J. Phys. Chem. 1996, 100, 14969-14977
14969
Structure and Dynamics at the Liquid Mercury-Water Interface Josef Bo1 cker,† Zinovii Gurskii,‡ and Karl Heinzinger*,† Max-Planck-Institut fu¨ r Chemie (Otto-Hahn-Institut), P.O. Box 3060, D-55020 Mainz, Germany, and Institute for Physics of Condensed Matter, Ukrainian Academy of Sciences, LViV 290005, Ukraine ReceiVed: April 9, 1996X
Molecular dynamics simulation studies of pure water near a liquid mercury surface are presented. The mercury-water and mercury-mercury interaction potentials developed recently are summarized. The structure perpendicular to the mercury-water interface is described in terms of density profiles, hydrogen bonding characteristics, energetics, and potential drop across the water phase. Radial distribution functions are calculated for different water and mercury layers. The orientations of different water molecule vectors are investigated. The structure parallel to the interface is described by the average positions and the trajectories of the interfacial atoms. The velocity autocorrelation functions, the self-diffusion coefficients, and the spectral densities of the hindered translational motions of the water molecules and the mercury atoms, parallel and perpendicular to the interface, are analyzed. The dynamical properties of water are further characterized by librational and vibrational modes for different water layers. The simulation results are compared with experimental and theoretical data.
Introduction It has been demonstrated some time ago that the use of the image charge model to describe the Coulombic part of the interactions between the partly charged oxygen and hydrogen atoms of a water molecule and a platinum wall led to contradictions with experimental results. Therefore, potentials have been introduced1 that are based on extended Hu¨ckel calculations of a five-atom platinum cluster and a water molecule.2 With these potentials a water lamella between Pt(100) surfaces has been simulated and the structural3 as well as dynamical4 properties of the water molecules have been reported. In a recent review paper all of the results achieved for water near the Pt(100) surface have been collected,5 including those that describe the effects of applied external electrical fields.6 In a more recent molecular dynamics (MD) study, employing a similar potential for the platinum-water interactions, Raghavan et al.7 extended the investigation to a Pt(111) surface. In a similar way a potential describing the interaction between water and a mercury surface has been derived on the basis of the ab initio calculations of Nazmutdinov et al.8 First results of MD simulations with this potential assuming a rigid mercury surface have been reported by us recently.9 To extend these investigations to a liquid mercury-water interface, a mercurymercury potential has been developed recently,10 which describes adequately different properties of pure liquid mercury. First results for the atomic density profiles perpendicular to the liquid mercury-water interface were briefly discussed in ref 11. In this presentation the structural, energetical, and dynamical properties of the interface are described in detail. The results are compared with those of a Pt(100)-water interface and with experimental data, especially those of a recently performed X-ray reflectivity study of the mercury liquid-vapor interface.12 The paper is organized as follows. The mercury-mercury pair potential and the mercury-water potential are described in the second section. The simulation procedures are outlined * Author to whom correspondence should be addressed. † Max-Plank-Insitute fu ¨ r Chemie. ‡ Ukrainian Academy of Sciences. X Abstract published in AdVance ACS Abstracts, August 1, 1996.
S0022-3654(96)01065-9 CCC: $12.00
in the third section. The structure of the liquid mercury-water interface is discussed in the fourth section in terms of the atomic density profiles, hydrogen bonding, and radial distribution functions, as well as the orientations of the water molecules as a function of distance from the surface and in terms of the atomic distributions in the surface layers. In the fifth section the dynamical properties of the interface are analyzed in terms of velocity autocorrelation functions, self-diffusion coefficients, and spectral densities of the translational, rotational, and vibrational motions. Finally, the simulation results are summarized in the sixth section. Interaction Potentials (a) Water-Water Potential. The water-water interactions are described by the flexible BJH water model.13 (b) Mercury-Mercury Potential. The pseudopotential method14 is employed to obtain the mercury-mercury pair potential. The total potential, which describes the effective interaction between two mercury ions, is given by the sum of an indirect and a direct part
VHg-Hg(r) ) Vind(r) + Vdir(r)
(1)
The potential of indirect interaction between two metal ions, Vind(r), induced by free electrons, is given in the theory of metals14,15 by the following equations
Vind(r) )
Ω2
∞ 4 2 q w (q) ∫ (2π) 3 0
1 - (q) sin qr dq qr (q)
(q) ) 1 + (4π/q2)Π(q) Π(q) )
Π0(q) )
Π0(q) 1 - 4πq-2G(q)Π0(q)
(
| |)
kF 1 4 - η2 2 + η ln 2-η π2 2 8η
with η ) q/kF
and kF ) (3π2z/Ω)1/3
© 1996 American Chemical Society
(2) (3) (4)
(5) (6)
14970 J. Phys. Chem., Vol. 100, No. 36, 1996
G(q) )
η2 1 2 η2 + 2[1 + 0.153(πk )-1]-1 F
Bo¨cker et al.
