Rotation driven translational diffusion of polyatomic ...

Rotation driven translational diffusion of polyatomic ions in water: A novel mechanism for breakdown of Stokes-Einstein relation Puja Banerjee, Subramanian Yashonath, and Biman Bagchi

Citation: The Journal of Chemical Physics 146, 164502 (2017); doi: 10.1063/1.4981257 View online: https://doi.org/10.1063/1.4981257 View Table of Contents: http://aip.scitation.org/toc/jcp/146/16 Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 146, 164502 (2017)

Rotation driven translational diffusion of polyatomic ions in water: A novel mechanism for breakdown of Stokes-Einstein relation Puja Banerjee, Subramanian Yashonath, and Biman Bagchia) Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, Karnataka 560012, India

(Received 17 October 2016; accepted 5 April 2017; published online 25 April 2017) While most of the existing theoretical and simulation studies have focused on simple, spherical, halide and alkali ions, many chemically, biologically, and industrially relevant electrolytes involve complex non-spherical polyatomic ions like nitrate, chlorate, and sulfate to name only a few. Interestingly, some polyatomic ions in spite of being larger in size show anomalously high diffusivity and therefore cause a breakdown of the venerable Stokes-Einstein (S-E) relation between the size and diffusivity. Here we report a detailed analysis of the dynamics of anions in aqueous potassium nitrate (KNO3 ) and aqueous potassium acetate (CH3 COOK) solutions. The two ions, nitrate (NO3 − ) and acetate (CH3 CO2 − ), with their similar size show a large difference in diffusivity values. We present evidence that the translational motion of these polyatomic ions is coupled to the rotational motion of the ion. We show that unlike the acetate ion, nitrate ion with a symmetric charge distribution among all periphery oxygen atoms shows a faster rotational motion with large amplitude rotational jumps which enhances its translational motion due to translational-rotational coupling. By creating a family of modifiedcharge model systems, we have analysed the rotational motion of asymmetric polyatomic ions and the contribution of it to the translational motion. These model systems help clarifying and establishing the relative contribution of rotational motion in enhancing the diffusivity of the nitrate ion over the value predicted by the S-E relation and also over the other polyatomic ions having asymmetric charge distribution like the acetate ion. In the latter case, reduced rotational motion results in lower diffusivity values than those with symmetric charge distribution. We propose translational-rotational coupling as a general mechanism of the breakdown of the S-E relation in the case of polyatomic ions. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4981257]

I. INTRODUCTION

temperature T, as given below

Electrolyte solutions are of immense importance in biology and chemistry and have remained a central area of research over several past decades.1–9 Apart from their many applications in chemical engineering, such as fertilizer production and oil recovery, salts are used in biological systems to stabilize or to precipitate out biomolecules as per the interaction of ions with macromolecules and nonpolar molecules. The structure and dynamics of polyatomic ions have remained the subject of discussion as in the Hofmeister series.10 But although the properties of electrolytic solutions containing monatomic ions have been widely explored, those of polyatomic ions, like NO3 − , SO4 2− , and PO4 3− , have not yet received sufficient systematic theoretical attention. The venerable Stokes-Einstein (S-E) relation11–13 between diffusion, size, and viscosity has been extensively studied and has been found to be generally valid under ambient conditions. For a particular solvent when the tagged solute molecule is changed, the Stokes-Einstein condition predicts an inverse dependence of the solute diffusion, D, on the solvent viscosity, η, and solute radius (assumed spherical) R at a constant

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0021-9606/2017/146(16)/164502/11/$30.00

D=

kB T . CηR

(1)

The constant C depends on the boundary condition, being 4π for slip and 6π for stick hydrodynamic boundary conditions. This is a widely used equation derived in the hydrodynamic limit where the solute is assumed to be large and move in a continuum medium of the solvent. In spite of these limiting approximations, it is found to work quite well even where the solute and solvent are of similar sizes, but known to break down in the supercooled liquid where general (hydrodynamic) assumptions employed to derive the Stokes-Einstein relation do not hold. In fact, whenever we find departure from the S-E relation, it points to special changes in the transport mechanism of the solute. The breakdown of the S-E relation has already been a subject of much discussion in both liquids and glasses for monatomic alkali cations and halide ions. Different studies attribute this breakdown to different phenomena. Some of the studies propose the phenomenological solvent-berg model14,15 which assumes the presence of a rigid solvation shell around an ion resulting in an effective ionic radius larger than the true radius of the ion. In some cases of observed deviations, the S-E relation is modified with the assumption of fractional viscosity dependence.16–20 It is important to note that the

