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Comparing the void space and long-range structure of an ionic liquid with a neutral mixture of similar sized molecules
Journal Pre-proof Comparing the void space and long-range structure of an ionic liquid with a neutral mixture of similar sized molecules
Ekaterina A. Shelepova, Dietmar Paschek, Ralf Ludwig, Nikolai N. Medvedev PII:
S0167-7322(19)34899-8
DOI:
https://doi.org/10.1016/j.molliq.2019.112121
Reference:
MOLLIQ 112121
To appear in:
Journal of Molecular Liquids
Received date:
2 September 2019
Revised date:
2 October 2019
Accepted date:
11 November 2019
Please cite this article as: E.A. Shelepova, D. Paschek, R. Ludwig, et al., Comparing the void space and long-range structure of an ionic liquid with a neutral mixture of similar sized molecules, Journal of Molecular Liquids(2019), https://doi.org/10.1016/ j.molliq.2019.112121
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© 2019 Published by Elsevier.
Journal Pre-proof COMPARING THE VOID SPACE AND LONG-RANGE STRUCTURE OF AN IONIC LIQUID WITH A NEUTRAL MIXTURE OF SIMILAR SIZED MOLECULES
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Ekaterina A. Shelepova1,2, Dietmar Paschek3, Ralf Ludwig3,4,5, Nikolai N. Medvedev1,2 1 Novosibirsk State University, Pirogova Street 2, Novosibirsk 630090, Russia; 2 Voevodsky Institute of Chemical Kinetics and Combustion, SB RAS, Novosibirsk 63090, Russia; 3 Institut für Chemie, Abteilung Physikalische und Theoretische Chemie, Universität Rostock, Dr.Lorenz-Weg 2, 18059 Rostock, Germany; 4 Department LL&M, Universität Rostock, Albert-Einstein-Straße 25, 18059 Rostock, Germany; 5 Leibniz-Institut für Katalyse an der Universität Rostock e.V., Albert-Einstein-Straße 29a, 18059 Rostock, Germany, E-mail: [email protected], [email protected]
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Abstract We study the void space and the structure of a coarse-grained molecular model of the ionic liquid 1butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] and the corresponding model of an uncharged liquid consisting of the same molecules but with zero charges as a function of temperature between 300 K and 600 K using molecular dynamics simulations. Long-distance radial correlation functions for representative sites and for interstitial spheres were calculated for the individual subsystems representing anions and cations. The geometry of the local structural arrangements was investigated using the technique of Delaunay simplices. As expected, the ionic liquid demonstrates a noticeable regularity with respect to the alternation of anions and cations, which is present in all studied distance-ranges. The structure of uncharged subsystems, however, demonstrates features typical for simple liquids, which are governed by geometrical laws of the dense disordered packing of impenetrable particles. A prominent structural feature of the studied ionic liquid is that the “mutually closest fours” of anions form configurations of pronounced tetrahedral shape with a cation being located in between them. Correspondingly, the cations (here represented by the position of their central site) also form tetrahedral configurations around an anion. In the uncharged system, such mutual arrangements are found to be absent. The analysis of the models for different temperatures demonstrates that the principal differences between the structures of the charged and uncharged subsystems remain present up to 600 K. Quite interestingly, the distribution of void spaces, as represented by the radii of interstitial spheres, is found to be consistently, albeit slightly broader for the charged model liquid.
1. Introduction Ionic liquids (ILs) are salts in their liquid state. ILs provide wide electrochemical windows, low vapour pressures and high thermal stabilities.[1-3] Meanwhile, they have changed from a lab curiosity to materials of tremendous academic and industrial interest. This category of new and remarkable liquid substances with unique and fascinating properties offer a phenomenal opportunity for new science and technology.[4-11] An important feature of these Coulomb fluids is that one can tune their physical properties by varying the size, shape and symmetry of cations and anions. Such a ‘design’ provides novel potential applications for this class of new liquid material. The molecular interactions are described by a delicate balance between Coulomb forces, hydrogen bonds, and
Journal Pre-proof dispersion forces and are thus crucial for understanding the properties of ILs.[7-9] All the important properties such as structure, diffusion, and viscosity depend on these interactions between cations and anions in ILs. The development of a fundamental understanding of these interactions at the molecular level is of current interest. [10]
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However, at the beginning of IL research, these Coulomb fluids were thought to fall within the traditional realm of molecular liquids as homogeneous, coherent, and essentially irregular systems. The IL bulk structure had been regarded as being similar to a highly concentrated salt solution or a molten salt. However, recent computer simulation models used to describe ILs have increasingly suggested that they should be considered as “structured solvents”, ranging from supramolecular to mesoscopic length scales.[12-24] The ability to design targeted liquid properties by forming binary and ternary mixtures has become also a focus of interest.[25-28] Beside mixtures of pure ILs, those of ILs with molecular liquids (MLs) are also studied.[29] The basic understanding of the differences in the structure of ILs and MLs is an essential requisite for preparing mixtures with desired properties. In this study, we suggest to focus on differences in the structure of an exemplary ionic liquid compared to a hypothetical neutral (uncharged) counterpart, which is composed of the same constituents with respect of molecular structure, size and van der Waals interactions. The detailed investigation of the structural features of ILs can be helpful to explain their structure and properties. A popular tool for analyzing the structure of ionic liquids is the computation of radial distribution functions (or structure factors). They are employed to judge the correlations in the mutual arrangement of ions, individual atoms, or groups of atoms (For example, see the review of ILs simulations [30], as well as the classic works [31, 32, 33]). An interesting feature of the structure of the ionic liquids is that anions and cations exhibit long-ranged spatial correlations. It was shown that oscillations of the radial distribution functions of ILs are visible at much larger distances than in neutral fluids [1]. At the same time, a spatial alternation of anions and cations is noted. For example, in Ref. [33] it was shown that anion-cation radial distribution function (calculated between the center of mass of the anion and the center of mass of the imidazolium ring) is in phase opposition with the cation-cation and anion-anion functions. This was demonstrated for two families of imidazolium-based ionic liquids. In Ref. [34] a coarse-grained model of the ionic liquid [C4mim] [PF-] was studied, similarly demonstrating the alternating structure of anions and cations. It was suggested that this alternation reflects the existence of a distorted interdigitated lattice, which exists in ILs due to the strong electrostatic interactions of ions. Beyond any doubt, well-defined electrical charges cause these specific spatial correlations [35, 36]. However, there is still no complete understanding of the general structural patterns in ionic liquids resulted from these interactions. A crystalline motif formed by parallel planes was previously discussed for the melts of zinc halides. Note, that structure factors of these halides exhibit a pre-peak (FSDP), suggesting the existence of an intermediate-range order in the fluid, which, in particular, could be explained by the presence of such parallel planes [37]. In a subsequent paper, these authors [38] detected such planes in a molecular dynamics simulation of molten ZnCl2. However, these planes were formed not by ions, but by the largest inter-ion voids (interstitial spheres of the largest radii). In addition, tetrahedral configurations of anions are found in such melts, namely, halogen atoms are located around the zinc at the vertices of the tetrahedron. This result was previously obtained experimentally [39, 40] and was recently studied in detail by molecular dynamics simulations [41,
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2.1. Models
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2. Models and Methods
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42]. Obviously, electrostatic interactions lead to some common structural features. It means they should occur in both molten simple salts and ionic liquids. Shirota and Castner [43] experimentally compared the dynamics of the pyrrolidinium ionic liquid and its neutral counterpart. It was shown that, compared with the neutral analogue, the ionic liquid has a 1.2-fold increased density and 30-fold larger shear viscosity, which is a consequence of the stronger interaction between the ions, compared with molecules in a neutral liquid. However, one can speak only hypothetically about an experimental “non-ionic” analogue of an ionic liquid. Therefore, the preparation and study of such a system in a computer simulation is of great interest. The models of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate, [C4mim][PF6] and its neutral analogue, consisting of the same molecules, but without charges, were obtained and studied using coarse-grained MD-simulations in Ref. [34]. A significant decrease in IL mobility compared to its neutral counterpart was shown. A fundamental structural difference between these systems was also obtained. However, only radial distribution functions were used for structural characterisation. In this paper, we present results from coarse-grained molecular dynamics simulations of [C4mim][PF6] and its neutral analogue using the model parameters outlined in Ref. [44]. We pay special attention to the study of local three-dimensional structural features based on a VoronoiDelaunay tessellation. This technique is well known for long time and widely used for structure investigation of computer models of different atomic and molecular systems, in particular ionic liquids [45-49]. Moreover, we compare our models for a wide temperature range (300 K to 600 K) to determine the influence of the temperature on the computed structural features.
