On the collective network of ionic liquid/water mixtures: III. Structural analysis of ionic liquids on the basis of Voronoi decomposition

Abstract

Three different mixtures of 1-butyl-3-methyl-imidazolium tetrafluoroborate with water have been studied by means of molecular dynamics simulations. Based on the classical Lopes-Padua force field trajectories of approximately 60 ns were computed. This is the third part of a series concerning the collective network of 1-butyl-3-methyl-imidazolium tetrafluoroborate/water mixtures. The first part [J. Chem. Phys. 127 (2007), 234503] dealt with the orientational structure and static dielectric constants. The second part [J. Chem. Phys. 129 (2008), 184501] was focused on the decomposition of the dielectric spectrum of these mixtures. In this work the focus lies on the characterization of the neighborhood of ionic liquids by means of the Voronoi decomposition. The Voronoi algorithm is a rational tool to uniquely decompose the space around a reference molecule without using any empirical parameters. Thus, neighborhood relations, direct and indirect ones, can be extracted and were used in combination with g-coefficients. These coefficients represent the generalization of the traditional radial distribution function in order to include the mutual positioning and orientation of anisotropic molecules. Furthermore, the Voronoi method provides, as a byproduct, the mutual coordination numbers of molecular species.

I. Introduction

In ionic systems a cation or an anion cannot be considered independently. While in the gas phase the thinking in terms of ion pairs is appropriate, things become much more complex in the condensed phase. The concept of an ion pair has to be replaced by an ion surrounded by shell of counterions and/or neutral solvent molecules. Although the central ion interacts with multiple counterions, the overall charge neutrality is fulfilled. This traditional picture of “charge clouds” is inherent to the well-known Debye-Hückel theory¹ which has already introduced such basic concepts like ion activity and the radius of the charge cloud. In the former case, the Coulomb interaction between the reference ion and its neighboring shell of counterions reduces the activity which can be calculated as a function of concentration, temperature and dielectric constant of the solvent. The very same parameters also enter the formula giving the radius of the ionic cloud. However, this concept of a radius can be hardly transferred to the situation of molecular ionic liquids with their inherent shape and charge anisotropy. This requires the development of new ideas for classifying neighborhood relations. Commonly in molecular dynamics simulations of ionic liquids, the investigation of the local structure is based on structure factors, radial distribution functions and three dimensional plots.^2–11 Augmenting these analysis tools by a specified distance criterion, neighborhood relations may be deduced. But, this inevitably necessitates the introduction of certain threshold below which particles are considered as neighbors. Consequently, these thresholds comprise a set of parameters which strongly influences the final results. This is already a problem in simple liquids becoming rather complex for molecular species with their anisotropy in shape and charge. For example, the distance threshold to be used depends on the reference atom to which neighborhood is defined. This parameter-based approach becomes even more difficult when dealing with mixtures of different molecular species. Consequently, a parameter-free method of spatial decomposition is highly desirable. Fortunately, computational geometry offers such a method: Voronoi polyhedra decompose space into cells, each of which consists of points closer to one particular reference point than to any others. This method has been reinvented, given different names, generalized, studied, and applied many times over in many different fields.^12,13 A first attempt to use this rational, parameter-free approach in the field can be found in Ref. 14. There, the ionic liquid cations were represented by a two sphere model whereas the anions are approximated by a sphere. This coarse-grained model was able to reproduce the dependence of density and transport properties on the charge delocalization. But the poor resolution of this coarse-grained model prohibits a detailed analysis of the neighborhood which is the aim of this work. Therefore, our Voronoi decomposition is performed at the full atomic resolution.

This work is part of a series of publications concerning the collective networks of 1-butyl-3-methyl-imidazolium tetrafluoroborate ${\text{BMIM}}^{+}{\text{BF}}_4^{-}$ with water: The first part concentrated on the structural analysis of the mutual networks of cations, anions and water.¹⁵ These networks were analyzed by atom-atom radial distribution functions as well as so-called g-coefficients, which reveal the mutual orientation and position of the molecules. In addition, the collectivity of the mixture was interpreted in terms of the Kirkwood G_k-factor and the static dielectric constant. The second part extended the analysis of the dielectric constant and compared the computational results of the frequency-dependent dielectric spectrum with experimental data.¹⁶ Both references observed an increased ordering of water with increasing content of ionic liquids in the mixture. This fact is probably due to the direct neighborhood to the ionic liquids. Therefore, this work focuses on a detailed analysis of this neighborhood, which can be unambiguously defined by a Voronoi decomposition. In other words, the Voronoi polyhedra of the neighbor molecule shares a common face with the reference polyhedron. Consequently, this method is particularly apt to cope with highly anisotropic reference sites, e.g. long tails of imidazolium based ionic liquids.

II. Theory and Methods

A. Voronoi decomposition

The Voronoi decomposition is simply defined by a non-degenerate set of points P_1…N and allocates all space amongst this set. Each point P_i [in our case atoms] is surrounded by an irregular polyhedron containing all space closer to its associated reference point than to any other point P_j of the given set. The sum of these non-overlapping polyhedra is space-filling and the faces of each of these polyhedra are constructed by planes perpendicular to the vectors between the associated reference point and its neighbor points. For example, in Fig. 1a the Voronoi cell of the point P₆ is the gray shaded area. Please note, that all sketches explaining the algorithm are drawn in two dimensions only for the sake of simplicity. However, this Voronoi decomposition into polyhedra corresponds geometrically to a Delaunay tessellation which has been used extensively in the design of efficient algorithms for interpolation, contouring or mesh generation.¹⁷ In principle, a Delaunay tessellation is an unique partitioning of the set P_1…N into “simplices”,¹³ e.g. P_iP_jP_kP_l. In three dimensional space such a simplex is an irregular tetrahedron and its four vertices [P_i, P_j, P_k and P_l] are a subset of P_1…N. These four vertices of a tetrahedron lie on the surface of a circumscribed sphere which does not contain any further [vertex] point P_i. This definition will be called “Delaunay criterion” subsequently. In two dimensions the circumscribed sphere and the tetrahedron reduces to a circumscribed circle and triangle, respectively. In Fig. 1b the circumscribed circle is marked bold for the triangle P₃P₄P₆. The center of this circle [or sphere in 3D] coincides with the vertex of the Voronoi cell. The edges [dashed lines] of the Delaunay simplex [triangle in 2D or tetrahedra in 3D] are the vectors connecting two points P_i and P_j being cut by the orthogonal Voronoi faces [solid line] midway.

