Electrode Models for Ionic Liquid-Based Capacitors - The Journal of

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Electrode Models for Ionic Liquid-Based Capacitors Konrad Breitsprecher, Kai Szuttor, and Christian Holm* Institute for Computational Physics, University of Stuttgart, Almandring 3, 70569 Stuttgart, Germany ABSTRACT: We investigate the influence of different electrode models on properties of a BMIM PF6 ionic liquidbased capacitor system using molecular dynamics simulations. We compared atomic graphene electrodes against smooth walls of equal average interaction energy. Furthermore, two different methods for simulating a constant potential drop between the electrodes were employed and compared to constant charge electrode models. In the evaluation of the voltage dependence of the ion configuration in the interfacial electrode regions, the influence of the electrode structure was determined by analyzing the in-plane radial distribution of the adsorbed ions, the interlayer ion exchange process, and the differential capacitance. We see similar results for the constant charge and constant potential approach for capacitance and structure of the adsorbed ions. We demonstrate that, due to the underlying ionic models, anions and cations adapt to the structural pattern of the graphene electrodes in different ways. This affects the charging process and leads to distinct asymmetric features in the differential capacitance.

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INTRODUCTION Ionic liquid (IL) based capacitors belong to the class of energy storage devices known predominantly as double layer capacitors or supercapacitors. They consist of a liquid electrolyte confined between two electrodes, a current collector connects the electrodes to the external circuit and a permeable nonconductive separator is placed in the bulk of the electrolyte to ensure that a short circuit does not occur. Ionic liquids are known to be good candidates for the electrolyte as they have some advantageous properties for the realization of capacitor devices:1 From a molecular perspective, their size and asymmetry in structure and charge distribution prevents the formation of regular patterns and shifts the melting point to lower values (often below room temperature) compared to other salts. Coulombic interaction prevents the evaporation of ILs and leads to a very low vapor pressure2one aspect why ILs are considered as electrolytes in supercapacitor applications. Compared to aqueous or organic electrolytes, the applied potential can be larger before molecular decomposition occurs, which leads to more energy being stored.3 The variety of synthesized ILs is such that a specific IL can be selected to achieve desired properties. The other major selection decision for supercapacitors is the type of electrode. Most favored are materials with large specific surface areas, which increase the energy density of devices. Examples for promising electrode materials4 are graphene sheets, carbon nanotubes or activated carbon, reaching surface areas of more than 1000 m2 g−1. The outstanding property of supercapacitors, however, is their excellent power density, this is primarily due to the nature of the charge storage mechanism. Charge is stored simply through ion separation and adsorption, this makes them superior over Faradaic devices as far as charging rates and cycle life are © 2015 American Chemical Society

concerned. Depending on the electrode material, the charging process of the capacitor can be dramatically different. In the case of planar electrodes with dense ionic electrolytes, charging occurs with particle exchange between strongly pronounced ion layers at the interface.5,6 This can happen faster than the mechanisms taking place in a porous network such as activated carbon, where the small size of pores causes the ions to become highly confined.7 A review about modeling electric double layer capacitors can be found in the work by Burt et al.8 Hughes and Walsh provided an overview about computational chemistry in graphene-based energy applications.9 In a recent experimental study by Elbourne at al.,10 the ion arrangement of the Stern layer was investigated via atomic force microscopy, showing potential dependent ion structures with molecular resolution. Asymmetric packing for opposite sign in applied potential, ion size and orientation effects are features that appear in both experiments and simulations. However, Elbourne at al. indicate that the potential dependent structure of the adsorbed ions is “even more complex than predicted”10 by simulations. In this study, we investigate the solid−liquid interface of a coarsegrained room temperature ionic liquid at planar graphite electrodes. A similar study has been performed by Merlet et al.11 followed by a discussion by Kornyshev and Qiao about double layer structures as 3D entities.12 This work complements the findings by highlighting the influence of an atomic carbon particle representation with hexagonal graphite structure versus a smooth one-dimensional wall potential. In addition, we show different ways of achieving a constant potential boundary Received: June 24, 2015 Revised: September 9, 2015 Published: September 10, 2015 22445

DOI: 10.1021/acs.jpcc.5b06046 J. Phys. Chem. C 2015, 119, 22445−22451

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The Journal of Physical Chemistry C condition and compare the results to constant charge simulations. The structure of the paper is as follows: First, we describe the details of short-range interaction and charge polarization of the electrode models under investigation. Then we compare the differential capacitance and discuss adsorption and charging mechanisms with the help of charge density profiles, cation orientation analysis, potential dependent first layer densities and radial distribution functions of the adsorbed counterions. In the conclusions, we summarize and classify the findings from a more general perspective.

