Composition Dependence of Dynamic Heterogeneity Time- and

Article pubs.acs.org/JPCB

Composition Dependence of Dynamic Heterogeneity Time- and Length Scales in [Omim][BF4]/Water Binary Mixtures: Molecular Dynamics Simulation Study Tamisra Pal and Ranjit Biswas* Department of Chemical, Biological and Macromolecular Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700098, India S Supporting Information *

ABSTRACT: Composition dependence of four-point dynamic susceptibilities, overlap functions, and other dynamic heterogeneity (DH) parameters have been investigated by using all-atom molecular dynamics simulations for aqueous solutions of the ionic liquid (IL), 1-octyl-3-methyl imidazolium tetrafluoroborate ([Omim][BF4]) covering the pure-to-pure range. Upon addition of water in the IL, the DH time scales become faster and the fourpoint dynamic susceptibility time scale softens. Evidences for jump motions for both water and ions have been found from the simulated single particle displacements that show strong deviation from Gaussian distribution. Estimated dynamic correlation length for water reflects effects of IL, whereas those for ions remain largely insensitive to the mixture composition. Simulated structural aspects and DH time scales provide microscopic explanations to the existing experimental observations from time-resolved fluorescence and Kerr spectroscopic measurements.

I. INTRODUCTION Dynamic heterogeneity (DH), heterogeneity due to spatially varying relaxation rates in a given medium, is a concept that has been used extensively in explaining nonexponential relaxation rates and viscosity-decoupling of transport properties in deeply supercooled liquids near glass transition.1−7 Interestingly, stretched or multiexponential relaxation characterizes the long-time dynamics of ionic liquids (ILs)8−29 and their binary mixtures with common molecular solvents30−38 at ambient condition. In addition, results from fluorescence microscopic studies39,40 and excitation energy dependent probe solvation and rotation measurements41−44 for a variety of room temperature ionic liquids have been explained in terms of spatially varying relaxation rates. Subsequent simulation studies on neat ionic liquids have revealed distribution of relaxation times and presence of DH via non-Gaussian single particle displacement distributions and van Hove correlation functions.45−49 Presence of nonhydrodynamic moves for ions such as jumps and the consequent decoupling of diffusion from viscosity have also been indicated in both experimental50 and simulation49 studies of neat room temperature ionic liquids. Although the heterogeneity in solvent dynamics has been investigated for several neat ILs via experiments39−44 and simulations,45−49,51 similar studies for IL/molecular solvent binary mixtures are extremely rare. In fact, information regarding DH time- and length-scales is still unavailable for binary mixtures containing ILs, and no rigorous measurements for the location dependent solute dynamics in IL/molecular solvent binary mixtures have been performed yet. This is © 2015 American Chemical Society

somewhat surprising given that viscosity coefficient (η) of an IL is known to dramatically reduce by the addition of common dipolar solvents due to screening of Coulomb interactions. Because η depends critically on structural organization and stress relaxation,52 modification of η upon addition of molecular solvents indicate significant alterations in both these spatial and temporal correlations. Optically heterodyne detected optical Kerr effect spectroscopic (OHD-OKE) measurements53 of [Omim][BF4]/water binary mixtures have reported water induced stiffening of alkyl tail−tail aggregation and slow time scale of ∼10 ns for cation reorientation diffusion. A femtosecond Raman-induced Kerr effect spectroscopic (fsRIKES) investigation of the same system54 has, on the other hand, revealed negligible effects of water on the low-frequency Kerr spectrum and weak perturbation of the interionic interaction. Neutron spin echo (NSE) measurements of relaxation dynamics at the nearest neighbor length scales have shown the presence of a nanosecond time scale for a few neat ILs at room temperature containing [Omim]+ with anions bis(trifluoromethylsulfonyl)imide, hexafluorophosphate, and chloride (abbreviated respectively as [TFSI]−, [PF6]−, and Cl−) and has been attributed to the center-of-mass motion of the ions.55 Interestingly, this slow time scale has also been found in time-resolved fluorescence measurements12 of the IL, [Omim][TFSI]. All these data motivate us to explore the Received: September 8, 2015 Revised: November 25, 2015 Published: December 2, 2015 15683

