to Model B studied in part I of the series, which has the same overall valency and excluded volume as Model C, but uses spherical cations with a centrally located charge.
Figure 2. Sketches and parameters of the generic IL models.
Model A: monovalent spheres with different ion size ratios ϑ = dcation /danion ∈ {1, 1.5, 2}; Model B: multivalent cations of volume Vcation = 3Vanion with charges q ∈ {+3, +2, +1}. Model C: spherical anions and triangular cations with different charge distributions on the triangle.
that the structure of the densely packed first layer of cations of Model C will very closely resemble the structure of Model A. The asymmetry of the Cd (U ) curves for Model C increases with decreasing charge on the cations and the maximum shifts towards positive potentials. The Cd (U ) on the cathode side decays faster with decreasing charge on the triangle. All these features have been observed in part I of our work [2], when the cation size was increased in Model A, or when the charge on the spherical cation was decreased in Model B. The Cd (U ) of Model C with q = +2 and q = +1 exhibits a clear shoulder on the anode side, in the range −1.1 V . U . −0.5 V. The Cd (U ) in the case of q = +2 in this range of potentials even becomes greater than the one for q = +3. No such behavior has been observed in the case of spherical particles with either asymmetry in size or asymmetry in valency. This
3. Results and discussion
In figure 3, we compare the capacitances Cd (U ) of all three triangular models. The plot shows that Cd (U ) for Model C, where all triangle constituents are charged (red curve in figure 3), almost coincides with the symmetric case of Model A studied in part I. This is not surprising, as the transition between these models is achieved by simply cutting the bonds in the triangle. Also, the close packing of equilateral triangles composed of spheres in a single layer is very similar to the close packing of non-bonded spherical particles. This implies 2
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Figure 3. Differential capacitance C d (U ) of Model C, with/without neutral constituents in the triangle, characterized by qtri . The similarities of the qtri = 3 case to the symmetric case of Model A and lower Cd (U ) (in the range U < 0) for decreasing qtri can be
explained with the charge per volume in the interfacial region.
effect is presumably related to orientational restructuring of the triangular cations. It is corroborated by the fact that it occurs before the shoulder at |U | ≈ 2 V, which was attributed in part I to first-layer saturation. In the following, we analyze in detail the density profiles and orientations of the triangles. 3.1. The anatomy of layers and triangle orientations
Like in part I, we examine now the density profiles of ions at the electrodes and try to link them to the features of Cd (U ). The number density profiles of particles which constitute the ions at various electrode surface charges do not exhibit any specific features in any of the variations of Model C. Therefore we omit them in the following figures and merely state that they exhibit peaks which reveal layering at each electrode and this layering diminishes towards the bulk. The height of the peaks increases while their width shrinks with increasing electrode surface charge, but their position does not change significantly. This is in contrast to the case for spherical ions with asymmetry in size or charge studied in the preceding work [2], which exhibited a significant shift, disappearance or reappearance of the total density peaks at different locations as the electrode surface charge was varied. We begin the discussion with the case qcat = 3, since its Cd (U ) is very close to that of the symmetric Model A from our preceding paper. Profiles of various quantities for this system are shown in figure 4. The integrated charge density in figure 4(a) exhibits a remarkable symmetry between the anode and cathode. This once again resembles the behavior of the symmetric Model A and is in line with the similarity of the capacitances Cd (U ) of Models A and C. The oscillating pattern in the charge density indicates a typical overcharging effect, which decays within a few layers. At higher surface charges the cathode exhibits a slightly stronger overcharging than the anode. At the cathode, the anions are expelled from the first and third layers and accumulate in the second layer (figure 4(b)), like for the symmetric Model A. The profile of the
Figure 4. Profiles of various quantities for qcat = 3: the integrated
charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes. 3
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Figure 5. Snapshot of the submodel with qtri = +3 at U = −0.32 V. For illustration, it is not the individual spheres of the triangles that are drawn but only a frame connecting the centers of the constituents. Also, only the first layer of cations is shown.
