Dynamic Heterogeneity in Room-Temperature Ionic Liquids

Ionic Liquids (ILs) are one of the most interesting and rapidly developing areas of modern physical chemistry, technologies and engineering. This book, consisting of 29 chapters gathered in 4 sections, reviews in detail and compiles information about some important physical-chemical properties of ILs and new practical approaches. This is the first book of a series of forthcoming publications on this field by this publisher. The first volume covers some aspects of synthesis, isolation, production, modification, the analysis methods and modeling to reveal the structures and properties of some room temperature ILs, as well as their new possible applications. The book will be of help to chemists, physicists, biologists, technologists and other experts in a variety of disciplines, both academic and industrial, as well as to students and PhD students. It may help to promote the progress in ILs development also.


Introduction
Room temperature ionic liquids (RTILs) are comprised of bulky organic cations and anions (Holbrey & Seddon, 1999;Weingärtner, 2008). Because of the intricate interplay of various inter-and intramolecular interactions, RTILs have rich dynamical properties and have found diverse applications. In particular, negligible vapor pressure of RTILs makes them a green alternative of conventional organic solvent. For understanding of RTIL's dynamical behavior at the molecular level, solvation and rotational dynamics have been studied experimentally Cang et al., 2003;Funston et al., 2007;Ingram et al., 2003;Jin et al., 2007;Karmakar & Samanta, 2002a;Lang et al., 2006) and theoretically Kobrak, 2006;2007;Kobrak & Znamenskiy, 2004;Shim et al., 2006;2007;Shim & Kim, 2009). It is found that the ultrafast relaxation in a subpicosecond time regime contributes substantially to solvation dynamics, disproving the diffusion-controlled solvation in RTILs. On the other hand, in the long time regime, solvation and rotational dynamics of RTILs show slow nonexponential relaxations, which is a characteristic of glassy liquids (Jeong et al., 2008;Shim et al., 2005b). In this chapter, we particularly focus on the dynamical properties of RTILs as a viscous liquid (Rodriguez & Brennecke, 2006). High viscosity of RTILs has two different aspects in nanoscale applications such as solar cells (Noda et al., 2003;Wang et al., 2003) and capacitors (Tsuda & Hussey, 2007): it is advantageous to preventing the leakage of electrolyte, but disadvantageous to enhancing transport properties. In order to realize the potential applications of RTILs, it is necessary to scrutinize the transport process and relaxation dynamics in RTILs microscopically via molecular dynamics (MD) simulations (Bhargava & Balasubramanian, 2005;Klähn et al., 2008;Popolo & Voth, 2004;Zhao et al., 2009). One notable feature observed in simulation studies of RTILs is so called dynamic heterogeneity (Habasaki & Ngai, 2008;Hu & Margulis, 2006), which refers to spatially inhomogeneous relaxation behavior (Ediger, 2000;Richert, 2002) in glassy or supercooled liquids (Debenedetti & Stillinger, 2001). Dynamic heterogeneity in RTILs has been observed in the fluorescence spectroscopy experiments (Hu & Margulis, 2006;Samanta, 2006). In this chapter, we present an overview on our recent MD studies of dynamic heterogeneity of RTILs employing a coarse-grained model of 1-ethyl-3-methylimidazolium hexafluorophsphate (EMI + PF 6 − ) Kim et al., 2010). Dynamic heterogeneity has been investigated numerically in various models of supercooled liquids and glass (Chakrabarti & Bagchi, 2006;Chaudhuri et al., 2007;2008;Kob et al., 1997;Lačević et al., 2003;Leonard & Berthier, 2005). Regardless of detailed description in modelling, observed in common are the nonexponential relaxation, breakdown of Stokes-Einstein (SE) or Debye-Stokes-Einstein relation, and decoupling of the exchange and persistence times (Hedges et al., 2007;Jung et al., 2004;. Our model of RTILs has molecular interactions which are strikingly different from those in supercooled liquids due to the Coulomb interactions. The dynamical properties mentioned above are also observed similarly in our model and suggested to be essential features of dynamic heterogeneity in RTILs. The heterogeneity in RTILs may imply structural heterogeneity in mesoscale such as the assembled structure of long alkyl chains (Wang et al., 2007;Wang & Voth, 2005). We point out that the cation in our model employs a united atom description for short alkyl chain. Thus, locally ordered structures and hydrogen bond networks are not expected to appear in our simulation results. However, structural influence on inhomogeneous dynamics is still an intriguing and open question in this simplified model. This chapter is organized as follows: In Sec. 2, a coarse-grained model of a RTIL is introduced. In Sec. 3, we present the MD results for glassy dynamics. Various dynamic properties manifesting dynamic heterogeneity are demonstrated in Sec. 4, while Sec. 5 is devoted to analyzing the dynamic propensity and Coulomb potential energy. Finally, we conclude in Sec. 6.