(see Figure 2). It is computational efficiency due to the small number of degrees of freedom allows to investigate the

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METHODS Short-Range Electrode Interaction Potential. In this work, we compared an explicit graphite structure to a computationally more efficient unstructured planar LennardJones (LJ) representation of the surface. The former is referred to as atomic; the latter we call smooth throughout the article. The atomic electrode model consisted of three layers of carbon atoms fixed in a hexagonal lattice structure with a C−C bond length of 1.42 Å and plane distances of 3.35 Å. For the carbon short-range interactions, we used the common 12−6 LennardJones parameters σC = 3.37 Å and ϵC = 0.23 kJ/mol13 with Lorentz−Berthelot mixing rules for ion-carbon interactions. To obtain the smooth representation, we averaged the short-range interaction of the atomic electrode model in the x−y-plane (parallel to the electrodes) for several distances z from the electrode and fitted the data with a Lennard-Jones function with noninteger exponents. As expected, (see Figure 1) a 9−3-

Figure 2. Coarse grained representation of the ionic liquid BMIM PF6.

extensive parameter space of applied voltages/charges and electrode models. After 3 ns of equilibration, we performed production runs in the NVT ensemble of 12 ns to 18 ns with a Langevin thermostat at 400 K, a fixed number of 320 ion pairs and a time step of 2 fs. For the systems with smooth electrodes, we used a simulation box size of 30 Å × 30 × 126.7 Å to obtain a molar volume of 2.247m3/mol in the bulk.17 The atomic graphene systems had a slightly adjusted box size of 27.2 Å × 30 Å × 147.83 Å (including space for the 3 graphene layers on each side) to account for the periodicity of the graphene pattern in the xy-plane. Each graphene layer consisted of 308 carbon atoms. All simulations were performed with the simulation package ESPResSo 3.2.20 Constant Potential (CP) with the ELCIC Method for Parallel Plates. We compared different approaches to model the dielectric interfaces. At the conducting boundaries, the effect of dynamic surface charge induction introduces an additional local attraction of the charged species toward the electrodes.21 To model the correct dynamical behavior of the charges and the electrodes, we used the Electrostatic Layer Correction with Image Charges (ELCIC) method which accounts for 2D-electrostatics and charge induction by evaluating the image charges in every time step.22−24 This results in a potential drop ΔΦind between the parallel electrodes. In the constant potential ensemble, we obtain the target potential ΔΦbat by superposing a homogeneous electric field created by

Figure 1. One dimensional functional representations of three layers of graphene. The atomic structure was probed and averaged in the plane parallel to the electrodes. Attempted fit functions are common surface potentials (9−3-LJ, Steele) and a LJ function with variable exponents. In this study, the latter is used for simulating smooth electrodes and compared to the atomic structure.

ΔΦapplied = −ΔΦind + ΔΦbat

The electric field in z-direction Ez of the induced charges 1 Ez = 2 Pz ϵL Lz (3)

Lennard-Jones function14 or the Steele potential15,16,39 common for liquid-surface interactions could not be fitted to the data since the potential is a superposition of three graphene layers. We found that ⎛⎛ σ ⎞9.32 ⎛ σ ⎞4.66⎞ V (z) = 4ϵP ⎜⎜⎜ P ⎟ − ⎜ P ⎟ ⎟⎟ ⎝z⎠ ⎠ ⎝⎝ z ⎠

(2)

with system size Lx = Ly = L and plate separation Lz is given by the global dipole moment Pz =

(1)

with the parameters σP = 3.58 Å and ϵP = 24.7 kJ/mol resulted in a precise representation of the data. Simulation Details. The IL selected was the well-known BMIM PF6 coarse grained model of Roy and Maroncelli17−19

∑ qizi i

(4)

of the ion charges qi at position zi. As Pz is already calculated by the ELCIC algorithm, the constant potential correction requires no additional computational effort. 22446