(7)
The following notations are introduced in eqs 2-7 written in atomic units: w(q) is the form factor of a bare pseudopotential, q is the wave vector, (q) is the static dielectric function, and Π(q) and Π0(q) are the polarization operators of the interacting and ideal free electrons, respectively. The local field function G(q) takes into account the correlation effects in the electron subsystem, z is the mercury ion valency, and Ω is the volume per ion. The Geldart-Vosko approach16 is used for G(q), and the Ashcroft model pseudopotential (MP) with the form factor
w(q) ) (4πz/Ωq2) cos(qrc)
(8)
and the parameter rc is employed; see ref 14 for details. The energy is written in atomic units. The values z ) 2 au, rc ) 0.915 au, and kF ) 0.7091 au determine the indirect ion-ion interaction in liquid mercury.14 It is an important feature of the indirect potential as given in eq 2 that it depends only weakly, via kF, on the average electron density. This means that Vind(r) need not be modified for the description of the indirect mercury-mercury interactions near the liquid mercury-water interface, although there are strong variations in the electron density between mercury in the bulk and at the surface. The direct potential is given by a sum of a Coulombic and a core overlap contribution (in the Born-Mayer form), written in atomic units, as
Vdir(r) ) z2/r + a exp(-br)
(9)
with a and b as adjustable parameters. The b value was chosen to be 1/rc. The simulation results for pure liquid mercury with the potential according to eq 1 and different parameter a values clearly indicated that a very good agreement with experimental data is obtained with an a value equivalent to about 2.8 × 103 kJ/mol.10 In Figure 1 the total effective mercury-mercury pair potential VHg-Hg(r), eq 1, is shown for a ) 2783 kJ/mol, which is used in the interface simulation. For further details of the mercury-mercury potential the reader is referred to ref 10. (c) Mercury-Water Potential. The mercury-water potential is derived from ab initio calculations of the adsorption energy of a water molecule on a mercury cluster.8 The following potential functions yield the best fit to the ab initio data
VHg-O(r, F) ) [25518 exp(-2.0829r) - 5508.2 exp(-1.3922r)]f(F) + 8813.2 exp(-2.1759r)[1 - f(F)] (10) with f(F) ) exp(-0.2213F2)
and F ) x∆x2 + ∆y2 (11)
VHg-H(r) ) 2603.6[exp(-2.2230r) + exp(-2.6737r)] (12) The energies are given in units of kilojoules per mole with the distances in angstroms. The total energy of one water molecule above the rhombohedral mercury surface was calculated for different relative positions and different orientations of the water dipole moment relative to the surface. The rhombohedral configuration was found to be a good model for describing the structure of pure liquid mercury.17 The relative positions of a water molecule on the mercury surface, denoted top, bridge, and hollow sites,
Figure 1. Total effective mercury-mercury pair potential. The inset shows the same function at an expanded scale.
Figure 2. Relative positions of a water molecule (oxygen atom) on the mercury surface.
are defined in Figure 2. In Figure 3 the potential energy between one water molecule and the infinitely extended rhombohedral mercury surface is shown as a function of the oxygen-surface distance z for different relative positions and orientations. The top site with the dipole moment of the water molecule pointing away from the surface, Figure 3a, with a minimum of -38.6 kJ/mol at z ) 2.87 Å, is energetically most favorable. The overall behavior of the mercury-water potential is similar to that of the platinum-water potential discussed in ref 3. The parametrized potential for Hg-O and Hg-H was employed in an MD simulation of water near a solid mercury surface.9 Details of the Simulations The interface simulation was started from a configuration where the basic periodic boxes from a pure mercury and a pure water simulation were combined at a distance of 3 Å, which is approximately the position of the potential minimum for the water-mercury interaction. During the equilibration period the dimensions of the combined box were changed until the average density at the center of the water as well as of the mercury phase showed the density of the pure liquids. The basic box contained 750 water molecules and 880 mercury atoms with side lengths of Lx ) 24.00 Å, Ly ) 25.98 Å, and Lz ) 69.80 Å and periodic boundary conditions in all three directions. For VHg-Hg a cutoff radius of 11.96 Å was chosen, which is the position near half the box size, where the pair potential has a maximum and the force is therefore zero. The cutoff radii for VHg-O and VHg-H were 11 Å. Those for the non-Coulombic parts of VO-O, VO-H, and VH-H were 9, 5, and 3 Å, respectively. The method of shifted force potentials18 was employed for all non-Coulombic interactions, except VHg-O. Image charges were not included in the simulations. The 2D Ewald summation method19 was used for calculating the long-range Coulombic interactions.
