^{1,3}, R. Cortes-Huerto

^{2}and P. Ballone

^{3}

### Abstract

The glass transition in prototypical room temperature ionic liquids has been investigated by molecular dynamics simulations based on an Amber-like empirical force field. Samples of [C_{4}mim][PF_{6}], [C_{4}mim][Tf_{2}N], and [C_{3}mim][Tf_{2}N] have been quenched from the liquid phase at *T* = 500 to a glassy state at *T* ∼ 0 K in discontinuous steps of 20 K every 1.2 ns. The glass temperature estimated by simulation (*T* _{ g } = 209 K for [C_{4}mim][PF_{6}], *T* _{ g } = 204 K for [C_{4}mim][Tf_{2}N], and *T* _{ g } = 196 K for [C_{3}mim][Tf_{2}N]) agrees semi-quantitatively with the experimental values (*T* _{ g } = 193÷196 K for [C_{4}mim][PF_{6}], *T* _{ g } = 186÷189 K for [C_{4}mim][Tf_{2}N], and *T* _{ g } = 183 K for [C_{3}mim][Tf_{2}N]). A model electron density is introduced to identify voids in the system. The temperature dependence of the size distribution of voids provided by simulation reproduce well the experimental results of positron annihilation lifetime spectroscopy reported in G. Dlubek, Y. Yu, R. Krause-Rehberg, W. Beichel, S. Bulut, N. Pogodina, I. Krossing, and Ch. Friedrich, J. Chem. Phys.133, 124502 (2010), with only one free parameter needed to fit the experimental data.

Simulations have been carried out on the High Performance Computing facilities of the Centre Interdisciplinaire de Nanoscience du Centre National de la Recherche Scientifique (CINaM-CNRS) at Aix-Marseille Université. One of us (P.B.) thanks the computational materials science of the Science and Technology Facilities Council (STFC, Rutherford Appleton Laboratory) for their kind hospitality during the final stages of this study.

I. INTRODUCTION

II. THE MODEL AND THE SIMULATION METHOD

III. THE COMPUTATION OF THE VOID DISTRIBUTION

IV. SIMULATION RESULTS

A. The glass transition

B. The distribution of voids

V. SUMMARY AND CONCLUSIVE REMARKS

### Key Topics

- Glass transitions
- 34.0
- Interpolation
- 13.0
- Molecular dynamics
- 9.0
- Probability theory
- 8.0
- Classical statistical mechanics
- 7.0

##### C21D1/62

## Figures

Schematic representation of the [C_{3}mim]^{+} and [C_{4}mim]^{+} cations and [Tf_{2}N]^{−}, [PF_{6}]^{−} anions considered in the present simulation study.

Schematic representation of the [C_{3}mim]^{+} and [C_{4}mim]^{+} cations and [Tf_{2}N]^{−}, [PF_{6}]^{−} anions considered in the present simulation study.

Comparison of the atomic and pseudo-electron charge density of carbon. The two radial functions match continuously and with continuous first derivative at the core radius *r* _{ c } = 2.2 a.u. The atomic charge density is computed within the local spin density approximation of density functional theory. Atomic and pseudo-atomic charge densities are assumed to be spherically symmetric.

Comparison of the atomic and pseudo-electron charge density of carbon. The two radial functions match continuously and with continuous first derivative at the core radius *r* _{ c } = 2.2 a.u. The atomic charge density is computed within the local spin density approximation of density functional theory. Atomic and pseudo-atomic charge densities are assumed to be spherically symmetric.

Comparison of the pseudo-electron charge density of carbon with its reconstruction from a Fourier representation with *E* _{ cut } = 69 Ry, corresponding to the lowest cut-off used in our analysis of voids.

Comparison of the pseudo-electron charge density of carbon with its reconstruction from a Fourier representation with *E* _{ cut } = 69 Ry, corresponding to the lowest cut-off used in our analysis of voids.

Temperature dependence of the average potential energy of [C_{4}mim][PF_{6}] (per ion pair) during the quench whose time schedule is described in the text. Dots: simulation results. Blue (full) line: linear interpolation of the 20 ⩽ *T* ⩽ 120 K range. Red (dash) line: linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. The error bar on the simulation data is less than the size of the dot.

Temperature dependence of the average potential energy of [C_{4}mim][PF_{6}] (per ion pair) during the quench whose time schedule is described in the text. Dots: simulation results. Blue (full) line: linear interpolation of the 20 ⩽ *T* ⩽ 120 K range. Red (dash) line: linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. The error bar on the simulation data is less than the size of the dot.

(a) Temperature dependence of Δ*U*(*T*) = *U*(*T*) − *a* _{ low } − *b* _{ low } *T* for [C_{4}mim][Tf_{2}N] during the quench whose time schedule is described in the text. *U*(*T*) is the average potential energy per atom pair, and *a* _{ low } + *b* _{ low } *T* is the linear interpolation to the 20 ⩽ *T* ⩽ 120 T data. Dots: simulation results. Dash line (blue): linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. The gray area points to the temperature range over which *U*(*T*) falls below the high temperature linear interpolation. (b) Same plot of part (a) for [C_{4}mim][PF_{6}].

