Dielectric relaxation in ionic liquids: Role of ion-ion and ion-dipole interactions, and effects of heterogeneity Snehasis Daschakraborty and Ranjit Biswas Citation: The Journal of Chemical Physics 140, 014504 (2014); doi: 10.1063/1.4860516 View online: http://dx.doi.org/10.1063/1.4860516 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of pressure on decoupling of ionic conductivity from structural relaxation in hydrated protic ionic liquid, lidocaine HCl J. Chem. Phys. 138, 204502 (2013); 10.1063/1.4807487 Fundamentals of ionic conductivity relaxation gained from study of procaine hydrochloride and procainamide hydrochloride at ambient and elevated pressure J. Chem. Phys. 136, 164507 (2012); 10.1063/1.4705274 A surface forces platform for dielectric measurements J. Chem. Phys. 119, 547 (2003); 10.1063/1.1568931 Decoupling of the dc conductivity and (α-) structural relaxation time in a fragile glass-forming liquid under high pressure J. Chem. Phys. 116, 9882 (2002); 10.1063/1.1473819 Effects of ions on the dielectric permittivity and relaxation rate and the decoupling of ionic diffusion from dielectric relaxation in supercooled liquid and glassy 1-propanol J. Chem. Phys. 116, 4192 (2002); 10.1063/1.1448289

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THE JOURNAL OF CHEMICAL PHYSICS 140, 014504 (2014)

Dielectric relaxation in ionic liquids: Role of ion-ion and ion-dipole interactions, and effects of heterogeneity Snehasis Daschakraborty and Ranjit Biswasa) Department of Chemical, Biological and Macromolecular Sciences, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700098, India

(Received 25 October 2013; accepted 17 December 2013; published online 7 January 2014) A semi-molecular theory for studying the dielectric relaxation (DR) dynamics in ionic liquids (ILs) has been developed here. The theory predicts triphasic relaxation of the generalized orientational correlation function in the collective limit. Relaxation process involves contributions from dipoledipole, ion-dipole, and ion-ion interactions. While the dipole-dipole and ion-ion interactions dictate the predicted three relaxation time constants, the relaxation amplitudes are determined by dipoledipole, ion-dipole, and ion-ion interactions. The ion-ion interaction produces a time constant in the range of 5-1000μs which parallels with the conductivity dominated dielectric loss peak observed in broadband dielectric measurements of ILs. Analytical expressions for two time constants originating from dipolar interactions in ILs match exactly with those derived earlier for dipolar solvents. The theory explores relations among single particle rotational time, collective rotational time, and DR time for ILs. Use of molecular volume for the rotating dipolar ion of a given IL leads to a predicted DR time constant much larger than the slowest DR time constant measured in experiments. In contrast, similar consideration for dipolar liquids produces semi-quantitative agreement between theory and experiments. This difference between ILs and common dipolar solvents has been understood in terms of extremely low effective rotational volume of dipolar ion, argued to arise from medium heterogeneity. Effective rotational volumes predicted by the present theory for ILs are in general agreement with estimates from experimental DR data and simulation results. Calculations at higher temperatures predict faster relaxation time constants reducing the difference between theory and experiments. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4860516] important and thus a modified expression has been derived15

I. INTRODUCTION

Dielectric relaxation is a well established experimental method and has been employed to understand the dynamical characteristics of neat ionic liquids (ILs) and (IL + common solvent) binary mixtures.1–12 These studies have revealed that the DR of ILs follows either Cole-Cole or Cole-Davidson process supplemented by faster Debye-type relaxation. These studies have investigated the probable microscopic origins of the DR timescales and also tested the hydrodynamic models of rotational dynamics in these systems. Experimental DR measurements associate with the collective (that is, kσ → 0, σ being diameter of the rotating species) polarization fluctuations and thus the collective response of the system is measured. There exists a macro-micro relation that connects single particle orientation time (τ u ) to the Debye relaxation time (τ D )13, 14 τu =

n2 + 2 τD , ε0 + 2

0021-9606/2014/140(1)/014504/11/$30.00

(2)

Note Eq. (3) includes molecularity of the solvent through the incorporation of the Kirkwood’s g factor18 which is a measure of short-range solvent-solvent correlations. g can be analytically expressed as the integration over the anisotropic part of the radial distribution function of the dipolar liquid. A further modified relation was derived later by including the dipole moment and a dynamic coupling parameter (g ):19, 20 τu =

a) Author to whom correspondence should be addressed. Electronic mail:

2ε0 + ε∞ τD , 3ε0

where ε∞ is the infinite-frequency dielectric constant of the solvent continuum. It is expected that for solvents with large ε0 , τ u ≈ τ D . Equation (2) is a special case of a more general relation,16, 17   2ε0 + ε∞ τD . (3) τu = 3ε0 g

(1)

where ε0 is the static dielectric constant and n is the refractive index of the solvent continuum, and assuming that such a continuum is characterized by a single relaxation time constant. For strongly polar solvents, however, medium polarizability is

[email protected]. Tel.: +91 33 2335 5706. Fax: +91 33 2335 3477.

