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Understanding Translational-Rotational Coupling in Liquid Water through Changes in Mass Distribution Dhivya Manogaran, and Yashonath Subramanian J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b05468 • Publication Date (Web): 09 Nov 2017 Downloaded from http://pubs.acs.org on November 20, 2017
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Understanding Translational-Rotational Coupling in Liquid Water through Changes in Mass Distribution Dhivya Manogaran and Yashonath Subramanian* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012, India. Email id:
[email protected] Phone number: +91-80-22932568
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Abstract A molecular dynamics study of liquid water and models of water has been carried out to understand the effect of changes in the mass distribution on molecular translation and rotation. Calculations on the motion of mH2O and H2nO where m and n vary over a range of values by varying the mass at the hydrogen and oxygen positions show that these form two distinct series. The two series exhibit different translational and rotational properties. Although, a decrease in diffusivity when compared to H2O is observed in both the series, in the case of mH2O series, an enhancement in the ratio of diffusivities {D[H2O]/D[ mH2O]} is found as compared to the square root of the inverse mass ratios while the effect of mass distribution for H2nO is seen to lead to a reduction in the ratio of diffusivities {D[H2O]/D[ H2nO]}with respect to the square root of the inverse mass ratios. However, the ratios of diffusivities in both the series deviate from the corresponding mass ratios which can be attributed to the translation-rotation coupling in liquid water.
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1.
Introduction
Water, in all its phases, is essential for life on earth. In addition to its biological relevance, water plays an important role in various fields of sciences including chemistry and materials science1-5. Recently, it has been found to exhibit some very interesting physical properties such as diffusion through a carbon nanosheet6. Given that the interesting properties of water stem from the water’s tendency of H-bonding and that H-bonding has effect on the rotational, translational and vibrational degrees of freedom of the system, understanding the molecular motion of a system like liquid water is important; in particular, the effect of rotational motion in facilitating/hindering the translation of the molecule is not very well understood and addressed. Several groups have studied rotational motion and hydrogen bond dynamics in pure water as well as salt solutions7-16. Bakker and co-workers7, Skinner8 as well as Fayer and co-workers9 have discussed the rotational motion of water in salt solutions as measured by ultrafast spectroscopy. Gaffney and co-workers10 have studied the hydrogen-bond dynamics in aqueous perchlorate solution with the help of polarization-selective multidimensional vibrational spectroscopy. Park and Fayer11 have investigated the NaBr solution by measuring the changes in the OD stretch by means of ultrafast 2D IR vibrational echo spectroscopy and polarization-selective IR pump–probe experiments. Evans and Clementi12 and Chen13 have investigated the relation between translation and rotation in pure liquid H2O through cross correlation functions and probability distributions based on molecular center of mass displacements as well as the angle of rotation, respectively. Laage and Hynes studied the jump mechanism of water during reorientation in pure water14,15. Lynden-Bell and coworkers17, have proposed and studied the properties of several non-water models obtained by varying intermolecular potentials and computed the liquid properties as a function of density and temperature.
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Apart from these, somewhat relevant to these studies are the experimental and theoretical studies on water with isotopic substitutions18-22. Guillot and Guissani18 found that the ratio of self-diffusivities (D[H2O]/D[D2O]) is the most affected property on going from water to heavy water. Hardy et al19 have investigated isotope effect in water and ammonia with the help of NMR measurements when hydrogen is substituted by deuterium. They found the diffusivity ratios of hydrogen containing to deuterium containing molecule to be higher than what is expected from the mass ratio. They attributed the deviations from mass ratio or ratio of moment of inertia to translation-rotation coupling and quantum effects. Holz et al20 have studied twelve molecular liquids with NMR to obtain the effect of increased mass. They also could not confirm the existence of any negative isotope effect, i.e., where the viscosity of the deuterated sample happens to be lower than the hydrogenated sample. Svishchev and Kusalik21 have made a comparative study of H2O, D2O and T2O with molecular dynamics (MD) technique. Wasler and co-workers22 too have tried to understand the dependence of diffusion coefficient on the system’s mass and have investigated the dynamic isotope effect. There have also been studies where the nuclear quantum effects have been obtained for several of the isotopic water systems23-24. In this work, we have tried to understand the translational-rotational coupling in liquid water by a systematic approach of varying the mass distribution. Results of classical molecular dynamics simulation of water, 2H2O and H218O are initially presented. This is then extended to include models of water with still heavier masses at H and O positions in order to obtain trends in various properties such as diffusivity. We show that two model water systems with identical total masses but different masses on the H and O atoms (for example, 4H2O and H222O with a total mass of 24 a.m.u each) show different translational diffusivities. We examine the effect of the mass distribution on various properties and the reasons for the observed differences. We have shown that the deviation of the diffusivity ratios as compared 4 ACS Paragon Plus Environment
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to the square root of the mass ratio can be higher or lower depending on the nature of the distribution of mass, i.e., a heavier H-atom or a heavier O-atom. The higher diffusivity ratio in comparison to the mass ratio in the case of heavy water has been discussed in earlier research available in the literature but to the best of our knowledge, no one has shown that increased mass at the O atom position in liquid water leads to reduced diffusivity ratios than what is conventionally expected from the mass ratios. We also report the jump amplitudes and the frequency of H-bond switches by modelling the water reorientation as described by Laage and Hynes14-16.