A coarse-grained model of ionic liquids
We introduce a coarse-grained models to study the dynamics of RTILs via MD simulations covering many orders of magnitude in time scale . Fast degrees of freedom which engage the hydrogen atom as well as the vibration of bonds are excluded by employing the united atom representation. Our coarse-grained model is based on the model studied by Kim and coworkers (Shim et al., 2005a). To be specific, the methyl group (M1) and the moieties of the ethyl group (E1, M3) in the cation, EMI + , were represented by united atoms, using the AMBER force field (Cornell et al., 1995) and the partial charge assignments proposed by Lynden-Bell and cowarkers. (Hanke et al., 2001) The anion PF 6 − was also described as a united atom. One further simplification employed in our model is that five atoms in the imidazole ring and three attached H atoms are represented by a united atom T1 positioned at the center of mass of 8 atoms. The Lennard-Jones (LJ) parameters of T1 were adjusted, so that our model reproduces the liquid structure of the model studied by Kim and coworkers. MD simulations of the coarse-grained model for EMI + PF − 6 are performed using the DL_POLY program. (Forster & Smith, 2001). Atoms i and j at positions r i and r j interact with each other through the LJ and Coulomb potentials: 10 5 10 4 10 3 10 2 10 1 10 0 10 −1  where r ij ≡| r i − r j | is the distance between the two atoms. The parameters of our coarse-grained model are compiled in Table 1. The model system comprises 512 pairs of rigid cations and anions. We performed simulations in the canonical ensemble using the Nosé-Hoover thermostat and at density ρ = 1.31 g/cm 3 at six different temperatures. Periodic and cubic boundary conditions were employed and long-range electrostatic interactions were computed via the Ewald method.

Glassy dynamics
Glassy dynamics of the coarse-grained RTIL is studied using MD simulation results. We analyze ion diffusion and structural relaxation with the mean square displacement, and self-intermediate scattering function, where ··· denotes the equilibrium ensemble average, N the number of ions, and q 0 the wave vector which corresponds to the position of the first peak in the static structure factor 10 5 10 4 10 3 10 2 10 1 10 0 10 −1 1 0.8 for all ions. In this model, q 0 is found to be 1.24 Å −1 regardless of the temperature. It is attributed to the constant density employed in our model, while the density increases slightly in real RTILs and so does q 0 , as the temperature is lowered. In Figs. 1 and 2, the ionic liquid exhibits subdiffusive behavior and nonexponential relaxation more apparently, as the temperature is lowered. Δ(t) shows sublinear dependence in the intermediate time scale, which is followed by a crossover to Fickian behavior eventually. In the time scale of non-Fickian behavior, we observe the plateau regime (β relaxation) in F s (q 0 , t), while the slow α relaxation follows in the long time regime. The latter is well described by a stretched exponential function, c exp[−(t/τ 0 ) β ]. At 300 K, the exponent β is found to be 0.64 and 0.59 for cations and anions, respectively, while 0.89 and 0.92 at 800 K. The exponent β being less than unity as well as the subdiffusion and β relaxation is known to be a good indicator of the glassy dynamics. The origin of the stretched exponential relaxation has been discussed to be either the superposition of different exponential relaxations which arise from heterogeneous dynamics or intrinsically nonexponential relaxation (Colmenero et al., 1999). The structural relaxation time τ α is determined by the relation F s (q 0 , τ α )=e −1 using the results in Fig 2. The temperature dependence of τ α is presented in Fig. 3. We observe that τ α does not follow the Arrhenius law τ α ∝ exp(d 2 /T), but rather a super-Arrhenius behavior given by τ α ∝ exp(d 1 /T 2 ). This indicates that our model of RTILs belongs to a fragile glass former. A super-Arrhenius behavior in a similar RTIL has been also observed in a recent experiment (Xu et al., 2003) For clarity, we note that we have assumed constant density at all temperatures in our model. If the variation of density were employed, the structural relaxation would be accelerated at high temperatures and decelerated at low temperatures. This would not change the fragile behavior of our model qualitatively.

