An ab initio molecular dynamics study of diffusion, orientational relaxation and hydrogen bond dynamics in acetone–water mixtures

Journal of Molecular Liquids 165 (2012) 1–6

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An ab initio molecular dynamics study of diffusion, orientational relaxation and hydrogen bond dynamics in acetone–water mixtures Rini Gupta 1, Amalendu Chandra ⁎ Department of Chemistry, Indian Institute of Technology, Kanpur 208016, India

a r t i c l e

i n f o

Article history: Received 17 August 2011 Accepted 14 September 2011 Available online 18 October 2011 Keywords: Water–acetone mixture Ab initio molecular dynamics Hydrogen bond dynamics Vibrational spectral diffusion

a b s t r a c t A theoretical study of the dynamics of water–acetone mixtures is carried out by using the methods of ab initio molecular dynamics for trajectory generation, time series analysis for frequency calculations and time correlation function approach for calculations of various dynamical quantities. It is found that diffusion coefficients and orientational relaxation times of water and acetone molecules change in a nonlinear manner with variation of composition of the mixture. The lifetimes of acetone–water hydrogen bonds are found to be shorter than that of water–water hydrogen bonds and both are found to increase with decrease of water concentration. The vibrational spectral diffusion of OD stretch modes of deuterated water molecules is also investigated for the equimolar mixture of water and acetone and the results are correlated with the dynamics of water– water and water–acetone hydrogen bonds and also of dangling OD modes. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Water–acetone mixtures are known to show significant nonideality in their equilibrium and dynamical properties with variation of composition [1–10]. The presence of both hydrophilic hydrogen bonded solvation of the carbonyl group and hydrophobic solvation of the methyl groups of acetone by water molecules is believed to give rise to very different water–acetone interactions as compared to those in the respective pure liquids. There have been a number of experimental [1–11] and theoretical [12–26] studies on various structural, energetic and dynamical aspects of these solutions by using a variety of experimental and theoretical techniques. For example, infrared spectroscopic studies on water–acetone systems [10] have revealed that the OH stretch band gets blue shifted as acetone is added to water. A recent classical molecular dynamics simulation study of water–acetone mixtures looked at the extent of nonideality in various single particle and pair dynamical properties of these solutions with varying compositions [18]. In recent years, there have also been a few studies on acetone– water systems through ab initio molecular dynamics [19–23] and combined quantum-classical simulations [24–26]. These calculations have primarily looked at solvent effects, intensity enhancement effects induced in aqueous solution of acetone and also vibrational spectral diffusion of water molecules in hydration shells of an acetone solute. However, these first principles simulation studies have not been carried out for a wide concentration range of water–acetone mixtures. ⁎ Corresponding author. Tel.: + 91 512 2597241; fax: + 91 512 2597436. E-mail address: [email protected] (A. Chandra). 1 Current address: Department of Chemistry, University of British Columbia, Vancouver, BC V6T 1Z1, Canada. 0167-7322/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2011.09.010

In the present work, we have carried out such a first principles theoretical study of structural and dynamical properties of water–acetone mixtures of three different compositions by employing the method of Car–Parrinello molecular dynamics [27,28]. Our main goal is to calculate various dynamical properties such as diffusion, orientational relaxation and hydrogen bond dynamics in these solutions. In addition to the calculations of diffusion, orientational relaxation and hydrogen bond relaxation of water–water and acetone–water pairs, we have also investigated the dynamics of vibrational spectral diffusion [29–35] of water molecules which are hydrogen bonded to either water or acetone molecules or remain free in the equimolar solution of water and acetone. These spectral diffusion calculations are done by means of a time series analysis and frequency time correlation functions. The frequency calculations are done by using method of wavelet analysis [35–38]. We have organized the rest of the paper as follows. In Section 2, we have discussed the details of ab initio molecular dynamics simulations. We have presented the results of diffusion coefficients and orientational relaxation of water and acetone molecules in Section 3. The results of hydrogen bond dynamics of water–acetone and water–water pairs are presented in Section 4. This section also deals with the dynamics of dangling OD modes of (deuterated) water molecules. In Section 5, we present our results of the dynamics of vibrational spectral diffusion of the equimolar water–acetone system. Our conclusions are briefly summarized in Section 6. 2. Ab initio molecular dynamics simulations The ab initio molecular dynamics simulations were carried out by employing the Car–Parrinello method [27] and the CPMD code [39].