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Stokes-Einstein relation can become inadequate in two ways. First, the most discussed is through the viscosity dependence. Second, it can breakdown through the dependence on the radius of the diffusing species. In an interesting paper, Zwanzig and Harrison brought these aspects together by suggesting that it is perhaps better to discuss the Stokes-Einstein relation in terms of an effective hydrodynamic radius that depends on solute-solvent interactions, rather than modifying the linear dependence of viscosity on the friction exerted by the solution.21 This argument is meant to capture the aspects of decoupling between the dynamics of the solute and the solvent that leads to a larger diffusion of the solute resulting in the breakdown of the S-E relation.22 Another related study has argued that depending on the size of the solute and the solvent, small solutes explore the solvent cages and the diffusion becomes faster which is known as the levitation effect.23,24 But the situation for polyatomic ions is more complex and has not been explored. A. Summary of experimental findings

Electrical conductivities of many polyatomic ions in water have been measured experimentally.25 Experimental diffusivity values (obtained from conductivity values by the NernstEinstein relation) of some singly charged polyatomic ions in water at infinite dilution (at 25 ◦ C) with their corresponding ionic radii are plotted in Figure 1. It shows two separate grouping of ions, although having a similar ionic radius: One group consists of acetate (CH3 CO2 − ), bicarbonate (HCO3 − ), iodate (IO3 − ) (having lower diffusivity) and the other group consists of nitrate (NO3 − ), nitrite (NO2 − ), chlorate (ClO3 − ), perchlorate (ClO4 − ) (having anomalously higher diffusivity in spite of being similar in size as the first group). We have calculated the diffusivity values of a solute having ionic radii the same as those polyatomic ions using the Stokes-Einstein relation (Equation (1)) at T = 298 K (25 ◦ C), η = 0.896 cP26 (for infinite dilute solution at 25 ◦ C). Figure 1 and Table I show a comparison between the experimental value of diffusivity and that predicted from the S-E relation. It is evident that for acetate (CH3 CO−2 ), HCO−3 and IO−3 , deviation of experimental diffusivity from the StokesEinstein predicted value is much smaller than that for the other 4 ions (shown in bold in Table I) which fall on the higher diffusivity regions in Figure 1 having around 50%-60% larger diffusivity values than the Stokes-Einstein predicted

FIG. 1. Diffusivity of chlorate (ClO3 − ), acetate, bicarbonate (HCO3 − ), iodate (IO3 − ), nitrate (NO3 − ), perchlorate (ClO4 − ), nitrite (NO2 − ) at infinite dilution at 25 ◦ C (data taken from Ref. 25). Values of diffusivity obtained by converting the experimental data of conductivity by the Nernst-Einstein equation. Brown circles show the diffusivity values predicted by the Stokes-Einstein relation at the same temperature and infinite dilution for a solute with those particular ionic radii.

values. In addition, while some ions like chlorate (ClO3 − ), nitrate (NO3 − ), perchlorate (ClO4 − ), and nitrite (NO2 − ) exhibit higher diffusivity values than S-E predicted diffusivity, the acetate ion shows a marginally lower diffusivity than S-E predicted D. It is therefore evident that there is a breakdown of the S-E relation for the systems of polyatomic ions in water. Wolynes and Wang analysed the enhanced diffusion due to the spontaneous collective motion of active matter in the systems of motor-driven active processes.27 They assumed that the motors generate a time series of isotropic kicks on the constituents of a many-body assembly and proposed an equation for an effective enhanced diffusion coefficient (Deff ), ! 1 κl2 , (2) Deff = D0 1 + 2d D0 where l is the kick step size and κ is the basal kicking rate for general spatial dimension d. In the case of some polyatomic ions, fluctuation of hydrogen bonding creates the jump reorientation motion of both the ion and water which results in kicks on the ion and enhances its diffusion coefficient over Stokes-Einstein diffusion. There are certain characteristic key features of polyatomic ions that make them distinct from rigid alkali cations and halide ions. First, the charge is often distributed among

TABLE I. Comparison of diffusivity of polyatomic ions (shown in Figure 1) from the experiment and that predicted from the Stokes-Einstein relation.