Fig. 1. Two dimensional sketches illustrating the Voronoi algorithm.

a) The dualism of Voronoi decomposition and Delaunay tessellation is shown. The gray shaded area is the Voronoi cell, the triangles with the dashed edges depict the Delaunay simplices. b) The Delaunay criterion [black circle] ensures that no further point lies within the circumscribed circle of the Delaunay triangle. c) The Delaunay triangle N₀ contain an image atoms P7’. This fact is saved by periodicity indices m. d) The new insertion point P₈ is located at the simplex P₄P₅P₆ which is called base (dark gray shaded area). The light gray shaded area shows a simplex which Delaunay criterion was violated by P₈. e) “Star-shaped” region around P₈. After deleting the old Delaunay distances, new distances are constructed in such a way that the Delaunay criterion is fulfilled for all simplex in the box.

In literature a plethora of tessellation algorithms can be found,^18–21 but in computer simulations of bulk media periodicity is commonly used to emulate infinite systems. Consequently, Delaunay algorithms^17,20 taking explicitly into account periodicity are of special importance. Thereby, periodicity is an essential part of the algorithm which excludes algorithms working on the primary cell augmented by its explicit images. Among the periodic Delaunay algorithms “insertion algorithms” have proven particularly successful and were adopted in various ways.^17,18,22 They start from a given tessellation of a subset of points and insert a new arbitrarily selected point into the tessellation until all points of the set are tessellated. Then, the finite set of unique simplices is called primary tessellation and can be completely described by specifying the four vertexes of each simplex. Consequently, each simplex is represented by a row in our data structure containing the indices of the vertex points [P_i, P_j, P_k and P_l]:

$P_i{m}_i^x{m}_i^y{m}_i^z$

$P_j{m}_j^x{m}_j^y{m}_j^z$

$P_k{m}_k^x{m}_k^y{m}_k^z$

$P_l{m}_l^x{m}_l^y{m}_l^z$

N1

N2

N3

N4

Furthermore, this row also contains the row number of all four adjacent simplices [N₁, N₂, N₃ and N₄] which share a face with the simplex P_iP_jP_kP_l. In Fig. 1c, the reference simplex is N₀ build up by P₁, P₂ and ${P^{\prime }}_7$. The first neighbor simplex N₁ [P₁P₂P₆] shares the face P₁P₂ with N₀. The storage of this neighbor information facilitates the overall computation. Since we are dealing with periodic tessellations, each vertex point information in that row is augmented by three indices $\left[m_1^x,m_1^y\text{and}{m}_1^z\right]$ indicating the periodic displacement of that vertex. For example, in our two dimensional sketch in Fig. 1c $m_1^y$ and $m_2^y$ equal zero because both points are located in the primary simulation box. ${P^{\prime }}_7$ is the image of P₇. Consequently, $m_7^y$ is +1 in this case.

Now, we turn to the description of our insertion algorithm. The initial tessellation is constructed from a randomly chosen point and seven of its images forming a cube. This cube is subdivided into six tetrahedra serving as initial simplices. The insertion of a new point is divided into several steps: First, one looks for the simplex containing this new point or one of its images. This simplex is called “base”. In our example in Fig. 1d the point P₈ is to be inserted. In this case the “base” [dark gray shaded area] is P₄P₅P₆. Unfortunately, the base location cannot be inferred directly from the current tessellation data nor the coordinate array. Therefore, a search starting from one tetrahedron must be performed efficiently to manage large tessellations.^17,22 In each tetrahedron along the path [arrows in Fig. 1d], one determines which of its four faces can be crossed to move closer to the insertion point and chooses one of these possible pathways randomly. Thereby, the search path may cross the periodic boundaries. If the located base is not in the primary cell it is periodically shifted there assuring that at least one vertex [P₁, P₂, P₃ or P₄] is in the primary cell. As a result, the primary tessellation contains no gaps and all tetrahedra are connected.

Second, all neighboring simplices failing the Delaunay criterion are detected. For example, in Fig. 1d the insertion point P₈ violates the criterion for the simplex P₃P₄P₆. The collection of these neighboring simplices plus the base is called “core” [gray shaded areas] and all points on its surface the “boundary surface”. In some cases the circumsphere of a neighboring simplex contains only an image of the insertion point. These tetrahedra are shifted in such a way that their circumsphere incloses the primary location of the insertion point. Afterwards, a visibility check is performed to check that all faces on the surface of the core are directly visible to the insertion point ensuring a “star-shaped” core region [c. f. Fig. 1e] and a non-negligible volume of each new tetrahedron.

Third, all simplices constituting the core are deleted and a new set of simplices is constructed by connecting the inserted point to all points on the boundary surface. Afterwards, a further new point is inserted and the whole procedure is continued iteratively. A more detailed description of each algorithm step in given in Ref. 17.

After completing the final tessellation of a trajectory frame, it can be used in a multitude of ways: A neighborhood list can be easily created by looking the rows in our data structure.In this senseIn this sense a “neighbor” is a point which shares at least one tetrahedron with the reference point. The ensemble of all these neighbors constitutes the first coordination shell. The neighbors of the neighbors form the second shell and so on. Alternatively, a shell may be defined as a set of particles which share a minimal Delaunay distance to the reference site. The term Delaunay distance is meant in a graph-theoretical sense. In other words, our decomposition provides a parameter-free definition of neighborhood which can be seen as a “direct interaction” rather than smallest distance criterion. This fact is also visualized by Fig. 2. There, a BMIM⁺ and two neighboring water molecules and their Voronoi polyhedra are depicted. The water molecule on the left bottom side has a small distance to the ring of the cation. Consequently, the interaction surface is bigger than the corresponding surface of the second water molecule at a longer distance. Nevertheless, both water molecules are direct neighbors of the cation.

Fig. 2.

Typical Voronoi polyhedron of BMIM⁺ and two neighboring water molecules. The tessellation was performed on an atomic resolution. The shaded areas show the interaction area of BMIM⁺ with each water molecule.

Another interesting feature of a Voronoi decomposition is the evaluation of volumes and surfaces of the Voronoi polyhedra. These properties can be calculated for each tessellation point exploiting the data structure described above. Each point shares exactly one area of its Voronoi polyhedron with one neighbor point. The corresponding centers of circumspheres used to check the Delaunay criterion in that case are the vertexes of this area. By rotating through the data structure each tetrahedron which shares this Voronoi surface area is found. Thereby, it might be necessary to check the boundary conditions again. In this manner, the Voronoi area can be determined by summing up the areas of the triangles constituting it. The volume of the Voronoi polyhedron is simply the sum of the selected tetrahedra. This rotation is repeated for each Voronoi surface belonging to the tessellation point under investigation. The volume of a molecule can be computed by the sum of the volumes of the tessellation points belonging to that molecule. In case of the surface, one has to keep in mind that there are inner and outer surface areas. The surface area is only build up by the latter ones.