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J. Phys. Chem., Vol. 100, No. 36, 1996 14971
Figure 4. Relative densities (left scale, thick lines) and running integration numbers (right scale, thin lines) of the mercury, oxygen, and hydrogen atoms as a function of the atom-surface distance z. The number of hydrogen atoms is divided by two. A fitted function for the mercury density profile is drawn additionally; see text for explanation.
Figure 3. Mercury-water interaction potential as a function of the oxygen-surface distance z for different relative positions and orientations: (a) top, bridge, and hollow site of a water molecule on the mercury surface with the water dipole moment pointing away from the surface; (b) top site with the water dipole moment pointing away from, parallel to, and toward the surface. Parallel a and parallel b are defined as orientations with the hydrogen-hydrogen vector parallel and perpendicular to the surface, respectively.
The system was equilibrated for 120 ps using only shifted force potentials and then for 15 ps employing the 2D Ewald summation. The constant temperature method according to Kast et al.20 was temporarily applied during the equilibration time. This method was incorporated into the Adams-Bashford predictor-corrector algorithm.21 The NVE ensemble without any velocity rescaling was employed during the last 39 ps. The chosen time step of 0.1 fs yielded a relative standard deviation in the total energy of about 9 × 10-4 for 10 ps. The last 33 ps are used to analyze the properties of the interface at an average temperature of 302 K. Structure at the Liquid Mercury-Water Interface The results for the two interfaces are averaged in all figures. All distance-dependent properties are collected in bins of 0.04 Å. The instantaneous average of the z coordinates of the mercury atoms in the mercury layer next to the water phase is used to define z ) 0. (a) Density Profiles Perpendicular to the Surface. The relative density profiles and the running integration numbers of the oxygen, hydrogen, and mercury atoms perpendicular to the interface are displayed in Figure 4. Both the oxygen and the hydrogen density profiles show two pronounced maxima and a small third one, indicating a strong layering of water molecules with enhanced densities up to about 10 Å. The first, well-pronounced oxygen peak with a maximum at 3.03 Å
contains 71 ( 1 water molecules. The first, strong hydrogen peak at 2.95 Å almost coincides with the position of the oxygen one. The positions of these maxima indicate that for most of the water molecules the dipole moment vector is orientated parallel to the mercury surface. The relative water density is 1.06 for the first water layer up to the first oxygen minimum at 4.45 Å, while all other layers have a relative density of nearly 1. A second maximum is located at 5.90 and 6.25 Å in the oxygen and hydrogen density profiles, respectively. The plateau between 4.0 and 5.2 Å in the hydrogen density profile is due to an overlap of hydrogen atoms belonging to molecules of the first and second oxygen layers. This indicates that water molecules at the larger distance side of the first oxygen maximum have orientations in which one hydrogen atom points away from the surface and that water molecules at the shorter distance side of the second oxygen maximum have orientations in which one hydrogen atom points toward the surface. The orientation of the water molecules is analyzed in more detail under part f of this section. The mercury density profile reveals far-ranging oscillations. It can be represented by the following analytical form:
F/F0 ) Ad + Rd exp(βdz) cos(γdz)
(13)
The best fit is achieved with Ad ) 1.010, Rd ) 1.173, βd ) 0.152 Å-1, and γd ) 2.403 Å-1. The parameter Ad can be taken as a quantitative measure of the mercury density in the center of the mercury phase. This value indicates that the bulk density of mercury is slightly higher than that of pure liquid mercury. The number of mercury atoms in the first layer near the water phase is 71 ( 1, whereas the number in all other layers is 67 ( 1. These numbers correspond to relative densities of 1.08 for the first layer and 1.01 for the consecutive layers. Therefore, only the first layer exhibits an increased density. This increased density is a consequence of the stronger interactions between water molecules and mercury atoms when compared with the mercury-mercury interactions, together with the shorter O-O distance compared to the Hg-Hg distance. The number of mercury atoms in the first layer is almost identical with the one of oxygen atoms in its first layer (almost identical relative densities), implying that each mercury atom adsorbs one water molecule. It is interesting to compare the mercury density profile with the one resulting from a recently performed X-ray reflectivity
14972 J. Phys. Chem., Vol. 100, No. 36, 1996
Bo¨cker et al.