center of mass of the triangular cations in figure 4(c) exhibits three peaks at the cathode. For the triangular cations we use the center of mass profile instead of accounting separately for the constituent particles in order to facilitate comparison between different triangular models and also to enable comparison of the density profiles with the order parameter profiles discussed below. The three peaks in figure 4(c) correspond to triangles lying flat on the electrode at z = 1 nm and triangles perpendicular√ to the electrode with two segments at its surface (z = 1 + 3/6 √ ≈ 1.3 nm) and with one segment at the surface (z = 1 + 3/3 ≈ 1.6 nm). At zero electrode surface charge, the two more distant peaks dominate, which is a consequence of the perpendicular orientations being entropically more favorable. With increasing surface charge, the portion of the flat-lying triangles grows and the third peak gradually disappears. The second peak starts to diminish at σ = 0.72 e nm−2 and eventually disappears at even higher σ (outside the range of σ in figure 4(c)). Further information about the orientations of the triangles can be conveniently expressed using an order parameter. For the angle α between the normal vector of the plane defined by a triangle and the z-axis, we define the order parameter as the second Legendre polynomial of cos(α): D E S = hP2 (cos α)i = 32 cos2 (α) − 21 . (1)
Figure 6. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +3 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange). There is a
strong trend towards flat alignment for increasing electrode charge σ with a small share of perpendicular orientation.
present, which is rather misleading. Therefore we propose that the overall tendency of cations for a particular orientational order at the cathode can be well characterized by the product of the order parameter and the number density of the center of mass S(z)ρcom (z) shown in figure 6. This has the advantage that it suppresses the peaks in S(z) in regions of low density, and enhances those in high density regions. Since the density is strictly positive, S(z)ρcom (z) preserves the information regarding the preference for perpendicular or flat alignment contained in the sign of S(z) and not in ρcom (z). With its scale zoomed to the close vicinity of the cathode, figure 6 reveals the increasing significance of the flat-lying cations with increasing σ in comparison with the almost σ -independent values for the perpendicular ones. Finally, at high surface charges, dense packing can be achieved only with flat-lying triangles. A further increase in electrode surface charge beyond the first-layer saturation might lead again to triangles perpendicular to the electrode, entirely filling up the first layer and protruding into the second layer. We observed signs of such extreme behavior at the highest electrode charges studied, but this is presumably of little practical significance since layer saturation is hard to exceed in experiments on real ILs [1]. The behavior at the anode as revealed from figure 4 is similar to but less complex than that at the cathode. At zero surface charge, the density profiles are mirror images of those at the cathode. Again, S(z) shows strong orientational ordering at 31 nm with very few cations present in this region. The double peak in cation density, which corresponds to the two possible perpendicular cation orientations, is transferred with increasing electrode charge to another double peak which corresponds to flat (≈30.3 nm) and perpendicular (≈29.8 nm)
By averaging over the available configurations and spatially resolving S along the z-axis, one can identify preferred orientations: triangles lying flat on the electrode are characterized by S → 1, while triangles perpendicular to the electrode (α = 0) yield S → 0.5. The order parameter profile as a function of the triangle center of mass position is shown in figure 4(d). The location of the peaks at the cathode correlates very well with the density profile in figure 4(c). Strong orientational ordering at z = 1 nm can be observed right from σ = 0, even though at low σ cations are scarce in this region. With increasing σ , the orientational ordering in the first layer reaches the limiting value of S = 1. In the second peak, a decrease in S with increasing σ shows that cations increasingly prefer perpendicular orientation, when one of their constituents is not directly at the electrode. This picture is further supported by a simulation snapshot in figure 5. The order parameter profile reveals strong orientational preference also in regions where almost no particles are 4
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orientation of triangles in the second layer. In this region, the anions are depleted and they accumulate around 29 nm. Let us now compare the behavior of Model C with qcat = 3 to those for the other two cases, qcat = 2 and qcat = 1. The profiles for the case qcat = 2 are shown in figure 7. The integrated charge density profile in figure 7(a) reveals quite some symmetry between the cathode and anode sides, like for qcat = 3. The profiles on the anode side are very similar to those for qcat = 3, so we will not discuss them in detail. We rather focus on the cathode, which reveals interesting information about the reorientation of cations. The anion density profile at the cathode (figure 7(b)) shows expulsion of anions from the first layer and their accumulation in the second, like for qcat = 3. In the cation center of mass density profile (figure 7(c)) on the cathode side, the third peak around 1.6 nm diminishes with increasing σ , but in contrast with the case for qcat = 3, the second peak at 1.3 nm is dominant over the first one at 1.0 nm. This suggests a different reorientation mechanism of the triangles with qcat = 2. Interestingly, the cation order parameter profile at the cathode varies very little with σ . Just the peak at 1.3 nm becomes enhanced with increasing σ , while the other peaks remain unaffected. At higher σ , a third peak appears around 3 nm in the cation density as well. The product S(z)ρcom (z) in figure 8 reveals a feature which is barely distinguishable in the density profile of figure 7(c): with increasing σ , the peak in S(z)ρcom (z) at 1.3 nm constantly increases, while that at 1.0 nm first grows but then diminishes again for U . −0.4 V. We recall that the shoulder