Correlated local excitations
A general feature in the diffusive dynamics of supercooled or viscous liquids is that particles are trapped in a cage for a long time because the thermal motions are not activated enough. This is also the case of our model of ionic liquids at low temperatures; an ion exhibits merely oscillatory motions, occasionally interrupted by significant movements. We monitor such large motions of each ion and thereby quantify local dynamics in the ionic liquid. In this study, local excitation events refer to the instances t 1 , t 2 , t 3 , ···, where the displacement of an ion i exceeds a threshold distance, i.e., |r ···, etc (Hedges et al., 2007). The more local excitation occurs frequently, the more the ion is mobile. The cut off distance d should be chosen appropriately in order to probe the local dynamics. We display the results for d = 3.0 Å, for example, and note that other choices of d on the order of the inter-ion distances do not alter our results qualitatively. An initial excitation of an ion may perturb the local environment and thereby lead to another excitation to the ion or neighboring ions. In the system where the excitations are sparse, the first excitation and subsequent ones indicate different physical circumstances (Jung et al., 2005). The molecular environment providing dynamic constraints persists until the first excitation occurs during fluctuations. Then, the excitation is followed by exchange events due to the dynamic correlations. To verify this, we define the persistence and exchange times to be the waiting times for the first excitation and following ones, and they are denoted by τ p and τ x , respectively. The facilitated dynamics would results in the decoupling of the exchange and persistence times. The distributions of persistence and exchange times at various temperature for cations and anions are displayed in Fig. 4. At 800 K, the two distributions almost coincide with each other, while they become separated as T is lowered. The correlated excitations bring about the decoupled distributions. This implies that the excitation events should not belong to the Poisson process. We remark that the above idea has been originally proposed in the kinetically-constrained model (KCM) (Garrahan & Chandler, 2002), which describes the dynamics of supercooled liquids successfully employing a two-state variable with kinetic constraint. Prior to our study of the ionic liquid, the decoupling of the exchange and persistence times has been shown in not only the KCM but also the WCA model (Hedges et al., 2007), which is an atomistic model for supercooled liquids.

Violation of Stokes-Einstein relation
The Stokes-Einstein(SE) relation is one of hydrodynamic relations for transport properties. In normal liquids, the relation is usually accurate. In this study, we use another equivalent relation, where τ α is the structural relaxation time assuming that τ α is proportional to η/T. We obtain the diffusion constant D from the mean square displacement . At low temperatures, sub-diffusive behavior is pronounced in the intermediate time scale. Thus, we find the time t 0 when the Fickian behavior appears in Δ[t − t 0 ] to specify the diffusion regime correctly (Szamel & Flenner, 2006). The structural relaxation time τ α has been obtained in Sec. 3. If Equation 5 holds, Dτ α would be a constant for all temperatures. We display the deviation of Dτ α for cations and anions in Fig. 5 (a), which means the structural relaxation decouples from diffusive dynamics as T is lowered. To characterize the decoupling behavior in comparison with supercooled liquids, we employ the scaling relation, ( 6 ) where ξ = 1 corresponds to the SE relation. The results ξ = 0.84 for cations and 0.93 for anions reveal a relatively weak SE violation in our model (Fig. 5 (b)), compared with supercooled liquids. In the KCM, τ α corresponds to the average persistence time τ p , while the mechanism of self-diffusion is related to the dynamic exchange events. Therefore, the violation of SE relation has a deep connection with the decoupling of the exchange and persistence times. In this sense, the violation of the SE relation is another manifestation of dynamic heterogeneity. It is also noteworthy that the exponent for the cation is less than that for the anion. This indicates that cations should contribute more significantly to the dynamic heterogeneity, which we turn to later.