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We have performed molecular dynamics simulations of 1-butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] using the coarse-grained rigid ILM2 Lennard-Jones/pointcharge model introduced in 2010 by Roy and Maroncelli [44] using three sites to represent the imidazolium cation and a single site for the hexafluorophosphate anion as indicated in Fig.1.
Fig. 1. Illustration of the coarse-grained ILM2 model: the anion (red) is represented by a single site, whereas and the cation (blue) is modelled using three sites. To better compare ionic and molecular liquids on a fundamental level we have prepared two principal variants of the ILM2 model. The first one is an ionic liquid where the anion and cation each carry the original ILM2 fractional charges of ± 0.78|e|, respectively [44]. In the second model,
Journal Pre-proof however, we have removed all the charges on the two constituents, turning it thus into a neutral molecular liquid carrying zero charges. All simulations were consisting of 1000 “ion pairs" applying periodic boundary conditions. The MD simulations of the original ILM2 ionic liquid were carried out under NPT conditions at a pressure of 1 bar, covering a broad temperature range between 300 K and 600 K. The obtained densities are listed in Table I and match the experimental data very well, as also shown in Fig. 2. The simulations of the zero-charge ILM2 model were carried out under constant-volume conditions at exactly the same densities obtained for the charged ILM2 model at each temperature. Production runs of 40 ns length were studied at each temperature following equilibration runs of 10 ns.
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ρ/ kg m-3 1359.1 1340.5 1321.2 1303.7 1285.1 1248.3 1211.4 1138.5
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T/K 300 325 350 375 400 450 500 600
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TABLE I: Temperature dependence of the mass densities obtained from MD simulations of the ILM2 model a p=1 bar.
Fig. 2. Temperature dependence of the mass densities obtained from MD simulations of the ILM2 model a p = 1 bar compared with experimental data of Machida et al. [50]. All simulations were performed with the Gromacs 4.5.3 program. [51] The preparation of topology files, as well as the data analysis were performed with the most recent version of the MOSCITO
Journal Pre-proof suite of programs.[52] Gromacs topology- and restart-files are available on request. Production runs of 20 ns length were employed for every temperature, starting from previously equilibrated configurations. The Nose-Hoover thermostat [53, 54] and the Parrinello-Rahman barostat [55,56] with coupling times τT =0:5 ps and τp=2:0 ps were used to control constant temperature and pressure (1 bar) conditions. The electrostatic interactions were treated by particle mesh Ewald summation.[57] A real space cutoff 1.6 nm was employed, and a mesh spacing of approximately 0.12nm (4th order interpolation) has been used to determine the reciprocal lattice contribution. The Ewald convergence parameter was set to a relative accuracy of the Ewald sum of 10 -6. LennardJones cutoff corrections for energy and pressure were considered. A 5 fs timestep was used, and every 50 steps a configuration was saved. Distance constraints were solved by the LINCS procedure.[58]
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2.2. Interstitial spheres. Interstitial spheres, i.e. the empty spheres inscribed between four atoms (touching their surfaces) characterise empty interatomic voids in a molecular system. Such spheres are determined from a Voronoi–Delaunay tessellation of the system. This tessellation is widely used in physics, chemistry and biology to analyse the structure of computer models of various molecular and atomic systems, for more details see Refs. [59,60,61]. An interstitial sphere is related with a Delaunay simplex (tetrahedron). Its vertices are defined by four mutually close atoms, between which an empty sphere is inscribed. To calculate the Voronoi-Delaunay tessellation we use our own homedeveloped computer programs [62,63,64]. Note we use the version of the tessellation, which is constructed relative to the surfaces of the atoms, so-called additively weighted [54] or Voronoi Stessellation [60, 62, 65]. The use of S-tessellation is important if atoms of the system have different size. In this case, the interstitial spheres are defined correctly, in contrast to the classical approach, which deals only with the centers of atoms [66, 67], or radical (power) tessellations, where the distance is measured along the tangent to the atomic surface [68]. Here, the atom diameters are usually defined by their Lennard-Jones parameters σ taken from the molecular-dynamics simulations. We calculated the Voronoi-Delaunay tessellation as for the whole systems as well as for the both subsystems (anionic and cationic). Fig. 3 illustrates the interstitial spheres for those cases in two dimensions.
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Fig. 3. Upper panel. Two-dimensional illustration of an ionic system and its subsystems. a) Entire system, anions are shown as red disks, and cations are blue; b) selected subsystem of anions (red); c) selected subsystem for C1-sites of cations (blue). Empty circles represent the interstitial spheres. Dashed line triangle shows a Delaunay simplex for each case. Lower panel. The same for the uncharged system. a) Entire system, the subsystem A (corresponded anions) are shown as dark disks, and the subsystem C (corresponded to cations) as grey disks; b) selected subsystem A (dark); c) selected subsystem C1 (dark).
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Note when we calculate the Voronoi-Delaunay tessellation for the entire system all coarse-grained sites are taken into account, as shown in Fig. 3a. For analyzing a subsystem, we only take into account the sites corresponding to a given subsystem, while ignoring all others (see Fig. 3b and Fig. 3c). Moreover, for analysing the cation subsystems we use only the central C1-site (see Fig.3c). Note that the interstitial spheres in disordered systems can overlap significantly. This occurs especially for relatively large cavities, which are surrounded by more than four coarse-grained sites (see, in particular, Fig.3). Thus, some spheres cover the same part of empty space. As a consequence, the volume of a cavity does not relate directly to the sum of volumes of its interstitial spheres. On the other hand, the volume of a cavity is equal to the sum of the empty volumes of the Delaunay simplices representing this cavity [60, 69]. Note that the volumes of the Delaunay simplices can differ significantly for these overlapping interstitial spheres, even their radii are close. Therefore, working with the interstitial spheres, we can consider them as itself, as well as taking into account the fact that they represent a different fraction of empty volumes. When we discuss empty volume in a system, we should keep in mind the last situation. To capture this, we may weight the interstitial spheres with the empty volume of their corresponding Delaunay simplices. 3. Results and discussions Following the nomenclature used in the caption of Fig. 3, we denote the anions in the ionic liquid and the corresponding particles in the uncharged mixture as subsystems A (charged or uncharged). The subsystems C are comprised of the cations and their neutral counterparts. When we deal with the central C1-site of the cations, we will denote them as subsystems C1 (charged or uncharged). The structural differences between the ionic and the uncharged systems become apparent by considering the cationic and anionic subsystems separately. Shown in Fig. 4 are snapshots for the charged and uncharged subsystems A from our models taken at T = 300 K. The structure in Fig. 4a depicting the ionic liquid appears to be much more uniform than the uncharged system shown in Fig. 4b, which includes “condensations” of sites and larger voids.