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R. Gupta, A. Chandra / Journal of Molecular Liquids 165 (2012) 1–6

Each simulation system contained 32 molecules in a cubic box. We have considered water–acetone mixtures of three different composition: xw = 0.03, 0.5 and 0.97 where xw is the mole fraction of water. System 1 contains one water and 31 acetone molecules in a cubic box of length 15.662 Å and System 2 contains an equimolar mixture of 16 water and 16 acetone molecules in a cubic box of length 13.324 Å. For System 3, we took the trajectory from our earlier work of Ref. [23] and subsequently ran the simulation for a longer period. This System consists of a single acetone in 31 water molecules in a cubic box of length 10.17 Å. The size of the simulation box for a given composition is determined by the experimental density [11] of water–acetone mixtures at that particular composition. The box was periodically replicated in three dimensions and the electronic structure of the extended systems was represented by the Kohn–Sham (KS) formulation [40] of DFT within a plane wave basis. The core electrons were treated via the atomic pseudo potentials of Troullier–Martins [41] and the plane wave expansion of the KS orbitals was truncated at a kinetic energy of 80 Ry. A fictitious mass of μ =800 a.u. was assigned to the electronic degrees of freedom and the coupled equations of motion describing the system dynamics was integrated by using a time step of 5 a.u. which is equal to about 0.12 fs. We used the BLYP [42] functional in the present simulations. We note that this functional was also employed in many of the earlier simulations of hydrogen bonded liquids such as water [43,44], methanol [45–47], ammonia [48,49] and also for mixed systems such as water–methanol mixture [50]. For Systems 1 and 2, the initial configurations of the water and acetone molecules were generated by carrying out classical molecular dynamics simulations using the empirical multisite interaction potentials. For water molecules, we used the SPC/E interaction potential [51] and the acetone molecules are modeled using the OPLS all atom model [52]. In ab initio molecular dynamics simulations, all hydrogen atoms were assigned the mass of deuterium to reduce the influence of quantum effects on the dynamical properties. Also, our choice of deuterium mass ensured that electronic adiabaticity and energy conservation were maintained throughout the simulations for the chosen values of the fictitious electronic mass parameter and time step. During ab initio molecular dynamics simulations, we equilibrated each system for 10 ps in NVT ensemble at 300 K using the Nose–Hoover chain method [53]. Upon equilibration, we continued the runs in NVE ensemble for another 50 ps for calculations of various structural and dynamical properties. For System 3, 60 ps trajectory was earlier generated through the work of Ref. [23]. We ran this system further to generate a total production trajectory of 70 ps and used this extended trajectory for calculations of various average properties for this system. 3. Diffusion coefficients and orientational relaxation of water and acetone molecules We have calculated the self-diffusion coefficients and orientational relaxation times of water and acetone molecules in all the systems. The diffusion coefficients are calculated from long time limit of the mean-square displacement (MSD) D ¼ limt→∞

< ½r ðt Þ−r ð0Þ2 > 6t

ð1Þ

where r(t) is the position of the center of mass of a molecule at time t and the average is carried out over the initial time. In Table 1, we have shown the results of the diffusion coefficients of water and acetone molecules in all the systems. It is found that diffusion coefficients of the 50:50 mixture are smaller than those of the (almost) neat liquids, i.e. Systems 1 and 3. When water is added to pure acetone, the water molecules are likely to form hydrogen bond complexes with acetone molecules. Also, the water molecules form hydrogen bonds between themselves. These hydrogen bond complexes