Our self-written program package GEPETTO constructs the sequence of Voronoi shells surrounding each reference site. The members of each shell as well as its volume and its surface are computed. Furthermore, it combines the traditional approach using common structural analyze tool, e.g. common radial distribution functions (RDF) g⁰⁰⁰(r) and orientational correlation functions [g¹¹⁰(r), g¹⁰¹(r) and g⁰¹¹(r)], with the method of Voronoi decomposition.

B. Combined g-coefficient/Voronoi analysis

The above mentioned g-coefficients are the distance dependent part of the orientational probability function g(r_ij, Ω_j, Ω_j) of molecular pairs. The orientation of a pair of molecules i and j is each described by a set of three Euler angles, denoted by Ω_i and Ω_j, respectively. The distance vector r_ij = r_j – r_i joining molecular centers is expressed in terms of polar coordinates, i.e., the intermolecular distance r_ij = |r_ij| and two polar angles, denoted here as Ω_ij. If one is only interested in the spatial distribution of molecular center irrespective of molecular orientations the general probability function reduces to g(r_ij, Ω_i,j).^23–25 Please note, these g(r_ij, Ω_ij) always refer to a specific molecule-fixed coordinate system. For example, in Ref. 23–25 the z-axis of the coordinate system coincides with the dipole vector of the reference water molecule. Quite generally, the coordinates of all neighbors of a reference molecule have to be projected to the molecule-fixed local coordinate system. This procedure is also inherent to all 3D histogram plot, e.g. Fig. 7 in our present work. On the contrary, the g-coefficient method is independent of the choice of the coordinate system and involves the complete set of rotational angles Ω_i, Ω_j and Ω_ij.

Fig. 7.

Schematic 3D vector diagram of the positioning and orientation of BMIM⁺ (blue), ${\text{BF}}_4^{-}$ (green) and water (red) in the neighborhood of a reference BMIM⁺. By construction the arrows represent the dipole moments averaged over all entries to the bin from which they emanate.

A detailed description of the expansion of the full angle dependent pair correlation function g(r_ij, Ω_i, Ω_j, Ω_ij) into a set of rotationally invariant basis functions can be found in Ref. 26. The expansion coefficients g^L_i,L_j,L_ij (r) depend on the mutual center-of-mass distance of the pair of molecules considered. For computational convenience, the angular basis functions are not expressed in the original Euler angles Ω_i, Ω_j and Ω_ij but are evaluated as Legendre polynomials of the cosines tabulated in Table I.²⁷ The upper indices (L_i, L_j and L_ij) specify the order of the angular momentum of the respective polar angles of the reference molecule i, its (direct or indirect) neighbor j and the distance vector r_ij. Principally, g^L_i,L_j,L_ij (r) looks like

$$g^{L_i,L_j,L_{ij}}\left(r\right)=\frac{1}{\rho 4\pi r^{2}dr}{\displaystyle \sum _j{\Phi }^{L_i,L_j,L_{ij}}\cdot \delta \left(r-\left|r_{ij}\right|\right).}$$ (1)

4πr²dr is the volume of a spherical shell of thickness dr and p is a global particle density. In practice, g^L_i,L_j,L_ij (r) is computed as a histogram accumulating entries of the angular function Φ^L_i,L_j,L_ij into a bin from r to r + dr. The bin selection is the computational analogue of the mathematical δ-function.

Table I.

List of the distance dependent g^L_i,L_j,L_ij(r)-coefficients with L ∈ {0,1,2}. μ_i and μ_j are the molecular dipole moments of molecule i and j being separated by the vector r_i,j.⁵¹ Higher orders of the angular momentum (L_i, L_j and L_ij) replace the cosine function by their respective Legendre polynomial. The latter functions allow a finer resolution of the angular space which can be seen in Fig. 3.

g-coefficient	ΦLi,Lj,Lij
g000(r)	1	radial distribution function
g110(r)	cos(μi, μj)	mutual orientation
g101(r)	cos(μi, rij)	position of j with respect to i
g011(r)	cos(μj, rij)	position of i with respect to j
g220(r)	$\frac{3}{2}\text{cos}\left({\mathit{\mu }}_{\mathit{\text{i}}},{\mathit{\mu }}_{\mathit{\text{j}}}\right)-\frac{1}{2}$	mutual orientation
g202(r)	$\frac{3}{2}\text{cos}\left({\mathit{\mu }}_{\mathit{\text{i}}},{\mathbf{\text{r}}}_{\mathbf{\text{ij}}}\right)-\frac{1}{2}$	position of j with respect to i
g022(r)	$\frac{3}{2}\text{cos}\left({\mathit{\mu }}_{\mathit{\text{j}}},{\mathbf{\text{r}}}_{\mathbf{\text{ij}}}\right)-\frac{1}{2}$	position of i with respect to j

In case of L_i = L_j = L_ij = 0, Φ^L_i,L_j,L_ij equals 1 and the corresponding g⁰⁰⁰(r) is identical to the well-known center-of-mass RDF. The extension to higher g^L_i,L_j,L_ij (r)-coefficients is straight forward: The entry of 1 is replaced by the value of the respective Legendre polynomial of the angular function Φ^L_i,L_j,L_ij.^28,29 These angular functions reflect the mutual position [g^L_i,0,L_ij(r) and g^0,L_j,L_ij(r)] and orientation [g^Li,Lj,0(r)] of a pair of molecules and may be seen as an extension of the traditional RDF, which is sufficient for simple liquids lacking molecular orientation. Additionally, in case of water neighbors g^0,L_j,L_ij (r) reveals the hydrogen bond donor/acceptor role of the reference site i.³⁰ An intuitive interpretation of Eq. 1 takes the value of a g^L_i,L_j,L_ij(r)-coefficient as the statistical average of the respective angular function at a fixed intermolecular distance r. This averaged value and in particular its sign may be attributed to the most populated angular regions. For the two types of angular functions, g^L_i,L_j,L_ij(r) and g^L_i,L_j,0(r), these angular regions are depicted in Fig.3. We have attached appropriate labels to regions in that figure in order to facilitate the subsequent discussion in literal terms. For the special case of L = 1 and L = 2, the pattern in sign discriminates three distinct regions: The two regions with a positive sign in the second Legendre polynomial, region 1 and 3, can be distinguished by the sign of the first Legendre polynomial. Region 2 is characterized by a negative sign of the second Legendre polynomial and a change of sign of the first Legendre polynomial.

Fig. 3.