τu =

βμ2 ρ0 g  τD , 3ε0 (ε0 − 1)

(4)

where β = 1 / kB T; kB is the Boltzmann constant and T the absolute temperature, μ the dipole moment of the liquid molecule, and ρ 0 the density of the dipolar liquid. The first molecular level theory was developed by Chandra and Bagchi (CB theory) for understanding DR in normal polar liquids. The CB theory was based on generalized Smoluchowski description for the temporal evolution of the

140, 014504-1

© 2014 AIP Publishing LLC

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position and orientation dependent density field,14, 21–28 and derived relations among τ D , τ u , and many body orientation relaxation time, τ M . Unlike the previous continuum model based theories,13, 16, 17, 19, 20 the CB theory was found to be valid over a large polarity range. The relation between τ D and τ u interpolates between Debye relation (Eq. (1)), valid at low dielectric constant, and Onsagar–Glarum relation (Eq. (2)). The theory also agreed well with the Madden and Kivelson conclusion regarding the relation between τ D and τ M , which stated that if the single particle orientation correlation function (Cu ) was single exponential, the many body orientation correlation function (CM ) would be bi-exponential. These authors also derived the microscopic form of frequency dependent dielectric function (ε(ω)) for dipolar liquids and obtained single Debye form for the associated relaxation. The macro-micro relations described above, though important, can be used only for correlating the different relaxation times in simple dipolar liquids. These relations are sometimes even used for ILs to relate the experimentally measured τ D with τ u to explore the applicability of hydrodynamics.29 These relations may have limited validity for ILs as the ion-ion and ion-dipole interactions can profoundly affect relaxation processes in ILs. A combined computational and experimental study has explored the validity of the Madden-Kivelson relation between single particle and collective motion in an IL, [Bmim] [BF4 ].30 Surprisingly, this study has found that the Madden–Kivelson relation is obeyed fairly well in simulations done over longer period (>100 ns), essentially suggesting that the collective rotational time for an IL can be predicted by the single particle orientation time and the static dipolar correlation factor. This finding is very important because it suggests negligible roles for the ion-ion and ion-dipole interactions on DR process in ILs. This interesting and important finding has not yet been investigated from a theoretical perspective. This constitutes the principal focus of the present work. It is now known that ILs are characterized by appreciable amount of spatial heterogeneity which can substantially affect the relaxation dynamics.31–40 It is seen that in common dipolar liquids τ D connects to medium viscosity (η) via τ u through the Stokes-Einstein-Debye (SED)3, 5, 7, 11, 41–45 relation, τu =

3Veff η , kB T

(5)

where Veff (=Cf Vm ) denotes the effective rotational volume of the dipolar species and is commonly determined by the molecular volume (Vm ), the shape factor (f) of the rotating particle, and a hydrodynamic friction coefficient (C). The importance of SED relation lies in its use for understanding the solute-solvent coupling in various media.46–48 Simulation studies have found that SED relation in ILs may not be valid.49 It is interesting to note that fluorescence Stokes shift dynamics results have suggested the validity of SED relation in ILs for the viscosity dependence of average rate of solvation of a dissolved probe.50–53 In DR measurements, the validity of SED relation in ILs has been verified by calculating τ u from experimentally measured τ D via macro-micro relations described in Eqs. (1)–(4). Interestingly, Veff , obtained by the above route for ILs, has been found to be much less than Vm .

For example, in case of aluminate ILs,11, 54, 55 estimated Veff from experimental DR data is only ∼ 4% of Vm . For imidazolium ILs, Veff is even less and within ∼ 1% of their respective Vm .5 This finding is important because it hints at either frictionless rotation in ILs or participation of a few particles in DR relaxation. In high viscous liquids such as these ILs, frictionless rotation is a highly idealized scenario and thus may not be operative at all. Then, non-diffusive angular motion, such as rotational jump,32, 56 and/or rotation of a few dipolar ionic species induced by the spatial heterogeneity might be the sources for producing such anomalously low Veff . In this context, we would like to mention that the failure of hydrodynamics in a low viscous liquid like water has been shown to arise from large angle jump motion of water molecules at the time of hydrogen bond switching between two adjacent oxygen atoms.57–59 In the present work, we have proposed a semi-molecular theory for the DR in ILs where the effects of ion-ion and ion-dipole interactions on DR time scales are investigated. We have also derived the relation among τ D , τ u , and τ M for ILs. The present work considers ion-dipole and ion-ion interactions in addition to dipole-dipole interactions while constructing the generalized Smoluchowski description for the time, position, and orientation dependent density field. The Smoluchowski equation contains dipole and ion density fluctuation terms, includes contributions from both rotational and translational motions. In the long wavelength (kσ → 0) limit, the translational contribution to the orientation correlation function disappears, although it is significant at intermediate length-scales. Heterogeneity aspect has been discussed in light of the validity of SED relation and Veff . Temperature dependence of DR time has also been explored for several ILs. II. THEORY AND CALCULATION DETAILS

We will discuss the theoretical details in Secs. II A and II B. In Sec. II A, we derive the Smoluchowski equation for IL and in Sec. II B, we use that to obtain the expressions for orientation relaxation function. Subsequently, microscopic expressions for various relaxation time constants relevant to IL systems have been derived.