Ion Acetate HCO3 − IO3 − NO3 − ClO3 − ClO4 − NO2 −

Size (pm)

Experimental diffusivity at 25 ◦ C and at infinite dilution (10 5 cm2 /s)

Predicted diffusivity from the Stokes-Einstein relation using viscosity value 0.896 cP26 for infinite dilution at 25 ◦ C (10 5 cm2 /s)

194 207 218 200 208 225 187

1.089 1.185 1.078 1.902 1.720 1.792 1.912

1.25 1.18 1.12 1.22 1.17 1.08 1.30

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different atoms or groups. In our computer simulation model, these are accommodated through fractional charges. Note that even in the water molecule we see the enormous consequence of such distributed charges. But water is electrically neutral, so the effect of the distributed charges is not felt too far from the tagged water molecule as the dipolar effect gets screened. Second, these ions can have certain symmetry, such as C2v and D3h , which makes the potential energy surface also symmetric with respect to rotation, thus facilitating large amplitude jumps (as the formation of any deep potential well is avoided). Often the configurations before the jump and after the jump are identical, but sometimes the process of jump may induce a change in the positions and orientations of the surrounding water molecules. At present, there is no general theory of ionic hydration for polyatomic ions and the main challenge is to deal with the distribution of charges. In this work, we demonstrate the importance of complex reorientational dynamics of polyatomic ions (e.g., nitrate) in enhancing its translational motion and therefore resulting in an anomalously high diffusivity which leads to the breakdown of the S-E relation. Already many experimental and theoretical studies have shown that the nitrate ion in water exhibits various interesting structural and dynamical features.28,29 Recently it has been demonstrated that the nitrate ion in water undergoes large amplitude rotational jumps which in turn coupled to the jump dynamics of neighboring water molecules. Now, it has been already discussed that a small-step-Brownian diffusive model is unable to describe the jump dynamics of water.30 The relationship between continuous diffusion and jump dynamics has been explored within the framework of mode-coupling theory in supercooled liquids.31 Jump motions were shown to reduce translational friction and thereby increase diffusion. Such jump motions could lead to deviations in the Stokes-Einstein relation that is derived on the assumption of continuous Brownian motion. The objective of the present study is to explore and analyse by detailed computer simulation studies the ion migration dynamics of KNO3 and CH3 COOK in water. The outline of this paper is as follows. At first, we validate our simulation models in Sections III A and III B. Then, in Section III C, we discuss the rotational motion of nitrate and acetate ions water. Section III D deals with translational and rotational dynamics of nitrate and acetate ions in water and also the coupling between rotation and translation. Sections III E and III F present the quantification of the translational-rotational coupling (trc) in the dynamics of both the ions in water by a time correlation function. Next, we model a system of the nitrate ion with modified charges to analyse the translational and rotational motions of asymmetric polyatomic ions and to verify the mechanism of translational-rotational coupling in those systems in Section III G. Then we conclude with a brief discussion of the salient results. II. SIMULATION DETAILS

Molecular dynamics simulations of potassium nitrate (KNO3 ) and potassium acetate (CH3 COOK) in water have been carried out using the DL POLY32 package. All

J. Chem. Phys. 146, 164502 (2017) TABLE II. Self-interaction parameters for KNO3 aqueous solution in SPC/E water. Species

Atom, i

σii (Å)

εii (kJ/mol)

qi (e)

References

Water Water Cation Nitrate anion Nitrate anion Acetate anion Acetate anion Acetate anion

Hw Ow K N O C1 (CO) C2 (CH3 ) O

0.000 000 3.166 000 4.934 630 3.150 000 2.850 000 3.750 000 3.910 000 2.960 000

0.000 000 0.650 000 0.001 372 0.711 300 0.836 800 0.439 000 0.732 000 0.879 000

+0.4238 −0.8476 1.000 1.118 −0.706 0.700 −0.100 −0.800

35 35 36 37 37 36 and 38 36 and 38 36 and 38

simulations have been carried out with periodic boundary conditions with a cutoff radius of 18 Å. The long-range forces have been computed with the Ewald summation.33,34 For all the simulations, rigid non-polarisable force field parameters have been used for water as well as for ions. We have employed the SPC/E model35 for water. The self interaction parameters consisting of Lennard-Jones and Coulombic terms are listed in Table II. Interactions between unlike species (ion-ion and ion-water interactions) are described by σ +σ using the Lorentz-Berthelot combination rule (σij = i 2 j and √ ε ij = ε i ε j ). For the simulation of aqueous alkali nitrate solution, 16 K+ and 16 NO3 − have been taken with 8756 water molecules in a cubic simulation cell of length 64.04 Å. This corresponds to 0.102M concentration of aqueous KNO3 solution and a density of 1.0065 g/cm3 . For the potassium ion, the OPLSAA36 force field and for the nitrate ion, the potential model as suggested by Vchirawongkwin et al. have been employed.37 Trajectory is propagated using a velocity Verlet integrator with a time step of 1 fs. The aqueous KNO3 system has been equilibrated for 300 ps at 300 K and then a 2 ns MD trajectory has been generated in the microcanonical (NVE) ensemble. The coordinates, velocities, and forces are stored every 5 fs for subsequent use for the evaluation of various properties. We have simulated the system of KNO3 in water at four different temperatures 250 K, 280 K, 320 K, 350 K to obtain the temperature dependence of diffusion of the nitrate ion in water. To study the role of symmetry of the charge distribution in the nitrate ion, we have modified the partial charges on the oxygen atoms of nitrate in a model ion, NOa (Ob )2 − , and carried out four simulations of different ratios of charges, qOa /qOb = 0.125, 0.5, 1.5, and 2 at 300 K. For all the cases, we have equilibrated the system at corresponding temperatures for 300 ps and 1 ns trajectory has been generated in the microcanonical ensemble. To simulate potassium acetate in water at 300 K, we have used the same system size and concentration as mentioned for the aqueous alkali nitrate solution. For the potassium ion, here also we have used the OPLS-AA36 force field and for the acetate ion, we have chosen the united atom OPLS force field.36,38 Similar to the simulation of KNO3 in water, here also we have equilibrated the system for 300 ps at 300 K and then a 2 ns MD trajectory has been generated in the microcanonical (NVE) ensemble.