Table 1 Values of diffusion coefficients and orientational relaxation times of water and acetone molecules in water–acetone mixtures. The diffusion coefficients and relaxation times are expressed in units of 10− 5 cm2 s− 1 and ps, respectively. System

xw

1 2 3

0.03 0.50 0.97

Diffusion

Coefficients

Orientational

Relaxation

Dw

DAc

τw

τAc

0.75 0.42 0.51

1.78 0.65 0.75

1.21 4.3 4.2

0.95 2.75 2.6

tend to decrease the rate of diffusion of water and acetone molecules in their mixtures as compared to the pure liquids. Also, at the high concentration of acetone, the hydrogen bonded network breaks down since acetone has only hydrogen bond accepting ability. Besides, acetone is a less polar solvent than water. These effects lead to higher diffusion coefficient in pure acetone as compared to pure water. The orientational motion of water and acetone molecules is analyzed by calculating the dipole orientational time correlation function defined by μ

Cl ðt Þ ¼


 < Pl eμ ðt Þ⋅eμ ð0Þ > ; μ μ < Pl ðe ð0Þ⋅e ð0ÞÞ >

ð2Þ

where Pl is the Legendre polynomial of rank l and e μ is the unit vector which points along the dipole axis of a molecule. The orientational correlation time τlμ is defined as the time integral of the orientational correlation function μ

∞

μ

τl ¼ ∫0 dtCl ðt Þ:

ð3Þ

In the present work, we calculated τlμ by explicit integration of the Clμ(t) from simulations until 9 ps and by calculating the integral for the tail part from the fitted exponential functions. In Table 1, we have shown the results of the second-rank orientational relaxation times for water and acetone molecules. For the three mixtures considered here, the maximum values of the orientational relaxation times are found at xw = 0.50 which is similar to the corresponding minimum obtained for diffusion of these molecules. We note that the overall trends of the composition dependence of diffusion coefficients and orientational relaxation times are found to be qualitatively similar to those found in experiments [7,8] of water–acetone mixtures. However, the absolute values of the diffusion coefficients are somewhat smaller and the orientational relaxation times are somewhat larger than the corresponding experimental values [7,8] which are likely due to the density functional and finite basis set cut-off that are employed in the present study. 4. Dynamics of hydrogen bonds and dangling OD modes We have calculated the relaxation of acetone–water and water– water hydrogen bonds by using the so-called population correlation function approach [35,54–61,63–65]. The existence of water–water and acetone–water hydrogen bonds is found by using the simple geometric criterion that D⋯O distance should be less than 2.5 Å and 2.6 Å, respectively. These cut-off distances correspond to the first minimum of the intermolecular radial distribution functions between water–water and acetone–water pair as shown in Fig. 1. To study the breaking dynamics of hydrogen bonds, we calculate the continuous hydrogen bond time correlation function, SHB(t), which describes the probability that an initially hydrogen bonded water–water or acetone– water pair remains bonded at all times up to time t [35,57–63]. The associated integrated relaxation time τHB gives the average lifetime of a hydrogen bond between an acetone and a water molecule or

R. Gupta, A. Chandra / Journal of Molecular Liquids 165 (2012) 1–6

4

1

a

O-O SHB (t)

g (r)

X W=0.03

W-W

X W=0.97

0.6

X W=0.5

0.4 0.2

X W=0.5 X W=0.97

1

a

0.8

3

2

3

0 1

0

b

Ac-W

0.8

b

SHB (t)

4

O-H

X W=0.97

0.6

X W=0.5

0.4

X W=0.03

3

g (r)

0.2 0

2

1 1

0

1

2

3

4

SDH (t)

0

c

0.8

5

r (Å)

X W=0.97

0.6

X W=0.5

0.4

X W=0.03

0.2 Fig. 1. The acetone–water radial distribution functions for varying composition of the water–acetone mixtures. The different panels are: (a) oxygen (acetone)–oxygen (water) and (b) oxygen (acetone)–hydrogen (water) radial distribution functions. The different curves in (b) are as in (a).

a

3

W-W

1

4

5

X W=0.97 X W=0.5

0.8 0.6 0.4 0.2 0

b

Ac-W

X W=0.03

1

ð4Þ

where kHB and k′HB are the forward and backward rate constants for hydrogen bond breaking. The inverse of kHB corresponds to the average lifetime of an acetone–water hydrogen bond and can be correlated with τHB obtained from the route of continuous hydrogen bond time correlation function. The results of the continuous and intermittent correlation functions are shown in Figs. 2 and 3 for both water–water and acetone–water hydrogen bonds. It is found that the decay of SHB(t) for water–water pairs is slower than that of acetone–water hydrogen bonds. In Table 2, we have shown the results of average lifetimes τHB, for both water–water and acetone–water hydrogen bonds. The acetone–water hydrogen bond lifetimes are found to be shorter than the lifetimes of water–

2

Fig. 2. The time dependence of the continuous hydrogen bond time correlation functions for (a) water–water (W–W) and (b) acetone–water (Ac–W) hydrogen bonds. (c) Time dependence of the correlation function for dangling OD modes of water molecules which are not hydrogen bonded initially.

