1. Introduction

Advances in the understandings of the structures of liquid metal-vapor and liquid metal-solid interfaces have been made in the past decades [1-6]. The progress has been made possible by a combination of theoretical computer simulation [7-25] and experimental studies of grazing incidence x-ray diffraction and reflection [26-34], including simple liquid metals, binary alloys [1, 11, 17-18, 20, 24 ], and ternary alloys [13, 29-30].

Advances in the understandings of the structures of liquid metal-vapor and liquid metal-solid interfaces have been made in the past decades ^{[1][2][3][4][5]}. The progress has been made possible by a combination of theoretical computer simulation ^{[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]} and experimental studies of grazing incidence x-ray diffraction and reflection [26-34], including simple liquid metals, binary alloys ^{[1][11][17][18][20]}, and ternary alloys ^[13][29][30].

The characteristic factors of the liquid metal-vapor interface are represented by the density distribution along the normal and the pair correlation function parallel to the liquid-vapor interface.  These factors can be used to predict the magnitude of segregation of solute in the liquid-vapor interfaces of alloys and the occurrence of two-dimensional crystallization in the segregated layer in the liquid-vapor interface in dilute alloy systems.

Due to the complexity of the inhomogeneous systems, there is no ideal solution to solve these systems. So far, the most successful theoretical approach [4-9] is the pseudo-potential representation that is established through a multi-component, disorderly distribution of ion cores and valence electrons. Then, based on the constructed pseudo-potential system Hamiltonian, a self-consistent Monte Carlo simulation is carried out to obtain the longitudinal density distribution.

Due to the complexity of the inhomogeneous systems, there is no ideal solution to solve these systems. So far, the most successful theoretical approach ^{[4][5][6][7][8][9]} is the pseudo-potential representation that is established through a multi-component, disorderly distribution of ion cores and valence electrons. Then, based on the constructed pseudo-potential system Hamiltonian, a self-consistent Monte Carlo simulation is carried out to obtain the longitudinal density distribution.

In many practical applications, local density approximations have been employed to approximate an inhomogeneous system. This is achieved by invoking the properties of a homogeneous fluid and using a local density to approximate the properties of an inhomogeneous system. Within many local density approaches, it appears that the local density with a pseudopotential approximation for ion interaction is quite satisfactory in the descriptions of the behavior of the transversal pair correlation function.

The reported liquid metal-vapor interface systems include both simple liquid metals and their alloys, e.g., alkaline metals, gallium (Ga), aluminum (Al), indium (In), thallium (Tl), mercury (Hg), tin (Sn), lead (Pb), InGa binary alloys, BiGa binary alloys, GaSn binary alloy, dilute Pb in Ga alloy, dilute Tl in Ga Alloy, dilute ternary alloy of Pb and Sn in Ga, and more.

2. The Density Distribution

2.1. Pseudo-potential Hamiltonian

The pseudo-potential system Hamiltonian is given as

where

R

is the distance between atom-

i

and atom-

j

,  is the effective pair interaction potential, The pseudo-potential is a structure independent contribution to the energy, including the electron-ion and the ion-ion interactions. The pseudo-potential depends only on the electron density  and a reference core density

  .

Fig. 1. A simulation slab at the outmost layer of the BiGa liquid metal-vapor interface [J. Chem. Phys. 1998, 108, 5055].

2.2. Monte Carlo Simulation

A common simulation model usually consists of a slab with N ions and

n

N electrons, where

n

is the valency of the metal. The particles are placed randomly on a slab parallel to the

x-y

plane within the 3-D boundaries

L₀

x

L₀

x

2L₀

in the

x

,

y

, and

z

directions.

L₀

can be selected such that the average density of ions in the slab matches the density of liquid metal for a given simulation temperature. A schematic representation of a simulation slab at the initial condition is presented in Fig. 1. It shows a liquid BiGa binary alloy with a dilute Bi (4%) in Ga (96%) at the liquid-vapor interface.

The total depth of the slab can be arranged from 10-17 layers. Its layer is about an atomic diameter in thickness. Each side of the slab at the interface can be about 5-7 layers. The initial configuration eliminates ion-core to ion-core overlaps by a force-biased Monte Carlo simulation with periodic boundary conditions.

Based on the given ion-ion interaction potential, several electron density profiles along the z-axis may be prepared before ion-core density simulation to achieve calculational efficiency. These profiles can be prepared around the bulk density of the liquid metal, from somewhat below to a little above. This may be achieved by applying a “jellium distribution” of a rigid electron-profile, then solve the Kohn-Sham equation to achieve local electroneutrality and to avoid excessive kinetic energy associated with increasing the curvature of the wave function of the electron. A schematic representation of the electron density profiles is presented in Fig. 2.

Fig. 2.

A normalized longitudinal electronic density profile of liquid Ga (oscillation) based on a “jellium distribution” (solid) [

Phys. Rev. E

.

1997,

56

, 7033].

3. Density Profiles and Discussion

3.1. The Transverse Pair Correlation Function

The normalized transverse pair correlation function can be calculated from a histogram of the separations of the paired particles in a thin slice of the interfacial region

where

N_T

is the total number of particles in the slice,

N(r, Δr)

is the average number of pairs of particles between

r

and (

r + Δr)

,

V_T

is the total volume of all the particles in the slice, and

V_s

is the average volume of the intersection of the spherical shell between

r

and (

r + Δr)

. A presentative of the air-correlation function of liquid metal (or alloy) is shown in Fig. 3.

Fig. 3. Pair-correlation functions of bulk liquid Ga at three different temperatures [Phys. Rev. E. 1997, 56, 7033].

3.2. The Longitudinal Density Distribution

The longitudinal density distributions in the liquid-vapor interface provide the most sensitive test of our calculations. Fig. 4 shows a normalized longitudinal density profile.

Fig. 4.

The longitudinal density distribution of liquid Ga with the corresponding electron density (dotted line) at the liquid-vapor interface [

Phys. Rev. E

.

1997,

56

, 7033].

Fig. 5 shows a representative comparison of the experimental observation of the density profile at the interface density to the simulation that shows that the calculated longitudinal density profile agrees well with the experimental density profile.

Fig. 5 shows a representative comparison of the experimental observation of the density profile at the interface density to the simulation that shows that the calculated longitudinal density profile agrees well with the experimental density profile.

For all the studied systems, as shown in Fig. 5, both the amplitudes and the peaks of the longitudinal density distribution are not quite sensitive to temperature. There is no noticeable difference between the simulation results for several different temperatures. Generally, the agreement between the simulated and the observed longitudinal density distributions agree well qualitatively, both for pure metals and for alloys.

For all the studied systems, as shown in Fig. 5, both the amplitudes and the peaks of the longitudinal density distribution are not quite sensitive to temperature. There is no noticeable difference between the simulation results for several different temperatures. Generally, the agreement between the simulated and the observed longitudinal density distributions agree well qualitatively, both for pure metals and for alloys.

Fig. 5.

A representative comparison of the density profile between the experiment (solid) and the simulation, using the BiGa binary alloy as an example [

J. Chem. Phys.

1998,

108, 5055].

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, 5055].