Density Profile of Liquid-Metal-Vapor Interface: Comparison
Please note this is a comparison between Version 1 by Meishan Zhao and Version 7 by Meishan Zhao.

Several metals and many alloys are in liquid form around room temperature, e.g., mercury (Hg, −38.8 °C), francium (Fr, 8.0 °C), cesium (Sc, 28.5 °C), gallium (Ga, 29.8 °C), the eutectic mercury-based alloys, and the eutectic gallium-based alloys. If eutectic, liquid metal alloys can be in liquid form that has been used in practical applications, replacing mercury. Liquid metals have high thermal and electric conductivity and have been used to conduct heat and electricity between non-metallic and metallic surfaces. They have also been used as thermal interface materials between coolers and processors. Concerning these metals, our understanding of the liquid-vapor interface is critical for proper applications. This entry summarizes the basic features of the density distribution of liquid metal-vapor interface, which are advanced based on pseudo-potential representation and numerical simulation at the University of Chicago.

Liquid metals are referred to as the metals (and alloys) that are in liquid form around room temperature. Alloys of liquid metals can also be liquid if they are eutectic. Gallium-based alloys have been used in various applications, replacing mercury. Due to their high thermal and electric conductivity, liquid metals have been used to conduct heat and electricity between non-metallic and metallic surfaces and have been used as thermal interface materials between coolers and processors. The various applications include wearable devices, medical devices, thermostats, switches, barometers, heat transfer systems, thermal cooling, heating designs, wetting to many non-metallic surfaces, and more. Understanding the structure of the liquid metal-vapor (and liquid metal-solid) interface is critical for proper applications. The density distribution of liquid metal at the interface is the key to the understanding of the interfacial properties.

  • Liquid metal and alloys
  • liquid metal-vapor interface
  • pair correlation function
  • liquid gallium
  • longitudinal density distribution
  • pseudo-potential representation
  • local density approximation
  • inhomogeneous liquids
  • self-consistent Monte Carlo simulation.

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 . The progress has been made possible by a combination of theoretical computer simulation and experimental studies of grazing incidence x-ray diffraction and reflection [26-34], including simple liquid metals, binary alloys , and ternary alloys .

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 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 by

whAdvanceres  isn the momentumunderstandings of the atomstructures of liquid metal-ivapor and wliquith masd metal-solid interfaces m;have been rmade isn the distance for an electron to the center of the atom. Rpast decades [1-6]. The progress has been made possible by a combination of theoretical computer simulation [7-25] and experimental studies of grazis the distance between atomng incidence x-ray diffraction and reflection [26-34], including simple liquid metals, binary alloys [1, 11, 17-i18, 20, 24 ], and ternatomry alloys [13, 29-j,30].

The  characteristic factors the effective paiof the liquid metal-vapor interaction potential, The pseudo-potentialface are represented by the density distribution along the normal and the pair correlation function parallel to the liquis a structure independent contribution to the energy, including the electron-iond-vapor interface.  These factors can be used to predict the magnitude of segregation of solute in the liquid-vapor interfaces of alloys and the ion-occurrence of two-dimensional crystallization interactions. The pseudo-potential d the segregated layer in the liquid-vapor interface in dilute alloy systems.

Duep to thends only on the electron density complexity of the inhomogeneous systems, there is no ideal solution to solve these systems.  So fandr, the a reference core density.

2.2. Electron Density Profile

Bmost successful theoretical approach [4-9] is the pseudo-potential representation that is establised on a given ion-ion interaction from the hed through a multi-component, disorderly distribution of ion cores and valence electrons. Then, based on the constructed pseudo-potential system Hamiltonian, several e a self-consistent Monte Carlo simulation is carried out to obtain the longitudinal density distribution.

In many practical appleictron ations, local density profiles may be prepared along the z-axis. This can be done before ion-coreapproximations have been employed to approximate an inhomogeneous system. This is achieved by invoking the properties of a homogeneous fluid and using a local density simulation to achieve calculational efficiency. These electron density profiles can bto 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 prepared around the bulk density of thorted liquid metal-vapor interface systems include both simple liquid metal. Based on a model “jes and their alloys, e.g., alkaline metals, gallium distribution” which is a rigid electron-profile, t(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 pselectron denudo-potential system Hamiltonian is given as

where R is ty profiles can be obthe distance between atom-i ained byd atom-j,  isolving the Kohn-Sham equation. A typicaleffective pair interaction potential, The pseudo-potential is a strepresentation ofucture independent contribution to the energy, including the electron density profiles is presented in Fig.1.