A. Derivation of the Smoluchowski equation

The present derivation of Smoluchowski equation60 is motivated by the CB theory21–28 for dipolar fluids. Like in CB theory here we assume that the relaxations of angular and spatial momenta are fast and thus focus on the continuity equation for the number density (ρ(r, , t))23, 61 ∂ρ(r, , t) = −∇ · J − ∇ · J , ∂t

(6)

where r and denote respectively the spatial position and orientation vectors. J and J are the spatial and angular fluxes, respectively. ∇ and ∇ represent the usual spatial and orientation gradients. In the over-damped limit, these fluxes are calculated from generalized free energy functional using the

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classical density functional theory,55, 62–65

B. Derivation of orientation relaxation function

Fourier transformation of Eq. (9) leads to the following expression:

βF [ρ(r, ), nα (r)]  = drd δρ(r, ) [ln ρ(r, ) − 1] − +

1 2



drd dr d  δρ(r, )δρ(r ,  )c(r, ; r ,  )

2  

drnα (r) [ln nα (r) − 1]

α=1

−

−

2 1  drdr cαβ (r, r )δnα (r)δnβ (r ) 2 α,β=1 2  

drdr d cdα (r, ; r )δρ(r, )δnα (r ),

∂ [δρ (k, , t)] ∂t = −DT k 2 δρ (k, , t) ρ   0 k 2 d  c(k, ,  )δρ(k,  , t) + DT 4π 2 ρ   0 [cdα (k, ) δnα (k, t)] k2 + DT 4π α=1 + DR ∇ 2 δρ (k, , t)  ρ  0 ∇ 2 d  c(k, ,  )δρ(k,  , t) − DR 4π 2 ρ   0 ∇ 2 + DR [cdα (k,  )δnα (k, t)]. (10) 4π α=1

(7)

α=1

where nα (r) is the ion number density. c(r, ; r ,  ), cαβ (r, r ), and cdα (r, ; r ) denote respectively the two particle dipoledipole, ion-ion, and dipole-ion static correlations. Note δ is used to represent the fluctuation over bulk value. In the overdamped limit, the fluxes can be written as,66–68 J = −DT ρ(r, , t)∇ ·

l,m

c(k, ,  ) =

δ F [{ρ(r, , t)}] δ ρ(r, , t)

δ F [{ρ(r, , t)}] , J = −DR ρ(r, , t)∇ · δ ρ(r, , t)

Using spherical harmonic expansion  cdα (k, ) = cα (lm; k) Ylm ( ) 

c (l1 l2 m; k) Yl1 m ( ) Yl2 m (  )

(11)

l1 l2 m

(8)

where DT and DR represent respectively the translational and rotation diffusion coefficients. Using Eqs. (6)–(8) we obtain, ∂ [δρ(r, , t)] ∂t = DT ∇ 2 δρ(r, , t)  ρ  0 ∇ 2 dr d  c(r, ; r ,  )δρ(r ,  , t) − DT 4π 2  ρ   0 2 ∇ − DT dr cdα (r, ; r )δnα (r , t) 4π α=1 + DR ∇ 2 δρ(r, , t)  ρ  0 2 ∇ dr d  c(r, ; r ,  )δρ(r ,  , t) − DR 4π 2  ρ   0 + DR ∇ 2 dr cdα (r, ; r )δnα (r , t). (9) 4π α=1 Equation (9) represents the Smoluchowski equation for dipolar ILs (ILs in which at least one of the ions possesses permanent dipole moment). Note in Eq. (9) both the fluctuating ion and dipolar density terms exist which will produce self (ion-ion and dipole-dipole) as well as cross (ion-dipole) terms while constructing the appropriate time correlation function. As a result, the DR in ILs will have contributions from dipoledipole, ion-dipole, and ion-ion interactions.

δρ(k,  , t) =



alm (k, t)Ylm (  ),

lm

and combining Eqs. (10) and (11), we obtain  ∂  alm (k, t) Ylm ( ) ∂t lm  = −DT k 2 alm (k, t)Ylm ( ) lm

+ DT ×

ρ  0

4π



d 



c (l1 l2 m; k) Yl1 m ( ) Yl2 m (  )

l1 l2 m

alm (k, t) Ylm (  )

lm

+ DT

 k2

ρ  0

 2  



k2 cα (lm; k) Ylm ( )δnα (k, t) 4π lm α=1  + DR ∇ 2 alm (k, t)Ylm ( ) lm

 ρ    ∇ 2 d  c (l1 l2 m; k) Yl1 m ( ) Yl2 m (  ) − DR 4π l1 l2 m  alm (k, t) Ylm (  ) × lm

 2  ρ    2 ∇ − DR cα (lm; k) Ylm ( )δnα (k, t) . 4π lm α=1 (12) Multiplying both sides of Eq. (12) by the complex conjugate of the spherical harmonics, we obtain

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 ∂  ∗ ( ) alm (k, t) Ylm ( ) Ylm ∂t lm = −DT k 2



∗ ( ) alm (k, t)Ylm ( ) Ylm

lm

+ DT

+ DT

ρ  0

4π ρ  0

4π

+ DR ∇ 2

 k2

k

2

2  

− DR

0

4π ρ  0

4π

c (l1 l2 m; k) Yl1 m ( ) Yl2 m (  )



∗ ( ) alm (k, t) Ylm (  )Ylm

lm



∗ cα (lm; k) Ylm ( )δnα (k, t) Ylm

( )

lm

α=1

∗ ( ) alm (k, t)Ylm ( ) Ylm

lm

− DR

 l1 l2 m





ρ 

d 

 ∇ 2

d  

∇ 2

 l1 l2 m

2   α=1

c (l1 l2 m; k) Yl1 m ( ) Yl2 m (  )



∗ ( ) alm (k, t) Ylm (  )Ylm

lm



∗ cα (lm; k) Ylm ( )δnα (k, t) Ylm

( ) .