Figure 6. Potential drop as a function of the oxygen-surface distance.
Figure 5. Number of nearest neighbors, nNN, and number of hydrogen bonds, nHB (a), and the total potential energy of a water molecule, EH2O-(H2O+Hg), and separately the interaction energies between one water molecule and all mercury atoms, EH2O-Hg, and all other water molecules, EH2O-H2O (b), as a function of the oxygen-surface distance z.
study of the mercury liquid-vapor interface.12 The authors state that their results provide the only conclusive experimental proof of surface layering in liquid metals to date. The electron density profile of the liquid phase reveals an oscillation length of 2.76 ( 0.20 Å from the second layer onward, which is in excellent agreement with our findings of 2.62 ( 0.04 Å. Also their relative densities of the different layers are very similar to our results. The values are 1.09 ( 0.12 and 1.08 ( 0.02 for the first layer and 1.00 ( 0.04 and 1.01 ( 0.02 for the other layers, from the X-ray and the MD study, respectively. The first layer of the liquid mercury phase near the vapor phase is much broader12 compared to the first layer of the mercury phase near the water phase. Another difference between our study and the X-ray one is that a faster decay of the mercury density is observed in the mercury liquid-vapor interface. These two differences might have to be attributed to the ordering effect of the water phase into the mercury phase. (b) Energetics Perpendicular to the Surface. The average number of nearest-neighbor water molecules, nNN, and the average number of hydrogen bonds per water molecule, nHB, are presented in Figure 5a as a function of the oxygen-surface distance z. nNN is defined as the number of oxygen atoms within a distance of less than 3.3 Å, which is the position of the first minimum in the oxygen-oxygen radial distribution function, gO-O(r) (see Figure 8). nHB is determined by counting the number of nearest neighbors with an interaction energy of less than -16.75 kJ/mol,22 which corresponds to the maximum of the distribution of the dimerization energies and is therefore a quite strict energetic criterion. The number of nearest neighbors nNN increases from the interface region onward up to about z ) 5 Å and then remains almost constant. It has a maximum around the region of the first minimum of the oxygen and hydrogen density profile, as a consequence of the higher density at shorter as well as at larger
distances. In the z region beyond 5 Å nNN ) 4.3, close to the value of 4.4 which was found for pure water.23 The number of hydrogen bonds is almost uniform for all z distances. Only very close to the interface is nHB slightly reduced due to the reduced nNN. This result underscores that the short-range structure of water is only slightly disturbed by the metal phase. A similar result has been found for water at the Pt(100) surface.3 The interaction energies between one water molecule and all mercury atoms, EH2O-Hg, one water molecule and all other water molecules, EH2O-H2O, and the total potential energy of a water molecule, EH2O-(H2O+Hg), are shown as a function of z in Figure 5b. The contribution of EH2O-Hg with an energy minimum of -27 kJ/mol at z ) 3.33 Å is only important for z values smaller than 4.5 Å, the region of the first water layer. The energy minimum is significantly less than the energetically most favorable value for a top position with the water dipole momemt pointing away from the surface but corresponds to a top position of a water molecule with a parallel dipole orientation (see Figure 3b), in agreement with the conclusions drawn from the density profiles. The experimental adsorption enthalpy of -29 kJ/mol is in good agreement with our result.24 The energy minimum of EH2O-(H2O+Hg) at z ) 3.39 Å has a value of -65 kJ/mol. Its z distance is slightly larger than the z position of the first maximum in the oxygen density profile. This shift is a consequence of the different water molecule orientations over the range of the first peak in the oxygen density profile (see above): The water molecules at the larger distance side of the first oxygen maximum have more often dipole moment vectors pointing away from the surface, and hence a more negative EH2O-Hg contribution, than the water molecules at positions shorter than the first oxygen maximum (see also Figure 11). The water-water interactions amount to more than 60% of the total interaction energy of an adsorbed water molecule. They are dominant for z values greater than about 4.5 Å, and EH2O-Hg is practically zero for z values larger than about 8 Å. The important role of the water-water interactions and the almost unchanged number of hydrogen bonds in the interface region (see Figure 5a) support the conclusion of Porter and Zinn25 that the structure of water at the mercury-water interface region is apparently dominated by hydrogen bonding. (c) Potential Drop. The surface potential drop in the liquid phase, δχH2O(z), is calculated from the charge density q(z) according to the equation
δχH2O(z) ) -(1/0) ∫0 q(z′)(z - z′) dz′ z
(14)
where 0 is the dielectric permittivity of the vacuum. In Figure 6 the potential drop is shown as a function of z. A value of
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J. Phys. Chem., Vol. 100, No. 36, 1996 14973
Figure 8. Mercury-mercury and oxygen-oxygen radial distribution functions (left scale, thick lines) for different layers and running integration numbers (right scale, thin lines) for the first mercury and the first water layer.