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III. RESULTS AND DISCUSSION A. Structure of the solvation shell

Figures 2(a) and 2(d) show the radial distribution functions (RDFs) of oxygen (Ow ) atoms of water around constituent atoms of nitrate and acetate ions in the aqueous solution at 300 K. The first maximum in the N–Ow RDF is at 3.5 Å (Figure 2(a)). This result is in good agreement with 3.4 Å from the work of Dang et al.,39 3.65 Å by Xie et al.40 , and 3.5 ± 0.311 Å obtained from the X-ray scattering experiment of Caminiti et al.41 The first maxima in the C1 –Ow RDF is at 3.51 Å (Figure 2(d)) which can be compared with the result of the previous ab initio QM/MM (quantum mechanics/molecular mechanics) study of Mo and Gao.42 From Figures 2(a) and 2(d), it is evident that the solvation shell structure around the acetate ion is not symmetric as that of the nitrate ion. There is no such discrete first solvation shell around C2 of the acetate ion. The first peak of C2 –Ow becomes merged with its second peak which is also superimposed on the second peak of O–Ow . This signifies that almost all water molecules in the first solvation shell of C1 of acetate are hydrogen-bonded with the two oxygen atoms. This asymmetry is caused by the different partial charges on carbon (C2 ) and two oxygen atoms in the acetate ion unlike the nitrate ion with symmetric charge distribution (Figures 2(b) and 2(e)). The probability distribution of water molecules around nitrate and acetate ions is shown in Figures 2(c) and

J. Chem. Phys. 146, 164502 (2017)

2(f). Red indicates higher probability zones. In these two figures, it is clear that the solvation shell around the nitrate ion is symmetric and that of the acetate ion is asymmetric and further this asymmetry is extended up to the second solvation shell. B. Diffusion of ions in aqueous solution

We compute the diffusion constant of NO3 − , DNO3 − , in the aqueous potassium nitrate solution from our simulations at five different temperatures through the ion mean-square displacement by Einstein’s diffusion equation D E D = lim 1/(6t) |r(t) − r(0)| 2 . (3) t→∞

1. Estimation of error

In order to obtain the error bar, we have used the wellknown method employed in many previous studies43,44 where the total trajectory time (in terms of the total number of MD steps, Nt ) is split into x number of blocks each with y number of MD steps such that Nt = xy. Next one computes diffusivity of ions within each block. This provides us with a distribution of diffusivity values with mean hDi and variance ∆D. In Figure 3, we have plotted the mean diffusivity (symbol: diamond) with the variance shown by vertical bars. Now, D as a function of inverse temperature (1/T ) could be described in terms of the Arrhenius relation

FIG. 2. (a) RDF of water oxygen atoms (Ow ) around the constituent atoms of the nitrate ion, (b) nitrate ion with partial charges on the constituent atoms. (c) Probability distribution function of water molecules around the nitrate ion. (d) RDF of water oxygen atoms (Ow ) around the constituent atoms of the acetate ion, (e) model of the united atom acetate ion with partial charges on the constituent atoms. (f) Probability distribution function of water molecules around the acetate ion.

