^{1}, C. Schröder

^{1}and O. Steinhauser

^{1,a)}

### Abstract

The relaxation of solvation shells is studied following a twofold strategy based on a direct analysis of simulated data as well as on a solution of a Markovian master equation. In both cases solvation shells are constructed by Voronoi decomposition or equivalent Delaunay tessellation. The theoretical framework is applied to two types of hydrated molecular ionic liquids, 1-butyl-3-methyl-imidazolium tetrafluoroborate and 1-ethyl-3-methyl-imidazolium trifluoromethylsulfonate, both mixed with water. Molecular dynamics simulations of both systems were performed at various mole fractions of water. A linear relationship between the mean residence time and the system’s viscosity is found from the direct analysis independent of the system’s type. The complex time behavior of shell relaxation can be modeled by a Kohlrausch–Williams–Watts function with an almost universal stretching parameter of 1/2 indicative of a square root time law. The probabilistic model enables an intuitive interpretation of essential motional parameters otherwise not accessible by direct analysis. Even more, incorporating the square root time law into the probabilistic model enables a quantitative prediction of shell relaxation from very short simulation studies. In particular, the viscosity of the respective systems can be predicted.

This work was supported by the Austrian Science Fund FWF (Project No. P19807). Furthermore, we would like to thank the Institute of Scientific Computing at the University of Vienna for a generous allocation of computer time.

I. INTRODUCTION

II. THEORY

A. Molecular dynamics of Voronoi shells

1. Tessellation and Voronoi shells

2. Time series and time correlation functions

3. Representation of TCF

B. Probabilistic approach

1. Markovian master equation

2. Probabilistic analog of TCF

3. Simple example

III. METHODS

A. Implementation of tessellation

B. Details of simulation

C. Construction of transition matrix

IV. RESULTS AND DISCUSSION

A. Motional parameters of Voronoi dynamics

1. Mean residence times

2. Distribution of relaxation times

B. Probabilistic approach

1. Reduction in spectral expansion to a system

2. Comparison of KWW and Markov amplitudes

3. Consistency check

4. Proximity resolved populations

5. Matching molecular dynamics and probabilistic approach

V. CONCLUSION

### Key Topics

- Viscosity
- 25.0
- Eigenvalues
- 19.0
- Relaxation times
- 16.0
- Correlation functions
- 11.0
- Polyhedra
- 5.0

## Figures

Duality of Voronoi and Delaunay tessellations. In this simplistic scheme, four points (A, B, C, D) are connected via their Delaunay (red, dashed-dotted) and Voronoi (green, dashed) diagrams. The two diagrams are dual, i.e., can be converted into each other without any further information. By connecting all points that share a common Voronoi face, the Delaunay diagram can be drawn. For example, points C and D share the common Voronoi face x-y (green) and are thus connected in the Delaunay diagram (red). The Voronoi nodes (x,y), in turn, coincide with the centers of Delaunay circumcircles. Given a Delaunay tessellation, the Voronoi nodes, e.g., x and y in the figure, are obtained in the following way. In two dimensions for each triangle, e.g., A-C-D, a circumcircle can be constructed. Its center coincides with the vertex x of a Voronoi polygon. In the three dimensional case a Voronoi vertex is identical to the center of the Delaunay tetrahedron’s circumsphere.

Duality of Voronoi and Delaunay tessellations. In this simplistic scheme, four points (A, B, C, D) are connected via their Delaunay (red, dashed-dotted) and Voronoi (green, dashed) diagrams. The two diagrams are dual, i.e., can be converted into each other without any further information. By connecting all points that share a common Voronoi face, the Delaunay diagram can be drawn. For example, points C and D share the common Voronoi face x-y (green) and are thus connected in the Delaunay diagram (red). The Voronoi nodes (x,y), in turn, coincide with the centers of Delaunay circumcircles. Given a Delaunay tessellation, the Voronoi nodes, e.g., x and y in the figure, are obtained in the following way. In two dimensions for each triangle, e.g., A-C-D, a circumcircle can be constructed. Its center coincides with the vertex x of a Voronoi polygon. In the three dimensional case a Voronoi vertex is identical to the center of the Delaunay tetrahedron’s circumsphere.

Proximity criterion. The two water molecules are both members of the first shell of the BMIM molecule but are assigned to two different states. The distance between water molecule A and the BMIM molecule is shortest at the butyl-terminal methyl group, thus molecule A is populating state seven of an eight state C-W matrix. Water molecule B is populating state four, as its proximity is maximal with respect to butyl hydrogen .

Proximity criterion. The two water molecules are both members of the first shell of the BMIM molecule but are assigned to two different states. The distance between water molecule A and the BMIM molecule is shortest at the butyl-terminal methyl group, thus molecule A is populating state seven of an eight state C-W matrix. Water molecule B is populating state four, as its proximity is maximal with respect to butyl hydrogen .

Residence correlation function for C-all of the system with (red solid line). The respective fit to a KWW function is given as a green dashed line. Please note the logarithmic time scale. The short time and long time limits are 26.3 and 1.02. This means that in the beginning the cation is surrounded by 26 other molecules of all three species. After 10 ns 25 particles have migrated from the cation, but one is still a resident.

Residence correlation function for C-all of the system with (red solid line). The respective fit to a KWW function is given as a green dashed line. Please note the logarithmic time scale. The short time and long time limits are 26.3 and 1.02. This means that in the beginning the cation is surrounded by 26 other molecules of all three species. After 10 ns 25 particles have migrated from the cation, but one is still a resident.

Plot of the system specific MRT vs the respective viscosity . The logarithmic stretching of both axes was chosen in order to cope with the wide spread of viscosity values. The three systems are denoted by red squares, the systems by blue circles.

Plot of the system specific MRT vs the respective viscosity . The logarithmic stretching of both axes was chosen in order to cope with the wide spread of viscosity values. The three systems are denoted by red squares, the systems by blue circles.

The dominant C-all Markov residence time is plotted vs the viscosity as a representative of the respective system. The three systems are denoted by red squares, the systems by blue circles. The straight line symbolizes the relation . Due to the wide spread of viscosity values a logarithmic scale stretching is used.

The dominant C-all Markov residence time is plotted vs the viscosity as a representative of the respective system. The three systems are denoted by red squares, the systems by blue circles. The straight line symbolizes the relation . Due to the wide spread of viscosity values a logarithmic scale stretching is used.

The dominant Markov residence time resolved for all 25 proximity sites of . Three combinations BMIM-BMIM (red diamonds), BMIM-TFB (green circles), and BMIM-water (blue squares).

The dominant Markov residence time resolved for all 25 proximity sites of . Three combinations BMIM-BMIM (red diamonds), BMIM-TFB (green circles), and BMIM-water (blue squares).

Modeling the simulated by the probabilistic approach for the case of with . The solid green line stands for the time evolution of the net shell population . The underlying transition matrix is for . The universal stretch parameter is . The residence correlation function is displayed as blue circles.

Modeling the simulated by the probabilistic approach for the case of with . The solid green line stands for the time evolution of the net shell population . The underlying transition matrix is for . The universal stretch parameter is . The residence correlation function is displayed as blue circles.

## Tables

Composition and coordination of the simulated MIL/water mixtures.

Composition and coordination of the simulated MIL/water mixtures.

Relaxation KWW parameters as obtained from the fit of the residence autocorrelation function .

Relaxation KWW parameters as obtained from the fit of the residence autocorrelation function .

Collection of relaxation amplitudes. The results from the KWW fit are compared with those from the Markovian master equation.

Collection of relaxation amplitudes. The results from the KWW fit are compared with those from the Markovian master equation.

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