in Cd (U ) in figure 3 occurs at the same potential. At low U both the flat alignment of the triangles and the perpendicular alignment, with two vertices at the electrode, are equivalently preferred. As long as void space is available at the electrode, both of these orientations provide two elementary charges in the first layer and hence equal interaction energy. When the first layer is filled up with cations, it can still accommodate more charge when the flat triangles change their orientation to perpendicular, moving the neutral constituent to the second layer and providing extra void space in the first layer to accommodate the charged vertices of other triangles. Although the triangles with qcat = 2 contain 2/3 of the charge of those with qcat = 3, they can achieve the same charge density in the first layer at the cost of extra orientational entropy. Therefore, at low U the Cd (U ) curve for qcat = 2 falls faster than that for qcat = 3. At higher U , when the extra entropy loss can be overcome, Cd (U ) for the divalent triangles exceeds that for the trivalent ones, so finally they end up with a comparable charge density in the first layer. The limiting situation then features all triangular cations standing perpendicular to the electrode, with the two charged constituents in the first layer and the neutral one in the second layer. This change in the polarization mechanism from simple accumulation to reorientation shows up as a shoulder around U = −0.4 V in Cd (U ) in figure 3. The two situations are illustrated by simulation snapshots in figure 9: at low electrode charge (top picture), both flat triangles and those perpendicular to the electrode with two vertices in the first layer are present. At higher electrode charge (bottom picture), the higher density close to the electrode due to the increased attraction
Figure 7. Profiles of various quantities for qcat = 2: the integrated
charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes. 5
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layers shifted in the x y-plane with respect to each other. Additionally, the tilted configurations are more degenerate and hence entropically more favored than the perpendicular ones. In agreement with that, the order parameter in figure 7(d) at z = 1.3 nm saturates around S ≈ −0.35 and does not increase with increasing σ . Indeed, if the first layer is almost filled with upright oriented triangles, they also occupy almost 50% of the available space in the second layer. The remaining space is left for the anions, which seems to be sufficient to avoid affecting the overcharging too much. Finally, we address the case with qcat = 1. Again, we omit a detailed discussion of the anode profiles which possess features very similar to those of the previous two cases, and focus only on the vicinity of the cathode. The integrated charge density profile of this system (figure 10(a)) is quite different from those for the previous two cases. It exhibits a pronounced asymmetry between the anode and cathode. On the cathode side, the first-layer saturation occurs already at σ ≈ 0.58 e nm−2 . In addition, the plateau in the integrated charge density, which indicated layer saturation in the spherical IL models studied in part I [2], now rises with further increase in σ . After the first-layer saturation, overcharging in further layers is strongly suppressed. Furthermore, anions are expelled from the first and second layers at the cathode and accumulate around 3 nm. Finally, in the cation density profile the first peak at 1.0 nm disappears at rather low σ , while the second peak at 1.3 nm increases and later the third peak at 1.6 nm reaches the height of the second one. Similarly, the first peak in the order parameter decreases with increasing σ while the latter two increase. This behavior can be understood by considering the packing of triangles with a single charged vertex, which maximizes the surface charge at the electrode. The maximum charge density is attained in a 1:1 mixture of triangles with one (charged) and two (one charged, one neutral) vertices at the electrode surface. The notion that this is the limiting structure at high σ is supported by the equal heights of the second and third peaks in figure 10(c), by the simulation snapshot in figure 11 and also by the product S(z)ρcom (z) in figure 12. Especially the latter reveals that the flat-lying triangles soon disappear, despite the strong peak in S(z), and the proportion of the other two orientations increases simultaneously. Finally, we note that the cation structure which maximizes the charge density at the cathode also entirely fills up the second layer with neutral particles. Since the plateau in the integrated charge density of figure 10(a) grows with increasing σ , we anticipate that at σ ≈ 0.58 e nm−2 all anions are expelled from the first two layers but still more charge density at the electrode can be gained by reorientation and more efficient packing of the cations. Because at this state the second layer is entirely filled with neutral particles and counterions can only occupy the third layer, overscreening effects are strongly suppressed, as can also be seen from figure 10(a).
Figure 8. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +2 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange): the triangle
gets tilted and the neutral constituent acts as a latent void for additional charges.
Figure 9. Snapshot of the submodel with qtri = +2 at U = −0.3 V
(top) and U = −1.5 V (bottom). Neutral parts of the triangles are colored in gray; charged ones are blue.
causes a change in structure: the previously flat aligned triangles rise, leaving the two charged vertices in the first layer and moving the neutral vertex away from the electrode. The final preferred orientation is not perpendicular (S → −0.5), but tilted with S(1.3 nm) ≈ −0.35, as can be observed in both snapshots of figure 9. With this tilt, more efficient close packing of the first and second layers can be achieved, which resembles the hexagonal close packed structure: two
4. Conclusions
In the present paper (part II) we extended the investigation of part I [2] on the behavior of a generic ionic liquid between two electrodes. While in part I we focused on spherical ions 6
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Figure 11. Snapshot of the submodel with qtri = +1 at U = −1.83 V.