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Fig. 4. Snapshots taken from molecular dynamics simulations at 300 K. Shown are only the particles representing the subsystems A for the ionic liquid (a) and for the uncharged model (b).
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We would like to emphasize that the structures representing the subsystems C1 show similar behaviour (not shown): again, the structure of charged subsystem appears to be much more homogeneous, while the neutral model is more distorted.
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3.1 Radial distribution functions. To characterize the intermolecular space we calculate distributions of the interstitial sphere radii. For a better representation of the volume of voids, we “weight” the interstitial spheres with the empty volume of the corresponding Delaunay simplices, as discussed in section 2.2. Thus, we present "volume-weighted” distributions of radii of the interstitial sphere. Fig. 5 compares the partial pair correlation functions gA-C1(r), gA-A(r) and gC1-C1(r) for the ionic and uncharged systems.
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Fig. 5. Partial pair correlation functions gA-C1(r), (a); gA-A(r), (b); and gC1-C1(r), (c); (red) and uncharged (blue) subsystems A and C1.
for ionic
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The oppositely charged ions tend to get close to each other, resulting in a higher first peak and a shift to smaller r in the ionic liquid compared with the uncharged system, Fig. 5a. Like-charged ions repel each other. Thus, the first peaks for these pair distribution functions are not well pronounced and lie further apart than what their van der Waals contact distances would suggest (see also the first peak for the uncharged systems, Fig. 5b and 5c). However, a principal difference between the charged and uncharged subsystems is observed at large distances. To illustrate it we calculate the function d(r) = r2(g(r)-1). The corresponding functions d A-C1(r), d A-A(r) and d C1-C1(r) are shown in Fig. 6.
Journal Pre-proof Fig. 6. Functions d(r) = r2(g(r)-1) for selected sites in the ionic liquids (a) and the uncharged system (b). For А-А (red), С1-С1 (blue), and А-С1 (black)
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Note that the decay of d(r) oscillations is much slower for the charged system. It indicates a longdistance correlation for the anions and cations in ILs. We also note that the maxima of the oscillations for the subsystems A and C1 completely coincide, see red and blue curves in Fig. 6a, and the oscillations of dA-C1(r) are shifted by a half the period (black curve). This observation is typical for ionic liquids [30] and demonstrates the alternating structure of positive and negative ions. Keep in mind that we use the central C1-site of the cations to calculate radial correlations, whereas other authors used the center of mass of molecules [31, 32, 34] or parts thereof. For example, in Ref. [33] the center of mass of the imidazolium ring was used. However, as we can see, there are no principal differences if one chooses the center of mass or a certain specific center located on the cation. The behavior of the radial distribution functions for the uncharged system is completely different from for the ILs, as it was first discussed in Ref. [34]. The curves in Fig. 6b resemble the ones obtained for simple liquids, both with respect to the area of the first peaks, as well as with the long-distance oscillations: there is always a well-defined first peak followed by rapidly fading oscillations. Having computed interstitial spheres for our models (see the illustration shown in Fig. 2), we can use their centers to calculate their corresponding radial distribution functions, as it was suggested in Refs. [70,71,72]. The functions d(r) for the centers of these spheres for charged and uncharged subsystems A and C1 are shown in Fig.7. Here, we again observe the distinctive weakly damped oscillations computed for charged subsystems and fundamentally different behavior for the uncharged systems.
Fig. 7. Functions h(r) d(r) = r2(g(r)-1) for interstitial spheres for subsystems A and C in the ionic liquid (a) and in the uncharged system (b). Thus, the greater regularity of the charged system also manifests itself in the long-distance correlations of intermolecular voids. In the remainder of this paper, we will focus on a local analysis
Journal Pre-proof of voids in our systems, particularly focusing on the radii of the interstitial spheres and the corresponding Delaunay simplices.
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3.2 Radii of the interstitial spheres To characterize intermolecular space in our systems we calculate the distributions of the interstitial sphere radii. For a better representation of the volume of voids, we “weight” the interstitial spheres with the empty volume of their corresponding Delaunay simplices, as discussed in section 2.2. Thus, we present "weighted” distributions of radii of the interstitial spheres. Fig. 8 shows the weighted distributions of the interstitial spheres radii for the subsystems A at 300 K. Note that the cations are explicitly not taken into account in this case, as illustrated in Fig. 3b. We observe a relatively narrow monomodal peak for anions, and a wide distribution with several maxima for the uncharged subsystem A.
Fig. 8. Weighted distributions of the interstitial spheres radii for the subsystem A: charged (red) and the uncharged (blue) at 300K. Typical configurations are also shown for the selected radii of the interstitial spheres (pointed out by numbers). A-sites are indicated by a red colour, Sites C1, C2 and C3 are depicted in blue.
By analysing snapshots of our molecular dynamics models we found typical configurations of the neighbouring A-sites having the chosen radii of the interstitial spheres (see Fig. 8). The narrow maximum in the distribution for the ionic liquid usually corresponds to a situation when four anions surround a cation rather homogeneously as indicated by configuration (1). The uncharged subsystem also contains similar configurations, however, their population is much lower (see the small maximum at 0.3 nm and one from the corresponding configurations (2.3)). The shoulder at small radii (≈ 0.1 nm) corresponds to the dense configurations of four neutral A-sites (2.2). We also found configurations like (2.1), which are the densest tetrahedral packing of four equal spheres. For the charged subsystem such situations are impossible because the charged A-sites repel each other.
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The uncharged system contains also a majority configuration with large radii of interstitial spheres. In this case, the A–sites are interdigitated with several “cations”, see configuration (2.4). We conducted the same analysis for the subsystems C1. In this case, the Voronoi-Delaunay tessellation was calculated ignoring all other coarse-grained sites. The interstitial sphere radii distribution for the charged subsystem C1 is also narrower than for the neutral one, as shown in Fig. 9. We found that four charged C1-sites always tend to be located around an anion. However, now the distribution is slightly splitting: the sharp peak at r = 0,28 nm corresponds to the closest arrangement of C1-sites around the anion, configuration (1.1), and the following shoulder contains all other situations (1.2).
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Fig. 9. Weighted distributions of the interstitial spheres radii for the subsystem C1: charged (red) and the neutral one (blue) at 300K. Typical configurations at the corresponding radii of the interstitial spheres (pointed out by numbers) are also shown: C1-sites are depicted dark blue, Asites are shown in red, while C2- and C3-sites are shown in light blue.
For the uncharged C1, we observe a broad uniform distribution. This means that C1-sites are located in space rather randomly (without any special features in the mutual arrangement). Note, unlike the uncharged A-particles, four neutral C1-sites never form the mutual contacts, i.e. do not form the densest configurations with the small radii of the interstitial spheres (around of 0.1 nm), as a subsystem of spherical A-particles. It is obvious because of the complex shape of the cation. 3.3 Shape of the Delaunay simplices. Recall, that an interstitial sphere is directly related to a corresponding Delaunay simplex. The analysis of the simplex shapes is used to study the structure of simple liquids, disordered packings of spheres, and molecular systems, see for example [60,74]. One of the most suitable characteristics
Journal Pre-proof of the simplex shape is the so-called tetrahedricity, T [74]. It is defined as the variation of the edge lengths of the Delaunay simplex:
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𝑇 = ∑𝑖≠𝑗 (𝑙𝑖 − 𝑙𝑗 )2 /15𝑙02, where 𝑙𝑖 and 𝑙𝑗 are the length of the i-th and j-th edges, and 𝑙0 is the mean edge length of a given simplex. A small T value means that the shape of this simplex is close to the perfect tetrahedron, for which all the edge length are the same. The Delaunay simplices are determined from the VoronoiDelaunay tessellation of the studied system. We compare the tetrahedricity distributions of the Delaunay simplices for the charged and uncharged subsystems A, Fig. 10. The narrow monomodal peak at small T values demonstrates that the anions in the ionic liquid are organized in configurations with relatively well-defined tetrahedral shapes.
