Hydrogen bond dynamics at vapour–water and metal–water interfaces

Chemical Physics Letters 386 (2004) 218–224

www.elsevier.com/locate/cplett

Hydrogen bond dynamics at vapour–water andmetal–water interfaces

Sandip Paul, Amalendu Chandra *

Department of Chemistry, Indian Institute of Technology, Kanpur 208016, India

Received 26 November 2003

Published online: 11 February 2004

Abstract

We have presented a molecular dynamics study of the dynamics of breaking and structural relaxation of hydrogen bonds at va-

pour–water and metal–water interfaces. For both interfaces, the relaxation of hydrogen bonds is found to occur at a slower rate than

that in bulk water. However, the dynamics of inter-region hydrogen bonds, where onemolecule of the bonded pair belongs to interface

and the other to its adjacent layer, are found to be faster than even the bulk hydrogen bonds. The results are explained in terms of the

energies of hydrogen bonds and the frictions that act on the rotational and translational motion of water molecules at interfaces.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

The dynamical behaviour of hydrogen bonds plays animportant role in determining the rates of many chem-

ical, physical and biochemical processes that occur in

aqueous media. In recent years, there have been a

number of experimental [1–5], theoretical [6,7] and

simulation [6–12] studies that focused on the relaxation

of hydrogen bonds in homogeneous bulk water and

aqueous solutions and now the attention is being shifted

to inhomogeneous media of aqueous interfaces [13–19].The present Letter is concerned with the dynamical

behaviour of hydrogen bonds at vapour–water and

metal–water interfaces. Both these interfaces are highly

inhomogeneous but the nature of inhomogeneity is very

different. In case of water–vapour interface, the density

decreases rather smoothly from liquid density to the

vapour density [20–23] whereas a water–metal interface

is characterized by highly oscillatory density profile dueto layered structure of such interfaces [24–27]. The ori-

entational structure of water molecules at these inter-

faces can also be quite different [20–29] and this varying

* Corresponding author.

E-mail address: [email protected] (A. Chandra).

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2003.12.120

inhomogeneity and orientational structure are expected

to influence both the structure and dynamics of hydro-

gen bonds at these interfaces. In fact, recent studies onvapour–water and metal–water interfaces have shown

significant differences of the single-particle translational

and orientational diffusion of interfacial molecules as

compared to that of bulk molecules [20,23,25–27].

However, we are not aware of any study on the dy-

namics of hydrogen bonds at these interfaces. In this

Letter, we present such a study by means of molecular

dynamics simulations.The outline of the present Letter is as follows. In

Section 2, we discuss the molecular models of the va-

pour–water and metal–water systems that are employed

in the present study and the simulation details are pre-

sented in Section 3. Sections 4 and 5 deal with the

interfacial density profiles and the distribution of hy-

drogen bonds, respectively. The dynamics of hydrogen

bonds at the interfaces and in the bulk phases are dis-cussed in Section 6 and our conclusions are summarized

in Section 7.

2. Models

For both vapour–water and metal–water systems, we

have used the classical SPC/E model of water where

mail to: [email protected]

S. Paul, A. Chandra / Chemical Physics Letters 386 (2004) 218–224 219

each water molecule is characterized by three interaction

sites located on oxygen and hydrogen atoms [30]. The

O–H bond distances are constrained at 1.0 �A and the

bond angle between two O–H bonds is fixed at 109.47�.The interaction between atomic sites of two watermolecules is expressed as

uðri; rjÞ ¼ 4�rrij

� �12"