First (solid line) and second (dotted line) Legendre polynomials of the cosine of the polar angle. The sign pattern of these functions reveal the position in g^L0L(r) and g^0LL(r) or the orientation in g^LL0(r).

While molecular anisotropy is reflected by the angular functions used to compute the g^L_i,L_j,L_ij (r)-coefficients the concept of radial bins still sticks to the picture of spherical shells. In our IL system, this description may be sufficient for the anion ${\text{BF}}_4^{-}$ and water but is certainly too limited for the cation BMIM⁺. As a first remedy BMIM⁺ may be splitted into three parts, head (methyl-group, H), ring (r) and tail(butyl-group, $\mathcal{T}$), for the analysis. This decomposition dates from Ref. 31, but there ℋ and ℛ were merged together. This still poses the problem of excluded volumes, i.e. spatial areas are occupied by other parts of the reference BMIM⁺ and not by its neighbors. This is again a drawback of the concept of spherical shells. In order to overcome this limitation we decided to combine the method of Voronoi decomposition with the g-coefficient analysis retaining the concept of head-ring-tail.

In the combined analysis the histogram of a particular g^L_i,L_j,L_ij (r)-coefficient is constructed as described above, but the set of neighbors j is restricted to a specific Voronoi shell. Thus, a g^L_i,L_j,L_ij (r)-coefficient is decomposed into a sum over all these Voronoi shell specific contributions. The advance of this combined method can be demonstrated in Fig.4. The solid line represents the overall g⁰⁰⁰(r). Starting with the radial distribution of BMIM⁺-tails around BMIM⁺-tails, the first peak of g⁰⁰⁰(r) in Fig.4a coincides with the peak of the first Voronoi shell. At first sight this assignment seems to be obvious but is not so straight forward for the second and third Voronoi shell since there is no further clear peak in the overall g⁰⁰⁰(r). Even more, the tail-tail distribution is a fortunate example for clear-cut assignments of the first peak. As visible in Fig. 4b the ring-ring distribution shows a clear peak at 8.5Å but this does not correspond to the first Voronoi shell. In other words, this peak is not constituted by the direct neighbors of the cationic ring. In fact, the direct neighbors are hidden in the foot of the peak at 6.5A indicated by the dotted line (first Voronoi shell). It is the second Voronoi shell containing only the neighbors of the direct neighbors which creates the first peak of the overall g⁰⁰⁰(r). The peak of the third Voronoi shell finally coincides with the first minimum of the overall g⁰⁰⁰(r) in contradiction to intuition. Another important point to mention is the very large extension of the third shell up to 20Ă This means that in our simulation box of 41.8Å only the first three Voronoi shells can be analyzed. We consider this as the minimum correlation length in order to permit a transition of structural oscillations to an uniform bulk property. Nevertheless, we are aware that the third shell is at the edge of this transition. In fact, there are numerous examples for pure ionic liquids where structural oscillations in radial distribution functions extend up to this threshold.^7,32–35 Smaller box sizes reduce the number of solvent layers around the cation to less than three. In systems with periodic boundary conditions this would favor spurious self-interactions between cations and its images.

Fig. 4.

Decomposition of the g⁰⁰⁰ (r)-coefficient into its contributions from the first (dotted), second (dashed) and third (dash-dotted line) Voronoi shells at _xH2O = 0.967: a) $\mathcal{T}-\mathcal{T}$ distribution, b) $ℛ-ℛ$ distribution. Here, the contributions of the first Voronoi shell at a mole fraction of x_H2O = 0.912 (green) and x_H2O = 0.768 (red) are displayed additionally.

C. Computational setup

Since this work is the third part of a series of paper, the detailed description of the molecular dynamics simulation setup is given in Ref. 15 and we will only briefly denoted here. Three different mixtures of BMIM⁺${\text{BF}}_4^{-}$ and TIP3P-water were simulated: The force field of the cation, anion and water were taken from Ref. 36,37, Ref. 38 and Ref. 39, respectively. Coulomb interactions were calculated by the Particle-Mesh Ewald method,^40,41 using a 10 Å cutoff and a κ of 0.41 Å⁻¹ for the real-space part interactions. All bond lengths were kept fixed by the SHAKE algorithm,⁴² whereas bond angles and dihedrals were left flexible. Trajectories were generated with the molecular dynamics program package CHARMM⁴³ under constant volume with a boxlength of 41.8 A and an average temperature of T = 300 K and an average pressure close to 1 atm. The trajectory was propagated for 62 ns with a time increment of ∆t =2 fs. Each trajectory frame was tessellated at an atomic level. A coarser graining representing the whole molecule by its center-of-mass was considered too crude since essential features of anisotropy are lost in this way. We want to emphasize that each tessellation was checked in a twofold way: First, the Delaunay criterion was carefully revisited in order to secure that pairs of particles recognized as direct neighbors are not separated by overseen interstitial molecules. Second, the sum of all molecular volumes had to equal the total volume of the simulation box.

Table II summarizes the compositions of the mixtures studied. Additionally, average molecular volumina in these liquid mixtures are provided. In combination with the particle numbers, the occupancy of each species (cation, anion and water) of the total simulation box can be computed. As the mole fraction x_H2O only counts particle numbers irrespective of the molecular volumes, we consider this occupancy a more intuitive measure of the composition. This is most evident in case of x_H2O =0.768 where the cations cover one half of the total volume leaving a quarter to water which in terms of mole fractions contributes three quarters. Seen in this way the mixtures studied are far from diluted systems as the might look at first sight.

Table II. Composition and coordination of the simulated ${\text{BMIM}}^{+}{\text{BF}}_4^{-}/\text{water}$ mixtures.

		xH2O
		0.768	0.912	0.967
particle	$N_{BMI{M}^{+}},N_{B{F}_4^{-}}$	166	111	55
numbers	NH2O	548	1147	1592
occupancy	NBMIM+· < VBMIM+ > /V	50.2%	33.0%	18.2%
	$N_{B{F}_4^{-}}\cdot <V_{B{F}_4^{-}}>/V$	23.9%	15.0%	7.3%
	NH2O· < VH2O > /V	25.9%	52.0%	74.5%
molecular volumina	< VBMIM+ > / Å3	270	270	285
	$<V_{B{F}_4^{-}}>/{\AA }^3$	63	59	59
	< VH2O > / Å3	33	32	33
reference site	neighbor	coordination number
cation	cation	9.4	5.5	2.5
	anion	5.6	3.8	2.1
	water	14.0	28.3	39.0
anion	cation	5.6	3.8	2.1
	anion	0.9	0.6	0.3
	water	5.3	10.6	14.9
water	cation	4.2	2.7	1.3
	anion	1.6	1.0	0.5
	water	4.3	8.4	12.2

The simultaneous application of the Voronoi algorithm and the g-coefficients combines the criterion of neighborhood with the the concept of distance spread. Thereby, we found that in a 41.8 Å simulation box only three Voronoi shells can be completely resolved. This is clear evidence that smaller systems are restricted to direct and indirect neighbors only thus missing any features of a bulk property.