The Journal of Physical Chemistry C

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Constant Potential with the ICC Method. Another approach to obtain the constant potential boundary condition is the Induced Charge Computation (ICC) method, which iteratively determines the induced charges given by a discretized, closed surface of point charges.25,26 In contrast to ELCIC, ICC was used with the 3D-periodic electrostatic solver p3m27,28 which by construction has no potential drop between the simulation boundaries. Here, only ΔΦbat has to be superimposed to obtain the target constant potential drop between the electrodes.21 For the sake of clarity, we do not display data for the ICC method in this discussion and only show the constant potential results obtained by the equivalent ELCIC method. Within the numerical errors, the ICC method and the ELCIC yield the same results. We used the carbon atoms of the honeycomb structure for the surface discretization of ICC which provided a sufficiently fine grid to obtain equal results. The ICC method differs from the charge induction method used by Siepmann and Sprink29 and later Reed et al.30 (CPM) by using point charges instead of Gaussians for the electrode charge discretization. Similar to CPM, ICC iteratively adapts the electrode charges in every time step to fulfill the constant potential boundary conditions. Constant Charge (CC) Simulations. In the constant charge simulations, the electrodes possess the surface charge ±(σind + σbat) . Because of the nonlinearity of the σ(ΔΦ) -dependence, the surface charge has to be obtained by constant potential simulations first for direct comparison between the two approaches. No charge induction methods have to be applied in this setup. The surface charge can either be modeled by an explicit charge lattice on the electrodes ormuch more efficient in the case of planar electrodesby a simple homogeneous electric field throughout the system.5,6 The maximum surface charge in our systems was σ = 10.4 μC cm−2, corresponding to an applied voltage of 6 V. This still resides in the multilayer regime with marked charge density oscillations.31,32 Differential Capacitance Analysis. For the constant charge simulations, the surface charge density σ is an input parameter and the differential capacitance Cd(ΔΦ) =

dσ(ΔΦ) dΔΦ

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RESULTS Effects of the Electrode Model on the Differential Capacitance. The comparison of Cd(U) for the different electrode models can be found in Figure 3. Qualitatively, the

Figure 3. Differential capacitance Cd measured for five different electrode models. Similar results are obtained for the constant charge and constant potential simulations. The data show major differences comparing smooth and atomic electrodes. Error crosses are omitted for clarity.

data is in line with other simulation studies of ionic liquids at graphite interfaces (e.g., united atom investigations34 or theoretical approaches35). The asymmetry of the positive and negative voltage regime of Cd yields the different adsorption behavior at the anode and cathode. In the voltage range under investigation, we obtain a similar shape of the Cd curves for the constant charge and constant potential methods within statistical precision. Also, explicitly charged carbon atoms lead to the same capacitance behavior as using the approximation of a homogeneously charged plane. A more pronounced difference can be seen comparing the differential capacitance of the atomic graphene system with the results of the smooth electrodes. There, the greatest deviation of about 1 μF cm−2 arises at 0 V. At higher voltages |U| > 1.5V, the curves start to align. This indicates that the influence of the graphene structure is most pronounced at low voltages and that its impact declines for increasing potential. Consequently, the arrangement of the adsorbed ions prevailing at higher voltages must dominate over the texturing influence coming from the graphene. Aside from the high voltage range, the differential capacitance for the two system classes shows a significant deviation. The ionic chargeand mass density profiles in Figures 4 and 5 show that the Lennard-Jones approximation of the graphite surface is not able to capture the exact distance of closest approach of the interface ions. We see a small decrease in distance from the electrode for the smooth wall at higher voltages. The smooth Lennard-Jones plane was fitted to match the average interaction potential of the atomic wall in the xy plane, it is origin is located at the zposition of the innermost graphene sheet toward the electrolyte. The change in length appears as a small outward shift of about 0.3 Å when comparing the first peak in the charge densities (Figure 4) or mass densities (Figure 5) for the two systems. The effect on the differential capacitance is profound, because it alters the distance between adsorbed first layer of charges and the onset of the metal boundary, which was kept in the same place for both models. If the ions can come closer to