Figure 1. A norm-ion and the ion-ion interactions. The pseudo-potentializ deped longitudinalnds only on the electronic density p and a roefile of liquiderence core density  .

Fig. 1. GaA (oscilsimulation) based on a “jellium distribution” (solid) slab at the outmost layer of the BiGa liquid metal-vapor interface [J. Chem. Phys. Rev. E 19978, 56108, 7503355].

 

2.32. Monte Carlo Simulation

Usually, aA common model of system ssimulation model usually consists of a slab that contains with N ions and nN electrons, where n is the valency of the metal. The particles are placed randomly placed on theon a slab that is parallel to the x-y plane. The simulation box has within the 3-D boundaries L0 x L0 x 2L0 in the x, y, and z directions. TheL0 sidecan of the box (L0) is be selected such that the average density of the ions in the slab matches the density of the liquid metal for a given simulation temperature. A schematic representation of a simulation slab at the initial condition is presented in Fig. 21. It shows a liquid BiGa binary alloy with a dilute Bi (4%) in Ga (96%) at the liquid-vapor interface.

Figure 2. A simulation slab at the outmost layer of the BiGa liquid metal-vapor interface where the XY-plane is parallel to the interface and the coordinates are in atomic units (au) [J. Chem. Phys. 1998, 108, 5055].

The total depth of the simulation slab mayb can be arranged from 10-17 layers which are usually slightly more than the characteristic. Its layering structure of the interface. Each layer i 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.

 

3. Atomic Density Profiles

ThBased most important feature of the physical structure in the liquid metal-vapor interfaceon the given ion-ion interaction potential, several electron density profiles along the z-axis may be understood from equilibrium correprepared before ion-core density simulation to achieve calculations of the particles both in the transverse aal efficiency. These profiles can be prepared around the longitudinal direction of the interfacbulk density of the liquid metal, from somewhat below to a little above.

3.1. The Transverse Density Profile

This may be pair correlation functionachieved by applying a “jellium distribution” of a system describes how the density changes concerning the separation between two particles in the system. For liquid-metal-vapor interfaces, the transverse pair correlrigid 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 function that describes how the densityof the electron density profiles is presented in Fig. 2.

Fig. 2. A vnormaries within alized longitudinal electronic density layer parallel to tprofile 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 normalinterfacezed transverse pair correlation function can be calculated from paia histogram of the separations of the paired particles in the a thin slice of the interfacial region

where V NT is the vtolume of all the tal number of particles in the layer. Assume that all atoms are in theslice, formN(r, Δr) of a spherical shell, thens the Vs is an average volume of the spherical shellnumber of pairs of particles between r and (r + Δr), NVT is the ntotal volumber of e of all the particles in a transverse density layerthe slice, and N(r, Δr)Vs is the average nvolumber of pairs of particlese 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.

where NT 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), VT is the total volume of all the particles in the slice, and Vs is the average volume of the intersection of the spherical shell between r and (r + Δr). A typical air-correlation function of liquid metal (or alloy) is shown in Fig. 3.

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

 

3.2. The Longitudinal Density PDistrofileibution

For a given system, tThe longitudinal density profile is a key characteristic of the system structure at the distributions in the liquid-vapor interface. It provides a the most sensitive test for theory in comparison to the experimental observof our calculations. Fig. 4 is a typical shows a normalized longitudinal density profile along with its corresponding electron density profile.

Figure. 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].

 

Forig. most of the reported systems of the liquid metal-vapor interface, either simple liquid metals or liquid metal alloys, pseudo-potential repre5 shows a representation provides a qualitative agreement between theoretical predictions andve comparison of the experimental observations. Fig. 5 shows a representative comparison between the experimental observation and the theoretical prediction from of the density profile at the interface density to the simulation based on pseudo-potential Hamiltonian. It that shows that the simcalculated longitudinal density profile agrees qualitatively well with the experimental obsedensity profile.

Forv ation.

Ill the studied systems, has been noted that the das shown in Fig. 5, both the amplitudes and the peaks of the longitudinal density distributions are usually not are not quite sensitive to temperature. There is hardly any obviousno noticeable difference between the density profiles at different temperatures. This is both for pure metals and for alloys.  