X W=0.5 X W=0.97

0.8

CHB (t)

−dCHB ðt Þ ′ ¼ kHB CHB ðt Þ−k HB NHB ðt Þ; dt

1

Time (ps)

CHB (t)

between two water molecules. The intermittent correlation function CHB(t) [35,54–61,63–65], on the other hand, does not depend on the continuous presence of a hydrogen bond and it describes the probability that an acetone–water hydrogen bond is intact at time t, given that it was intact at time t = 0, independent of possible breaking in the interim time. Since, the correlation function CHB(t) allows recrossing of the diving surface that separates the bonded and hydrogen bonded states, its relaxation describes the structural relaxation of hydrogen bonds. After a hydrogen bond is broken, the water–water or acetone–water pair can remain as nearest neighbors for some time before either the bond is reformed or the molecule diffuses away from each other. We also calculate a third probability function NHB (t) [35,54–57] which describes the time-dependent probability that a hydrogen bond is broken at time zero, but the two molecules remain in the vicinity of each other, i.e. as nearest neighbors, but not hydrogen bonded at time t. Then, following previous work [54], one can write a simple rate equation for the intermittent correlations

0 0

0.6 0.4 0.2 0

0

1

2

3

4

5

Time (ps) Fig. 3. The time dependence of the intermittent hydrogen bond time correlation functions for (a) water–water (W–W) and (b) acetone–water (Ac–W) hydrogen bonds for different mole fraction of water.

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R. Gupta, A. Chandra / Journal of Molecular Liquids 165 (2012) 1–6

Table 2 Lifetimes of hydrogen bonds between water–acetone and water–water pairs in water– acetone mixtures. All the time constants are expressed in ps. System

1 2 3

xw

0.03 0.50 0.97

Water–acetone

is expressed in terms of basis functions which are constructed as translations and dilations of a mother wavelet ψ [37] −1=2

Water–water

ψa;b ðt Þ ¼ a

τHB

1/kHB ; short

1/kHB ; long

τHB

1/kHB ; short

1/kHB ; long

1.75 1.62 1.40

1.66 1.25 1.2

11.5 10.9 10.5

– 2.6 2.3

– 2.5 2.0

– 14.1 13.0

  t−b ; ψ a

and the coefficients of this expansion are given by the wavelet transform of f(t), which is defined as −1=2

Lψ f ða; bÞ ¼ a

5. Vibrational spectral diffusion

   t−b ∞ dt; ∫−∞ f ðt Þ ψ a

ð6Þ

for a > 0 and b real. We have used the Morlet–Grossman wavelet as the mother wavelet [69] ψðt Þ ¼

1 2πiλt −t 2 =2σ 2 pffiffiffiffiffiffi e e ; σ 2π

ð7Þ

with λ = 1 and σ = 2, and i represents imaginary number. The inverse of the scale factor a is proportional to the frequency and thus the wavelet transform of Eq. (6) at each b gives the frequency content of f(t) over a time window about b. Since we are interested in the OD stretch frequencies, the time dependent function f(t) for a given OD bond is constructed to be a complex function with its real and imaginary parts corresponding to the instantaneous fluctuations in OD distance and the corresponding momentum along the OD bond at time t. The stretch frequency of this bond at a given time is then determined from the scale a that maximizes the modulus of the corresponding wavelet transform at that time. We have calculated the time correlations of fluctuating OD stretch frequencies of water molecules that are initially hydrogen bonded to the water and acetone molecules, respectively. We have also calculated the time correlations for free OD modes of water molecules which are neither hydrogen bonded to water nor to acetone molecules at the initial time. We note that this frequency time correlation function serves as a key dynamical quantity in the studies of vibrational spectral diffusion. This correlation function is defined as [29–35] 2