� rrij

� �6#þ qiqj

rij; ð1Þ

where the first term is the Lennard-Jones interaction

which is calculated only between the oxygen sites and

the second term represents the electrostatic interaction.qi is the charge of the ith atom. The values of the po-

tential parameters r, � and qi for SPC/E model are given

in [30]. For the metal–water system, the metal surface is

treated quantum mechanically by using the jellium

model. The water–surface interaction uwðzÞ is expressedin the following form [27]

uwðzÞ ¼33=2�w2

rw

z

� �9�

� rw

z

� �3�� EðzÞ � li; ð2Þ

where z ¼ jzi � z0j, �w and rw are the 9–3 Lennard-Joneswell-depth parameter and diameter which characterize

the short range part of the metal–water interaction and

EðzÞ is the electric field generated by the metallic sur-

faces. The parameters of the short range metal–surface

interaction are: rw ¼ 2:474 �A and �w ¼ 1:936 kJ/mol

[27]. In order to calculate the electrostatic field ðEðzÞÞ ofa metal surface, we note that EðzÞ ¼ �ðo=ozÞV ðzÞ, whereV ðzÞ is the metal electrostatic potential. V ðzÞ satisfies thePoisson equation

d2

dz2V ðzÞ ¼ �4pqcðzÞ; ð3Þ

where qcðzÞ is the charge density of the metal. An ex-plicit modelling of the electronic structure of the metal is

now necessary in order to calculate the charge density

and the metal field. Following earlier work [27], we

model the metal walls by semi-infinite jellium slabs of

width 2zw. The jellium model consists of a uniform

background of positive charge density qþ which repre-

sents the metal nuclei and core electrons and the asso-

ciated valence electron density qeðzÞ. The valenceelectron density is calculated by using density functional

theory [31,32]. In this approach, the electron density is

calculated by solving the effective one-electron Schro-

dinger equation

� �h2

2me

d2

dz2wnðz0Þ þ Veffðz0Þwnðz0Þ ¼ �nwnðz0Þ; ð4Þ

where wn and �n are the one-electron normalized

eigenfunction and energy eigenvalue for the nth state

and me is the mass of an electron. z0 denotes the z-coordinate with origin at the centre of the metal slab.

Veffðz0Þ is the effective potential which is given by

Veffðz0Þ ¼ Vjelðz0Þ þ Vxcðz0Þ þ Vsolðz0Þ; ð5Þ

where Vjelðz0Þ represents instantaneous interaction of

an electron with the field of the jellium, Vxcðz0Þ is the

exchange and correlation potential and Vsolðz0Þ is the

average interaction energy of the electron with the polar

solvent.In the present work, we have used the local density

approximation with Wigner�s expression for the ex-

change and correlation energy [33]. The jellium potential

is obtained through a solution of the Poisson equation

involving the electron density. Vsolðz0Þ depends on the

solvent density near the surface and its expression is

available in [27]. We note that the metal potential de-

pends on the solvent density which, in turn, depends onthe potential of the metal surface. Thus, the metal po-

tential from density functional theory and the solvent

structure from simulation are obtained self-consistently

through iteration.

3. Details of simulations

For construction of the vapour–water interfaces, we

first carried out a bulk simulation in a cubic box of 500

water molecules periodically replicated in all three di-

mensions. The box length L was adjusted according to

the experimental density of water at 298 K. After this

bulk solution was properly equilibrated, two empty

boxes of equal size were added on either side of the

original simulation box along the z-dimension and thislarger rectangular box was taken as the simulation box

in the next phase of the simulation run. The system was

reequilibrated by imposing periodic boundary condi-

tions in all three dimensions. This resulted in a lamella

of approximate width L with vapour–water interfaces on

both sides of the lamella.

The simulation of the metal–water system is done in

a rectangular box with wall origins at �11.86 �A, andperiodic boundary conditions are set at �9.27 �A in xand y directions. The simulations are carried out with

a total of 256 water molecules. Initially, the metal

potential is calculated by replacing the water by vac-

uum. The liquid was then introduced and the solvent

density was found out. After this initial calculation,

Vsolðz0Þ was evaluated and the new electron distribution

was calculated by solving the density functionalequations. The metal potential was calculated from the

new electron distribution and the corresponding metal

field was used in the next set of simulations and this

iterative process was continued until convergence was

attained.