III. Results and Discussion

In principle, the Voronoi decomposition of g-coefficients may be applied to any pair of species or moieties, e.g. anion ($\mathcal{A}$), water ($\mathcal{W}$), head ($\mathcal{H}$), ring ($\mathcal{R}$) and tail($\mathcal{T}$) ending up with 15 possible pairs for each g-coefficient. As our box size permits the construction of three complete Voronoi shells, each g-coefficient can be further decomposed into its Voronoi shell contributions. As we use g⁰⁰⁰ (r), g^L_i,0,L_ij (r), g^0,L_j,L_ij (r) and g^L_i,L_j,0(r) to describe the mutual position and orientation of these pairs, the complete data set covering all this information is already extremely large and is even more enlarged when considering all three compositions of the IL/water mixtures. Nevertheless, we have analyzed the whole data field in order to get a consistent picture. For the presentation in this paper, however, we have reduced the information to the essential features.

A. The hydrophobic cationic reference site

Turning back to the $\mathcal{T}$-$\mathcal{T}$ configuration displayed in Fig. 4a the height of the peak of the first Voronoi-shell increases from 1.14 over 1.24 to 1.38 at the mole fractions x_H2O =0.967, 0.912 and 0.768, respectively. An even more pronounced enhancement of the local density can be achieved by lengthening the tail to an octyl chain.³¹ This automatically increases the molecular volume of the cation. As we have learnt from Table II this enhanced cationic volume leads to very high occupancies. In other words, the total cationic volume overwhelms all other species. Therefore, the enhanced hydrophobicity is merely a matter of occupancy. In the cited systems, the local density is increased to a maximum value of roughly 3. However, the position of the peak remains unchanged. In our systems, the peak of the second Voronoi shell located around 10Å is almost unaffected by the variation of the water content. The increased local density of the first Voronoi shell is in accordance with the increased number of cations in the solution. However, the enhancement of the mass density does not result in smaller distances between the tails of the cations. Nevertheless, approximately 60% of the cation–cation coordination number in Table II involves tail contributions. As a result, the $\mathcal{T}$-$\mathcal{T}$ configuration is the preferred configuration between two adjacent cations and support the thesis of “nonpolar” domains.³⁴ The g^L_i,0,L_ij(r) with L = 1,2 in Fig.5 show a preferred position in the proximity of the terminal CH₃-group. The change in sign of g²⁰²(r) indicates a splitting into two subgroups: Tails closer to the center-of-mass of the reference tail cover the right part of region 2 in Fig.3, i.e. they are aligned sideways with respect to the reference tail. The more distant tails above 6ÅA are in region 3 corresponding to position behind the terminal CH₃-group. The fluctuating sign of g¹¹⁰(r) and the clear sign pattern of g²²⁰(r) indicate an overall orthogonal orientation of tails pointing away from the reference tail. The second and third Voronoi shell show no preferred orientation.

Fig. 5.

First (a) and second (b) $g_{00}^{L0L}$ (r) of $\mathcal{T}-\mathcal{T}$ (tail-tail) angular correlation of BMIM⁺BF^–₄ in water at a mole fraction x_H2O = 0.768. The dotted, dashed and dash-dotted line correspond to the first, second and third Voronoi shell, respectively.

The first peak of the RDF g⁰⁰⁰(r) of $\mathcal{T}$-$\mathcal{A}$ (tail-anion) pair is entirely determined by the first Voronoi shell with a peak height of approximately 2.0. In all these RDFs the variation with water content is marginal. This value is lower compared to that of $\mathcal{H}$-$\mathcal{A}$ (head-anion) and $\mathcal{R}$-$\mathcal{A}$ (ring-anion). A similar trend is found for $\mathcal{T}$-$\mathcal{W}$,$\mathcal{R}$-$\mathcal{W}$ and $\mathcal{H}$-$\mathcal{W}$ in accordance with the expected hydrophobicity of the tails. From the combined analysis of g¹⁰¹ and g²⁰² a clear preference for the region 1 is observed. In other words, anion and water are actually in the proximity of the ring which is in accordance with simulated 1-octyl-3methylimidazolium nitrate water mixtures. ³¹

B. The hydrophilic cationic reference site

The local density of water in the first Voronoi shell of the imidazolium ring is a function of the mole fraction x_H2O. It exceeds the global density ρ by a mere 20% in case of x_H2O=0.967 but increases to 80% above ρ in case of x_H2O=0.768. One reason for this behavior might be the decreased number of water molecules in the simulation box. Consequently, the probability to find an other water molecule in the vicinity of the reference water decreases as these sites are now occupied by the ions. However, this stoichiometric effect is not sufficient to explain the excess of 80%. Rather, the enhanced viscosity of the mixture sharpens the structure. In molecular terms one would say the electric fields exerted by the additional ions orient the water dipoles. For $\mathcal{R}$-$\mathcal{W}$ this effect is not restricted to the raise of the first peak but also shows up in additional oscillations of g⁰⁰⁰(r) in Fig. 6a for the lowest mole fraction of x_H2O=0.768 extending up to 14Å. The decomposition into Voronoi shells reveals the origin of the small peak at 5.7Å: It comes from the superposition of the first and second Voronoi shell and does not correspond to a single discrete region or shell. This occurrence of virtual shells can only be detected by Voronoi decomposition technique. At this distance the molecules are either direct neighbors of the ring (first Voronoi shell) or indirect ones (second Voronoi shell) mediated by first neighbors. The rather different character of these Voronoi shell can be demonstrated by the hydrogen bond donor/acceptor function g⁰¹¹(r) in Fig.6b. Up to a distance of 5Å the imidazolium ring acts as a hydrogen bond donor for the direct water neighbors. In other words the water dipoles point away from the imidazolium ring. On the one hand, this is due to the higher affinity of the water oxygen to the hydrogen of the imidazolium ring. On the other hand, the water hydrogens are partially repelled by the hydrogens of the butyl chain attached to the ring. Beyond the distance of 5Å the donor behavior is reversed visible in negative values of g⁰¹¹(r). This remarkable change from a donor to an acceptor role of the ring can be seen in all three Voronoi shells in Fig.6b. As the Voronoi shells are shifted relative to each other on a distance scale the donor region of one shell overlaps with the acceptor region of the preceding shell. While g⁰¹¹ (r) describes the orientation of the water dipoles relative to the distance vector from the center of the ring, their position relative to the ring is characterized by g¹⁰¹ (r). Fig.6c shows a clear preference of 𝒲 in the first Voronoi shell to the region in the proximity of the H₂ hydrogen of the imidazolium ring. Positioning near H₄ and H₅ is less favorable in the first Voronoi shell. However, water molecules at approximately 5.9A in the second Voronoi shell are located here.