(5)

is calculated via the potential drop ΔΦ from electrode to the bulk-like region. The electrostatic potential Φ(z) in the system is obtained from the charge density profiles. A detailed description of the analysis method can be found in ref 5. In the constant potential simulations, the average surface charge is an observable of the electrode algorithm. Although the total potential difference between the electrodes is constant, one of the individual potential drops ΔΦ from cathode/anode to the bulk-like region has to be obtained via the charge density profiles due to the asymmetry of the ionic liquid species. Another approach we did not follow in this work is to obtain the differential capacitance from the fluctuations of the surface charge.33 The error in the abscissa of the differential capacitance arises from the fluctuations of Φ(z) in the bulklike region. In the constant potential simulations, the fluctuating induced surface charge introduces another error in σ. Both are propagated to the differential capacitance values when calculating the numerical derivative via eq 5.5,6 22447


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Figure 4. Ionic charge density profiles for smooth and atomic electrodes at different voltages. A stronger influence of the electrode structure at low voltages is observed.

Figure 6. Averaged lateral cation orientation at the cathode shown 3 1 with the Second Legendre polynomial S = 2 cos2 α − 2 , where α is the angle between the z-axis and the normal vector of the plane defined by the three beads of the cation. Complete parallel alignment is characterized by S = 1, perpendicular orientation results in S = −0.5. Comparison between smooth and atomic electrodes at different voltages.

the effect is largest at 0 V, supporting the assumption that the structured wall has the most influence at low potentials. The voltage behavior of the mass density has different trends for anode and cathode, consistently for both electrode models: in the context of the mass density, the separation of charges in the interface region appears at the cathode as a reductionand at the anode as an increase of the first peak in the total mass density. This is due to the increased excluded volume of BMIM compared to PF6 and the similar molecular weights of the two components: At the cathode, the replacement of less dense anions with cations reduced the density whereas at the anode, the opposite is observed. The increased ion layer adsorption induced by the graphene structure at low to intermediate voltages can be seen in a more systematic way by analyzing the ion transfer from the first to the second layer. Figure 7 shows the voltage dependence of the number of ions in the first layer for the two electrode structure models.36 The ’first layer’ is not simply a constant range in z-direction, but defined as the range from the electrode to the first local minimum in the total density ρan(z) + ρcat(z) (see Figure 5). This kind of analysis averages out rotational features and ion arrangement in the xyplane, but captures the voltage dependence of the ion adsorption at the interface. Comparing the growth of the electrode attracted ion species in Figure 7 assuming linear behavior, we notice a higher slope for the anions (12.8 × 10−4 Å −2V−1) than for the cations (6.9 × 10−4 Å −2V−1). Regarding the electrode repelled ion species, the anion emigration (upper subplot, lower curves) has a strong nonlinear potential behavior and is lower compared to the cation emigration (lower subplot, lower curves). This shows that during charging in the first layer, both electrodes favor the migration of the spherical anions over the more bulky cations. Furthermore, Figure 7 shows that accumulation and transfer of ions at the interface generally

Figure 5. Ionic mass density profiles for smooth and atomic electrodes at different voltages. An increased ion adsorption on the structured electrode appears at all investigated voltages.

the electrodes, the global dipole moment and therefore the induced charge is higher for the same applied potential, which is equivalent to an increase in differential capacitance. Because of the voltage dependence of the shift it is difficult to separate the quantitative influence on the differential capacitance. The strong disagreement in the differential capacitance in Figure 3 on the cathode side (U < 0 V) is due to the different behavior of the cation reorientation for the two electrode structures which can be supported by comparing the cation orientation profiles shown in Figure 6. At low potentials, the data shows an enhanced parallel alignment of the cations for the graphene electrode. To some degree, a parallel lying cation will adapt to the graphene surface, allowing less change in orientation than for a smooth interface. Also, the first peak in the charge density profile next to the surface (Figure 4, cathode) is sharper than in the case of the smooth walls, indicating that all three charges of the BMIM model are more likely to be found in-plane. Consequently, this leads to the more pronounced second layer of anions at 0 V. The mass density in Figure 5 shows the enhanced ion adsorption on the structured surface: for both ion species and at all voltages we see a higher first layer peak in the atomic graphite system than in the case of smooth walls. Again, 22448