DOI: 10.1021/acs.jpcb.5b08763 J. Phys. Chem. B 2015, 119, 15683−15695

Article

The Journal of Physical Chemistry B

the ab initio calculations at the HF/6-31+G(d) level of theory and setting the cation and anion charges at +1 and −1, respectively. Partial atomic charges (generated via Gaussian package for cation) and van der Waals parameters εi and σi were taken from literature.70 These are provided in Appendix 1. Interaction between two different atom types were obtained via the Lorentz−Berthelot combination rules,71 εij = (εiεj)1/2 and σij = (σi + σj)/2. For rigid water, the well-known SPC/E model72 was used where the bonds were constrained using the SHAKE algorithm. Nose-Hoover thermostat73,74 was employed with a coupling constant of 200 fs employing the canonical (NPT) ensemble. Verlet Leapfrog algorithm71 with 2 fs time step was used to solve classical equations of motion. The cutoff radius for the short-range interaction was set to 16 Å. Electrostatic interactions were dealt with via the Ewald summation71 technique. Equilibration and production run lengths were of 20 and 100 ns (with a time gap of 100 fs), respectively. Systems were first equilibrated in NPT ensemble and then transferred to NVT ensemble for further equilibration and production run. At all mole fractions, the simulated density were within ∼1−2% of the experimental profile,54 presented in Figure S2. This level of accuracy provides a fidelity check to the force fields used in our simulations. Initially, each system was equilibrated at 450 K and then cooled down in a stepwise process (using a 50 K step) to generate the minimum energy configuration at 298 K. These pre-equilibrated configurations were subsequently utilized for the final NVT run that includes equilibration followed by production.

interconnection between the DH time- and length scales on one hand, as well as the slow time scale in the nanosecond and beyond observed in different relaxation measurements on the other. The proposed study has been performed by carrying out molecular dynamics simulations of neat 1-octyl-3-methylimidazolium tetrafluoroborate ([Omim][BF4]), and its binary mixtures with water (H2O) at 298 K. A total of five different mixture compositions have been considered that included both the neat liquids. DH characteristics have been followed via computing the non-Gaussian (NG)56−58 and new nonGaussian (NNG)59,60 parameters, single particle displacement distributions,61 four-point dynamic susceptibilities, overlap functions, and dynamic correlation lengths.62−67 Composition dependence of all these DH parameters provides microscopic insights to the interaction of water with the IL, and the subsequent modification of the dynamics, and brings out the microscopic origin of time and length scales observed in a variety of experiments with these Coulomb fluids.

II. COMPUTATIONAL DETAILS Molecular dynamics simulation of mixtures of 3-site rigid SPC/ E H2O model and 42-site flexible [Omim][BF4] covering the whole composition range were performed using the Mdynamix package.68 The mole-fractions of IL investigated were f = 0.0, 0.2, 0.4, 0.6, and 1.0. Numbers of ion-pairs required to achieve the desired mole-fraction (f) in the system with their respective box-lengths are summarized in Table 1. Chemical structures of

III. RESULTS AND DISCUSSION A. Structural Overview: Water-Induced Modifications. We designate the ring carbon-atom flanked by the two nitrogen atoms as CR (see Figure S1) and consider this as the head. Methyl groups bearing CT-carbon atoms construct the tail. In this IL then the tail−tail interaction corresponds to the interaction between the nonpolar groups, and the head−anion interaction to the interaction between the polar groups. Modulation of these interactions upon addition of water as well as interaction of ions with the added water molecules may be followed via the appropriate radial distribution functions, (RDF, g(r)). Figure 1 summarizes the effects of addition of water where the IL mole fraction dependent g(r) for various components of interactions are shown. The following aspects are to be noted. First, no dramatic modification in the head− head RDF (upper left panel) is seen, even when the system changes from molten electrolyte (that is, f = 1) to dilute aqueous electrolyte solution (that is, f = 0.2). The only change we see here is a tiny alteration in the g(r) peak height and softening of the shoulder (at ∼3.5 Å) upon successive addition of water in the IL, which eventually disappears at f = 0.2 through broadening. The shoulder at ∼3.5 Å may be arising from a side-to-end arrangement (cross configuration) of the head groups.75−77 At higher dilution such as at f = 0.2, increased interaction with water molecules and lower number density of ions does not probably support the cross configuration mentioned above. The weak dependence of the g(r) main peak height (at ∼5.6 Å) on water concentration provides a molecular basis to the suggestion of OHD-RIKES results54 that water has little influence on the microscopic structure and interionic interaction in this head region. The tail−tail RDF (lower left panel), on the other hand, shows a significant change where the first peak height of the g(r) nearly doubles compared to that either at f = 0.4 or at f = 1.