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Fig. 10. Tetrahedricity distribution of the Delaunay simplices for the subsystems A at 300K: charged (red), uncharged (blue).
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Note that the calculated distributions of tetrahedricity complement our conclusions of the analysis of the interstitial sphere radii. Now we can state that the local configurations of the anions are similar in size, and their shape is close to the regular tetrahedron. In Fig. 11 a configuration is shown, extracted from a molecular dynamics trajectory, demonstrating a typical tetrahedral local arrangement of anions in the ionic liquid with a cation being placed among them.
Journal Pre-proof Fig.11. Typical anion (red) arrangement in the ionic liquid. The lines show the Delaunay simplex of the subsystem A. Cations are shown by green. The configuration is extracted from the molecular dynamics model at 300K. The situation is different for the uncharged subsystem A, as shown in Fig. 10. In this case, the tetrahedricity-distribution is much broader and shifted towards larger T values, indicating that the shape of the simplices is more distorted. On the other hand, a shoulder (plateau) appears at small T values. This suggests that simplices with a very well-defined tetrahedral shape do also exist, although in significantly reduced numbers. They correspond to the densest local configurations with the small interstitial sphere radii, as discussed above and shown in Fig.8. The shape of those corresponding Delaunay simplices is close to the perfect tetrahedron [74].
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To investigate the the relation between tetrahedricity and interstitial sphere radius for the uncharged subsystem A in more detail, we calculated the tetrahedricity distributions for Delaunay simplices corresponding to the various maxima given in the distributions shown in Fig 8. The distributions shown in Fig. 12 demonstrate that the smallest radii of the interstitial spheres (around 0.1 nm) are related to a narrow maximum of the near-perfect tetrahedricity close to zero. For the other selected radii (around 0.3 nm and around 0.45 nm), we observe very different shapes of the simplices, which are far from the perfect shape. However, the tetrahedricity of Delaunay simplices and the radius of the interstitial sphere reflect different properties of a local configuration.
Fig. 12. Tetrahedricity distributions of Delaunay simplices corresponding the interstitial spheres radii from the intervals: 0.09-0.11 nm (blue line), 0.29-0.31 nm (red), 0.44 - 0.46 nm (green) on the distribution for the uncharged subsystem A in Fig.8. For the subsystems С1, the tetrahedricity distributions are shown in Fig. 13. For the charged system, the Delaunay simplices are found to be also more tetrahedral. This is not unexpected since the cations in the ionic liquid are interlocked in a common structural motif with the anions due to their electrostatic interactions.
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Fig. 13. Tetrahedricity distributions of the Delaunay simplices for the subsystem C1: charged (red), uncharged (blue) at 300K.
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3.4 Temperature influence
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However, in the case of the cations, the tetrahedricity-distribution is shifted towards large T values and is broader than for the subsystems A shown in Fig.10. This shift and additional distortion for the C1 subsystem is likely caused by the more complex shape of the cation.
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The distributions of the interstitial spheres radii for the entire ionic and uncharged system at 300 K and 600 K are compared in Fig. 14. Here, we do not divide our systems into the subsystems, and interstitial spheres are calculated as it was shown in Fig. 3a.
Fig. 14. Weighted distributions of the interstitial spheres radii for the entire models of the ionic liquid (red curves) and for the uncharged system (blue) at 300K and 600K. Note that the difference between the charged and neutral systems is small at each temperature. The increasing leads to a decreasing density, as shown in Fig.2. Thus the intermolecular voids increase, and the distributions move to larger radii and became broader. However, the distributions remain very similar for a given temperature, for both, the charged and uncharged systems. We performed the calculation also for the intermediate temperatures and obtained the smooth transition from the
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initial to the final distributions shown in Fig. 14. Position of the maximum of the distributions is plotted as a function of temperature in Fig.15.
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Fig. 15. Positions of the maxima of the interstitial spheres radii distribution as a function of temperature for the entire ionic (red) and uncharged (blue) systems.
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Practically linear and parallel lines give a reason to say that both systems are affected by the temperature in the same way. In other words, we do not see a different influence of the temperature on the ionic and uncharged systems. For the subsystems, we found a similar shift and broadening with temperature. At the same time, structural consistency between charged and uncharged systems is also preserved. The distributions of the interstitial spheres radii for subsystem A at 600K are shown in Fig. 16.
Fig. 16. Distributions of the interstitial spheres radii for the subsystem A: charged (red) and the uncharged (blue) at 600K. Data for 300K (from Fig. 8) is shown by dashed lines.
The charged subsystem A again demonstrates a narrow monomodal peak as it was for the low temperature. It is only slightly broadened and shifted toward larger radii. The uncharged subsystem A distribution at high temperature is also much wider than for the charged one. It is now more smoothed because the heating distortion also increases. Figure 17 shows the tetrahedricity distribution of the Delaunay simplices in the charged and uncharged subsystems A at 600K.
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Fig.17. Tetrahedricity distribution of the Delaunay simplices for the subsystems A at 600K: charged (red), uncharged (blue). Data for 300K (from Fig. 10) is shown by dashed lines.
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The difference between the charged and neutral systems is very similar to that at 300K, Fig. 10. Thus, the temperature increase retains existing structural difference of the ionic liquid and the uncharged system.
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4 Conclusion
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We have performed molecular dynamics simulation of a coarse-grained model of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate and a system consisting of the same molecules but without charges at various temperatures. The ionic liquid is here represented by the ILM2 force field proposed by Roy and Maroncelli in 2010 [44]. Partial radial distribution functions for the interaction sites and for computed interstitial spheres representing the void space were calculated for the long-distance interval up to 3 nm. In addition, the intermolecular voids and shape of the local configurations of the anions and cations are examined using the Voronoi-Delaunay method. The structures of the anion and cation subsystems of our models substantially differ from the structures of their uncharged analogues. As expected, the partial radial distribution functions for anion and cation subsystems demonstrate the spatial alternation of positively and negatively charged particles, typically observed for salts. Such an alternation pattern is absent for the uncharged subsystems, where we observe a common structural motif for both subsystems governed by geometrical laws of the dense disordered packing of impenetrable particles. The use of the Voronoi-Delaunay tessellation for each of the subsystems allows us to determine the geometrical features of the local configurations for anions and cations separately. We have obtained that the radii of spheres inscribed between anions have a narrow distribution, i.e. the size of the “holes” between the mutually nearest anions are approximately the same. For the uncharged counterpart, however, there is a wide distribution of the radii of such spheres. A similar scenario is observed for the cation-subsystem. However, in this case, the distributions of the interstitial sphere radii are more diffuse due to the more complex shape of the cation. By analyzing the shape of the computed Delaunay simplices we found that the fours of the nearest anions in the
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ionic liquid represent configurations with a shape close to a regular tetrahedron, whilst a counterion is located in between these anions. The same can also be said about cations. This is in consistent with the observed spatial alternation of ions in the ionic liquid. We would like to emphasize that such tetrahedral configurations of ions have been previously observed for simple salt melts, for example, in molten zinc halides. For the case of the uncharged subsystems these pronouncedly tetrahedral features are found to be absent. Finally, we also investigated the temperature effect on the observed structural differences between the charged and uncharged subsystems. In both cases with increasing temperature, we only observe a general increase in the structural disorder. Principal structural features of the subsystems, however, remain the same. In particular, the local configurations of anions (cations) in the ionic liquid have keep their close to perfectly tetrahedral shape for the all investigated temperatures between 300 K and 600 K. For the uncharged subsystems, the similar configurations are found to be also much more distorted.