(13)

lm

Integrating both sides of Eq. (13) over orientation space and using the normalization rule of spherical harmonics, we obtain the following equation for the time derivative of alm (k, t), ρ   ∂ 0 [alm (k, t)] = −DT k 2 alm (k, t) + DT k 2 (−1)m c (ll2 m; k)al2 m (k, t) ∂t 4π l 2

+ DT k 2

ρ

0

4π

2 

cα (lm; k)δnα (k, t) −DR l (l + 1) alm (k, t)

α=1

+ DR l (l + 1) (−1)m

 ρ  0 4π

c (ll2 m; k)al2 m (k, t) + DR l (l + 1)

l2

2  ρ  0

4π

cα (lm; k)δnα (k, t) . (14)

α=1

If l = 1, Eq. (14) will be the equation of motion relevant to DR in ILs. Here we use mean spherical approximation (MSA) theory for obtaining c(k, ,  ) which predicts that the only contributing c(l1 , l2 , m; k)’s are c(110; k), and c(111; k).69–71 Eq. (14) then reduces to a simpler equation as follows: 

∂ [alm (k, t)] = − DR l (l + 1) + DT k 2 alm (k, t) ∂t ρ 

 0 + DR l (l + 1) + DT k 2 (−1)m c (llm; k) alm (k, t) [δl,1 + δl,0 ] 4π 2

  ρ0   + DR l (l + 1) + DT k 2 (15) cα (lm; k) δnα (k, t). 4π α=1 Rearrangement of Eq. (15) leads to the following equation: ρ 

 ∂ 0 [alm (k, t)] = − DR l (l + 1) + DT k 2 alm (k, t) 1 − (−1)m c (llm; k) [δl,1 + δl,0 ] ∂t 4π 2

  ρ0   + DR l (l + 1) + DT k 2 cα (lm; k) δnα (k, t). 4π α=1

Note Eq. (16) is a first order differential equation with respect to time. This equation can easily be solved to obtain the wave number, rank, and time dependent solution of alm (k, t). Here one important point should be mentioned. The last term of the Eq. (16), originated from ion-dipole interaction, contains ion

(16)

density fluctuation term (δnα (k, t)), which is also wave number and time dependent. For solving the equation for alm (k, t), we use the approximation, like in our theoretical studies of Stokes shift dynamics in ILs,63, 64 that δnα (k, t) is much slower than that of δρ(k, , t) and thus remain unchanged during

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J. Chem. Phys. 140, 014504 (2014)

the timescale of orientational relaxation. This is a critical approximation whose validity should be examined via suitable simulation studies. The expression for the generalized orientational correlation function can be written as71 ψ (k, t) = alm (k, 0) alm (−k, t) .

(17)

At kσ → 0 limit, ψ(k, t) corresponds to CM , CM = Lt [ψ (k, t)] k→0

(18)

ion (k, t) = δnα (k; 0)δnβ (−k; t) . Sαβ

and at kσ → ∞ limit, ψ(k, t) corresponds to Cu , Cu = Lt [ψ (k, t)] . k→∞

(19)

Now by solving Eq. (16), we obtain the following functional form for alm (k, t): alm (k, t) =

[{Aalm (k, 0) + C}] exp (At) − Bt , A

It is clear that in Eq. (23) Bt is still a time dependent function. An inspection of Eqs. (20)–(23) reveals that generalized relaxation in ILs contains both translational and rotational contributions, and DR time constant would be affected by ion translation. However, full coupling between the rotation and translation has been missed due to the approximation made while solving Eq. (16). For the calculation of ψ(k, t) using Eq. (17), we need the ion (k, t), defined as isotropic ion dynamic structure factor, (Sαβ

(20)

where



A = − DR l(l + 1) + DT k 2  ρ   0 c (llm; k) [δl,1 + δl,0 ] , (21) × 1 − (−1)m 4π

 2  ρ0 

2 C = DR l(l + 1)+DT k cα (lm; k) δnα (k, 0) , 4π α=1 (22) and  2  ρ0 

2 Bt = DR l(l + 1)+DT k cα (lm; k) δnα (k, t) . 4π α=1 (23)

ion (k, t) has been assumed to be given by,70 Sαβ

 ion ion (k, t) = Sαβ (k) exp −DT k 2 t / Sαα (k) Sαβ

(24)

(25)

ion with Sαβ (k)is defined by Eq. (11) of Ref. 64. Using Eqs. (20)–(25) in Eq. (17), we obtain the following expression for ψ(k, t):

ψ(k, t) = X exp(−t/τ1 ) + Y exp(−t/τ2 ) + Z exp(−t/τ3 ). (26) Equation (26) is the main result of this paper. Equation (26) suggests that the generalized orientational correlation function for ILs is a tri-exponential function of time. For dipolar liquids, this correlation function was found to be biexponential.21–28 Note that while the first two time constants (τ 1 and τ 2 ) of Eq. (26) have originated from pure dipoledipole interaction, the third one (τ 3 ) arises from pure ion-ion interaction. This is evident in Eq. (30) described later. However, each of the coefficients (X, Y, and Z) of Eq. (26) contains contributions from dipole-dipole, ion-ion, and ion-dipole interactions via the static spatial correlations. This is shown in Eqs. (27), (28), and (29) described below:

ρ 

ρ  2 0 0 c (110; k) 1 − c (110; −k) a10 (k, 0) a10 (−k, 0)

2DR + DT k 2 1 − 4π 4π 2

   2 2 ρ0 [cα (10; k) cα (10; −k) δnα (k, 0) δnα (−k, 0) ] + 2DR + DT k 4π α=1

ρ 

ρ  X= ,

2 0 0 c (110; k) 1 − c (110; −k) 2DR + DT k 2 1 − 4π 4π



ρ 

ρ  2 0 0 c (111; k) 1 + c (111; −k) a11 (k, 0) a11 (−k, 0)

2DR + DT k 2 1 + 4π 4π 2

   2 2 ρ0 [cα (10; k) cα (10; −k) δnα (k, 0) δnα (−k, 0) ] + 2DR + DT k 4π α=1

ρ 

ρ  Y = ,

2 0 0 c (111; k) 1 + c (111; −k) 2DR + DT k 2 1 + 4π 4π

(27)



(28)

and

Z=

2 2  ρ0 2  [cα (10; k) cα (10; −k) δnα (k, 0) δnα (−k, 0) ] 4π α=1

ρ 

 .