Table 1. Water Molecules and Ions Employed in Simulations at Different IL Mole Fractions (f) no. of [Omim]+ molecules

f

no. of water molecules H2O

no. of [BF4]−molecules

0.0

256

0

0

0.2

204

52

52

0.4

154

102

102

0.6

102

154

154

1.0

0

128

128

simulation cell (X,Y,Z) Å 19.75, 19.75, 19.75 30.29, 30.29, 30.29 36.09, 36.09, 36.09 40.86, 40.86, 40.86 38.01, 38.01, 38.01

the cation and anion of IL along with the force field atom types are shown in Figure S1. All the atoms in the flexible IL molecule interacted via AMBER-type69,70 force field as already mentioned in our previous work.48 The total potential energy functional form is as follows: Vtot =

∑

k b(r − r0)2 +

bonds

∑ dihedrals

∑

kθ(θ − θ0)2

angles

kχ 2

[1 + cos(nχ − δ)]

12 ⎧ ⎡⎛ ⎛ rmin, ij ⎞6 ⎤ qiqj ⎫ ⎪ ⎪ ⎢ rmin, ij ⎞ ⎥ ⎟⎟ − ⎜⎜ ⎟⎟ + ⎬ ∑ ⎨εij⎢⎜⎜ ⎥ r rij ⎪ ⎝ rij ⎠ ⎦ j>i ⎪ ⎭ ⎩ ⎣⎝ ij ⎠

N−1 N

+

∑ i=1

(1)

Simulations were carried out in a cubic box employing periodic boundary conditions. The minimum energy geometry of the [Omim]+ and [BF4]− were determined via performing 15684


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Figure 1. Simulated composition-dependent radial distribution function, RDF, [g(r)] for head−head, water−water, tail−tail, and anion−anion at 298 K. The mole fraction of the IL in this binary mixture is represented by f. Note the terminal methyl (CT) has been considered as the tail group, while the CR atom on the polar imidazole ring has been marked as the headgroup. Representations are colorcoded.

Figure 2. Simulated various site−site RDFs at various compositions for [Omim][BF4]/water binary mixtures at 298 K. Representations are, as before, color-coded. For discussion, see the main text.

interaction of water, either via the oxygen atom or the hydrogen atom, with the ions increases even though water concentration decreases. In addition, uniformly higher peak value (first peak) for the F(a)-H(w) g(r) than that for H(5)c-O(w) g(r) at all IL mole fractions indicates stronger anion−water interaction than cation−water interaction. Relatively smaller size of the anion facilitating stronger ion−dipole interaction over that for the cation may be responsible for such a difference. The H-bonding interaction between water molecules are described via the g(r) between the oxygen atom of one water molecule (O(w)) and the hydrogen atom of another (H(w)) in the lower right panel. Interestingly, O(w)-H(w) interaction increases upon decreasing the water concentration, suggesting formation of water clusters H bonded to both anion and among themselves. Figure 3 depicts such a proposed structure which, although overly simplified, may represent a probable precursor to micelle formation87,88 in this mixture at appropriate IL concentration. B. Mean Square Displacements (MSDs): Signature of Water-Mediated Cage Softening. Translational motions of all particles in this IL/water mixture were tracked by following the center-of-mass MSDs defined as,49,71

This indicates enhancement of alkyl chain−alkyl chain interaction at dilute aqueous solutions of this IL and support the view of water-assisted alkyl chain association from OHDOKE measurements.53 A reverse scenario, that is, increased water−water interaction, is expected at high IL concentrations because several studies involving aqueous solutions of amphiphiles strongly suggests such a possibility.78−80 Composition-dependent water−water RDF (involving oxygen atoms) shown in the upper right panel does indicate a sharp increase in water−water interaction upon increasing the IL mole fraction in the aqueous mixture. Note the increase of g(r) main peak height is ∼5 times for a composition change from f = 0 to f = 0.6, suggesting water association in the IL-rich solutions. Composition dependent anion−anion RDF is shown next in the lower right panel where the first maxima appear at a distance where approximately the first minima of the cation− cation RDFs are located. This is a reflection of charge alternation,81−83 which extends in the dilute solution regime with IL mole fraction as low as f = 0.2. Next we explore in Figure 2 the H-bonding interaction between various species in this IL/water binary system. The interaction between the most acidic hydrogen on the cation (H5(c)) and one of the fluorine atoms of the anion (F(a)) described in the upper left panel of this figure shows that the g(r) peak height (first peak) decreases upon decreasing the IL mole fraction in the aqueous solution. This is expected as successive addition of water in the mixture reduces the number density of both the ions. Moreover, gradual decrease of the g(r) peak height upon addition of water suggests successive screening of electrostatic interaction between the ions of opposite charges by water molecules, reducing the probability of formation of ion-pairs84−86 upon dilution. The upper right panel shows the composition-dependent RDF describing the interaction between the H5(c) and the oxygen atom of the water molecules. The H-bonding interaction between the anion fluoride (F(a)) and water hydrogen (H(w)) are shown via g(r) in the lower left panel. Note in both these panels the