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Acknowledgements NNM gratefully acknowledges financial support from grants RFFI (No. 18-03-00045).
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Ekaterina A. Shelepova, Dietmar Paschek, Ralf Ludwig, Nikolai N. Medvedev PII:
S0167-7322(19)34899-8
DOI:
https://doi.org/10.1016/j.molliq.2019.112121
Reference:
MOLLIQ 112121
To appear in:
Journal of Molecular Liquids
Received date:
2 September 2019
Revised date:
2 October 2019
Accepted date:
11 November 2019
Please cite this article as: E.A. Shelepova, D. Paschek, R. Ludwig, et al., Comparing the void space and long-range structure of an ionic liquid with a neutral mixture of similar sized molecules, Journal of Molecular Liquids(2019), https://doi.org/10.1016/ j.molliq.2019.112121
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Journal Pre-proof COMPARING THE VOID SPACE AND LONG-RANGE STRUCTURE OF AN IONIC LIQUID WITH A NEUTRAL MIXTURE OF SIMILAR SIZED MOLECULES
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Ekaterina A. Shelepova1,2, Dietmar Paschek3, Ralf Ludwig3,4,5, Nikolai N. Medvedev1,2 1 Novosibirsk State University, Pirogova Street 2, Novosibirsk 630090, Russia; 2 Voevodsky Institute of Chemical Kinetics and Combustion, SB RAS, Novosibirsk 63090, Russia; 3 Institut für Chemie, Abteilung Physikalische und Theoretische Chemie, Universität Rostock, Dr.Lorenz-Weg 2, 18059 Rostock, Germany; 4 Department LL&M, Universität Rostock, Albert-Einstein-Straße 25, 18059 Rostock, Germany; 5 Leibniz-Institut für Katalyse an der Universität Rostock e.V., Albert-Einstein-Straße 29a, 18059 Rostock, Germany, E-mail: [email protected], [email protected]
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Abstract We study the void space and the structure of a coarse-grained molecular model of the ionic liquid 1butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] and the corresponding model of an uncharged liquid consisting of the same molecules but with zero charges as a function of temperature between 300 K and 600 K using molecular dynamics simulations. Long-distance radial correlation functions for representative sites and for interstitial spheres were calculated for the individual subsystems representing anions and cations. The geometry of the local structural arrangements was investigated using the technique of Delaunay simplices. As expected, the ionic liquid demonstrates a noticeable regularity with respect to the alternation of anions and cations, which is present in all studied distance-ranges. The structure of uncharged subsystems, however, demonstrates features typical for simple liquids, which are governed by geometrical laws of the dense disordered packing of impenetrable particles. A prominent structural feature of the studied ionic liquid is that the “mutually closest fours” of anions form configurations of pronounced tetrahedral shape with a cation being located in between them. Correspondingly, the cations (here represented by the position of their central site) also form tetrahedral configurations around an anion. In the uncharged system, such mutual arrangements are found to be absent. The analysis of the models for different temperatures demonstrates that the principal differences between the structures of the charged and uncharged subsystems remain present up to 600 K. Quite interestingly, the distribution of void spaces, as represented by the radii of interstitial spheres, is found to be consistently, albeit slightly broader for the charged model liquid.
1. Introduction Ionic liquids (ILs) are salts in their liquid state. ILs provide wide electrochemical windows, low vapour pressures and high thermal stabilities.[1-3] Meanwhile, they have changed from a lab curiosity to materials of tremendous academic and industrial interest. This category of new and remarkable liquid substances with unique and fascinating properties offer a phenomenal opportunity for new science and technology.[4-11] An important feature of these Coulomb fluids is that one can tune their physical properties by varying the size, shape and symmetry of cations and anions. Such a ‘design’ provides novel potential applications for this class of new liquid material. The molecular interactions are described by a delicate balance between Coulomb forces, hydrogen bonds, and
Journal Pre-proof dispersion forces and are thus crucial for understanding the properties of ILs.[7-9] All the important properties such as structure, diffusion, and viscosity depend on these interactions between cations and anions in ILs. The development of a fundamental understanding of these interactions at the molecular level is of current interest. [10]
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However, at the beginning of IL research, these Coulomb fluids were thought to fall within the traditional realm of molecular liquids as homogeneous, coherent, and essentially irregular systems. The IL bulk structure had been regarded as being similar to a highly concentrated salt solution or a molten salt. However, recent computer simulation models used to describe ILs have increasingly suggested that they should be considered as “structured solvents”, ranging from supramolecular to mesoscopic length scales.[12-24] The ability to design targeted liquid properties by forming binary and ternary mixtures has become also a focus of interest.[25-28] Beside mixtures of pure ILs, those of ILs with molecular liquids (MLs) are also studied.[29] The basic understanding of the differences in the structure of ILs and MLs is an essential requisite for preparing mixtures with desired properties. In this study, we suggest to focus on differences in the structure of an exemplary ionic liquid compared to a hypothetical neutral (uncharged) counterpart, which is composed of the same constituents with respect of molecular structure, size and van der Waals interactions. The detailed investigation of the structural features of ILs can be helpful to explain their structure and properties. A popular tool for analyzing the structure of ionic liquids is the computation of radial distribution functions (or structure factors). They are employed to judge the correlations in the mutual arrangement of ions, individual atoms, or groups of atoms (For example, see the review of ILs simulations [30], as well as the classic works [31, 32, 33]). An interesting feature of the structure of the ionic liquids is that anions and cations exhibit long-ranged spatial correlations. It was shown that oscillations of the radial distribution functions of ILs are visible at much larger distances than in neutral fluids [1]. At the same time, a spatial alternation of anions and cations is noted. For example, in Ref. [33] it was shown that anion-cation radial distribution function (calculated between the center of mass of the anion and the center of mass of the imidazolium ring) is in phase opposition with the cation-cation and anion-anion functions. This was demonstrated for two families of imidazolium-based ionic liquids. In Ref. [34] a coarse-grained model of the ionic liquid [C4mim] [PF-] was studied, similarly demonstrating the alternating structure of anions and cations. It was suggested that this alternation reflects the existence of a distorted interdigitated lattice, which exists in ILs due to the strong electrostatic interactions of ions. Beyond any doubt, well-defined electrical charges cause these specific spatial correlations [35, 36]. However, there is still no complete understanding of the general structural patterns in ionic liquids resulted from these interactions. A crystalline motif formed by parallel planes was previously discussed for the melts of zinc halides. Note, that structure factors of these halides exhibit a pre-peak (FSDP), suggesting the existence of an intermediate-range order in the fluid, which, in particular, could be explained by the presence of such parallel planes [37]. In a subsequent paper, these authors [38] detected such planes in a molecular dynamics simulation of molten ZnCl2. However, these planes were formed not by ions, but by the largest inter-ion voids (interstitial spheres of the largest radii). In addition, tetrahedral configurations of anions are found in such melts, namely, halogen atoms are located around the zinc at the vertices of the tetrahedron. This result was previously obtained experimentally [39, 40] and was recently studied in detail by molecular dynamics simulations [41,
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2.1. Models
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42]. Obviously, electrostatic interactions lead to some common structural features. It means they should occur in both molten simple salts and ionic liquids. Shirota and Castner [43] experimentally compared the dynamics of the pyrrolidinium ionic liquid and its neutral counterpart. It was shown that, compared with the neutral analogue, the ionic liquid has a 1.2-fold increased density and 30-fold larger shear viscosity, which is a consequence of the stronger interaction between the ions, compared with molecules in a neutral liquid. However, one can speak only hypothetically about an experimental “non-ionic” analogue of an ionic liquid. Therefore, the preparation and study of such a system in a computer simulation is of great interest. The models of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate, [C4mim][PF6] and its neutral analogue, consisting of the same molecules, but without charges, were obtained and studied using coarse-grained MD-simulations in Ref. [34]. A significant decrease in IL mobility compared to its neutral counterpart was shown. A fundamental structural difference between these systems was also obtained. However, only radial distribution functions were used for structural characterisation. In this paper, we present results from coarse-grained molecular dynamics simulations of [C4mim][PF6] and its neutral analogue using the model parameters outlined in Ref. [44]. We pay special attention to the study of local three-dimensional structural features based on a VoronoiDelaunay tessellation. This technique is well known for long time and widely used for structure investigation of computer models of different atomic and molecular systems, in particular ionic liquids [45-49]. Moreover, we compare our models for a wide temperature range (300 K to 600 K) to determine the influence of the temperature on the computed structural features.