Figure 12. Combined cation orientation and density plots S(z)ρcom (z) for the submodel with qtri = +1 in the range from σ = 0.07 e nm−2 (black) to σ = 0.58 e nm−2 (orange): triangles
start with flat alignment and get tilted with increasing surface charge. An efficient packing in the first layer has to include neutral constituents and is achieved by a mixture of two perpendicular orientations (sketched bottom right).
asymmetric in size (Model A) and valency (Model B), we investigated in part II the asymmetry in shape which we introduced by connecting three spherical particles to form a single triangular cation. One, two or all three particles of the triangle can carry an elementary charge and hence can produce an asymmetry in charge localization. The fully charged triangular cation showed similar behavior of Cd (U ) to the symmetric spheres of Model A. This was due to the like behavior of the first electrode layers produced by the same dense packing in these two models, supported by the fact that a triangle close to the electrodes will most likely be oriented with its normal vector perpendicular to the electrode plane. The similarity in layer filling leads to the observed similarity of the Cd (U ) curves. A general trend that
Figure 10. Profiles of various quantities for qcat = 1: the integrated charge density (a), anion density (b), density of centers of mass of the cations (c) and order parameter for cation orientations (d). Note the different scales on the axes.
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Acknowledgments
came along with the replacement of charged components of the triangular molecule with neutral ones was the overall reduction of Cd (U ) on the cathode side of the curves—similar to the results from Model B, where the valency was varied. As the potential increased, the triangles were restructured such that the charged constituents became located closest to the electrodes. With a single neutral particle in the triangle, the preferred alignment changed from flat to perpendicular orientation for increasing potential. Such potential-dependent reorientations could result in a local enhancement of the Cd (U ) curves, if the rearrangement led to an increase of charge close to the electrode. Finally, the case of two neutral components and a single charge left on the triangles resulted in a mixture of two perpendicular alignments that most efficiently occupied the available space at the electrode. Our analysis showed that the charging mechanism of ionic liquid-based capacitors included (re)orientation effects, which were affected by the polarity of the underlying molecules. At low voltages, displacement of cations by anions at the cathode and the opposite behavior at the anode were the dominant mechanisms of the capacitor charging. In a simplified argument the orientational rearrangement brought a charged particle closer to the electrode and moved a neutral particle further away, which is just half of the energy gain when charges of opposite signs are exchanged. The orientational rearrangement occurred at higher voltages and appeared as a shoulder in the Cd (U ) dependence. This behavior could not be observed in a generic model with spherical particles but should be present in most real ILs, since they typically contain bulky cations with the charge localized in a smaller region. In line with our earlier investigation in part I, we observed also here that the general shape of Cd (U ) was mainly controlled by the charge per occupied volume in the very first layer.
The authors acknowledge financial support by the DFG through the SimTech cluster of excellence and SFB 716. PK thanks the MSMT of the Czech Republic for financial support under grant LK 21302. References [1] Bazant M Z, Kilic M S, Storey B D and Ajdari A 2009 Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions Adv. Colloid Interface Sci. 152 48–88 [2] Breitsprecher K, Koˇsovan P and Holm C 2014 Coarse grained simulations of an ionic liquid-based capacitor I: density, ion size, and valency effects J. Phys.: Condens. Matter 26 284108 [3] Fedorov M V, Georgi N and Kornyshev A A 2010 Double layer in ionic liquids: the nature of the camel shape of capacitance Electrochem. Commun. 12 296–9 [4] Fedorov M V and Kornyshev A A 2008 Ionic liquid near a charged wall: structure and capacitance of electrical double layer J. Phys. Chem. B 112 11868–72 [5] Georgi N, Kornyshev A A and Fedorov M V 2010 The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: electrostriction versus lattice saturation J. Electroanal. Chem. 649 261–667 [6] Grest G S and Kremer K 1986 Molecular dynamics simulation for polymers in the presence of a heat bath Phys. Rev. A 33 3628–31 [7] Kornyshev A A 2007 Double-layer in ionic liquids: paradigm change? J. Phys. Chem. B 111 5545–57 [8] Lauw Y, Horne M D, Rodopoulos T, Nelson A and Leermakers F A M 2010 Electrical double-layer capacitance in room temperature ionic liquids: ion-size and specific adsorption effects J. Phys. Chem. B 114 11149–54
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