N

⟨δr 2(t )⟩ = ⟨N −1 ∑ |Δri c(0, t )|2 ⟩ i=1

and the corresponding translational diffusion coefficients obtained by using the formula, ⎡1 ⎤ D = ⎢ (⟨δr 2(t )⟩)⎥ ⎣ 6t ⎦t →∞

Figure 4 presents the composition-dependent MSDs for water (upper panel), cation (middle panel), and anion (lower panel) simulated at 298 K. At higher IL concentrations (f > 0.2), three different regimes comprised of inertial motion at extremely short time, followed by intermediate subdiffusive terrain and then diffusive regime can be seen for both water and ions. The subdiffusive regime is reflected in the nearly flat region (MSDs possessing a tα dependence with α ≪ 1) of these curves arising from rattling in a cage51,57,90,91 for a significant duration of time without producing appreciable displacements for the corresponding particles. Note the extent of this 15685


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Figure 3. A simplified model showing hydrogen-bonding between cation (1-octyl-3-methyl imidazolium), anion(tetrafluoroborate), and water molecules which may be the precursor for the formation micellar structure in [Omim][BF4]/water binary mixtures.

subdiffusive plateau regime reduces upon addition of water in this mixture, reflecting water-induced cage softening. The diffusive regime starts once the cage melts away, allowing the structural relaxation to take place. Approximate time scales for the onset of the diffusive regime on these curves are indicated by bullet marks and also summarized as insets. As expected, these diffusion on-set time scales become increasingly shorter upon addition of water in [Omim][BF4]. C. Softening of Dynamic Heterogeneity (DH): Composition Dependent Non-Gaussian (NG) and New Non-Gaussian (NNG) Parameters. As rattling in a cage motion suggests the presence of the varying relaxation rates and thus the existence for the DH, cage softening upon addition of water in [Omim][BF4] is expected to have marked effects on the DH characteristics. DH can be easily tracked by following the temporal profiles of the NG (α2)57,58and the NNG (γ)49,60 parameters defined as follows: α2(t ) =

Figure 4. Composition-dependent translational mean squared displacements (MSDs) for water (upper panel), cation (middle panel), and anion (lower panel) at 298 K. Bullet marks indicate approximately the diffusion onset time scales.

∼0.2 predicted for homogeneous liquids.57 For neat water, in contrast, τNG and τNNG are approximately within a picosecond with the α2(t) peak height ∼0.2. This indicates that the neat [Omim][BF4] itself is far more dynamically heterogeneous than neat water. Moreover, the simulated τNG and τNNG are much longer than those obtained earlier49 for [Bmim][PF6], and this variation parallels with the viscosity difference28,92 between these two neat ILs. The composition dependencies of α2(t) and γ(t) are presented in Figures 6 and 7, respectively, for the [Omim][BF4]/water system. α2(t) shown in the upper panel of Figure 6 clearly demonstrates that both the NG peak height and time scale for water undergo a dramatic increase as the system changes from neat water (f = 0.0) to concentrated aqueous IL solution ( f = 0.6). In fact, a severe slowing down in τNG occurs which changes from ∼1 ps at f = 0.0 to ∼300 ps at f = 0.6 along with more than an order-of-magnitude increase in the peak height. Such a strong composition dependence is, however, missing for both the cation (middle panel) and the anion (lower panel). Here τNG slows down by a factor of ∼4−5 with an overall ∼10−30% change in the peak height upon changing the composition from f = 0.2 to f = 1. In brief, subnanosecond to a few nanosecond time scale from α2(t) appears consistently

3⟨δr 4(t )⟩ −1 5⟨δr 2(t )⟩2

and γ (t ) =

1 2 1 ⟨δr (t )⟩ 3 δr 2(t )

−1

Both α2 and γ depict nonmonotonic time-dependence and are linked to the slow time scales present in a given system. In addition, α2 follows the relatively faster particles than those tracked by γ. Consequently, the NNG time scale (τNNG), which corresponds to the peak of the γ(t) profile, is slower than the peak time scale (τNG) associated with the NG parameter. Figure 5 shows the simulated α2(t) and γ(t) for the neat IL (upper panel) and the neat water (lower panel) at 298 K. In fact, the upper panel depicts these NG and NNG parameters separately for the constituent ions, [Omim]+ and [BF4]−. For both the ions, the peak time scales, τNG and τNNg, range between ∼3−4 ns and ∼6−7 ns, respectively. Interestingly, the peak heights of α2(t) for both the ions are much higher than 15686