In both the simulations, the long range electrostatic

interactions were treated by using the three dimensional

Ewald method [34]. The short range Lennard-Jones in-teractions were calculated by using a spherical cut-off at

0 4 8 12 160

0.3

0.6

0.9

1.2

Den

sity

0 2 4 6 8 10 1 2z (Å)

0

2

4

6

8

Den

sity

(a)

(b)

220 S. Paul, A. Chandra / Chemical Physics Letters 386 (2004) 218–224

distance L=2. We employed the quaternion formulation

of the equations of rotational motion and the leap-frog

algorithm for the integration over time with a time step

of 10�15 s (1 fs).

For the vapour–water interfacial system, MD runs of300 ps were used to equilibrate the system in the bulk

phase and then MD runs of 400 ps were used to equil-

ibrate the liquid–vapour interfacial system. During the

equilibration, the temperature of the simulation system

was kept at 298 K through rescaling of the velocities.

The simulations of the vapour–water interface was

then continued in microcanonical ensemble for another

600 ps for the calculation of hydrogen bond properties.For the metal–water system, we equilibrated the system

for 200 ps and then continued the simulation for another

800 ps for calculation of various equilibrium and dy-

namical quantities.

Fig. 1. The number density profiles of water molecules of (a) vapour–

water and (b) metal–water systems. The results shown are for half of

the simulation box along z-direction.
4. Inhomogeneous structure at the interfaces: the distri-
bution of number density

The interfaces of both the systems were found to be

stable over the simulation time for all the systems

studied here. In order to characterize the location of the

interfacial regions, we calculated the number density

profiles of water both at vapour–water and metal water

interfaces and the results are shown in Fig. 1. In this

figure, the results are normalized by the correspondingbulk densities. It is seen that the water density at the

surfaces is highly inhomogeneous. For the purpose of

analysis of various interfacial and bulk properties, we

decompose the whole water system into two or three

regions. We define the interfacial region of a liquid–

vapour interface as the distance over which the number

density decreases from 90% to 10% of the bulk liquid

density [20,23]. We call this interfacial region as region Iand the rest of the system on the liquid side as region II.

For the metal–water system, the contact interface or

region Ia consists of molecules in the first layer near the

surfaces, the region Ib or the diffuse interface includes

water molecules in the second layer near the surfaces

and the rest of the water molecules belong to region II or

the bulk-like region.

Table 1

The percentage of water molecules having n number of hydrogen bonds and

regions of vapour–water and metal–water systems

System Region f1 f2

Vapour–water I 2.45 24.25

II 0.28 4.55

Metal–water Ia 0.11 2.6

Ib 0.16 3.05

II 0.24 4.12

5. Hydrogen bond distribution at interfaces

In order to define a hydrogen bond, we have used a

set of geometric criteria where two water molecules

are taken to be hydrogen bonded if their interoxygen

distance is less than 3.5 �A and simultaneously hydrogen–

oxygen distance is less than 2.45 �A and the oxygen–

oxygen–hydrogen angle is less than a cut-off value hc.We note that the critical distances of 3.5 and 2.45 �A are

essentially the positions of the first minimum in the ox-ygen–oxygen and oxygen–hydrogen radial distribution

functions, respectively. The angular criterion reflects the

directional character of hydrogen bonds. We have used a

value of 45� for hc which corresponds to a �less strict�definition of the hydrogen bonds [7]. The quantities of

interest are the percentages fn of water molecules that

engage in n hydrogen bonds and the average number of

hydrogen bonds per water molecule nHB. The values offn (n ¼ 1; . . . ; 5) and nHB are included in Table 1 for the

interfacial and bulk regions of both the systems.

In the bulk phase of water, majority of water mole-

cules participate in four hydrogen bonds whereas, in the

liquid–vapour interfacial region, most of the molecules

the average number of hydrogen bonds per water molecule in different

f3 f4 f5 nHB

49.9 19.18 4.0 2.97

25.6 58.6 10.8 3.74

23.6 58.45 14.6 3.83

21.65 58.45 15.5 3.82

24.8 58.05 12.2 3.76

-3

-2

-1

0

ln S

HB(t

)

-3

-2

-1

0

ln S

HB(t

)

(a)

(b)


are found to have either three or two hydrogen bonds.