Fig. 6. Several g-coefficients describe the characteristics of the ℛ-𝒲 interaction.

a) The local density g⁰⁰⁰ is displayed for x_H2O = 0.967 (black), 0.912 (red) and 0.768 (green). In case of x_H2O = 0.768 the first (green dotted), the second (green dashed) and the third (green dash-dotted) Voronoi shell is shown. b) g⁰¹¹ reveals the donor/acceptor role of the im-idazolium hydrogens for the first (dotted), second (dashed) and third (dash-dotted) Voronoi shells at x⊓_2θ = 0.768. c) g¹⁰¹ indicates the position of the water molecules with respect to the reference imidazolium ring for the first (dotted), second (dashed) and third (dash-dotted) Voronoi shells at x_H2O = 0.768.

In contrast, the anions, which are direct neighbors of the ring, populate all regions near the imidazolium hydrogens as shown in Fig. 8. Anions closer than 4.9Å to the ring center are found near H₄ and H₅ complementary to nearest water molecules. Beyond that distance anion and water compete for the H₂ region irrespective of the Voronoi shell to which they belong. Fig.8 shows a characteristic sign pattern of g¹⁰¹(r) that occurs in all three Voronoi shells: A negative region at lower distances is followed by a positive one. This alternation in sign corresponds to a population of the H₄ and H₅ region first and the H₂ region afterwards. Since the first peak in the RDF g⁰⁰⁰(r) of $\mathcal{R}$-$\mathcal{A}$ is made up by the first Voronoi shell, other authors find similar locations for the first shell in case of pure BMIM⁺${\text{PF}}_6^{-}$.^3,7,33,44 But the second Voronoi shell which contain all neighbors of the direct neighbors of the BMIM⁺ has its maximum at approximately 8Å which coincides with the first minimum of the overall g⁰⁰⁰ (r). Consequently, this shell cannot be detected by a distance based threshold as used by the cited literature above. In fact, the second maximum in g⁰⁰⁰(r) originates from a superposition of the second and third Voronoi shell. Since the anions and water molecules occupy the regions near the imidazolium hydrogens, the neighboring cations are forced to reside above and below the ring. This can also be deduced from the $\mathcal{R}$-$\mathcal{T}$ g¹⁰¹ (r) (data not shown).

Fig. 8.

g¹⁰¹-coefficient of $ℛ-\mathcal{A}$ at a mole fraction of x_H2O = 0.768. The solid, dotted, dashed and dash-dotted line represent the overall, first, second and third Voronoi shell, respectively.

The ℋ-𝒜 RDF in Fig. 9a displays a significant aggregation of anions at 4.2ÅA which increases with decreasing water content. This effect seems to be more pronounced if the tail of the cation is elongated to an octyl chain.³¹ The corresponding g²⁰²(r) is negative at this distance which we attributed to region 2 in Fig. 3. This means that the anions prefer an equatorial position in accordance with the findings of the $\mathcal{R}$-$\mathcal{A}$ pair correlations. It seems that the affinity of the imidazolium hydrogens for the anions is stronger compared to the methyl hydrogens. Consequently, anions in the vicinity of the methyl-group $\mathcal{H}$ are always near the imidazolium hydrogens, too. This result in the equatorial position of BF-with respect to $\mathcal{H}$. This feature seems common for all ionic liquids since it was found in numerous molecular dynamics simulations in literature.^3,5,33,44 Furthermore, it is supported by density functional theory calculations on a 6-31++G(d,p) basis set.⁴⁵ In principle, all the observations concerning $\mathcal{H}$-$\mathcal{A}$ are also valid for $\mathcal{H}$-$\mathcal{W}$ in our mixtures but the local density, positioning and orientation is slightly reduced.

Fig. 9.

Local density (a) and position (b) of the anions with respect to the reference methyl-group (ℋ) of the cations.

As a first summary we present a schematic 3D vector diagram of cationic neighborhood in Fig. 7. The tail $\mathcal{T}$ of that reference BMIM⁺ is located in the lower left corner, the head in the upper left. The viewpoint is towards H₄ and H₅ of the imidazolium ring. The blue arrows stand for the dipole moments of the adjacent cations and represent their average orientation. The positioning of the cations is given by the 3D bin from which the arrow emanates. Only bins with population of more than 2.5 times the average density are shown. In case of the anions the threshold was doubled. One has to bear in mind that the arrows do not represent individual cations but display the favored bins. The length and direction of the arrow was calculated by averaging all dipole vectors of that bin. If there exists no preferred dipolar orientation, the resulting arrow is very small, e.g. some blue arrows in the vicinity of the tail in lower left corner. In the vicinity of the ring, the direction of cationic dipoles is already more pronounced. In case of a preferred dipolar orientation, the direction is confined to an angular cone and the arrow is of high length. This is indeed the case for water dipoles near ring $\mathcal{R}$ (red arrows). Since ${\text{BF}}_4^{-}$ exhibits small temporarily fluctuating dipoles, only their position is depicted by green surfaces.

Many features discussed on the basis of g-functions can be found in Fig. 7 again. The preferred positions of cations around the reference BMIM⁺ are at the tail region as well as above and under the ring. The cationic dipole vectors point away as showed by the respective g¹¹⁰(r). From the g-coefficient analysis anion and water molecules are located near the imidazolium ring. Interestingly, all three imidazolium hydrogens in the 3D picture 7 appear to be beset with anions and water molecules. At first sight, this seems to be in contradiction to the findings from the g-coefficients, which postulate a strong affinity of $\mathcal{W}$ to the imidazolium H₂. A more detailed inspection, however, shows that the water molecules in the proximity of H₄ and H₅ are not first Voronoi shell members, but belong to the second shell due to the respective negative sign of $\mathcal{R}$ - $\mathcal{W}$ g¹⁰¹(r) (dashed line in Fig.6c). The distance of 6AÅ appears to be rather short for second neighbors as it demands an interstitial partner. However, this needs not to be a complete molecule. Based on our atomwise tessellation an interstitial fluorine atom of ${\text{BF}}_4^{-}$ is sufficient. Altogether we arrive at the conclusion that second neighbors to the H₄ and H₅ (curved green surface in front of the BMIM⁺ in Fig.7) do not deserve the attribute “hydrogen bonded”. The confined angular freedom of the water dipoles is intuitively visible in the long red arrows of Fig. 7 and is rationally found in the high peak of$\mathcal{R}$-$\mathcal{W}$ g⁰¹¹(r) in Fig.6b.