2 ρ0  0 c (111; k) 1 + c (111; −k) 2DR + DT k 2 1 + 4π 4π

2DR + DT k 2

(29)

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τ1 =

−1

−1 

ρ  ρ  

  0 0 c (110; k) c (110; k) 2DR + DT k 2 1 − = (2DR )−1 1 + p (kσ )2 1 − 4π 4π

τ2 =

−1

−1 

ρ  ρ  

  0 0 c (111; k) c (111; k) 2DR + DT k 2 1 + = (2DR )−1 1 + p (kσ )2 1 + , 4π 4π

(30)

−1   2  2 



  2 −1 2 Sαα (k) / DT k Sαα (k) / p (kσ ) , = (2DR ) τ3 = α=1

α=1

where p = DT / (2DR σ 2 ). For dipolar liquids, the CB treatment21–28 will produce exactly the same expressions for τ 1 and τ 2 but the associated coefficients (X and Y) will be different from the present ones. These two time constants (τ 1 and τ 2 ) will be calculated for several dipolar liquids using the expressions given above and compared with relevant experiments. In our calculations, the wavenumber dependent static orientational correlations, c(110; k), and c(111; k), have been obtained by assuming the IL molecules as neutral dipolar species, neglecting completely the charge and shape (deviation from being spherical) aspects.63–66 Also, spatial heterogeneity of ILs has not been considered while obtaining these orientational correlations. In addition, proper corrections both at k → 0 limit (guided by measured ε0 ) and k → ∞ limit (single particle limit) have been employed for these correlations.63 Note that the inaccuracy involved in this approximate method of calculating the wavenumber dependent static correlations may not critically influence our predicted relaxation parameters because DR, such as Stokes shift dynamics, is a collective phenomenon where details of the solvent structure can assume secondary role. The wavenumber dependent dipole-ion static correlations, cα (10, k), have been determined again in the homogeneous limit approximated above and calculated by using Eq. (13) of Ref. 64. DT and DR are supplied from hydrodynamics using stick boundary condition. IL densities, viscosity coefficients, dipole moments, and diameters required for these calculations are provided in Table S1 of the supplementary material.72

where τD = τM2 = ε0 τM1 τDii = τM3 τM1 = Lt τ1 (k) k→0

τM2 = Lt τ2 (k)

(31)

.

k→0

τM3 = Lt τ3 (k) k→0

Hence τD = Lt

k→0

  (2DR )−1 [1 + p(kσ )2 ]

−1  ρ  0 c(111; k) × 1+ 4π and

 τDii = Lt

k→0

 (2DR )−1

2 

(32)

 [Sαα (k)/p(kσ )2 ]

.

(33)

α=1

As mentioned above, τ D in Eq. (32) originates from dipoledipole interaction and τDii in Eq. (33) from ion-ion interaction. Now from the definition of τ u we know, τu =

1 . L (L + 1) DR

(34)

For dielectric relaxation, L = 1, τu = (2DR )−1 .

(35)

Use of Eq. (35) then establishes the relations among τ D , τ u , and τ M via Eqs. (29)–(33). III. NUMERICAL RESULTS AND DISCUSSION

C. Relationship among relaxation time constants

The relation between τ D and τ M has already been established by the CB theory for dipolar liquids.25 It is known that for dipolar liquids, τ D is equal to the transverse polarization relaxation time, and also equal to the longitudinal polarization relaxation time multiplied by the static dielectric constant. In ILs, where CM (≡ (kσ → 0, t)) is tri-exponential with time constants originating from dipole-dipole and ionion interactions, the relation between τ D and τ M can be separated into two parts. One of these dissected parts would be related to dipole-dipole interaction and the other one would be to ion-ion interaction. The relations between τ D and τ M are as follows,25

We have chosen six ILs here for our theoretical study and these are 1-Butyl-3-methylimidazolium tetrafluoroborate ([Bmim][BF4 ]), 1-Butyl-3-methylimidazolium 1-Ethyl-3hexafluorophaosphate ([Bmim][PF6 ]), methylimidazolium dicyanamide ([Emim][DCA]), 1Hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl) 1-Ethyl-3-methylimidazolium imide ([Hmim][NTf2 ]), tetrafluoroborate ([Emim][BF4 ]), and sodium 2,5,8,11tetraoxatridecan-13-oate ([Na][TOTO]). These six ILs have been chosen because temperature-dependent experimental DR data for them are available and these data are required for verification of our theoretical results. Various physical parameters of these ILs, needed as input for calculations, are provided in Table S1 of the supplementary material.72

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In addition, we have used the present theory to calculate the dielectric relaxation time scales for five dipolar liquids, water (H2 O), methanol (CH3 OH), ethanol (CH3 CH2 OH), n-propanol (CH3 CH2 CH2 OH), and acetonitrile (CH3 CN). Calculations for these conventional solvents assist in examining the performance of the present formalism for common dipolar liquids as well as highlight the possible role of spatial heterogeneity for DR in ILs. A. Orientational relaxation: Dependence on wavenumber and ion translation