2. Methods 2.1 Details of Molecular Dynamics Classical molecular dynamics simulations have been performed in the microcanonical ensemble using DL_POLY CLASSIC Package25,26 with velocity Verlet algorithm. Calculations have been carried out on water, heavy water and H218O and on other water models with heavier masses. Table 1 lists the model systems as well as the atomic and molecular masses. Substitution at the hydrogen position in mH2O (m = 2, 3, 4, 6 and 8) leads to the species in the order: 2H2O, 3H2O, 4H2O, 6H2O and 8H2O and substitution at the oxygen position in H2nO (n = 18, 20, 22, 26, 30) leads to the systems H218O, H220O, H222O, H226O, H230O, respectively. SPC/E model potential27 for water has been employed in all studies irrespective of the mass of the atom as the chosen model potential is known to give a value of diffusion coefficient in good agreement with experiment28. Ewald sum was employed to include the long range forces29,30.
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Table 1. Water and model water systems are listed. Note the sets of identical total masses (TM) in the last column. MH and MO denote the masses of hydrogen and oxygen atoms, respectively.
S.No
Systems
MH
MO
TM
(1)
H2O
1.0
16.0
18.0
(2)
2
H2O
2.0
16.0
20.0
(3)
H218O
1.0
18.0
20.0
(4)
3
H2O
3.0
16.0
22.0
(5)
H220O
1.0
20.0
22.0
(6)
4
H2O
4.0
16.0
24.0
(7)
H222O
1.0
22.0
24.0
(8)
6
H2O
6.0
16.0
28.0
(9)
H226O
1.0
26.0
28.0
(10)
8
H2O
8.0
16.0
32.0
(11)
H230O
1.0
30.0
32.0
2.2 Computer Simulation details For H2O system, the simulation was carried out on 216 molecules in a simulation cell of length 18.67 Å corresponding to a density of 0.992 g/cm3 which lies in the range of densities of liquid water reported in the literature31. The experimental density of liquid water32 at 300 K is ~0.996 g/cm3. The corresponding number density at which the calculations have been carried out is 0.03319/Å3. The computed trends seen here are likely to remain the same for any change in the density at the third decimal place. System was equilibrated for 50 ps and
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production run carried out for 1 ns with a time step of 1 fs. Temperature was adjusted by velocity scaling at every time step during the equilibration period to obtain the desired simulation temperature and later turned off. This is the usual way of adjusting the temperature in the microcanonical simulation30 . Simulations were mostly carried out at 300 K unless stated otherwise. We also note that the average temperatures of the runs are close to 300 K (Table S1 in SI lists the average temperatures for the model water systems). For mean square displacement, positions and their derivatives were stored every 100 fs while the other properties were averaged over 200 ps with data stored at every time step. Cubic periodic boundary conditions were imposed. Systems at temperatures other than 300 K as well as the model water systems described in Table 1 were all simulated at the same number density using similar simulation procedure (molecular geometry, model water potential, simulation cell size, etc). For few selected systems, runs have also been carried out for larger system sizes consisting of, apart from 216 molecules of water, 432 and 2035 at 300 K. Only translational diffusivities have been computed for these runs.
2.3 Calculation of Properties The diffusion coefficient has been calculated by (1) integrating the linear velocity auto correlation function (VACF) and (2) from the slope of the mean square displacement (MSD) of the molecules30, using the following expressions:
= < 0 ∙ >
-------------
(1)
where, i(t) is the velocity of the center of mass of the molecule and in practice, the upper limit of the integration is replaced with a sufficiently long time by which the correlation function has decayed completely. 7 ACS Paragon Plus Environment
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2 =
< | − 0 | >
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-------------- (2)
Here, is the position of the center of mass of the ith molecule at time t. ‘d=3’, refers to the dimensionality. ‘< >’ indicate averaging over different molecules as well as different time origins. Angular velocity33 for a molecule containing, n = 3 atoms, in this case: H2O, was computed in the molecular frame using equation 3.
rrj ×v vj
= ∑nj=1
|rrj |2
-------------------- (3)
The value |# | is added in the denominator to account for the dimensions of (1/time) for angular velocity. The normalized angular velocity autocorrelation function (AVACF) can then be obtained using
$% ∙% ' ( [ $% ],
∙% ( &
&
&
&
where ‘j’ denotes the atom while ‘i’ denotes the
molecule. See Supplementary Information (SI) for more details on the calculation of AVACF. For the calculation of activation energies, we used the Arrhenius equation34: = )
*+,/ -.
--------------------- (4)
where, D is the translational diffusion coefficient, D0 is the pre-exponential factor, Ea is the activation energy, R is the gas constant and T is the absolute temperature. Using the angular velocity and center of mass linear velocity values in the principal axes frame [see SI for transformation of values to principal axes frame], the cross correlation function (ccf) can be defined as 12:
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001 =
$2 .456 7 (
------------------ (5)
$2 8 (9/8 $456 8 (9/8
Here ‘α’ and ‘;’ refer to the directional component x/y/z. Thus, the ccf matrix comprises of nine elements. The rotational correlation function C2(t)14 was calculated as: ?@ 0 . ?@ A > ------------------- (6) where, P2 is a second order legendre polynomial given by
30CD ∅ − 1 , ∅ being the angle
between the OH bond vector at time t = 0 and at time t = t. The jump amplitude and the frequencies of H-bond switches for a given short timescale was calculated by selecting one water molecule at a time and following its H-bond switching pattern over the chosen timescale. For example, see Figure 1, HD originally H-bonded to OA1 reorients through molecular rotation to H-bond with OA2. The jump amplitude is given by the
Figure 1. HD originally H-bonded to OA1 reorients through molecular rotation to H-bond with OA2. The H-bonds are shown in dashed purple line. The jump amplitude is given by the angle OA1-OD-OA2 (θ). Note that water molecule II is not protonated water but the same atom HD is shown before and after molecular rotation.
angle OA1-OD-OA2 indicated as θ. Two water molecules were considered to be H-bonded if they followed the following widely accepted H-bonding criteria:
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HD-OA1/HD-OA2 bond distances