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analysis of the radial distribution functions we have to keep in mind that the charging process not only changes the ratio of anions and cations in the first layer, but also the occupied volume and the position of the first global minimum in the density profile, i.e., the thickness of the layer. For low voltages, the presence of both ionic species results in a mixed-ion structure whereas at higher voltages, a single ion species dominates and the layer thickness decreases. At this state, ion− ion charge repulsion contributes to the final in-plane structure by maximizing the distances between the same ionic species. Because of the asymmetry of the ions in shape and size, the depletion of (larger) cations at the anode during charging has a different contribution to the change in occupied volume and total mass density than the depletion of (smaller) anions at the cathode (see also Figure 5). In addition, the adjacent border which is either a smooth or structured will have an impact on the ion arrangement. Figure 8 shows the g(r) data of the first layer, respective bulk data is included. We start with the observation that the

Figure 7. Voltage dependence of the number of ions in the first ion layer for smooth and atomic walls. In the case of structured electrodes, the migration of co-ions is shifted to higher potentials. This effect is more pronounced for the spherical anion.

proceeds in a similar fashion for the atomic and smooth electrodes. An exception can be found for co-ion species (anions at the cathode and vice versa): in the case of the atomic graphene electrode, a higher potential is necessary to initiate a transition into the next layer. This effect is more distinct for anions at the cathode. This observation is in line with the expectation that the structured graphene surface will contribute to the binding of the ions, as the particles can become slightly trapped by the hexagonal rings. The increased ion adsorption for the graphene electrode also reflects in the differential capacitance. In a simplified picture, the applied voltage has to reach a certain level to overcome the additional adsorption of the graphene structure. Only then, the co-ions can get pulled away from their adsorption sites. The double-hump like increase in the differential capacitance of the graphene systems may be attributed to this effect. Radial Distribution Functions of Counterions. In the following section, we will investigate the in-plane layer structure by analyzing the radial distribution functions g(r) in the first ion layer at the electrodes. We compare the radial distribution functions of the same ionic species at their corresponding electrode (gAA(r) at the anode and gCC(r) at the cathode) for three different voltages or surface charges, depending on the electrode model. This analysis was performed to elucidate whether the electrode structure models (smooth vs atomic), or the electrode charge methods (constant charge vs constant potential), lead to a different planar ordering of the ionic liquid. By comparing the g(r) at different voltages, we also gain insight into the structural transitions of the adsorbed ions that happen during charging. We calculate the g(r) in the range in zdirection from the electrode to the first global minimum of the counterion density. In the analysis of gCC(r), we use the center of masses of the BMIM-cation and attribute adsorbed cations with a tilted orientation also to the first layer. Note that only the volume section of the spheres lying in the z-range has to be taken into account in the normalization of the data. For the

Figure 8. In-plane radial distribution functions of the adsorbed ions in the first layer for three different voltages, smooth and atomic walls, and constant charge (CC) and constant potential (CP) methods. Dashed lines show the bulk radial distribution functions.

electrode charge method does not significantly change the inplane radial distribution functions (Figure 8 a vs c, b vs d, e vs g, and f vs h). We use the average of the fluctuating surface charge occurring in the constant potential method for the constant charge simulations. Up to 4.8 V, the radial distribution function is not affected by the fluctuating charges. Next, we discuss the results concerning the electrode structure: The comparison of atomic and smooth walls (a vs e and b vs f within the constant charge results, c vs g and d vs h within the 22449


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The findings related to the surface structure of the graphene electrode showed that both the lateral behavior of the ion layers, as well as the in-plane structure of the ionic liquid are affected by the choice of the electrode model. It is clear that the ions will adapt to the local structure of the surface and that the oscillating interfacial layers known to be present in ILs will transfer this influence. The comparison of the in-plane radial distributions of the first ionic layer revealed the details of this effect: The direct comparison between the anions and cations showed that a more anisotropic ion model disturbs the adaption to the electrode structure. The in-plane order of the simple spherical anions is much more affected by the electrode structure than the anisotropic, three-bead cations. At higher voltages, we observed that the influence of the underlying ion model on the resulting configuration increased because the first layer was shifted toward the aforementioned closely packed, single ion species dominated domain. This could also be seen in the differential capacitance for both structured and unstructured electrodes. The orders of magnitude were equal and the asymptotic behaviors were also very close, but the capacitance in the low voltage range differed for the two electrode systems which can be explained by the fundamental differences in the electrode models: Because the ionic liquid adsorbed on the smooth wall shows less order at low potentials and therefore has a greater capability to rearrange and carry more charge toward the electrode, the differential capacitance is higher compared to the graphene system in the low voltage range. The two maxima of the DC curves of the graphene systems were explained by a “delayed” ion rearrangement at higher voltages due the fact that the applied potential has to overcome the increased adsorption of ions on the honeycomb pattern. The two different approaches to account for the electrode polarization showed no significant influence on the capacitance or the first layer structure. However, Merlet et al.37 and Wang et al.38 showed that with sufficiently high voltage, local charge induction can have a strong effect on the ionic structure at the interface. The studies showed that energetically more favorable configurations of the double layer with a patterned surface charge and altered ion structures are possible and only accessible using the constant potential approach with adaptive electrode charges. With this method, electrode “hotspots” are able to carry charges above the mean surface charge density. The energy gain of creating such a pair of charge hotspot and adsorbed ion may break local particle structures like a favorable ion orientation or weak ion coordination. When simulating an IL double-layer, this interplay is very complex as the in-plane ion structure itself heavily depends on the force-field and applied potential. For the BMIM PF6 ionic liquid model at flat electrodes, our results point in the direction that within the voltage range reasonable for capacitor applications, the constant charge approach yields very similar adsorption behavior compared to the constant potential method. Investigations concerning dynamical properties like charging time or applied alternating voltage, however, require the electrode surface charge to be adaptive.