The average number of hydrogen bonds per water

molecule is also significantly smaller at the liquid–

vapour interface than that in the bulk phase. This

smaller number of hydrogen bonds at the vapour–waterinterfaces is likely due to the lower density and the

presence of vapour (essentially vacuum) on one side of

the liquid. In case of metal–water system, however, the

distributions of hydrogen bonds in the interfacial and

bulk regions are found to be not very different from each

other. The interfacial molecules near the metal surfaces

are found to participate in slightly more number of

hydrogen bonds which might be due to higher density ofwater near the metal surfaces.

0 1 2 3Time (ps)

Fig. 2. The time dependence of the continuous hydrogen bond corre-

lation function SHBðtÞ in different regions of (a) vapour–water and (b)

metal–water systems. In (a), the solid, dashed and the dotted-dashed

curves correspond to I–I, I–II and II–II hydrogen bonds and, in (b),

the solid, dashed and dotted-dashed curves correspond to Ia–Ia, Ia–Ib

and II–II hydrogen bonds.

6. Dynamics of hydrogen bonds

The dynamics of hydrogen bonds in the interfacial

and bulk regions are investigated by calculating the

average lifetime, the rate constant of hydrogen bondbreaking and also the structural relaxation time of hy-

drogen bonds in these regions. We have calculated these

dynamical properties for hydrogen bonds between two

molecules of the same region, interface or bulk, and also

for inter-region hydrogen bonds where one molecule of

the bonded pair is in the interface and the other in the

adjacent layer on the bulk side.

The calculations of the above dynamical quantitiesare done by means of different hydrogen bond time

correlation functions. In order to define these functions,

we first define two hydrogen bond population variables

hðtÞ and HðtÞ: hðtÞ is unity when a particular tagged pair

of water molecules is hydrogen bonded at time t, ac-cording to the adopted definition as described in Section

5 and zero otherwise. The function HðtÞ is unity if the

tagged pair of water molecules remains continuouslyhydrogen bonded from t ¼ 0 to time t and it is zero

otherwise. We define the continuous hydrogen bond

time correlation function SHBðtÞ as [9,11–14]SHBðtÞ ¼ hhð0ÞHðtÞi=hhi; ð6Þwhere h� � �i denotes an average over all hydrogen bonds

that are present at t ¼ 0. Clearly, SHBðtÞ describes the

Table 2

The dynamical properties and the energies of hydrogen bonds between wate

System Region sHB

Vapour–water I–I 1.50

I–II 0.80

II–II 1.35

Metal–water Ia–Ia 2.0

Ia–Ib 1.15

II–II 1.3

Region A–B means one molecule of the hydrogen bonded pair is in region

rate constant are expressed in units of ps and the hydrogen bond energies a

probability that a water pair, which was hydrogen

bonded at t ¼ 0, remains continuously bonded up totime t. The time integral of this function describes the

average time that a hydrogen bond survives after it is

chosen at time t ¼ 0. We denote the integral by sHB and

call it the average hydrogen bond lifetime. However,

strictly speaking, the time constant sHB corresponds to

the lifetime of a hydrogen bond if only those hydrogen

bonds are chosen which are created at time t ¼ 0. In the

present study, the hydrogen bonds are chosen randomlywithout keeping any condition on when they were cre-

ated and, therefore, the integral of the present SHBðtÞshould better be called the average persistence time of a

randomly chosen hydrogen bond [7].

In Fig. 2, we have shown the decay of SHBðtÞ for

different regions of the vapour–water and metal–water

systems and the corresponding results of sHB are in-

cluded in Table 2. We note that both fast librational andslower diffusional motion can contribute to the decay of

SHBðtÞ. Since the librational motion occurs on a faster

r pairs

EHB sR 1=k

)19.74 10.5 3.2

)18.65 4.5 1.1

)18.72 7.1 2.4

)17.82 22.0 4.1

)16.94 7.45 2.0

)18.60 7.6 2.4

A and the other one is in region B. The relaxation times and the inverse

re expressed in units of kJ/mol.