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Figure 5. Comparison between the simulated NG and NNG parameters, α2(t) and γ(t), respectively, at 298 K for [Omim]+ (upper panel) and H2O (lower panel) with those for [BF4]− in the inset of the upper panel. While the solid lines denote α2(t), dashed lines represent γ(t). Vertical lines indicate peak times (approximate) for the NG and the NNG parameters.

for this mixture at all nonzero IL mole fractions where all the species, water, cation, and anion contribute. Interestingly, nanosecond and even slower time scale appears for water when we examine the corresponding γ(t) profile at various IL mole fractions presented in the upper panel of Figure 7. In comparison to neat water where τNNG ∼ 1 ps, the NNG time scale for water at f = 0.6 is ∼7 ns, registering a slowing down of ∼7000 times over the neat value. For ions, τNNG remains within a couple to several nanoseconds, showing an increase of ∼3−4 times for changing the composition from f = 0.2 to f = 1. A comparison between the composition dependent τNG and τNNG are shown in Figure S3, quantifying the above discussion. The occurrence of nanosecond or slower time scales in relaxation studies (for example, inelastic neutron scattering or time-resolved fluorescence Stokes shift measurements) with these binary mixtures would therefore be natural and thus should be tested against the experiments. D. Single Particle Displacement Distributions: Signature of Jump. For homogeneous liquids, the single particle displacement distribution, P[log10(δr);t)], is supposed to be Gaussian at any given time.52 However, the distribution deviates from being Gaussian in the presence of DH, and this can be explored via the simulated self-part of the van Hove correlation function, Gs(δr,t), by using the relation,59,61,93−95 P[log10(δr);t) = ln(10)4πδr3Gs(δr,t). Figure 8 displays the simulated displacement distributions at τNG for water (upper panel), cation (middle panel), and anion (lower panel) at various IL mole fractions. For a Gaussian G s (δr,t), P[log10(δr);t)] becomes independent of time59 and assumes the peak height of ∼2.13. This is indicated by showing in each

Figure 6. Mixture composition dependence of the simulated nonGaussian parameter, α2(t), for water (upper panel), cation (middle panel), and anion (lower panel) at 298 K.

panel displacement distribution for a Gaussian Gs(δr,t) at a given IL mole fraction. It is interesting to note that P[log10(δr);t = τNG)] is bimodal only for water, with the major “vibration” peak at ∼0.3σw. In fact, a hint of bimodality for water first appears at f = 0.2 with a shoulder (a minor peak at other higher IL mole fractions) occurring at a single water diameter (σw). This shoulder at σw becomes more prominent as IL mole fraction increases. This suggests that water molecules, upon successive addition of IL in the mixture, increasingly access to jump modes for translation in addition to the conventional stochastic Brownian moves. Note P[log10(δr);t = τNG)] for neat water (that is at f = 0.0) shows a moderate deviation from being Gaussian, suggesting neat water is also dynamically heterogeneous to some extent.96 Compositiondependent displacement distributions for both the ions possess qualitatively the similar shape without any hint of ion jump but much deviated from being Gaussian. This is somewhat different from our earlier observation for [Bmim][PF6] where jump was detected for [PF6]− at 298 K.49 We further examined the displacement distribution characteristics at τNNG, a time scale that is connected to the “immobile” particles where jump motion is relatively more likely. Figure 9 displays the composition dependent P[log10(δr);t = τNNG)] for water (upper panel), [Omim]+ (middle panel), and [BF4]− (lower panel). As before, calculated Gaussian displacement 15687


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Figure 7. Composition dependence of simulated new non-Gaussian parameter, γ(t), for water (upper panel), cation (middle panel), and anion (lower panel) at 298 K.

Figure 8. Composition dependence of the simulated single particle displacement distribution, P[log10(δr);t], in this IL/water mixture for water (top panel), cation (middle panel), and anion (bottom panel). Note these distributions were calculated at t = τNG. The labeled black dotted line denotes the displacement profile corresponding to a Gaussian distribution. Horizontal broken lines represent the peak height (∼2.13) for such a Gaussian distribution of particle displacements.