constant potential results) shows that the overall behavior is similar (e.g., the position of extrema does not change significantly). A closer analysis of the graphene systems reveals additional features at distinct positions in the radial distribution functions. For the adsorbed anions at the anode, these features (a, r = 8−10 Å) persist throughout all three voltages whereas with cations at the cathode (b, r = 11 Å), only the low voltage data shows significant deviations. An explanation can be found in the model of the IL: the spherical anions can be patterned more easily by the hexagonal graphene structure than the anisotropic cations. The cations however are affected by rotational constraints enforcing parallel alignment to the planar surface, no matter if the electrode is structured or not. So we observe that only in the low voltage regime where we still find a mixed ion state, the graphene pattern leaves an imprint in the cation−cation radial distribution functions. The direct comparison between anode and cathode in Figure 8 shows that the voltage transition behavior is much more distinct at the anode. In Figure 8 a, the average distance between the anions drops as a result of the ion exchange process during charging. The aforementioned effect of cation reorientation helps understanding the cathode behavior in b: rather than pushing additional cations in the first layer, the system reacts with the reorientation of tilted cations toward parallel alignment, which does not strongly reduce the average distance of the cation’s center of mass. This results in potential independence of cation−cation g(r) and almost no decrease in the position of the main peak with increasing voltage.

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CONCLUSIONS We performed MD simulations on supercapacitor systems with different electrode models and a coarse-grained representation of the ionic liquid BMIM PF6. The differential capacitance behavior is determined by local ion reorganization effects at the interfacial region. Our results highlight that not only the force field of the ionic liquid, but also the electrode-ion interaction is an important part of the supercapacitor model. The comparison between structured and unstructured planar electrodes showed remarkable distinctions in differential capacitance, ion profiles, and properties of the adsorbed ions. We tried to understand the processes taking place in supercapacitors by analyzing the response to an applied voltage in the close proximity of the electrodes. With increasing applied voltage, the charge alternating layers at the interface exchange ions and adapt to the change in electrode surface charge. During this process, the first layer becomes more populated with a single ion species. The charge contributions affecting the in-plane structure shifts from a 2D ion lattice with both anions and cations to a counterion dominated configuration. Because of the nature of ionic liquids, this effect can be seen as ’universal’ for ILs at planar interfaces although its details will depend on the ion model, partial charge configuration or molecular polarization properties. Another main contribution to the in-plane structure is the “entropic” part including ion shape- and size-effects, which is set by the underlying ion model. Here, it is difficult to make universal, model independent conclusions with the exception of some local effects, which have to appear in both coarse-grained and united- or all-atom systems. One such local effect is the voltage induced reorganization of the BMIM cation, where the flat geometry and asymmetric charge distribution causes the molecule to readjust to the applied electrical field in order to obtain closer charge packing.

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AUTHOR INFORMATION

Corresponding Author

*(C.H.) Telephone: +49(0)71168563593. Fax: +49(0)71168563658. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 22450


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ACKNOWLEDGMENTS We acknowledge financial support through the DFG via the SimTech Cluster of Excellence and the SFB 716. Furthermore, we acknowledge the HLRS for the computing time granted on the Cray XE6 (Hermit), where most of the simulations were carried out.

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