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We have performed molecular dynamics simulations of 1-butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] using the coarse-grained rigid ILM2 Lennard-Jones/pointcharge model introduced in 2010 by Roy and Maroncelli [44] using three sites to represent the imidazolium cation and a single site for the hexafluorophosphate anion as indicated in Fig.1.
Fig. 1. Illustration of the coarse-grained ILM2 model: the anion (red) is represented by a single site, whereas and the cation (blue) is modelled using three sites. To better compare ionic and molecular liquids on a fundamental level we have prepared two principal variants of the ILM2 model. The first one is an ionic liquid where the anion and cation each carry the original ILM2 fractional charges of ± 0.78|e|, respectively [44]. In the second model,
Journal Pre-proof however, we have removed all the charges on the two constituents, turning it thus into a neutral molecular liquid carrying zero charges. All simulations were consisting of 1000 “ion pairs" applying periodic boundary conditions. The MD simulations of the original ILM2 ionic liquid were carried out under NPT conditions at a pressure of 1 bar, covering a broad temperature range between 300 K and 600 K. The obtained densities are listed in Table I and match the experimental data very well, as also shown in Fig. 2. The simulations of the zero-charge ILM2 model were carried out under constant-volume conditions at exactly the same densities obtained for the charged ILM2 model at each temperature. Production runs of 40 ns length were studied at each temperature following equilibration runs of 10 ns.
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ρ/ kg m-3 1359.1 1340.5 1321.2 1303.7 1285.1 1248.3 1211.4 1138.5
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TABLE I: Temperature dependence of the mass densities obtained from MD simulations of the ILM2 model a p=1 bar.
Fig. 2. Temperature dependence of the mass densities obtained from MD simulations of the ILM2 model a p = 1 bar compared with experimental data of Machida et al. [50]. All simulations were performed with the Gromacs 4.5.3 program. [51] The preparation of topology files, as well as the data analysis were performed with the most recent version of the MOSCITO
Journal Pre-proof suite of programs.[52] Gromacs topology- and restart-files are available on request. Production runs of 20 ns length were employed for every temperature, starting from previously equilibrated configurations. The Nose-Hoover thermostat [53, 54] and the Parrinello-Rahman barostat [55,56] with coupling times τT =0:5 ps and τp=2:0 ps were used to control constant temperature and pressure (1 bar) conditions. The electrostatic interactions were treated by particle mesh Ewald summation.[57] A real space cutoff 1.6 nm was employed, and a mesh spacing of approximately 0.12nm (4th order interpolation) has been used to determine the reciprocal lattice contribution. The Ewald convergence parameter was set to a relative accuracy of the Ewald sum of 10 -6. LennardJones cutoff corrections for energy and pressure were considered. A 5 fs timestep was used, and every 50 steps a configuration was saved. Distance constraints were solved by the LINCS procedure.[58]
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2.2. Interstitial spheres. Interstitial spheres, i.e. the empty spheres inscribed between four atoms (touching their surfaces) characterise empty interatomic voids in a molecular system. Such spheres are determined from a Voronoi–Delaunay tessellation of the system. This tessellation is widely used in physics, chemistry and biology to analyse the structure of computer models of various molecular and atomic systems, for more details see Refs. [59,60,61]. An interstitial sphere is related with a Delaunay simplex (tetrahedron). Its vertices are defined by four mutually close atoms, between which an empty sphere is inscribed. To calculate the Voronoi-Delaunay tessellation we use our own homedeveloped computer programs [62,63,64]. Note we use the version of the tessellation, which is constructed relative to the surfaces of the atoms, so-called additively weighted [54] or Voronoi Stessellation [60, 62, 65]. The use of S-tessellation is important if atoms of the system have different size. In this case, the interstitial spheres are defined correctly, in contrast to the classical approach, which deals only with the centers of atoms [66, 67], or radical (power) tessellations, where the distance is measured along the tangent to the atomic surface [68]. Here, the atom diameters are usually defined by their Lennard-Jones parameters σ taken from the molecular-dynamics simulations. We calculated the Voronoi-Delaunay tessellation as for the whole systems as well as for the both subsystems (anionic and cationic). Fig. 3 illustrates the interstitial spheres for those cases in two dimensions.
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Fig. 3. Upper panel. Two-dimensional illustration of an ionic system and its subsystems. a) Entire system, anions are shown as red disks, and cations are blue; b) selected subsystem of anions (red); c) selected subsystem for C1-sites of cations (blue). Empty circles represent the interstitial spheres. Dashed line triangle shows a Delaunay simplex for each case. Lower panel. The same for the uncharged system. a) Entire system, the subsystem A (corresponded anions) are shown as dark disks, and the subsystem C (corresponded to cations) as grey disks; b) selected subsystem A (dark); c) selected subsystem C1 (dark).
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Note when we calculate the Voronoi-Delaunay tessellation for the entire system all coarse-grained sites are taken into account, as shown in Fig. 3a. For analyzing a subsystem, we only take into account the sites corresponding to a given subsystem, while ignoring all others (see Fig. 3b and Fig. 3c). Moreover, for analysing the cation subsystems we use only the central C1-site (see Fig.3c). Note that the interstitial spheres in disordered systems can overlap significantly. This occurs especially for relatively large cavities, which are surrounded by more than four coarse-grained sites (see, in particular, Fig.3). Thus, some spheres cover the same part of empty space. As a consequence, the volume of a cavity does not relate directly to the sum of volumes of its interstitial spheres. On the other hand, the volume of a cavity is equal to the sum of the empty volumes of the Delaunay simplices representing this cavity [60, 69]. Note that the volumes of the Delaunay simplices can differ significantly for these overlapping interstitial spheres, even their radii are close. Therefore, working with the interstitial spheres, we can consider them as itself, as well as taking into account the fact that they represent a different fraction of empty volumes. When we discuss empty volume in a system, we should keep in mind the last situation. To capture this, we may weight the interstitial spheres with the empty volume of their corresponding Delaunay simplices. 3. Results and discussions Following the nomenclature used in the caption of Fig. 3, we denote the anions in the ionic liquid and the corresponding particles in the uncharged mixture as subsystems A (charged or uncharged). The subsystems C are comprised of the cations and their neutral counterparts. When we deal with the central C1-site of the cations, we will denote them as subsystems C1 (charged or uncharged). The structural differences between the ionic and the uncharged systems become apparent by considering the cationic and anionic subsystems separately. Shown in Fig. 4 are snapshots for the charged and uncharged subsystems A from our models taken at T = 300 K. The structure in Fig. 4a depicting the ionic liquid appears to be much more uniform than the uncharged system shown in Fig. 4b, which includes “condensations” of sites and larger voids.
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Fig. 4. Snapshots taken from molecular dynamics simulations at 300 K. Shown are only the particles representing the subsystems A for the ionic liquid (a) and for the uncharged model (b).