C. The anion–water network

So far our analysis has mainly dealt with the correlations of the cations with the anion-water network.^15,46 Now, we turn to the anion-water network itself. The RDFs of this network decomposes into two classes:$\mathcal{W}$-$\mathcal{W}$ and $\mathcal{A}$-$\mathcal{W}$ look similar and show a pronounced peak at 2.8 and 3.7Å, respectively. This peak can be exclusively attributed to the first Voronoi-shell. It increases with decreasing water content which is in accordance with simulation results of dimethylimidazolium chloride mixtures with water.⁴⁷ Beyond this peak the local density is very similar to the global one. The common feature of both interaction pairs is the dipolar interaction, where in case of $\mathcal{A}$-$\mathcal{W}$ the anionic dipole is induced by the distortion of the tetrahedral geometry of ${\text{BF}}_4^{-}.$ This is a consequence of the strong hydrogen bond acceptor role of the anion which is also reflected in a strong negative peak of g⁰¹¹(r) of anion and water (data not shown). On contrast, the $\mathcal{A}-\mathcal{A}$ RDF is dominated by charge–charge interaction as well as by the larger effective radius of the anion. Consequently, the first Voronoi peak in Fig. 10a-c is shifted to distances around 5.9Å. Moreover, the position and the height of this peak does not change with x_H2O. The repulsion of the anions strongly reduces the possibility for direct neighbors. Hence, the first Voronoi peak achieves only 80% of the global density, even the second Voronoi peak exceeds the global density by a mere of 20%. This second peak in Fig. 10a-c exhibits a double shape which can be attributed to an interstitial neighbor. At short distances, this neighbor is probably water whereas at longer distances cations are possible, too. The double shape of the second peak is also responsible for the peak structure of the overall RDF. While its peak constitutes the second peak of the overall RDF, its shoulder together with the first Voronoi peak gives as a superposition the first peak of the overall RDF which resembles the corresponding g⁰⁰⁰(r) of ${\text{NO}}_3^{-}$ in a simulation of a 1-octyl-3-methyl-imidazolium nitrate/water mixture.³¹

Fig. 10.

Local density of the anions near anions at different x_H2O = 0.768 (a), 0.912 (b) and 0.967 (c). Additionally, the first (dotted) and second (dashed) Voronoi shells are visible.

The first negative peak in g¹⁰¹(r) of $\mathcal{W}-\mathcal{W}$ occurs at 2.8Å, i.e. at the same distance as the corresponding RDF g⁰⁰⁰ (r). As this represents the shortest possible oxygen–oxygen distance a neighboring water molecule can only attack the oxygen site of the reference water and not its hydrogens. The dipole of the reference water points in the direction of the above mentioned hydrogens. Therefore, the angle between this dipole vector and the distance vector pointing to the neighbor waters is almost 180° resulting in the negative sign of the peak in g¹⁰¹(r) (region 3). At larger distance but still in the first Voronoi shell the region around the hydrogens is populated. Consequently, the sign of g¹⁰¹(r) is reversed. This reversal in sign within the very same Voronoi shell can be observed in all three Voronoi shells.

Altogether, the features found and analyzed in $\mathcal{A}-\mathcal{A}$, $\mathcal{A}-\mathcal{W}$ and $\mathcal{W}-\mathcal{W}$ correlations are indicative for a strong anion-water network. This network strengthens with decreasing water content. In other words, in the competition of the ions for water, the anion strongly overrules the cations thus expelling the cations from the anion-water network. The cations organize themselves in a hydrophobic $\mathcal{T}-\mathcal{T}$ network. Consequently, the coordination number of BMIM⁺ around BMIM⁺ increases from 2.5 at x_H2O=0.967 to 9.4 at x_H2O=0.768. This effect is not caused by the stoichiometry since the coordination number of the anions increases much slower from 2.1 to 5.6 in Table II at the above mentioned water contents. This clustering of tails was already found in the beginning of this discussion, but now we observe even an indirect influence on ℋ-ℋ and ℛ-ℛ correlations. This can be directly seen in Fig. 4b for the first Voronoi shell of the ℛ-ℛ packing. The peak raises from 20% to 45% of the global density with decreasing water content and its position is shifted to shorter distances. This behavior may be interpreted as ring-ring stacking. This stacking is not restricted to mixtures between ionic liquids and water but can be also found in simulations of pure ionic liquids.^6,7

D. Coordination numbers

Traditionally, the coordination number c.n. is calculated by integrating the RDF over spherical shells up to a certain threshold r_cut:

$$c.n.=4\pi \frac{N_s}{V}{\displaystyle \underset{0}{\overset{r_{cut}}{\int }}{g}^{000}\left(r\right)r^{2}dr}$$ (2)

where N_s is the number of molecules of species s and V the total volume of the sample. The basic problem of this procedure is the choice of the threshold r_cut: In case of several species, r_cut has to be defined for each possible pair. For example, Ref. 48 operates with five r_cut in the range from 3.804Å to 5.551Å. Even the three imidazolium hydrogens have a spread of 1.2Å. These values are defined as the first minimum of the respective g⁰⁰⁰(r). Usually the distance spread of Voronoi shells is larger than 5Å as visible in the Fig. 4, 6, 10 and 11. As a result, the coordination numbers derived from them are bigger than those computed by the integration of the radial distribution function up to a more or less arbitrary limit.

Fig. 11.

Local density of the water near water at different x_H2O = 0.768 (a), 0.912 (b) and 0.967 (c). Additionally, the first (dotted) and second (dashed) Voronoi shells are visible. The first Voronoi shell lasts longer than the first minimum of the overall g⁰⁰⁰ (r).

In this paper we have found numerous examples rendering the r_cut-method questionable: First, the first shell does not need to coincide with the first peak in g⁰⁰⁰(r), e.g. ℛ-ℛ in Fig.4b. Second, the overlap of two shells on a radial scale may create superposition peaks, e.g. Fig.10a-c. Third, in extreme cases the minimum of g⁰⁰⁰(r) coincides with the maximum of a shell, e.g. the third Voronoi shell in Fig.4b. Fortunately, the Voronoi decomposition provides coordination numbers without any further calculation by simply counting the members of the first Voronoi shell. Therefore, it is free of integration errors due to low statistics and/or inappropriate integration limits. Of course, if one applies Eq. 2 to the g⁰⁰⁰(r) of the first Voronoi shell, the very same Voronoi c.n. is obtained.