The upper panel of Fig. 1 depicts the time-dependent decay of the generalized orientational relaxation, ψ(kσ , t), at three different wavenumbers, kσ ∼ 0, kσ ∼ 2π , and kσ ∼ 4π for [Bmim][BF4 ] at 298 K with p = 1. Note the decay becomes faster at large wavenumbers. This has also been observed for dipolar liquids and understood in terms of participation of nearest neighbour molecules.21 The decay at the collective limit (kσ ∼ 0) is extremely sluggish and suggests presence of a very slow relaxation timescale. The lower panel shows the full decay of ψ(kσ , t) at the collective limit along with the time constants and normalized values for the coefficients. Note the slowest time constant van-

FIG. 1. Plot for generalized orientation correlation as a function of time at three magnitudes of wavevectors for [Bmim][BF4 ] as a representative IL at T = 298 K. The translation parameter (p = DT /2DR ) has been set to unity. Lower panel presents the full decay of the correlation function at the collective limit (kσ ∼ 0). Time constants and amplitudes are also shown inside the panel.

J. Chem. Phys. 140, 014504 (2014)

ishes at larger wavenumbers (kσ ∼ 2π and kσ ∼ 4π ) because the translational mode becomes too efficient at these wavenumbers.21 The increased efficiency arises because of the inverse quadratic dependence on wavenumber of τ 3 (see Eq. (33)). Note the slowest time constant associated with the decay of ψ(kσ → 0, t) is ∼20 μs for [Bmim][BF4 ] at 298 K and originates from ion-ion interaction. Interestingly, broadband dielectric measurements of [Bmim][BF4 ] at 280 K with frequency coverage, 0.1 Hz ≤ ν ≤ 1.8 GHz, have revealed peaks for the imaginary parts of complex dielectric function and conductivity at frequency corresponding to 100 μs timescale.73 This similarity in timescale suggests that ion translation does contribute to dielectric relaxation in ILs at low frequencies, and explains analytically the emergence of conductivity and dielectric loss peaks at similar frequencies in measurements with ILs at low frequencies. Effects of translational diffusion coefficient on the wavenumber dependent relaxation time constants τ 1 , τ 2 , and τ 3 are shown in Fig. 2 for [Bmim][BF4 ] at 298 K. The calculations are done for four p(= DT / 2DR σ 2 ) values of 0.0, 0.2, 0.4, and 1.0. While the upper panel shows the effects of translation on time constants originating from dipoledipole interaction, the same on time constant arising from ionion interaction is presented in the lower panel. It is evident from Fig. 2 that time constants become faster as translation

FIG. 2. Wavenumber (kσ ) and ion translation (p = DT / 2DR ) dependence of three relaxation time constants of generalized orientation correlation for [Bmim][BF4 ].

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contribution increases. In addition, time constants originating from dipolar interaction (τ 1 and τ 2 ) do not show any dependence on the translational parameter (p) at the collective limit. In the absence of translation (that is, at p = 0), τ 1 slows down considerably at intermediate wavenumbers. For dipolar liquids, this slowing down has been explained21 in terms of de Gennes’ narrowing74–76 of the dynamic structure factor of a dense liquid at intermediate wave vectors. This slowing down of the relaxation is less prominent for τ 2 because of weak wavenumber dependence of the transverse component of the orientational static structure factor. At a very large wavenumber, τ 1 and τ 2 becomes nearly equal but very small. The third and the slowest relaxation time (τ 3 ), arising from ion-ion interaction, behaves quite differently from τ 1 andτ 2 . At the collective limit, τ 3 is much larger than τ 1 and τ 2 and thus dominates the relaxation process. This is the source for the conductivity dominated dielectric loss peak observed in broadband dielectric relaxation measurements in the 0.1 Hz1.8 GHz frequency range for [Bmim][BF4 ] at 280 K.73

TABLE II. Comparison between the experimental and theoretical DR times for ILs and common dipolar liquids. ILs System [Bmim][BF4 ] [Bmim][PF6 ] [Emim][DCA] [Hmim][NTf2 ] [Emim][BF4 ] [Na][TOTO] Dipolar liquids H2 O CH3 OH CH3 CN CH3 CH2 OH CH3 CH2 CH2 OH

T (K) η (poise) 298 298 298 298 298 294

0.996 2.496 0.210 0.678 0.372 3108

298 298 298 298 298

0.001 0.005 0.003 0.011 0.022

τ D (ps)

exp t.

τD

15 463 284 40 547 1178 2734 31 15 567 233 4223 46.6 5.6 × 107 1.1 × 107 8 50 10 60 84

exp t.