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FIG. 3. Arrhenius plot of the activation energy for the diffusion of the nitrate ion in water. Diffusivity values of NO3 − ion in aqueous KNO3 solution at five different temperatures are also shown in the inset. Diamonds: simulated values with error bars, pink curve: linear fitted curve. Error bars represent standard deviation.

ln D = ln D0 − Ea /RT,

(4)

where Ea is the activation energy, D0 is the pre-exponential coefficient, and R is the gas constant. When ln D versus (1/T) curve is fitted to Equation (4), the obtained value of the activation energy of diffusion (Ea ) for NO3 is Ea,NO3 − = 14.015 kJ/mol. This may be compared with the experimental value of 15 ± 0.7 kJ/mol from the work of Hashitani and Tanaka.45 The diffusion constant of the acetate ion in water, DAc -, is also computed from our simulation at 300 K using the method described earlier. A comparison of our results with that of experiments shown in Table III suggests that there is a good agreement between our results and experimental measurements. Thus, the force field we have used for the simulation seems to validate the intermolecular potential for both the ions in water. At the same time, our simulation also establishes the discrepancy between the diffusivity values of nitrate and acetate ions in water, the simulated D value of nitrate being 77% larger than the acetate ion with a similar ionic radius which is also found from the experiment (shown in Figure 1). C. Rotational motion of ions in water: Orientational time correlation function

To analyse the reorientation dynamics of nitrate and acetate ions in water, we compute the second-order time correlation function, C2 (t), which is defined as D − −n (t + t)E , C2 (t) = P2 (→ n (t0 ) · → (5) 0

FIG. 4. Orientational time correlation function of the nitrate ion and acetate ion in water. TABLE IV. Decay constants of the orientational relaxation of nitrate and acetate ions in water.

Nitrate ion Acetate ion

a1

τ1 (ps)

a2

τ2 (ps)

0.202 0.141

0.194 0.159

0.798 0.859

2.497 6.498

where P2 (x) is the second order Legendre polynomial defined as P2 (x) =

1 2 (3x − 1). 2

−n is the unit vector along the bond vectors of nitrate and Here → acetate ions: 3 N–O bond vectors of the nitrate ion and 2 C–O and one C–C bond vectors for the acetate ion. Figure 4 shows the average orientational time correlation function (C2 (t)) of bond vectors of the nitrate ion and acetate ion in the aqueous solution. A faster decay of C2 (t) of the nitrate ion than the acetate ion clearly demonstrates that the rotational motion of the nitrate ion in water is faster than the acetate ion. These curves are fitted to the biexponential decay function: y = a0 + a1 exp(−x/τ1 ) + a2 exp(−x/τ2 ). Here two time constants correspond to libration motion (τ1 ) and full reorientation (τ2 ) as also seen for water reorientation. Table IV shows time constants derived from the fit. D. Coupled translational-rotational motion of nitrate ion in water

To understand the molecular motion of nitrate and acetate ions in water, we first plot the square displacement (SD) of the centre of mass of a particular nitrate ion and a particular acetate ion in water. Then we analyse the rotational motion of that ion in a time window where it exhibits sudden jump in the SD value.

TABLE III. Comparison between the values of DNO3 − , Ea,NO3 − , and DAc− from our simulation and from experiments.

DNO3 − (cm2 /s) Ea,NO3 − (kJ/mol) DAc− (cm2 /s)

(6)

Experimental value

Our work (0.1M aqueous salt solution at 27 ◦ C)

1.902 × 10 5 (at infinite dilution at 25 ◦ C)25 15 ± 0.7 (0.1M aqueous KNO3 solution)45 1.089 × 10 5 (at infinite dilution at 25 ◦ C)25

(1.67 ± 0.15) × 10 5 14.015 (0.94 ± 0.06) × 10 5

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FIG. 5. (a) Square displacement of a particular nitrate ion from the initial time of the trajectory, (b) time evolution of the quaternions of that chosen nitrate ion, (c) and (d) belongs to the acetate ion.

To analyse the rotational motion of these polyatomic ions, we determine the time evolution of the quaternions of these two ions. At first, we define three suitable orthonormal basis vectors as the reference frame for the nitrate ion and acetate ion. Then from the rotational matrix constituted of direction cosines, we compute quaternions from the matrix elements.46,47 In Figure 5(a), we see that a chosen nitrate ion shows a sudden increase in square displacement around 80 ps. Therefore, we analysed the rotational motion of that nitrate ion in that time window (Figure 5(b)). It is evident that except q4 , all three quaternions exhibit a jump around 80 ps which results in an enhancement of its translational motion. A similar situation is seen in the case of a chosen acetate ion (Figures 5(c) and 5(d)). This finding suggests that the rotational and translational motions of both the ions are strongly correlated. Figure 6 (Multimedia view) shows a frame of the movie that depicts the translational-rotational coupling of a nitrate ion in water.

(t0 + τ), the angle between a N–O bond vector at time t0 + τ (~rNO (t0 + τ)) and d~r(τ) (as shown in Figure 7). The quantity, |θi (t0 ) θf (t0 + τ)|(=θ(τ)) gives the amount of rotation of a N–O bond vector along the direction of the displacement of the centre of mass of the nitrate ion for a time interval τ. In the same way, we have measured d~r(τ) and θ(τ) for the acetate ion also.