The difference of c.n. derived by Voronoi decomposition and r_cut-method can be demonstrated for the $\mathcal{W}-\mathcal{W}$ correlation. A threshold of r_cut =3.5Å representing the minimum of oxygen-oxygen RDF is well established in the literature.^30,47,49,50 However, the first Voronoi shell for all mixtures x_H2O = 0.768, 0.912 and 0.967 is depicted in Fig.11a-c as dotted line. The first peak of g⁰⁰⁰(r) is exclusively made up by the first Voronoi shell, but this shell does not end at 3.5Å. It has a shoulder gradually declining up to 6Å. The second shell starts at approximately 4Å The traditional threshold r_cut ensures that no indirect neighbors are taken into account but misses some direct neighbors: The c.n. by means of r_cut are 2.0, 3.2 and 3.8 for x_H2O = 0.768, 0.912 and 0.967, respectively. The Voronoi method (including the shoulder) gives alternative values of 4.3, 8.4 and 12.2 in Table II. It is clear that water molecules beyond 3.5Å are not “hydrogen-bonded” to the reference. Nevertheless, they are direct neighbors filling up the free spaces left by the “hydrogen-bonded” neighbors in order to complete the first shell. Although not “hydrogen-bonded” these space-filling direct neighbors still have strong electrostatic interactions with the reference water due to high value of the local partial charges. We emphasize that the used term “hydrogen-bonded” is solely defined on a distance criterion and implies no preferred orientations between water molecules which is normally necessary to completely define an hydrogen bond.

The normalization of c.n. by N_s is better suited than the absolute c.n. for the comparison of direct neighborhood and RDFs because the influence of the stoichiometry is ruled out as can be seen in Eq. 2. For well equilibrated and sufficiently sampled mixtures a constant ratio c.n./N_s should be achieved. In other words, global particle ratios should be also found on a local scale. Dividing the c.n.s by the total number of water molecules, the Voronoi c.n. values are indeed fairly constant while the r_cut values decrease considerably. This rather different behavior comes from the gradual transformation of “hydrogen-bonded” neighbors to space-filling direct neighbors with increasing water content. In Fig.11a-c a decrement of the first peak height is more or less perfectly balanced by an increment of the accompanying shoulder. With increasing water content the stabilizing effect of the ionic field is weakened. Consequently, “hydrogen bonding” is destabilized in some cases. Nevertheless, the respective water molecules remain direct neighbors.

Table II gives the absolute c.n. for all pairs of species in order to be comparable with literature data. For example, the cation-anion c.n. of 5.6 at x_H2O=0.768 corresponds well to the value reported in Ref. 45 in case of EMIM⁺${\text{BF}}_4^{-}$. There, the coordination number of six was attributed to a somehow dynamical coordination mechanism. The static quantum-mechanical calculations in Ref. 45 suggest a c.n. of four ${\text{BF}}_4^{-}$ near the cation which was found by a molecular dynamics study as well.⁴⁶ On the one hand, the higher c.n. in our simulation may be due to our cation BMIM⁺ which is significantly larger than EMIM⁺. On the other hand, water is also present in vicinity of the cation. Therefore, repulsive forces of two coordinating anions may be screened by interstitial waters. However, as discussed above, our main interpretation is based on the ratio of c.n./N_s. This has the additional benefit that the mutual c.n.(AB)/N_B equals c.n.(BA)/N_A. Furthermore, these ratios are rather independent of the water mole fraction x_H2O and comparable to the g-coefficients. From the cations point of view the preferred neighbor is also a cation in accordance with the hydrophobic association of tails. The anionic ratio is approximately 30% higher compared to water. This is to be expected from electrostatic interactions as well as the repulsion of cations from the anion-water network. Since the simpler anions lack an hydrophobic part, their mutual electrostatic repulsion strongly reduces the c.n./N_s ratio. Analogously, electrostatic attraction leads to the highest ratio with anions as reference sites. In case of water as reference site, many neighbors exceed by far the reference water in size. As a result, the favorite position of neighbors changes drastically and may explain the correlation of the c.n./N_s ratios with the square of the peak position of the respective g⁰⁰⁰(r) of the first Voronoi shell. Therefore, we favor in this case the interpretation in terms of g⁰⁰⁰(r) rather than mere coordination numbers.

IV. Conclusion

The Voronoi decomposition is a rational concept of proximity or neighborhood. It can be applied to a variety of analyses either spatial or temporal. In this pilot study we have combined the Voronoi decomposition with structural g-coefficients: In case of rather disjoint Voronoi shells, the peak pattern of the respective radial distribution function g⁰⁰⁰(r) is in accordance with the sequence of Voronoi shells. In other words, the first peak of g⁰⁰⁰(r) mainly consists of molecules in the first Voronoi shell, the second peak is constituted by the second Voronoi shell and so on … An example for this case is the tail-tail distribution of the cations in this work. In other cases, peaks in the radial distribution function are a superposition of different Voronoi shells. Hence, particles located around this distance are of hybrid nature incorporating two different neighborhoods, e.g. the first peak of the anion-anion radial distribution. In extreme cases, the first peak of the radial distribution does not represent the direct neighbors of the reference site, but their neighbors, i.e. the second Voronoi shell. A typical example is the ring-ring distribution. This has severe consequence for the interpretation of 3D occupancy plots which are based on a certain threshold. If this threshold is above the maximum of the first Voronoi shell the occupancy plot show only indirect neighbors and misses the direct ones. Lowering the threshold blurs the picture as it mixes direct and indirect neighbors.

Besides these general findings exceeding traditional methods the combined Voronoi-g-coefficient analysis reveals the well established features of ionic liquid/water mixtures: The hydrophobicity of the tails, the enhanced presence of water and anions in the proximity of the ring as well as the existence of a strong anion–water network. In a more detailed view, the first Voronoi shell plays a special role trying to compensate the major part of anisotropy of the reference site. A typical example are water molecules in the direct neighborhood of a water molecule. Some of these direct neighbors are “hydrogen-bonded” whereas others complete the first shell to a more isotropic body. Hence, the first Voronoi shell is most often characterized by an uniform sign of the respective g⁰¹¹(r) and g ¹⁰¹(r)-coefficients. Despite their strong coupling in an anion–water network both species compete for favorable positions in the proximity of the imidazolium hydrogens. While water seems to be restricted to the H₂ region, the anion is more ubiquitous. The most important feature of our coordination number analysis is the independence of the ratio c.n./N_s of the water mole fraction.