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We would like to emphasize that the structures representing the subsystems C1 show similar behaviour (not shown): again, the structure of charged subsystem appears to be much more homogeneous, while the neutral model is more distorted.
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3.1 Radial distribution functions. To characterize the intermolecular space we calculate distributions of the interstitial sphere radii. For a better representation of the volume of voids, we “weight” the interstitial spheres with the empty volume of the corresponding Delaunay simplices, as discussed in section 2.2. Thus, we present "volume-weighted” distributions of radii of the interstitial sphere. Fig. 5 compares the partial pair correlation functions gA-C1(r), gA-A(r) and gC1-C1(r) for the ionic and uncharged systems.
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Fig. 5. Partial pair correlation functions gA-C1(r), (a); gA-A(r), (b); and gC1-C1(r), (c); (red) and uncharged (blue) subsystems A and C1.
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The oppositely charged ions tend to get close to each other, resulting in a higher first peak and a shift to smaller r in the ionic liquid compared with the uncharged system, Fig. 5a. Like-charged ions repel each other. Thus, the first peaks for these pair distribution functions are not well pronounced and lie further apart than what their van der Waals contact distances would suggest (see also the first peak for the uncharged systems, Fig. 5b and 5c). However, a principal difference between the charged and uncharged subsystems is observed at large distances. To illustrate it we calculate the function d(r) = r2(g(r)-1). The corresponding functions d A-C1(r), d A-A(r) and d C1-C1(r) are shown in Fig. 6.
Journal Pre-proof Fig. 6. Functions d(r) = r2(g(r)-1) for selected sites in the ionic liquids (a) and the uncharged system (b). For А-А (red), С1-С1 (blue), and А-С1 (black)
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Note that the decay of d(r) oscillations is much slower for the charged system. It indicates a longdistance correlation for the anions and cations in ILs. We also note that the maxima of the oscillations for the subsystems A and C1 completely coincide, see red and blue curves in Fig. 6a, and the oscillations of dA-C1(r) are shifted by a half the period (black curve). This observation is typical for ionic liquids [30] and demonstrates the alternating structure of positive and negative ions. Keep in mind that we use the central C1-site of the cations to calculate radial correlations, whereas other authors used the center of mass of molecules [31, 32, 34] or parts thereof. For example, in Ref. [33] the center of mass of the imidazolium ring was used. However, as we can see, there are no principal differences if one chooses the center of mass or a certain specific center located on the cation. The behavior of the radial distribution functions for the uncharged system is completely different from for the ILs, as it was first discussed in Ref. [34]. The curves in Fig. 6b resemble the ones obtained for simple liquids, both with respect to the area of the first peaks, as well as with the long-distance oscillations: there is always a well-defined first peak followed by rapidly fading oscillations. Having computed interstitial spheres for our models (see the illustration shown in Fig. 2), we can use their centers to calculate their corresponding radial distribution functions, as it was suggested in Refs. [70,71,72]. The functions d(r) for the centers of these spheres for charged and uncharged subsystems A and C1 are shown in Fig.7. Here, we again observe the distinctive weakly damped oscillations computed for charged subsystems and fundamentally different behavior for the uncharged systems.
Fig. 7. Functions h(r) d(r) = r2(g(r)-1) for interstitial spheres for subsystems A and C in the ionic liquid (a) and in the uncharged system (b). Thus, the greater regularity of the charged system also manifests itself in the long-distance correlations of intermolecular voids. In the remainder of this paper, we will focus on a local analysis
Journal Pre-proof of voids in our systems, particularly focusing on the radii of the interstitial spheres and the corresponding Delaunay simplices.
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3.2 Radii of the interstitial spheres To characterize intermolecular space in our systems we calculate the distributions of the interstitial sphere radii. For a better representation of the volume of voids, we “weight” the interstitial spheres with the empty volume of their corresponding Delaunay simplices, as discussed in section 2.2. Thus, we present "weighted” distributions of radii of the interstitial spheres. Fig. 8 shows the weighted distributions of the interstitial spheres radii for the subsystems A at 300 K. Note that the cations are explicitly not taken into account in this case, as illustrated in Fig. 3b. We observe a relatively narrow monomodal peak for anions, and a wide distribution with several maxima for the uncharged subsystem A.
Fig. 8. Weighted distributions of the interstitial spheres radii for the subsystem A: charged (red) and the uncharged (blue) at 300K. Typical configurations are also shown for the selected radii of the interstitial spheres (pointed out by numbers). A-sites are indicated by a red colour, Sites C1, C2 and C3 are depicted in blue.
By analysing snapshots of our molecular dynamics models we found typical configurations of the neighbouring A-sites having the chosen radii of the interstitial spheres (see Fig. 8). The narrow maximum in the distribution for the ionic liquid usually corresponds to a situation when four anions surround a cation rather homogeneously as indicated by configuration (1). The uncharged subsystem also contains similar configurations, however, their population is much lower (see the small maximum at 0.3 nm and one from the corresponding configurations (2.3)). The shoulder at small radii (≈ 0.1 nm) corresponds to the dense configurations of four neutral A-sites (2.2). We also found configurations like (2.1), which are the densest tetrahedral packing of four equal spheres. For the charged subsystem such situations are impossible because the charged A-sites repel each other.
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The uncharged system contains also a majority configuration with large radii of interstitial spheres. In this case, the A–sites are interdigitated with several “cations”, see configuration (2.4). We conducted the same analysis for the subsystems C1. In this case, the Voronoi-Delaunay tessellation was calculated ignoring all other coarse-grained sites. The interstitial sphere radii distribution for the charged subsystem C1 is also narrower than for the neutral one, as shown in Fig. 9. We found that four charged C1-sites always tend to be located around an anion. However, now the distribution is slightly splitting: the sharp peak at r = 0,28 nm corresponds to the closest arrangement of C1-sites around the anion, configuration (1.1), and the following shoulder contains all other situations (1.2).
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Fig. 9. Weighted distributions of the interstitial spheres radii for the subsystem C1: charged (red) and the neutral one (blue) at 300K. Typical configurations at the corresponding radii of the interstitial spheres (pointed out by numbers) are also shown: C1-sites are depicted dark blue, Asites are shown in red, while C2- and C3-sites are shown in light blue.
For the uncharged C1, we observe a broad uniform distribution. This means that C1-sites are located in space rather randomly (without any special features in the mutual arrangement). Note, unlike the uncharged A-particles, four neutral C1-sites never form the mutual contacts, i.e. do not form the densest configurations with the small radii of the interstitial spheres (around of 0.1 nm), as a subsystem of spherical A-particles. It is obvious because of the complex shape of the cation. 3.3 Shape of the Delaunay simplices. Recall, that an interstitial sphere is directly related to a corresponding Delaunay simplex. The analysis of the simplex shapes is used to study the structure of simple liquids, disordered packings of spheres, and molecular systems, see for example [60,74]. One of the most suitable characteristics
Journal Pre-proof of the simplex shape is the so-called tetrahedricity, T [74]. It is defined as the variation of the edge lengths of the Delaunay simplex:
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𝑇 = ∑𝑖≠𝑗 (𝑙𝑖 − 𝑙𝑗 )2 /15𝑙02, where 𝑙𝑖 and 𝑙𝑗 are the length of the i-th and j-th edges, and 𝑙0 is the mean edge length of a given simplex. A small T value means that the shape of this simplex is close to the perfect tetrahedron, for which all the edge length are the same. The Delaunay simplices are determined from the VoronoiDelaunay tessellation of the studied system. We compare the tetrahedricity distributions of the Delaunay simplices for the charged and uncharged subsystems A, Fig. 10. The narrow monomodal peak at small T values demonstrates that the anions in the ionic liquid are organized in configurations with relatively well-defined tetrahedral shapes.