E. Quantification of translational-rotational coupling

To quantify the correlation between translational and rotational motions of the nitrate ion, we compute the displacement of the nitrate ion caused by the rotational jump motion along the direction of displacement. For that, we choose a time interval τ and determine the displacement vector of center of mass of the nitrate ion, d~r(τ), for that time interval. Next, we compute the following angles: (1) θi (t0 ), the angle between a N–O bond vector at time t0 (~rNO (t0 )) and d~r(τ) and (2) θf

FIG. 6. Nitrate ion in water. The water molecules in the 1st solvation shell of the nitrate ion are highlighted in yellow, green, and purple. Green and purple water molecules are shown in different colours to describe the translationalrotational coupling in the movie. They break the hydrogen bond with nitrate in the time duration of the movie, which is followed by the rotational jump motion of both nitrate and water and this rotational jump results in the translational motion of the centre of mass of the nitrate ion. Within the actual time duration of the movie (4.035 ps), centre-of-mass of the nitrate ion shows a significant displacement of 3.505 Å. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4981257.1]

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FIG. 7. Schematic model to demonstrate the displacement vector of the centre of mass of the nitrate ion (d~r) and rotational angle θ(=|θi θf |). These three snapshots with different orientations are at different time instants.

In Figure 8, we have shown contour plots of the average displacement (dr) versus the angle of rotation of the nitrate ion and acetate ion in water at 3 time intervals (2 ps, 5 ps, and 10 ps). By comparing the contours for nitrate and acetate ions in difference maps (Figures 8(c), 8(f), and 8(i)), we find that for a given time interval, such as 2 ps, the probability of rotation to a higher rotational angle (θ) is greater for nitrate than acetate ion in water (shown by the blue region). This finding agrees well with the results of orientational time correlation function decay, discussed in Section III C, where we have shown that the nitrate ion rotates faster than the acetate ion in water.

FIG. 9. Average displacement of the centre of mass of the nitrate ion (blue curve) and acetate ion (pink curve) as a function of the rotation angle (θ) along the displacement vector.

Furthermore, this figure shows that both the displacement (dr) and angle of rotation along the displacement (θ) increase with time for both the ions. This signifies that the rotational and translational motions are strongly coupled for both the ions. Therefore, with a slower rotational motion of the acetate ion in water, it shows a reduced translational motion also. In Figure 9, we have plotted the displacement versus angle of

FIG. 8. (a)–(c) are the contour plots of translation-rotation probability distribution of the nitrate ion in water (Pnitrate (dr, θ,τ)) and acetate ion in water (Pacetate (dr, θ, τ)) and the difference of probability distribution of the nitrate ion from that of the acetate ion (Pnitrate (dr, θ, τ) Pacetate (dr, θ, τ)), respectively, at a time interval (τ) 2 ps, (d)–(f) correspond to the same quantities at a time interval (τ) 5 ps and (g)–(i) are at 10 ps.

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rotation averaged with respect to time intervals (τ, taken as 2 ps). We have marked points on the curves with their corresponding τ values. Although for both the ions displacement increases with the angle of rotation, at a particular time interval, the nitrate ion shows both dr and θ values higher than the acetate ion and we see that at 5*τ, which is equal to 10 ps, the average displacement of the nitrate ion is 0.5 Å higher than the acetate ion. F. Time correlation function for translational-rotational coupling (trc)

In order to unravel the mechanistic aspect of translationalrotational coupling (trc), we employ a new time correlation function Ctrc (t) which is defined as Ctrc (t) = hH [θ(τ) − a] ∗ H [b − dr(τ + t)]i,

(7)

where θ(τ) is the rotational angle θ (as defined earlier) at a time interval τ and dr(τ + t) is the displacement of the centre of mass of the nitrate ion at a time interval τ + t. a and b are the values of the rotational angle (θ) and displacement (dr) chosen from Figure 9 and are, respectively, 49.75◦ and 2 Å for the nitrate ion and 39◦ and 2 Å for the acetate ion. H[ ] is the Heaviside theta function defined by ( 0, x < 0 H [x] = . (8) 1, x ≥ 0 For the nitrate ion, when the rotational angle θ becomes greater than a (where a = 49.75◦ ), first H[x] gives 1 (since x > 0) and until the displacement (dr) becomes greater than b (where b = 2 Å), second H[x] also gives 1. But once dr(τ + t) becomes greater than b, second H[x] gives 0 (since x < 0) and the correlation function, Ctrc (t), becomes 0. Therefore, Ctrc (t) measures the correlation time of the displacement to occur after a rotational jump motion of the nitrate ion. Figure 10 is obtained by averaging over all ions present in the system. Similarly we have obtained Ctrc (t) for the acetate ion. Both nitrate and acetate ions show a similar decay of the correlation function, Ctrc (t). Average decay constants (τavg ) obtained by integrating the time correlation functions for both the ions are 0.84 ps for the nitrate ion and 0.8 ps for the acetate ion. This behavior puts in evidence that as rotational motions of these two ions are coupled to the translational motion with a similar coupling strength, with a reduced rotational motion, the acetate ion shows less diffusion than the