-3

-2

-1

0

ln C

HB(t

)

0 3 6 9 1 2Time (ps)

-3

-2

-1

0

ln C

HB(t

)

(a)

(b)

Fig. 3. The time dependence of the intermittent hydrogen bond cor-

relation function CHBðtÞ in different regions of (a) vapour–water and

(b) metal–water systems. The different curves are as in Fig. 2.


time scale and since the correlation function SHBðtÞ doesnot allow any reformation event, the dynamics of SHBðtÞprimarily reveal the dynamics of hydrogen bond

breaking due to fast librational motion. The relaxation

of SHBðtÞ is found to be slower at both vapour–waterand metal–water interfaces compared to the corre-

sponding relaxation in the bulk phases. An insight into

this different relaxation behaviour of interfacial and

bulk hydrogen bonds can be obtained from the ener-

getics of these hydrogen bonds which are also included

in Table 2. For the metal–water system, the interfacial

hydrogen bonds are found to be of relatively higher

energy than those of the bulk and this would lead to afaster breaking of interfacial hydrogen bonds. But there

is a second effect which is the additional friction that is

exerted by the metal surface on the rotational and

translational motion of water molecules in its vicinity.

For metal–water system, this frictional effect wins over

the hydrogen bond energetic effect and the net result is a

slowing down on the hydrogen bond dynamics in the

vicinity of the metal surface. For vapour–liquid inter-faces, however, there is no such external field that gives

rise to an additional friction on the motion of water

molecules. The hydrogen bond energies are )19.7 and

)18.7 kJ/mol at vapour–liquid interface and in the bulk

phase, respectively. Thus, although, the number of hy-

drogen bonds in the vapour–water interfacial region is

less as reported in Table 1, the hydrogen bonds in this

region are found to be relatively stronger and hence livelonger.

A different way to analyze the hydrogen bond dy-

namics is to calculate the intermittent hydrogen bond

correlation function [6–15,35]

CHBðtÞ ¼ hhð0ÞhðtÞi=hhi; ð7Þ

which describes the probability that a hydrogen bond is

intact at time t, given it was intact at time zero, inde-

pendent of possible breaking in the interim time.

Clearly, bonds which were briefly �broken� by fast li-

brational motions would continue to contribute to thecorrelation function at later times and this leads to a

much slower decay of CHBðtÞ at longer times. The re-

laxation time sR of this function is usually called the

structural relaxation time of hydrogen bonds. In Fig. 3,

we have shown the results of CHB and the values of sR,which are obtained by assuming an exponential decay of

CHBðtÞ as described in [11], are included in Table 2.

Again, the relaxation of hydrogen bonds at interfaces(i.e. regions I and Ia) is found to be slower than that

in the bulk for both vapour–water and metal–water

systems.

We note that after a hydrogen bond is broken, the

two water molecules can remain in the vicinity of each

other for some time before either the bond is reformed

or the molecules diffuse away from each other. We de-

fine NHBðtÞ as 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. asnearest neighbours but not hydrogen bonded at time t.Following previous work [6–8,35], we write a simple rate

equation for the �reactive flux� �dCHB=dt in terms of

CHBðtÞ and NHBðtÞ

� dCHBðtÞdt

¼ kCHBðtÞ � k0NHBðtÞ; ð8Þ

where k and k0 are the forward and backward rate

constants for hydrogen bond breaking. The inverse of kcan be interpreted as the average lifetime of a hydrogenbond. The probability function NHBðtÞ can be calculated

from the simulation trajectories through the following

correlation function approach [8]

NHBðtÞ ¼ hhð0Þ½1� hðtÞ�h0ðtÞi=hhi; ð9Þwhere h0ðtÞ is unity if the interoxygen distance of the pair

of water molecules is less than 3.5 �A at time t and it is

zero otherwise. The results of NHBðtÞ are shown in Fig. 4.