distribution at a given IL mole fraction is also shown in each panel for a comparison. Quite interestingly, displacement distribution for water becomes more structured at t = τNNG with the “hop” peak at ∼1.5σw more intense than the “vibration” peak at ∼0.3σw. This may be construed as longer waiting time leads to larger jumps. In addition, the splitting of hop peak for water at f = 0.4 and 0.6 into multiple peaks suggest presence of multiple jump length-scales. Note for the pure IL, a tiny “foot” at ∼0.5 σ+ for [Omim]+ and a similar one for [BF4]− at ∼σ− appear. These we interpret as evidence for cation and anion jumps, and the associated jump length-scales corroborate well with results from broadband dielectric spectroscopy (BDS) and pulsed field gradient nuclear magnetic resonance (PFG NMR) measurements.97 E. Self-Intermediate Scattering Function at the Nearest Neighbor and Overlap Function: Search for the Slowest Relaxation. The self-intermediate scattering function, Fs(k,t) = ⟨ρs(k,t)ρs(−k,0)⟩, tracks the density fluctuation at a given space point and is amenable to incoherent neutron scattering measurements. The 1/e decay time of Fs(k,t) relaxation in the limit of the nearest neighbor wavenumber (kσ → 2π, σ being the appropriate diameter) provides an estimate for the α-relaxation time scale, τα. In the present study, Fs(k,t) has been simulated from the real part of the scattering function,52,60,98 Fs(k,t) = N1−∑i⟨cos k × [ri(t) − ri(0)]⟩. Simulated relaxation profiles of Fs(kσ → 2π,t) at various mixture compositions are shown in Figure 10, where bullet

marks indicate the respective 1/e decay times, τ1/e F . As before, the mixture composition dependence is relatively stronger for water (upper panel) than [Omim]+ (middle panel) and [BF4]− (lower panel). Expectedly, the relaxation becomes faster as water is added, and at f = 0.0 (that is, neat water), τ1/e F becomes comparable to the diffusion onset time scale shown in Figure 4. Moreover, a comparison between τ1/e F and τNNG in Table 2 shows that the structural relaxation time scales (τ1/e F ) for water do not reflect the slowest time scales available to it in these aqueous mixtures. It is interesting to note that ∼60% of the early relaxation of Fs(kσ → 2π,t) for [Omim]+ at f = 1.0 (neat IL) remains the fastest among those at other compositions studied. Fs(kσ → 2π,t) relaxation for [BF4]− during some intermediate times (∼0.1−3 ns) at f = 1.0 also remains faster than those at f = 0.4 and 0.6. This probably arises from waterinduced stiffening of aggregated structures.53 τF1/e values summarized in Table 2 for these ions also do not match with the corresponding τNNG, reflecting complexity of these IL/water mixtures. 15688


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Figure 10. Composition-dependent normalized self-part of the intermediate scattering function, Fs(k,t), for [Omim][BF4]/water binary mixture at 298 K. Each simulated curve corresponds to the nearest neighbor (kσ → 2π) mode. Bullets indicate the respective 1/e decay times (τF1/e).

Figure 9. Same as shown in Figure 8, but the distributions are calculated at t = τNNG.

Next, we investigate the composition dependence of the overlap function Q(k,t) defined as,65,66,99,100

Table 2. Comparison of 1/e Times Obtained from Various Simulated DH Parameters for [Omim][BF4]/Water Mixtures at Different IL Mole Fractions (f)

N

Q (k , t ) =

∑ Wi (a , t )exp[i k·ri(0)] i=1

where a Heaviside step function defines Wi(a,t) as follows: Wi (a , t ) = Θ(a − |ri(t ) − ri(0)|)

Following an earlier suggestion,66 we have used a = 0.3σ, with σ being the particle diameter. It has been shown earlier66 that the relaxation of Q(t) = N−1⟨∑Ni Wi(a,t)⟩, where a = 0.3σ produces a relaxation profile very similar to that of Fs(k → 2π,t). This arises mainly from the choice of the relaxation length scale (a = 0.3σ) for Q(t), and as a result, comparable 1/e decay times are expected. However, there are examples where the relaxation time scales for Q(t) and Fs(k → 2π,t) differ.49 Figure 11 presents the composition-dependent Q(t) decays for water, [Omim]+, and [BF4]− with bullet marks representing the respective 1/e decay times, τ1/e Q . Interestingly, Q(t) decays for water and [BF4]− show significant composition dependence, which is somewhat different from what we had seen for the Fs(k → 2π,t) decays in Figure 10. τ1/e Q values summarized in Table 2 indicate that slow time scales in the range of a few tens of nanosecond may arise in these systems from the relaxation of

f

τNNG (ns)

0.0 0.2 0.4 0.6

0.001 0.3 2.5 6

0.2 0.4 0.6 1.0

1.7 2 4 6

0.2 0.4 0.6 1.0

1.8 2 6.5 8.7

τF1/e (ns) H2O 0.001 0.02 0.23 0.39 [Omim]+ 4300 6000 6800 7000 [BF4]− 870 5000 6000 12500