(ps) ξ = τD / τD

8 60 11 90 134

54 34 88 67 90 5 1 0.8 0.9 0.7 0.6

Three collective orientation relaxation time constants (τM1 , τM2 , and τM3 ) calculated from the present theory for six ILs at 298 K are provided in Table I. Note for all these ILs, time constants are well separated and the microsecond component constitutes nearly the half of the total relaxation. In addition, time constants obtained for [Na][TOTO] are much larger than those for imidazolium ILs. This is because of much larger viscosity of [Na][TOTO] than that for imidazolium ILs at this temperature. Since τM2 = τD , it is worth comparing the calculated dielectric relaxation times (τ D ) with those from experiments for these ILs. Since multiple relaxation times are reported in measurements of these ILs, (see Table S2 of the supplementary material),72 we compare our calculated τ D with the slowest ones from measurements. This is done in Table II where calculated and experimental (slowest) times77, 78 for several dipolar liquids at 298 K are also compared. Note that calculated τ D for these ILs are much larger than the measured values. However, the agreement between calculated and experimental time constants is favourable for dipolar liquids. The closeness between theory and experiments is parameterized via the ratio between the calculated and experimental time constants, ξ , shown in the last column of Table II. Note ξ varies between 0.6 and 1.0 for dipolar solvents but for ILs the range is 5–90. More inter-

estingly, the best prediction has been made for [Na][TOTO] which is the most viscous of the ILs considered here. This suggests that viscosity is not the reason for the poor agreement between theory and experiments observed here. The above is further highlighted in Table III where calculated τ D at various temperatures for four ILs have been compared with the available experimental data. Temperature dependent densities (ρ) and viscosities (η)8, 64 are also tabulated which were used in our calculations. Clearly, both calculated and experimental times decrease with temperature as viscosity decreases but the ratio (ξ ) does not show a smooth decrease. In addition, for imidazolium ILs at 338 K where η lies between 8 and 20 cP and is at least two orders of magnitude smaller than that of [Na][TOTO] at comparable temperature, the extent of disagreement is much larger for imidazolium ILs than for [Na][TOTO]. Consideration of these results leads us to suggest that non-diffusive rotational mechanism might be the reason for the experimental relaxation time being faster than the hydrodynamic predictions. In fact, molecular dynamics simulations for an imidazolium IL have already indicated limited validity of hydrodynamic relations between viscosity and diffusion, and the deviation connected to medium temporal heterogeneity.49 Note temperature dependent DR measurements5, 8 of these ILs have reported substantially stretched relaxation dynamics even at a temperature as large as ∼338 K. Simulations studies have supported these observations by showing the presence of substantial temporal heterogeneity at higher temperatures.79–81 These and the studies revealing spatial heterogeneity in ILs34, 36–40 support us to

TABLE I. Collective orientation correlation time constants and their respective coefficients.

TABLE III. Comparison between the calculated effective volume and molecular volumes of the rotating ion.

IL

ILs

B. Comparison between theory and experiments: Effects of heterogeneity

[Bmim][BF4 ] [Bmim][PF6 ] [Emim][DCA] [Hmim][NTf2 ] [Emim][BF4 ] [Na][TOTO]

X

τM1 (ns)

Y

τM2 (ns)

Z

τM3 (μs)

0.05 0.02 0.07 0.02 0.03 0.04

4.53 7.04 0.91 2.63 0.99 1.4 × 104

0.48 0.49 0.47 0.49 0.49 0.48

15.5 40.6 27.4 156 4.22 5.6 × 104

0.47 0.49 0.46 0.49 0.48 0.48

21 29 4.6 19 5.1 1050

[Emim][DCA] [Bmim][BF4 ] [Bmim][PF6 ] [Hmim][NTf2 ] [Emim][BF4 ] [Na][TOTO]

 dip  Veff 10−30 m3

 dip  Vmol 10−30 m3

χ = Veff / Vmol

0.7 1.4 5.3 2.2 0.8 55

149 163 163 191 149 300

0.005 0.009 0.033 0.012 0.005 0.180

dip

dip

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TABLE IV. Temperature dependence of theoretical and measured DR time. exp t.

exp t.

ξ = τD /τD

T(K)

ρ(g/cm3 )

η(P)

τ D (ps)

[Bmim][BF4 ]

278.15 288.15 298.15 308.15 318.15 328.15 338.15

1.217 1.209 1.202 1.195 1.189 1.183 1.177

3.192 1.947 0.996 0.582 0.378 0.290 0.208

52 411 31 072 15 463 8799 5570 4168 2919

670 351 284 140 93.7 59.4 52.5

78 89 54 63 59 71 56

[Emim][DCA]

278.15 288.15 298.15 308.15 318.15 328.15 338.15

1.098 1.082 1.066 1.050 1.034 1.018 1.002

0.403 0.288 0.210 0.156 0.119 0.091 0.072

5554 3856 2734 1977 1470 1096 846

46.4 34.5 30.7 25.2 18.9 16.7 16.1

120 112 89 78 77 64 53

[Emim][BF4 ]

278.15 288.15 298.15 308.15 318.15 328.15 338.15

1.295 1.288 1.279 1.272 1.265 1.258 1.250

0.761 0.532 0.372 0.256 0.188 0.127 0.089

9164 6216 4223 2826 2020 1330 909

99.3 60.7 46.6 36.6 21.9 18.4 15.5

92 102 91 77 92 72 59

[Na][TOTO]

254 264 274 284 294 304 314 324 334 344

1.268 1.263 1.259 1.254 1.249 1.245 1.240 1.235 1.230 1.226

2 855 724 768 12 245 323 31 374 22 563 3108 662 191 69 30 14

5.8 × 1013 2.4 × 1011 6.0 × 109 4.2 × 108 5.6 × 107 1.2 × 107 3.2 × 106 1.1 × 106 1.4 × 105 2.2 × 105

2.2 × 109 2.4 × 109 2.4 × 108 4.6 × 107 1.1 × 107 3.3 × 106 1.1 × 106 4.0 × 105 1.6 × 105 7.0 × 104