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Fig. 10. Tetrahedricity distribution of the Delaunay simplices for the subsystems A at 300K: charged (red), uncharged (blue).
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Note that the calculated distributions of tetrahedricity complement our conclusions of the analysis of the interstitial sphere radii. Now we can state that the local configurations of the anions are similar in size, and their shape is close to the regular tetrahedron. In Fig. 11 a configuration is shown, extracted from a molecular dynamics trajectory, demonstrating a typical tetrahedral local arrangement of anions in the ionic liquid with a cation being placed among them.
Journal Pre-proof Fig.11. Typical anion (red) arrangement in the ionic liquid. The lines show the Delaunay simplex of the subsystem A. Cations are shown by green. The configuration is extracted from the molecular dynamics model at 300K. The situation is different for the uncharged subsystem A, as shown in Fig. 10. In this case, the tetrahedricity-distribution is much broader and shifted towards larger T values, indicating that the shape of the simplices is more distorted. On the other hand, a shoulder (plateau) appears at small T values. This suggests that simplices with a very well-defined tetrahedral shape do also exist, although in significantly reduced numbers. They correspond to the densest local configurations with the small interstitial sphere radii, as discussed above and shown in Fig.8. The shape of those corresponding Delaunay simplices is close to the perfect tetrahedron [74].
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To investigate the the relation between tetrahedricity and interstitial sphere radius for the uncharged subsystem A in more detail, we calculated the tetrahedricity distributions for Delaunay simplices corresponding to the various maxima given in the distributions shown in Fig 8. The distributions shown in Fig. 12 demonstrate that the smallest radii of the interstitial spheres (around 0.1 nm) are related to a narrow maximum of the near-perfect tetrahedricity close to zero. For the other selected radii (around 0.3 nm and around 0.45 nm), we observe very different shapes of the simplices, which are far from the perfect shape. However, the tetrahedricity of Delaunay simplices and the radius of the interstitial sphere reflect different properties of a local configuration.
Fig. 12. Tetrahedricity distributions of Delaunay simplices corresponding the interstitial spheres radii from the intervals: 0.09-0.11 nm (blue line), 0.29-0.31 nm (red), 0.44 - 0.46 nm (green) on the distribution for the uncharged subsystem A in Fig.8. For the subsystems С1, the tetrahedricity distributions are shown in Fig. 13. For the charged system, the Delaunay simplices are found to be also more tetrahedral. This is not unexpected since the cations in the ionic liquid are interlocked in a common structural motif with the anions due to their electrostatic interactions.
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Fig. 13. Tetrahedricity distributions of the Delaunay simplices for the subsystem C1: charged (red), uncharged (blue) at 300K.
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However, in the case of the cations, the tetrahedricity-distribution is shifted towards large T values and is broader than for the subsystems A shown in Fig.10. This shift and additional distortion for the C1 subsystem is likely caused by the more complex shape of the cation.
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The distributions of the interstitial spheres radii for the entire ionic and uncharged system at 300 K and 600 K are compared in Fig. 14. Here, we do not divide our systems into the subsystems, and interstitial spheres are calculated as it was shown in Fig. 3a.
Fig. 14. Weighted distributions of the interstitial spheres radii for the entire models of the ionic liquid (red curves) and for the uncharged system (blue) at 300K and 600K. Note that the difference between the charged and neutral systems is small at each temperature. The increasing leads to a decreasing density, as shown in Fig.2. Thus the intermolecular voids increase, and the distributions move to larger radii and became broader. However, the distributions remain very similar for a given temperature, for both, the charged and uncharged systems. We performed the calculation also for the intermediate temperatures and obtained the smooth transition from the
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initial to the final distributions shown in Fig. 14. Position of the maximum of the distributions is plotted as a function of temperature in Fig.15.
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Fig. 15. Positions of the maxima of the interstitial spheres radii distribution as a function of temperature for the entire ionic (red) and uncharged (blue) systems.
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Practically linear and parallel lines give a reason to say that both systems are affected by the temperature in the same way. In other words, we do not see a different influence of the temperature on the ionic and uncharged systems. For the subsystems, we found a similar shift and broadening with temperature. At the same time, structural consistency between charged and uncharged systems is also preserved. The distributions of the interstitial spheres radii for subsystem A at 600K are shown in Fig. 16.
Fig. 16. Distributions of the interstitial spheres radii for the subsystem A: charged (red) and the uncharged (blue) at 600K. Data for 300K (from Fig. 8) is shown by dashed lines.
The charged subsystem A again demonstrates a narrow monomodal peak as it was for the low temperature. It is only slightly broadened and shifted toward larger radii. The uncharged subsystem A distribution at high temperature is also much wider than for the charged one. It is now more smoothed because the heating distortion also increases. Figure 17 shows the tetrahedricity distribution of the Delaunay simplices in the charged and uncharged subsystems A at 600K.
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Fig.17. Tetrahedricity distribution of the Delaunay simplices for the subsystems A at 600K: charged (red), uncharged (blue). Data for 300K (from Fig. 10) is shown by dashed lines.
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The difference between the charged and neutral systems is very similar to that at 300K, Fig. 10. Thus, the temperature increase retains existing structural difference of the ionic liquid and the uncharged system.
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We have performed molecular dynamics simulation of a coarse-grained model of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate and a system consisting of the same molecules but without charges at various temperatures. The ionic liquid is here represented by the ILM2 force field proposed by Roy and Maroncelli in 2010 [44]. Partial radial distribution functions for the interaction sites and for computed interstitial spheres representing the void space were calculated for the long-distance interval up to 3 nm. In addition, the intermolecular voids and shape of the local configurations of the anions and cations are examined using the Voronoi-Delaunay method. The structures of the anion and cation subsystems of our models substantially differ from the structures of their uncharged analogues. As expected, the partial radial distribution functions for anion and cation subsystems demonstrate the spatial alternation of positively and negatively charged particles, typically observed for salts. Such an alternation pattern is absent for the uncharged subsystems, where we observe a common structural motif for both subsystems governed by geometrical laws of the dense disordered packing of impenetrable particles. The use of the Voronoi-Delaunay tessellation for each of the subsystems allows us to determine the geometrical features of the local configurations for anions and cations separately. We have obtained that the radii of spheres inscribed between anions have a narrow distribution, i.e. the size of the “holes” between the mutually nearest anions are approximately the same. For the uncharged counterpart, however, there is a wide distribution of the radii of such spheres. A similar scenario is observed for the cation-subsystem. However, in this case, the distributions of the interstitial sphere radii are more diffuse due to the more complex shape of the cation. By analyzing the shape of the computed Delaunay simplices we found that the fours of the nearest anions in the
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ionic liquid represent configurations with a shape close to a regular tetrahedron, whilst a counterion is located in between these anions. The same can also be said about cations. This is in consistent with the observed spatial alternation of ions in the ionic liquid. We would like to emphasize that such tetrahedral configurations of ions have been previously observed for simple salt melts, for example, in molten zinc halides. For the case of the uncharged subsystems these pronouncedly tetrahedral features are found to be absent. Finally, we also investigated the temperature effect on the observed structural differences between the charged and uncharged subsystems. In both cases with increasing temperature, we only observe a general increase in the structural disorder. Principal structural features of the subsystems, however, remain the same. In particular, the local configurations of anions (cations) in the ionic liquid have keep their close to perfectly tetrahedral shape for the all investigated temperatures between 300 K and 600 K. For the uncharged subsystems, the similar configurations are found to be also much more distorted.
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Acknowledgements NNM gratefully acknowledges financial support from grants RFFI (No. 18-03-00045).
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