Heterogeneity relaxation
We use the excitations defined in Sec. 4.1 to characterize the dynamic heterogeneity by quantifying the local dynamics. We count excitations during a time interval to specify mobile and immobile regions. The simulation box is divided into 7 × 7 × 7 cells. The mobility m k (t) is defined to be the number of excitations which occur at time between t − τ cg /2 and t + τ cg /2 in the k-th cell. We hereby obtain the mobility as a function of the position and time which are both coarse-grained. The time interval τ cg is determined to accommodate three excitations on average and depends on the temperature accordingly. Figure 6 (a) describe the spatial heterogeneity of the mobility at 300 K, where the radius of the spheres centered at the k-th cell is proportional to m k . We observe the clustering of mobile and immobile regions. In contrast, the mobility appears relatively more homogeneous at 800 K, in Fig. 6 (b). We confirm this by observing the distributions of the mobility in Fig. 7. At 800 K, P(m k ) exhibits a Gaussian distribution. As the temperature is lowered, P(m k ) becomes asymmetric and develops a tail at large m k demonstrating a clustering of excitations. Note that the the average of m k over all cells is set to be 3.0 by adjusting τ cg . The heterogeneity of the mobility is expected to relax with a long time scale. We obtain the time scale of the heterogeneity relaxation by computing the normalized time correlation function given by where the overline and ··· denote the averages over k and the equilibrium ensemble, respectively. The results for C(t) at T = 300 ∼ 800 K are displayed in Fig. 8. C(t) shows rapid decay for t τ cg followed by nonexponential decay thereafter. scale τ m of the heterogeneity is determined by the relation C(τ m )=0.1. Figure 9 shows the temperature dependences of τ m and τ cg . The increase in τ cg with a decrease in the temperature indicates the slowing down of the dynamics. In particular, τ m increases faster than that of τ cg as the temperature is lowered. At low temperatures, the correlation of the mobility maintains for a significantly long time, e.g., about ten nanoseconds at 300 K which is about a hundred times longer than the time interval for measuring the mobility.

Structural heterogeneity
Dynamic properties presented in Sec. 4 demonstrate that local dynamic constraint in fluctuation-dominated dynamics is responsible for the dynamic heterogeneity in our coarse-grained ionic liquid. To consider the structural origin producing the dynamic constraint in RTILs, we examine dynamic propensity to probe the role of initial structure in Sec. 5.1. Then, we analyze the Coulomb potential energy which represent the heterogeneous structures of RTILs in Sec. 5.2.

Dynamic propensity
Dynamic propensity has been introduced to study structural influences on the heterogeneous dynamics of supercooled liquids (Rodriguez Fris et al., 2009;Widmer-Cooper & Harrowell, 2007;Widmer-Cooper et al., 2004). Dynamic propensity of the ion i, denoted by p i ,i s defined as the mean squared displacement of the ion i for t * , which is averaged over the trajectories starting from a given initial configuration with different initial momenta, i.e., the isoconfigurational (IC) ensemble, The initial configuration is taken from the equilibrium ensemble and the initial momenta are chosen to follow the Maxwell-Boltzmann distribution corresponding to the temperature. In this study, we repeat simulations 200 times for each initial structure. The time interval t * is chosen to be 1.5τ α at each temperature, where τ α 's have been determined in Sec. 3. The averages over the IC ensemble exclude the influence of the direction and magnitude of initial momenta on the resultant dynamics. Thus, p i would reveal how the ion i tends to be mobile, resulted by, if any, the initial configuration. Previous studies of supercooled liquids have found that several local structural quantities and the potential energy do not have any direct correlations with the propensity of individual particle, but a general overlap of spatial distributions in a large lengh scale (Berthier & Jack, 2007;Widmer-Cooper & Harrowell, 2006;Widmer-Cooper et al., 2004;2008). Figure 10 shows the probability distributions P(p i ) of dynamic propensities for cations and anions at five temperatures. The distributions are obtained by performing averages over the results from five uncorrelated initial configurations at each temperature. At 800 K, the highest temperature we studied, P(p i ) follow a narrow Gaussian distribution for both cations and anions. As the temperature is lowered, P(p i ) becomes more broad and asymmetric. The Gaussian shape of P(p i ) implies that each ion has an identical environment statistically for t * according to the central limit theorem. Thus, a substantial deviation from the Gaussian distribution means that each ion is located in heterogeneous environments. Though not specified yet, there might be relatively more restrictive structures resulting in small propensity, while less restrictive structures facilitate large propensity. Thus, the change of the distributions with the decrease of the temperature indicates the development of the dynamic heterogeneity. One interesting feature in Fig. 10 is that the tails of P(p i ) at large propensity become more distinct in case of the cations, as the temperature is lowered. It implies that cations make more dominant contributions to the heterogeneous dynamics in our model. In Sec. 3, we have observed that τ α for anions is much longer than that of cations, especially at low temperatures, and so is t * . Therefore, most of surrounding cations for an anion undergo complete structural relaxation, resulting in relatively homogeneous environment for the anion. It is consistent with the result in Sec. 4.2, where the breakdown of SE relation is prominent in case of cations.