Figure 7. Projections of the mercury and oxygen atom trajectories in the first layers onto the xy plane and their average positions (Hg, squares; O, crosses) from a 3 ps simulation period.
about 0.6 V is found in the center of the water phase (z > 8 Å). The main contribution results from the first adsorbed water layer. A value of -0.26 V for the mercury (uncharged)-water contact potential difference, δχ, has been derived from Volta potential measurements by Randles.26 This difference consists of zdependent contributions from the charge distributions of the mercury ions and its free electrons, δχHg, and the water charge distribution, δχH2O, and is defined by δχ ) δχHg - δχH2O, where δχi is positive when the positive end of the dipole layer is directed toward the interior of the specified phase. These two individual parts are both estimated to be relatively small, e.g. δχHg ) -0.33 V and δχH2O ) -0.07 V.27 Other estimates for δχH2O reach from 0 to ∼0.3 V,28 but there are no experimental data available. The δχHg contribution is zero in our simulation
because the mercury free electrons are not treated explicitly in our model. (d) Surface Layer Trajectories. The projections of the trajectories of the mercury and oxygen atoms in their first layers onto the xy plane are shown in Figure 7. They are calculated from a 3 ps time interval of the simulation. This short time span was chosen because after about 3 ps it can happen that one of the interfacial atoms moves from the first to the second layer or vice versa. This movement perpendicular to the interface is found for the water molecules as well as for the mercury atoms. The interval chosen from the simulation seems to be representative for the whole simulation run. It can be seen that the water molecules (correctly the oxygen atoms) move significantly more quickly than the much heavier mercury atoms. There are no exchanges between the positions in the xy plane. Of course, the movements in the direction perpendicular to the interface cannot be seen in the projections. The trajectories cover an area up to several square angstroms. In addition to the trajectories of the water molecules and the mercury atoms in the surface layer shown in Figure 7, their average positions over the same time period of 3 ps are depicted in Figure 7. There are 72 mercury and 72 oxygen atoms in the interface region. It is seen that some oxygen atoms are located on top of a mercury atom and most of them are close to this relative position. This position is energetically the most favorable one. The probability for finding different relative positions is not regularly distributed over the interface. Instead, there are small domains with very similar relative positions. Furthermore, it can be seen that the interfacial atoms are arranged in a distorted hexagonal pattern. (e) Radial Distribution Functions. The mercury-mercury, gHg-Hg, the oxygen-oxygen, gO-O, the oxygen-hydrogen, gO-H,
14974 J. Phys. Chem., Vol. 100, No. 36, 1996
Figure 9. Oxygen-hydrogen and hydrogen-hydrogen radial distribution functions (left scale, thick lines) for different layers and running integration numbers (right scale, thin lines) for the first mercury and the first water layer.
Figure 10. Mercury-oxygen and mercury-hydrogen radial distribution functions (left scale, thick lines) and running integration numbers (right scale, thin lines) for the first mercury and the first water layer.
the hydrogen-hydrogen, gH-H, the mercury-oxygen, gHg-O, and the mercury-hydrogen, gHg-H, radial distribution functions (RDFs) are shown for different layers in Figures 8-10. The running integration numbers, n, are shown additionally for the first mercury and the first water layer. To take into account the restricted volume of a layer, the mercury and oxygen densities are not related to a complete spherical volume but to the volume of the specified layer. The layers are defined by two successive minima of the corresponding atomic densities (see Figure 4). The bulk regions for water and mercury are chosen as the regions 12.4 < z