FIG. 10. Time correlation function, Ctrc (t), that measures translationalrotational coupling.

nitrate ion where faster rotation enhances the translational motion. G. Study of modified nitrate ion model: Verification of translational-rotational coupling

To understand the properties of a system, modeling the system has been an aim of molecular simulation since its early days.48,49 There exist several previous studies that employed different model systems to explore the effects of translationalrotational coupling in molecular liquids.50–52 Here, we have made a modified-charge family of models (Figure 11(b)) to further verify the mechanism of translational-rotational coupling. We have introduced asymmetry in the charge distribution of the nitrate ion. In the model NOa (Ob )2 − , there are two types of oxygen atoms Oa and Ob having all the interaction parameters the same, but different charges, keeping the total charge of NOa (Ob )2 − fixed ( 1). We have modified the charge distribution of the nitrate ion here to understand the effect of symmetry in charge distribution on diffusivity. These models are similar to the slower diffusing ions which also have asymmetric charge distributions (shown in Figure 1). 1. Rotational motion of model systems, NOa (Ob )2 −

We investigate the reorientational motion of this model by computing the orientational time correlation function (C2 (t)) that is defined as

  C2 (t) = P2 ~u(0) · ~u(t) , (9) where ~u is the unit vector along N–Oa and N–Ob bonds of model ions and P2 is the second rank Legendre polynomial. Figure 12(a) shows C2 (t) of NO3 − and NOa (Ob )2 − (averaged over N–Oa and N–Ob bond vectors of the model ion). It clearly shows that the reorientational motion becomes slower when the model ion has asymmetric charge distribution as compared to symmetric ions such as nitrate. The interesting fact to note from this result is that the decay of rotational relaxation is slower for the models with qOa /qOb = 0.125 and 0.5 compared to the models with qOa /qOb = 1.5 and 2. In addition, we have analysed the rotational motion of N–Oa and N–Ob bond vectors separately and the C2 (t) values are plotted in Figures 12(b) and 12(c) for the models with qOa /qOb = 0.125 and 2.0, respectively. It is clearly shown that the reorientation motion of the two different bonds in the asymmetric model ions is different. In the case of the model with qOa /qOb = 2, for the contribution of a slower reorienting N–Oa bond vector, overall reorientation motion of the model system

FIG. 11. Representation of the standard nitrate model (a) NO3 − and modified nitrate model (b) NOa (Ob )2 − .

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FIG. 12. (a) Second order time correlation function of the nitrate ion and the model systems (NOa (Ob )2 − ) with different qOa /qOb ratios. Different contributions of N–Oa and N–Ob bond vectors to the average rotational relaxation of NOa (Ob )2 − for (b) qOa /qOb = 0.125, (c) qOa /qOb = 2.0.

becomes slower. These results explain the enhancement of the diffusivity of the nitrite ion, NO2 − (shown in Figure 1), with no N–Oa type bond. While for the model with qOa /qOb = 0.125, N–Ob bond vectors rotate slower than N–Oa bond vectors. For all the models with different qOa /qOb ratios, decay constants of the orientational time correlation function are shown in Table V. Therefore, among asymmetric ions, like NOa (Ob )2 − , some charge distribution facilitates diffusion over others depending on the fact that there is any minima in the potential energy surface with respect to rotation. And if it exists, then the hydrogen bond strength of water with those groups of polyatomic ions who can rotate decides the rotational as well as translational motion. Here in the model NOa (Ob )2 − with the C2 axis of rotation, two N–Ob bonds can rotate. As in the model with qOa /qOb = 2 and 1.5, Ob atoms posses less charge than Oa , the hydrogen bond strength of Ob with water molecules surrounding it is less than Oa . But in the model with

qOa /qOb = 0.125, Ob atoms have higher partial charges than Oa . This difference in charge distribution results in a difference in the solvation shell structures around the two types of model ions having qOa /qOb > 1 and qOa /qOb < 1. In Figure 13, we have shown two contour plots showing the probability distribution of water molecules in the solvation shells of these two types of model ions. In the first model having qOa /qOb = 0.125, the probability of finding water molecules is higher around two Ob atoms compared to Oa atoms while in the second one the situation is just the reverse; Oa atom is strongly hydrated compared to two Ob atoms. Therefore, in the model having qOa /qOb = 0.125, due to stronger hydrogen bonding at the two oxygen sites (Ob ) having the C2 axis of rotation (Figure 13(a)), the overall rotational motions become hindered compared to the other model as shown in the rotational relaxation study (Figure 12(a)).