(a) Temperature dependence of Δ*U*(*T*) = *U*(*T*) − *a* _{ low } − *b* _{ low } *T* for [C_{4}mim][Tf_{2}N] during the quench whose time schedule is described in the text. *U*(*T*) is the average potential energy per atom pair, and *a* _{ low } + *b* _{ low } *T* is the linear interpolation to the 20 ⩽ *T* ⩽ 120 T data. Dots: simulation results. Dash line (blue): linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. The gray area points to the temperature range over which *U*(*T*) falls below the high temperature linear interpolation. (b) Same plot of part (a) for [C_{4}mim][PF_{6}].

Constant pressure (*P* = 1 atm) specific heat *C* _{ p } of [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}], computed by differentiating the Padé interpolation of the system enthalpy (see text). *N* _{ a } is the total number of atoms and *k* _{ B } is the Boltzmann constant.

Constant pressure (*P* = 1 atm) specific heat *C* _{ p } of [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}], computed by differentiating the Padé interpolation of the system enthalpy (see text). *N* _{ a } is the total number of atoms and *k* _{ B } is the Boltzmann constant.

Temperature dependence of the average volume of [C_{4}mim][Tf_{2}N] during the quench whose time schedule is described in the text. Dots: simulation results. Full line (blue): linear interpolation of the 20 ⩽ *T* ⩽ 120 K range. Dash line (red): linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. *P* = 1 atm.

Temperature dependence of the average volume of [C_{4}mim][Tf_{2}N] during the quench whose time schedule is described in the text. Dots: simulation results. Full line (blue): linear interpolation of the 20 ⩽ *T* ⩽ 120 K range. Dash line (red): linear interpolation of the 340 ⩽ *T* ⩽ 500 K range. *P* = 1 atm.

Radial distribution functions for [C_{4}mim][Tf_{2}N] computed in the point-particle representation of cations and anions (see text). Upper panel: *T* = 320 K, in the equilibrium liquid phase; lower panel: *T* = 140 K, in the amorphous phase.

Radial distribution functions for [C_{4}mim][Tf_{2}N] computed in the point-particle representation of cations and anions (see text). Upper panel: *T* = 320 K, in the equilibrium liquid phase; lower panel: *T* = 140 K, in the amorphous phase.

Arrhenius plot of the diffusion coefficient of cations and anions in [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}]. The full (straight) lines are a guide to the eye. In each panel, the vertical straight lines identify the glass transition temperature *T* _{ g }.

Arrhenius plot of the diffusion coefficient of cations and anions in [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}]. The full (straight) lines are a guide to the eye. In each panel, the vertical straight lines identify the glass transition temperature *T* _{ g }.

Volume fraction of voids (in percent of the total volume) as a function of *T* for [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}].

Volume fraction of voids (in percent of the total volume) as a function of *T* for [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}].

Simulation snapshot of voids in [C_{3}mim][Tf_{2}N] at *T* = 300 K. Clusters have been painted in six different colours to identify them, and to highlight their size.

Simulation snapshot of voids in [C_{3}mim][Tf_{2}N] at *T* = 300 K. Clusters have been painted in six different colours to identify them, and to highlight their size.

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } in [C_{3}mim][Tf_{2}N], computed at three different temperatures from the voids' volume upon assuming a spherical hole geometry (see text).

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } in [C_{3}mim][Tf_{2}N], computed at three different temperatures from the voids' volume upon assuming a spherical hole geometry (see text).

Volume fraction of voids (in percent of the total volume) for [C_{3}mim][Tf_{2}N] as a function of *T*, computed at (*r* _{ s } = 8), and at (*r* _{ s } = 6), for [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}].

Volume fraction of voids (in percent of the total volume) for [C_{3}mim][Tf_{2}N] as a function of *T*, computed at (*r* _{ s } = 8), and at (*r* _{ s } = 6), for [C_{3}mim][Tf_{2}N], [C_{4}mim][Tf_{2}N], and [C_{4}mim][PF_{6}].

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } in [C_{3}mim][Tf_{2}N] at *T* = 360 K computed at two different values of the cut-off density (see text).

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } in [C_{3}mim][Tf_{2}N] at *T* = 360 K computed at two different values of the cut-off density (see text).

Average volume ⟨*v* _{ h }⟩ of voids in [C_{3}mim][Tf_{2}N] as a function of temperature. The simulation data for ⟨*v* _{ h }⟩ have been rescaled by the factor ⟨*n*⟩ accounting for the coalescence of primary holes (see text).

Average volume ⟨*v* _{ h }⟩ of voids in [C_{3}mim][Tf_{2}N] as a function of temperature. The simulation data for ⟨*v* _{ h }⟩ have been rescaled by the factor ⟨*n*⟩ accounting for the coalescence of primary holes (see text).

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } from the [C_{3}mim][Tf_{2}N] simulations shown on a semi-logarithmic scale.

Probability distribution *p*(*r* _{ v }) of the voids' radius *r* _{ v } from the [C_{3}mim][Tf_{2}N] simulations shown on a semi-logarithmic scale.

## Tables

Cut-off radii (atomic units) used to generate the pseudo-charge densities (see text).

Cut-off radii (atomic units) used to generate the pseudo-charge densities (see text).

Glass transition temperature estimated by simulation and measured in experiments.

Glass transition temperature estimated by simulation and measured in experiments.

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