Coulomb potential energy
To examine a heterogeneous enviornment which may affects motions of individual ion, we calculate the Coulomb potential energy U i at the position of the ion i using the Ewald method (Forster & Smith, 2001), where q i is the charge of the ion i and α is a parameter which determines a Gaussian distribution for screening and compensating charges in the Ewald method. Due to the inertial dynamics of ions, U i fluctuates with large amplitude. We take average of U i during a coarse-graining time interval τ cg to probe only the structural aspect. The coarse-grained Coulomb potential energy for the ion i is given by We study how the distribution of U cg i depends on τ cg here. With the increase of τ cg ,t h e variance of U cg i over ions, σ 2 [U cg ], should decrease in general. However, Figure 11 shows that σ 2 [U cg ] does not decrease substantially as τ cg increases at a low temperature 350 K. It is because the values of τ cg employed in Fig. 11 are shorter than the structural relaxation time τ α , which are found to be 1960 ps and 4840 ps for cations and anions, respectively, in Sec. 3. On the other hand, σ 2 [U cg ] decreases faster at T = 800 K, where τ α are 3.4 ps for cations and 3.9 ps for anions. Specifically, the variance scales as σ[U cg ] ∼ τ −1 cg at 800 K, as it should be when U cg i follows an identical distribution for all i, according to the central limit theorem. We thus consider the weak decreasing of σ 2 [U cg ] with τ cg at 350 K as a result of heterogeneous structures. If we further increase τ cg to the time much longer than τ α , σ 2 [U cg ] would start to decrease substantially, and the heterogeneity would be averaged out, eventually. On this account, we consider U cg i as a quantity which has structural information for individual ion at a low temperature. To verify the environment which facilitates ions to be mobile or immobile, we consider U cg i and the mobility m i , which is defined as the number of excitations of the ion i during τ cg . We obtain conditional probability distributions of U cg i on the conditions that the ion i has the mobility m i larger and smaller than m c . At 300 K, τ cg are set to be 1 ns and 2.5 ns for cations and anions, respectively, while 15 ps and 18 ps at 800 K. We choose m c = 2 for cations at 350 K, otherwise, m c = 1. In Fig. 12, the conditional probability distributions for m i > m c are observed to shift to higher energies for all cases. We first point out that mobile ions have higher U cg i on average at 800 K as well as 350 K. At 800 K, we have not observed the heterogeneity of m i or U cg i which persists significantly. Therefore, higher mobility would not be facilitated by higher U cg i . It is conjectured that the shift of P(U cg i ) should be mainly attributed to the perturbed structures caused by local excitations. Moreover, two distributions exhibit considerable overlaps. Ions can have relatively lower U cg i even though the ions are mobile in a variety of situations made by complicated dynamics in liquids. In this sense, larger values of U cg i cannot be a decisive factor to make the ion mobile. However, we also observe more distinct discrepancy between two distributions for cations at 350 K in Fig. 12 (a), compared to other cases. This might be a contribution from heterogeneous structures, which will be specified further in detail as a future study. It is also necessary to examine if these structures act as mobile or immobile environments.

Conclusions
To summarize the chapter, we have shown various dynamic properties manifesting the dynamic heterogeneity in a coarse-grained model of EMI + PF 6 − . As the temperature is lowered, our model system exhibits nonexponential relaxation, subdiffusive behavior, the breakdown of the Stokes-Einstein relation, and the decoupling of persistence and exchange times. We point out that all the properties have been observed in models of supercooled liquids before. The dynamical similarity regardless of strikingly different molecular details is of particular interest. We essentially attribute the universality to the dynamic correlations existed in glassy liquids. Furthermore, the dynamic heterogeneity of our model have been verified by introducing the mobiity, which we claim a convenient quantity to describe local dynamic states. We have also studied how the dynamic correlations are developed, specifically whether they are structurally originated, by computing the dynamic propensity and Coulomb potential energy. In conclusion, the dynamic propensity and Coulomb potential energy demonstrate the dynamic heterogeneity well. However, they do not seem to affect the dynamics decisively. The dynamics of RTILs resolve itself into the fluctuation-dominated dynamics. Nevertheless, it is still necessary to improve microscopic understanding of the diffusion mechanism in terms of the Coulomb interactions. More detailed analyses on the Coulomb potential energy are in progress.