τQ1/e (ns)

tmax (ns) 4

0.002 0.08 0.7 1.5

0.001 0.07 0.6

6.8 9.1 32.3 28.5

8 25 − 26

0.7 5.7 11.1 32.7

2 17.7 − 33

closely packed ions. In addition, these slow time scales, like τNG, τNNG, and τF1/e, become slower upon increasing the IL concentration in the mixture. 15689


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Subsequently, composition-dependent χ4(k,t) at the nearest neighbor wavenumber have been calculated, and the correlated time scale, tmax 4 , obtained from the location of the maximum on time axis in each of the nonmonotonic temporal profiles simulated. The dynamic correlation length, ξ, at t = tmax has been 4 estimated from the overlap function by using the Ornstein− Zernike relation,63,66,101,102 S4(k , t ) 1 = χ4 (k → 0, t ) 1 + [kξ(t )]p

(2)

−1

where S4(k,t) = N ⟨Q(k,t)Q(−k,t)⟩, χ4(k → 0,t) = S4(k → 0,t) 2 2 ∞ = M∫ ∞ 0 dr r g(r), and S4(k,t) = M∫ 0 dr r (sin kr/kr)g(r) with 2 M = 4πρ[Q(t)/N] . We have used p = 2, although different values have been used in the past65 for the same purpose. Figure 12 displays the simulated χ4(k,t) at the nearest neighbor wavenumber at four representative mixture compo-

Figure 11. Composition-dependent simulated overlap function, Q(t), for water (top panel), cation (middle panel), and anion (bottom panel). Bullets denote the respective 1/e decays times (τQ1/e).

Figure 12. A comparison of the simulated four-point dynamic susceptibility, χ4(k,t), among different components at a specific IL mole fraction at 298 K. The vertical lines denoted the respective peak times (approximately), tmax 4 .

F. Four-Point Correlations, Correlated Time- and Length-Scales: Composition Dependence. In slow viscous systems, density fluctuations at two different space points (say, r1 and r2) may remain correlated over a certain period of time. This correlation may also extend over a certain length scale covering a few to several molecular diameters even at ambient condition. This means that several particles at two different places in a given system may have similar mobility for a certain duration of time. This length is known as the dynamic correlation length, ξ, and the maximum time over which the mobility fluctuations at two different space points remain correlated is designated as tmax 4 . Four-point density correlation function, G4(r,t), provides an avenue to access these time- and length scales.62,66,100 G4(r,t) therefore probes the cooperative dynamics, and is given by67

sitions for water, [Omim]+, and [BF4]− at 298 K. At f = 0.0 where neat water is the system, the peak height is ∼0.02 with tmax ∼ 1.5 ps. This time scale for neat water is similar to those 4 obtained from other analyses (see Table 2). However, complexities of the IL/water mixtures start showing up as we move to f = 0.2, where not only the peak heights increase by a factor of ∼5 for all the species (water, [Omim]+, and [BF4]−) but tmax also shifts to much longer times. These time scales are 4 provided in Table 1, which indicate tmax for water at f = 0.2 4 becomes ∼50 times longer than that at f = 0.0. For ions at this composition, tmax reaches ∼2−8 ns and shows some qualitative 4 similarity with the corresponding τ1/e Q . As the IL mole fraction is further increased to f = 0.4, χ4(k,t) attains higher height with further lengthening of tmax 4 for all the species. In neat IL (that is, at f = 1.0), tmax becomes the longest, falling in the ∼25−35 ns 4 region. Presence of such a slow correlated domain motion has been indicated earlier in time-resolved fluorescence measurements using a dipolar probe in several other neat ILs.103 Also note that respective peak heights at f = 0.4 are higher than those at f = 0.2 and f = 1.0, suggesting the dynamic correlations will be the maximum somewhere around the equimolar composition.

G4(r, t ) = ⟨ρ(0, 0)ρ(0, t)ρ(r, 0)ρ(r, t)⟩ − ⟨ρ(0, 0)ρ(0, t)⟩⟨ρ(r, 0)ρ(r, t)⟩

where ρ(r,t) denotes the time-dependent density at a position r. Similar information can also be obtained via calculating the variance of Fs(k,t) via the four point dynamic susceptibility, defined as63−67 χ4 (k, t) = N⌊⟨(Fs(k , t )2 ⟩ − ⟨Fs(k , t )⟩2 ⌋ 15690


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The Journal of Physical Chemistry B Next, we estimate ξ by using eq 2 so that a qualitative sense of correlated length scales can be built up for these aqueous mixtures. Figure 13 shows the estimates via numerical fits for