Cω ðt Þ ¼< δωðt Þδωð0Þ > = < δωð0Þ >

ð8Þ

where δω(t) is the fluctuation from the average frequency at time t. The average of Eq. (8) is over the initial time and over all the OD groups of the system. The results of the frequency time correlation for all three types of OD modes which are present in System 2 are shown in Figs. 4–6. We observe a fast decay and a weak oscillation 1

Frequency correlation 0.8

0.6

Cw (t)

water hydrogen bonds in all systems which is similar to our previous classical results [18]. We have used a least-square fit of the simulation results of CHB(t) and NHB(t) to Eq. (4) to obtain the hydrogen bond lifetimes from the route of intermittent correlations. It is found that the entire decay could not be fitted well by single constant values of kHB and k′HB. Hence, we have considered two different rate constants: the one with a higher value that is applicable in the faster short-time part of the dynamics and a smaller value of the rate constants which is applicable in the longer-time part of the relaxation. In an attempt to find out these short-time and long-time rate constants, we performed the fitting separately in the short-time region of 0 < t < 3 ps and in the longer time region of 3 < t < 12 ps. The inverses of the corresponding forward rate constants are denoted as 1/kHB ; short and 1/kHB ; long and their values are presented in Table 2. The values of 1/kHB ; short are similar to τHB obtained from the route of continuous correlation functions. We note that both water–water and acetone–water hydrogen bond lifetimes decrease with increase of water concentration. This is due to an enhancement of cooperativity through more number of hydrogen bonds on addition of water to acetone. Also, it can be seen from oxygen–oxygen and oxygen–hydrogen radial distribution functions for water–water and acetone–water pairs that their peak heights increase with decrease of water concentration. This implies that the effective hydrogen bond correlations between water–water or acetone–water pairs decrease on addition of water and, as a result, the corresponding hydrogen bond lifetimes also decrease. We have also looked at lifetimes of dangling (i.e. free) OD modes which neither bonded to water nor to acetone molecules in water– acetone systems. We constructed the continuous dangling OD correlation SDH(t) defined as SDH(t) = < hDH(0)HDH(t) >/ where, hDH(t) is unity when a particular OD mode of water is not hydrogen bonded either to water or to an acetone molecule at time t according to the adopted definition and zero otherwise and H(t) = 1, if the OD mode remains continuously free from t = 0 to time t and it is zero otherwise. The associated integrated relaxation time τDH gives the average lifetime of free OD modes. The relaxation of SDH(t) is shown in Fig. 2(c). The results of the lifetimes of free OD modes are found to be 0.3, 0.25 and 0.1 ps, respectively, for water–acetone mixtures with xw = 0.03, 0.50 and 0.97. We note that the result for xw = 0.97 is essentially the same as found earlier for pure water [35].

ð5Þ

0.4

The vibrational frequency of an OD bond of water molecules in these water–acetone solutions fluctuates due to fluctuations in its interactions with the environment caused by continual motion of surrounding molecules. A quantitative calculation of the time dependent vibrational frequencies of OD bonds can be carried out through a time series analysis of the ab initio molecular dynamics trajectories using the wavelet method [37]. This method has already been used for calculations of fluctuating frequencies of water and aqueous solutions of ions and other solutes from simulated trajectories [23,35,66–68]. Here we apply the method to calculate time dependent OD frequencies of the equimolar mixture of water and acetone. In this method, a time dependent function f(t)

0.2

0

0

1

2

3

4

5

Time (ps) Fig. 4. The time correlation function of OD fluctuating frequencies averaged over water molecules which are hydrogen bonded to water oxygen at the initial time. The results are for System 2. The solid curve represents the fit by the function as given by Eq. (9).

R. Gupta, A. Chandra / Journal of Molecular Liquids 165 (2012) 1–6

1

5

Table 3 Values of the various time constants (ps), frequency (cm− 1) and weights of frequency time correlations for OD modes of the equimolar water–acetone mixture.