We calculated the derivative of the intermittent hy-drogen bond correlation of Eq. (7) from the simulation

results of CHBðtÞ that are presented in Fig. 3 and we

used a least-squares fit of Eq. (7) to the simulation re-

sults of the CHBðtÞ, its time derivative and NHBðtÞ to

produce the forward and backward rate constants. The

inverse of the corresponding forward rate constant,

which correspond to the average hydrogen bond life-

time, is included in Table 2. The values of 1=k alsoreveal a slower dynamics of the hydrogen bonds at

both vapour–water and metal–water interfaces. We

note that, for all the regions, the values of 1=k are found

to be somewhat longer than sHB obtained from the

continuous hydrogen bond correlation function SHBðtÞ.This is not unexpected because SHBðtÞ primarily cap-

tures the hydrogen bond �breaking� dynamics due to fast

0

0.05

0.1

0.15N

HB(t

)

0 3 6 9 12Time (ps)

0

0.05

0.1

0.15

NH

B(t

)(a)

(b)

Fig. 4. The time dependence of the probability function NHBðtÞ, definedby Eq. (9), in different regions of (a) vapour–water and (b) metal–water

systems. The different curves are as in Fig. 2.


librational and rotational motion whereas the quantity

1=k includes contributions from fast librational, rota-tional and also from slower translational diffusional

motion of water molecules. In fact, when k is obtained

only from the short time part of Eq. (8), the value of 1=kdoes appear to be close to sHB obtained from SHBðtÞ. We

also note that the value of 1=k for bulk water as re-

ported in Table 2 agree well with the results of [8]

considering the fact that the fitting of simulation data to

Eq. (8) was done over a longer time (25 ps) in [8] thanthat in the present work (12 ps).

Till now we have discussed the dynamics of hydrogen

bonds where both molecules of a bonded pair either

belong to the interface (regions I or Ia) or to the bulk

(region II). An interesting dynamical behaviour is found

for the dynamics of those hydrogen bonds where one

molecule of the bonded pair belongs to the interface

(I or Ia) and the second one to its adjacent region, i.e.region II for vapour–water or region Ib for metal–water

systems. Both the lifetime and the structural relaxation

time of these inter-region I–II or Ia–Ib hydrogen bonds

are found to be faster than even those in the bulk phase.

It is seen from Table 1 that the hydrogen bond envi-

ronment of different regions are different to some extent.

The energetics data of Table 2 show that the energies of

these inter-region or inter-environment hydrogen bondsare higher than the corresponding intra-region hydrogen

bonds for the bulk or interfacial zones for both the va-

pour–water and metal–water systems. Thus, the hydro-

gen bonds that connect water molecules of two different

regions or environments are found to be relatively

weaker and hence relax at a faster rate than those be-

longing to a single region, either interface or the bulk.

We note that, for the vapour–water system, the some-what lower energy of the interfacial (I–I) hydrogen

bonds as compared to that of the bulk phase was also

found in an earlier study of hydrogen bonding at va-

pour–water interfaces [36].

7. Conclusion

In this Letter, we have investigated the dynamics of

water–water hydrogen bonds at vapour–water and me-

tal–water interfaces by means of molecular dynamics

simulations. The inhomogeneous density and also thedistribution of hydrogen bonds in interfacial regions are

also calculated which help to understand the structural

aspects of these interfaces and their effects on the dy-

namical properties. For both interfaces, the relaxation

of hydrogen bonds is found to occur at a slower rate

than that in bulk water. However, the dynamics of those

hydrogen bonds which connect interfacial molecules to

its adjacent layer on the bulk side is found to be fasterthan even the bulk hydrogen bonds. The results are

explained in terms of the energies of hydrogen bonds

and also the frictions that act on the rotational and

translational motion of water molecules in different re-

gions of the vapour–water and metal–water systems.

Acknowledgements

We gratefully acknowledge the financial support from

BRNS, Department of Atomic Energy and Council of

Scientific and Industrial Research, Government of In-

dia.

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Hydrogen bond dynamics at vapour–water and metal–water interfaces

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Transcript of Hydrogen bond dynamics at vapour–water and metal–water interfaces