26364 100 25 9 5 4 3 3 1 3

ascribe the current disagreement between theory and experiments for ILs to the non-diffusive rotation of the dipolar ion induced by the medium heterogeneity. If the non-diffusive rotation involves large angle jumps,82 then the solute-medium coupling is expected to be much weaker than that envisaged in hydrodynamics. Such a weaker coupling may be approximately considered as being hydrodynamic rotation of a species with a smaller volume. Effective rotation volumes arising from this consideration for the rotating dipolar ions of these ILs have been predicted by the current calculations by assuming equality between the calculated and measured times exp t. (that is, τD = τD ). Table IV summarizes the predicted efdip fective rotation volumes (Veff ) for dipolar ions at 298 K and dip

compares with the corresponding molecular volumes (Vmol ). dip dip The ratio between these two volumes (χ = Veff / Vmol ) are also presented in the last column which indicates that an extremely small volume is associated with the dipolar ion rotation if analyzed using the conventional hydrodynamic arguments. The prediction that ∼0.5%-1% of ionic molecular volumes is associated with rotation has already been found in simulation studies with an imidazolium IL at 303 K. Analyses of measured DR data using the macro-micro relation between τ D and τ u in conjunction with SED relation have also suggested similar low effective rotation volumes for imida-

τD

(ps)

zolium cations.5, 11 We therefore suggest that the observed disagreement between the predicted and measured DR time constants in ILs is originating from heterogeneity-induced nondiffusive rotational mechanism for dipolar ions in these ILs.

IV. CONCLUSION

In conclusion, the first semi-molecular theory presented here for understanding dielectric relaxation in ILs suggests that the time constants associated with triphasic relaxation of the collective generalized orientational relaxation emerge from dipole-dipole and ion-ion interactions. The ion-ion interaction produces a timescale which compares favourably with the experimental frequency of the conductivity dominated dielectric loss peak, providing an analytical explanation for the relevant measurements. Interestingly, even though the relaxation time constants can be separated into pure dipole-dipole and ion-ion contributions, no such separation is possible for the relaxation amplitudes as each of the amplitudes derive contributions from all the three interaction contributions— dipole-dipole, ion-ion, and ion-dipole. The relation among single particle rotational time, collective rotational time, and DR time has been studied for ILs. The expressions derived here for time constants originating from pure dipolar

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interactions in ILs match exactly with the expressions obtained earlier for common dipolar liquids. The DR time constants predicted for dipolar liquids agree well with the experimental data. This and the disagreement for ILs are considered here as arising from heterogeneity-induced nonhydrodynamic rotation mechanism. Attempt to understand the heterogeneity effects on dipolar ion orientation in these ILs in terms of conventional hydrodynamics leads to extremely low ion rotation volumes, an observation has already been made in earlier simulation and experimental studies with ILs. While describing the merits of this semi-molecular theory we would like to mention that the spatial heterogeneity aspect needs to be incorporated in order for rigorous comparison between predicted and measured DR times. This can be done via the spatial structural and orientational correlations but unfortunately analytical scheme for calculating them in presence of microscopic heterogeneity is not available yet. One can of course employ computer simulations for supplying them and carry out the necessary calculations. Another improvement would be to consider the solvent inertia while developing the theory for predicting the complete DR. This would be an arduous task as that would require correct analytical description for the frequency dependent frictional response of the medium. These caveats notwithstanding, it would be interesting to see how the current theoretical framework can be expanded to understand the dielectric relaxations in (IL+polar solvent) binary mixtures,83, 84 (amide + electrolyte) deep eutectics,85–88 and subsequently, the Stokes shift dynamics in them. Interestingly, the separation between dipolar and ionic timescales revealed by the present theory seems to support the attempt of describing solvation timescales in dipolar ILs in terms of separated dipolar and ionic rearrangement timescales.63–66, 89, 90 However, just like in collective orientational relaxation, the amplitudes associated with solvation response function will have inseparable contributions from dipole-dipole, ion-ion, and ion-dipole interactions. While modelling ionic liquids in this study as composed of ions with point dipoles embedded at the respective centres (for dipolar ions) and the resultant interactions as dipoledipole, ion-ion, and ion-dipole components, there could be a danger in such approximate description. This is because the dipole moment of an ion is a coordinate dependent quantity and attempts to define it based on a centre of mass coordinate scheme can induce serious error.91–93 For example, longer alkyl chain associated with [Hmim] relative to [Emim] can shift the centre of mass away from the imidazolium ring, producing a larger calculated dipole moment at the centre of mass and making [Hmim] appear to be more polar than [Emim]. This is energetically incorrect as various experiments indicate stronger electrostatic solute-solvent interactions for [Emim]. However, the present calculations are free from this error as dipole moments used to calculate orientational static correlations have been obtained from experimental static dielectric constants5 via the MSA theory. However, the inherent coupling between the ion rotation and translation neglected in the present work may have impact on the conclusion that the anomalously low ion rotation volume has arisen solely from large angle jumps. This is because a very recent instantaneous normal mode (INM) study of model ILs94 have shown that

J. Chem. Phys. 140, 014504 (2014)

details of the charge distribution might strongly influence the extent of coupling between translational and rotational motions in real ILs, substantially modifying the solute-medium interaction viewed from traditional hydrodynamics. Further analytical works suitably aided by computational study are therefore required to construct an accurate molecular description of orientational relaxation of these complex yet fascinating solvent systems.

ACKNOWLEDGMENTS

We thank Professor A. Chandra for suggesting this problem. S.D. thanks the Council of Scientific and Industrial Research (CSIR), India for providing a research fellowship. It is our pleasure to dedicate this work to Professor B. Bagchi who is turning sixty in January 2014. 1 A.

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