Molecular dynamics simulations have been performed to study the complex and heterogeneous dynamics of ions in ionic liquids. The dynamics of cations and anions in 1-ethyl-3-methyl imidazolium nitrate (EMIM-NO(3)) are characterized by van Hove functions and the corresponding intermediate scattering functions F(s)(k,t) and elucidated by the trajectories augmented by the use of singular spectrum analysis (SSA). Several time regions are found in the mean squared displacement of the ions. Change in the slope in a plot of the diffusion coefficient against temperature is found at around 410 K in the simulation. Heterogeneous dynamics with the presence of both localized ions and fast ions capable of successive jumps were observed at long time scales in the self-part of the van Hove functions and in the trajectories. Non-Gaussian dynamics are evidenced by the self-part of the van Hove functions and wave number dependence of F(s)(k,t) and characterized as Levy flights. Successive motion of some ions can continue even after several nanoseconds at 370 K, which is longer than the onset time of diffusive motion, t(dif). Structure of the long time dynamics of fast ions is clarified by the phase space plot of the successive motion using the denoised data by SSA. The continual dynamics are shown to have a long term memory, and therefore local structure is not enough to explain the heterogeneity. The motion connecting localized regions at about 370 K is jumplike, but there is no typical one due to local structural changes during jump motion. With the local motion, mutual diffusion between cation and anion occurs. On decreasing temperature, mutual diffusion is suppressed, which results in slowing down of the dynamics. This "mixing effect of cation and anion" is compared with the "mixed alkali effect" found in the ionics in the ionically conducting glasses, where the interception of paths by different alkali metal ions causes the large reduction in the dynamics [J. Habasaki and K. L. Ngai, Phys. Chem. Chem. Phys. 9, 4673 (2007), and references herein]. Although a similar mechanism of the slowing down is observed, strong coupling of the motion of cation and anion prevents complete interception unless deeply supercooled, and this explains the wide temperature region of the existence of the liquid and supercooled liquid states in the ionic liquid.
By now it is well established that the structural α-relaxation time, τ(α), of non-associated small molecular and polymeric glass-formers obey thermodynamic scaling. In other words, τ(α) is a function Φ of the product variable, ρ(γ)/T, where ρ is the density and T the temperature. The constant γ as well as the function, τ(α) = Φ(ρ(γ)/T), is material dependent. Actually this dependence of τ(α) on ρ(γ)/T originates from the dependence on the same product variable of the Johari-Goldstein β-relaxation time, τ(β), or the primitive relaxation time, τ(0), of the coupling model. To support this assertion, we give evidences from various sources itemized as follows. (1) The invariance of the relation between τ(α) and τ(β) or τ(0) to widely different combinations of pressure and temperature. (2) Experimental dielectric and viscosity data of glass-forming van der Waals liquids and polymer. (3) Molecular dynamics simulations of binary Lennard-Jones (LJ) models, the Lewis-Wahnström model of ortho-terphenyl, 1,4 polybutadiene, a room temperature ionic liquid, 1-ethyl-3-methylimidazolium nitrate, and a molten salt 2Ca(NO(3))(2)·3KNO(3) (CKN). (4) Both diffusivity and structural relaxation time, as well as the breakdown of Stokes-Einstein relation in CKN obey thermodynamic scaling by ρ(γ)/T with the same γ. (5) In polymers, the chain normal mode relaxation time, τ(N), is another function of ρ(γ)/T with the same γ as segmental relaxation time τ(α). (6) While the data of τ(α) from simulations for the full LJ binary mixture obey very well the thermodynamic scaling, it is strongly violated when the LJ interaction potential is truncated beyond typical inter-particle distance, although in both cases the repulsive pair potentials coincide for some distances.
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1.0 , corresponding to long-range diffusion. Both t  and t 1.0 terms have strong temperature dependence and they are the analogs of the ac conductivity ͓()ϰ 1Ϫ ͔ and dc conductivity of hopping ions. The MD results in conjunction with the coupling model support the following proposed interpretation for conductivity relaxation of ionic conductors: ͑1͒ the NCL originates from very slow initial decay of the cage with time caused by few independent hops of the ions because t x1 Ӷ o , where o is the independent hop relaxation time; ͑2͒ the broad crossover from the NCL to the cooperative ion hopping conductivity ()ϰ 1Ϫ occurs when the cage decays more rapidly starting at t x1 ; ͑3͒ ()ϰ 1Ϫ is fully established at a time t x2 comparable to o when the cage has decayed to such an extent that thereafter all ions participate in the slowed dynamics of cooperative jump motion; and ͑4͒ finally, at long times ͑͒ becomes frequency independent, i.e., the dc conductivity. MD simulations show the non-Gaussian parameter peaks at approximately t x2 and the motion of the Li ϩ ions is dynamically heterogeneous. Roughly divided into two categories of slow ͑A͒ and fast ͑B͒ moving ions, their mean square displacements ͗r A 2 ͘ and ͗r B 2 ͘ are about the same for tϽt x2 , but ͗r B 2 ͘ of the fast ions increases much more rapidly for tϾt x2 . The self-part of the van Hove function of Li ϩ reveals that first jumps for some Li ϩ ions, which are apparently independent free jumps, have taken place before t x2 . While after t x2 the angle between the first jump and the next is affected by the other ions, again indicating cooperative jump motion. The dynamic properties are analogous to those found in supercooled colloidal particle suspension by confocal microscopy. I. BACKGROUNDThe most commonly used experimental technique to probe the ions is electrical conductivity relaxation that measures the macroscopic dielectric response of a sample as a function of frequency. The conductance and capacitance of the sample are usually measured and from the results the complex dielectric susceptibility *() and complex conductivity *() are obtained ͓1-10͔, or directly the complex electric modulus M *() is obtained ͓11,12͔. The frequency dependence of data in the complex conductivity representation usually is well described by using Jonscher's expression ͓7͔where 0 is the dc conductivity, p a characteristic relaxation frequency, and n J a fractional exponent. On the other hand, the same data in the electric modulus representation are also well described by the one-sided Fourier transform,of the Kohlrausch stretched exponential function ͓1-5,13͔⌽͑t ͒ϭexp͓Ϫ͑ t/ ͒ 1Ϫn ͔. ͑3͒The Ј() obtained from Eqs. ͑2͒ and ͑3͒ is similar to the Jonscher's expression in having the dc conductivity at low frequencies and increases as a power law () n at high frequencies.
We have examined the relaxation behavior of alkali metal ions in lithium metasilicate glasses by means of molecular dynamics simulation. We have observed a change of slope of the mean squared displacement at ϳ300 ps. In shorter time regions, localized motion of lithium ions within neighboring sites is observed, which is caused by the small fracton dimension ͑fracton excitation͒. On the other hand, an accelerated motion of particles due to cooperative jumps is found, which characterizes the diffusion and conduction mechanisms of the alkali metal ions in longer time regions. The dynamics of accelerated motion is discussed in relation to Lévy flight dynamics. ͓S0163-1829͑97͒03510-8͔
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