TABLE V. Average rotational relaxation time of the model ions NOa (Ob )2 − for different values of qOa /qOb . Time constants are obtained by fitting the corresponding time correlation function (C2 (t)) to a standard biexponential decay curve. Here we report only the time constants for full rotations (not for libration motion).

In Figure 14, we have plotted the diffusivity values of the nitrate ion and that of modified nitrate ions with error bars that we have calculated by the method discussed in Section III B. The figure shows that with increasing asymmetry in charge distribution, the diffusivity of the model system is decreased. Asymmetry in charge distribution disturbs the rotational motion (as discussed in Sec. III G 1). Thus, due to translational-rotational coupling, ions with asymmetric charge distribution with their reduced rotational motion exhibit lower diffusivity. This seems to clearly establish the promoting role of the rotational motion in augmenting translational diffusion.

qOa /qOb qOa /qOb qOa /qOb qOa /qOb qOa /qOb

=1 = 0.125 = 0.5 = 1.5 =2

τAverage (ps)

τN–Oa (ps)

τN–Ob (ps)

2.497 7.802 4.394 2.936 3.745

7.008 3.61 3.789 5.73

8.122 4.672 2.407 2.318

2. Translational diffusion of model systems, NOa (Ob )2 −

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The acetate ion can be modeled with the NOa (Ob )2 − model with qOa /qOb = 0.125. Our analysis explains why it has such a lower diffusivity compared to the nitrate ion with a similar ionic radius. First, the acetate ion has asymmetric charge distribution compared to the nitrate ion. Second, although two C–O bonds of acetate can rotate as it possesses a C2 axis of rotation, but, oxygen atoms have higher partial charges ( 0.8 e) than carbon atoms (C2 ) ( 0.1 e) (Figure 1(e)). Thus, they become strongly hydrogen bonded with water which again makes its rotational motion more hindered. As we have shown that translational and rotational motions for these ions are strongly coupled, reduced rotational motion lowers the diffusion of the acetate ion to a large extent than the nitrate ion. IV. CONCLUSIONS

FIG. 13. Probability distribution of water molecules around the model ions: (a) qOa /qOb = 0.125, (b) qOa /qOb = 2. Red indicates higher probability regions.

Therefore, the symmetrical charge distribution of the nitrate ion plays an important role to enhance its diffusivity compared to other lower diffusivity ions, like iodate, acetate, HCO3 − , with asymmetric charge distribution shown in Figure 1. Among the asymmetric ions also, the nature of charge distribution decides its rotation and in turn translational motion. As discussed earlier, the models with qOa /qOb = 2 and 1.5 show faster rotational motion than the models with qOa /qOb = 0.125 and 0.5. Therefore, diffusivity also shows a similar trend and the model systems with qOa /qOb = 2 and 1.5 show a higher diffusivity compared to those with qOa /qOb = 0.125 and 0.5 if we compare with respect to the extent of asymmetry (Figure 14).

FIG. 14. Diffusivity of the nitrate ion (qOa /qOb = 1) and the model systems: NOa (Ob )2 − with qOa /qOb = 0.125, 0.5, 1.5, and 2. Error bars represent standard deviation.

Based on the work present here, we conclude that in addition to several different phenomenological and microscopic models used earlier to explain the breakdown of the StokesEinstein relation, in various systems, the role of translationalrotational coupling should be considered as an additional mechanism. We note that this mechanism has not been discussed in any detail for the system of aqueous polyatomic ions. This coupling could be a highly effective mechanism in the case of polyatomic ions. Depending on the symmetry, shape, and interaction of a polyatomic ion with surrounding water molecules in the aqueous solution, this mechanism selectively facilitates the diffusion of some of the ions with favorable (that is, symmetric) charge distribution over others, thus giving rise to higher diffusivity than other ions of comparable, or even smaller sizes. Further studies are needed in this direction to enhance our understanding of anomalous ionic mobility of aqueous polyatomic ions. ACKNOWLEDGMENTS

We thank the Department of Science and Technology (DST, India), Council of Scientific and Industrial Research (CSIR, India), and Sir J. C. Bose Fellowship to Professor B. Bagchi for providing partial financial support. 1 P.

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