Frequency correlation 0.8

Cw (t)

0.6

Initial hydrogen bonded state of OD modes of water

τ0

Hydrogen bonded to water Hydrogen bonded to acetone Free

0.14 0.17 2.6 116.78 0.12 0.55 0.15 0.11 2.1 114.85 0.14 0.5 0.062 0.52 1.85 99.77 0.26 0.69

τ1

τ2

ωs

a0

a1

0.4

6. Conclusions 0.2

0 0

1

2

3

4

5

Time (ps) Fig. 5. The time correlation functions of OD fluctuating frequencies averaged over water molecules which are hydrogen bonded to acetone oxygen at the initial time. The results are for System 2. The solid curve represents the fit by the function as given by Eq. (9).

at short times followed by slower decay extending to a few ps for both types of water molecules which are hydrogen bonded to water and acetone molecules, respectively, as shown in Figs. 4–5. We used the following function including a damped oscillatory function to fit the calculated results of spectral diffusion [34,35]

f ðt Þ ¼ a0 cosωs te

−t=τ0

−t=τ1

þ a1 e

−t=τ2

þ ð1−a0 −a1 Þe

ð9Þ

The values of time constants, frequency of damped oscillation and the weights are included in Table 3 for all three types of OD modes. It is clear from the results of the previous section that the slower relaxation times of spectral diffusion correspond to the lifetimes of water–water and acetone–water hydrogen bonds. Generally, the faster relaxation times correspond to the dynamics of intact hydrogen bonds and also to the lifetimes of dangling OD modes in the water–acetone mixture. In a dynamical equilibrium, hydrogen bonds between water–water and acetone–water pairs continuously break and reform and also a water molecule can leave or enter the hydration shells of another water or acetone. These dynamical processes alter the OD frequencies and induce characteristic time scales in the dynamics of spectral diffusion of both hydrogen bonded and free water molecules.

1

Frequency correlation

We presented a first principles theoretical study of water–acetone mixtures of three different concentrations to investigate the structural and dynamical properties of these solutions. Our calculations are based on ab initio molecular dynamics simulations for trajectory generation. Unlike classical simulations, no empirical potential is used for water and acetone molecules in the present ab initio simulation study. We calculated the diffusion coefficients and orientational relaxation times of water and acetone molecules for each system and found that they change in a non monotonic manner with composition. The non monotonic behavior of the diffusion and orientational relaxation with composition is qualitatively similar to our earlier classical results [18] and also of experiments [7,8]. We have also calculated the lifetimes for water–acetone and water–water hydrogen bonds and found that the water–water hydrogen bonds have longer lifetimes than those of acetone–water hydrogen bonds. The lifetimes of both water–water and acetone–water hydrogen bonds are found to decrease with increase of water concentration. This is attributed to an enhancement of cooperativity through more number of hydrogen bonds with increase of water concentration in the mixtures. We have also studied the vibrational spectral diffusion in the water–acetone mixture containing equal number of water and acetone molecules. The dynamics of vibrational spectral diffusion of OD modes of deuterated water molecules have been investigated through frequency time correlation functions. A fast decay and a weak oscillation at short times followed by slower decay extending to a few ps are found for both types of water molecules which are hydrogen bonded or dangling initially. We note that similar short and long time decays of the frequency correlation functions were also found for pure water in earlier studies although the time scales were different quantitatively [33–35]. In the present case, the longer time constants correspond to the lifetimes of water–water and acetone–water hydrogen bonds while the shorter time scales correspond to the dynamics of intact hydrogen bonds and also lifetimes of dangling OD modes. We note that generally vibrational spectral diffusion of OD modes have contributions from both hydrogen bonded as well as non-hydrogen bonded molecules. Hence, the relaxation of both types of molecules can alter different parts of the overall dynamics of vibrational spectral diffusion.

0.8

Acknowledgment

Cw (t)

0.6

We gratefully acknowledge the financial support from Department of Science and Technology (DST) and Council of Scientific and Industrial Research (CSIR), Government of India.

0.4 0.2

References

0 −0.2

0

1

2

3

4

5

Time (ps) Fig. 6. The time correlation functions of OD fluctuating frequencies averaged over water molecules which are not hydrogen bonded to water or acetone molecules. The results are for System 2. The solid curve represents the fit by the function as given by Eq. (9).

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