Figure 13. Estimation of the dynamic correlation length, ξ, for water in this IL/water binary mixture at various compositions. Red ○ represent the simulated data, whereas the black dotted lines represent the nonlinear least-squares fits of the simulated data to the OZ expression shown inside the panel. Correlation coefficients to these fits were found to be ∼0.96.

water at neat condition as well as in the presence of the IL at two different mole fractions. Note ξ has been estimated to be ∼5 Å at f = 0.0 (neat water) and ∼8 Å at f = 0.2 and 0.4, which roughly corresponds to 2 and 3 water diameters, respectively. Although estimation of such a length scale requires a much larger system size (because fluctuations at kσ → 0 modes are involved here) and thus may suffer from inadequacy, the increase in ξ for water upon addition of IL is expected if formation of micellar structure is considered. Figure 14 presents the relevant fits for cations and anions for neat IL ( f = 1.0) and f = 0.2 and 0.4. For both the cation and the anion at all these compositions, ξ has been found to be ∼8 Å, showing no composition dependence. This length scale is ∼1−2 ionic diameters for the present IL.

Figure 14. Estimation of ξ at a few representative IL mole fractions at 298 K for both the cation and the anion in this [Omim][BF4]/water binary mixture. While the main panels show the results for the cation, insets represent the results for the anion.

remains independent of mixture composition, although for water it changes upon addition of IL. Similar studies for deep eutectics104−111 and alcohol/water mixtures112−115 would be useful to fully understand the role of DH time scales in regulating various relaxation processes in these complex multicomponent mixtures. Simulation study exploring formation of ion pairs in the neat IL and subsequent change of its magnitude upon successive addition of water116,117 is necessary for understanding the role of water in modifying ion−ion interactions in IL/water binary mixtures and the dynamics of these systems. It would also be informative to investigate the system-size dependence of the simulated DH parameters, particularly for the dynamic correlation length, ξ, as it involves density fluctuations at the long wavelength limit. However, simulations of very large systems possessing ion−ion interactions employing all-atom description for interaction potentials are prohibitively difficult (because of the computation cost), limiting a thorough study of the system size dependence. Before we end, it would be worthwhile to mention that use of SPC/E as the water model in the present study may appear inappropriate as Amber force field (employed for the IL) was designed to work with the TIP3P water model and, as a result,

IV. CONCLUDING REMARKS In summary, the present study provides simulation evidence for enhanced alkyl chain-alkyl chain static correlation in the presence of water and increased water−water interaction in the presence of the IL. Various simulated radial distribution functions suggest formation of micellar structure in the [Omim][BF4]/water mixture where the anion resides at the close proximity of the charged headgroup (the imidazolium cation). Presence of IL severely affects the water diffusivity, particularly in the IL-rich mixtures. Accordingly, diffusion onset time scales for [Omim]+ and [BF4]− become faster upon successive addition of water in the mixture. DH time scales obtained from NG and NNG parameters show substantial composition dependence and slow down as IL mole fraction in the mixture is increased. Evidences of hopping motion for both water and ions at very long times have been found where nonGaussian displacement distribution for all the species characterizes the particle movements. Relaxation time scales from intermediate scattering function at the nearest neighbor wavenumber, overlap function, and four-point dynamic susceptibility become significantly faster upon addition of water into the IL. Estimated dynamic correlation length for ions 15691


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may not provide the most reasonable description of dynamics. However, it is well-documented118 that the SPC/E model reproduces the experimental diffusion coefficient for water119,120 much better than the TIP3P model. Our own simulations employing the TIP3P and SPC/E water models (results not shown here) also agree to this literature view.118 Therefore, we believe that our simulation results on DH for this ionic liquid/water system will not undergo any qualitative changes if other models for water are used. Also note a qualitative characterization of the composition dependence of the simulated dynamic heterogeneity parameters is only possible because of the nonavailability of the relevant experimental data. In addition, huge computation cost required by all-atom ionic liquid simulations (as done here) also compels us to settle for the qualitative description of the DH parameters.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b08763. Figures showing group identities and chemical structures of the ions constituting the IL, a comparison between the composition-dependent simulated and experimental densities and between the NG and NNG times, and force field parameters charges on each atom are provided (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: 91 33 2335 3477. Tel: 91 33 2335 5706. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We acknowledge the help of Dr. P. K. Ghorai, IISER, Kolkata, for the calculations of partial charges on the IL ions via the Gaussian package. We thank an anonymous reviewer for a critical review that helped improve the quality of the manuscript. Computational facilities provided by a TUE-CMS project at the Centre (SR/NM/NS-29/2011(G)) are utilized for the simulations presented here.

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REFERENCES

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