Figure 5. A reprsimulation results for sesentative comparison of the density profile between the experiment (solid) and the simulation, using the BiGa binary alloy as an example [J. Chem. Phys. 1998ral different temperatures. Generally, 108, 5055].

 

4. Discussion and Perspective

Tthe theoagretical description of pseudo-potential Hamiltonian with self-consistent Monte Carlo ement between the simulation has been successful in comparison to experimentaled and the observations. Studies on the liquid-vapor interface have provided us an improved understanding of these systems. In turn, it leads to successful applicaed longitudinal density distributions in daily life. Contrarily, our understanding of the structures of liquid metal-solid interfaces has been very limited. This is mainly because of the limited experimental and theoreticalagree well qualitatively, both for pure metals and for alloys.

Fig. 5. studiesA reported so far.

Thresere have been only a few studies of liquid metal-solid interfaces, as reported in the literature [35-39]. One tative comparison of the first studies was an experimental observation and a self-consistent Monte Carlo simulation concerning the liquid Hg−Sapphire interface [35-36]. In that system, liquid Hg had an interface on the Sapphire (001) surface. We need focused studies on the liquid metal-solid interface to have a better understanding.

We are interested to know, based on odensity profile between the experiment (solid) and the simur knowledge of the liquid metal-vapor interface, how to improve our understanding of liquid metal-solid interfaces. Specifically, we would like to know the differences between liquid metal-solid interfaces and liquid metal-vapor interfaces and what are the key elements that lead toation, using the BiGa binary alloy as an example [J. Chem. Phys. 1998, these108, differences5055].

 

References

  1. Rice, S. A. Structure of the liquid-vapor interfaces of metals and binary alloys. Non-Cryst. Solids 1996, 207, 755.
  2. Yang, B.; Gidalevitz, D.; Li, D.; Huang, Z.; Rice, S. A. Two-dimensional freezing in the liquid-vapor interface of a dilute Pb:Ga alloy. PNAS 1999, 96(23),
  3. Zhao, M.; Rice, S. A. Density Distribution in the Liquid Hg-sapphire Interface. Phys. Chem A 2011, 115, 3859–3866.
  4. Woo, C. H.; Wang, S.; Matsuura, M. Electronic structure of metals. I. Energy independent model pseudopotential formalism. Phys. F Metal Phys. 1975, 5, 1836.
  5. Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Rev. A 1965, 140, 1133.
  6. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Rev. B 1964, 136, 864.
  7. D’Evelyn, M. P; Rice, S. A. Structure in the density profile at the liquid-metal-vapor interface. Rev. Lett. 1981, 47, 1844.
  8. D’Evelyn, M. P.; Rice, S. A. A study of the liquid-vapor interface of mercury: Computer simulation results. Chem. Phys. 1983, 78, 5081.
  9. D’Evelyn, M. P.; Rice, S. A. A Pseudoatom theory for the liquid-vapor interface of simple metals: Computer simulation studies of sodium and cesium. Chem. Phys. 1983, 78, 5225.
  10. Harris, J. G.; Gryko, J.; Rice, S. A. Self‐consistent Monte Carlo simulations of the electron and ion distributions of inhomogeneous liquid alkali metals. I. Longitudinal and transverse density distributions in the liquid-vapor interface of a one‐component system. Chem. Phys. 1987, 87, 3069.
  11. Harris, J. G.; Gryko, J.; Rice, S. A. Self-consistent Monte Carlo simulations of the electron and ion distributions of inhomogeneous liquid alkali metals. II. Longitudinal and transverse density distributions in the liquid-vapor interface of binary metallic alloys. Stat. Phys. 1987, 48, 1109.
  12. Jiang, X.; A. Rice, S. A. A theoretical study of the structure of the liquid Ga-diamond (111) interface. Chem. Phys. 2005, 123, 104703.
  13. Jiang, X.; Zhao, M.; Rice, S. A. Theoretical study of the longitudinal density distribution in the liquid-vapor interface of a dilute ternary alloy: Pb and Sn in Ga. Rev. B 2005, 72, 942011-942017.
  14. Jiang, X.; Zhao, M.; Rice, S. A. Longitudinal Density Distribution in the Liquid-Vapor Interface of a Dilute Tl in Ga Alloy. Rev. B 2005, 71, 1042031.
  15. Zhao, M; Rice, S. A. Density Distribution in the Liquid-Vapor Interface of a Dilute Pb in Ga. Rev. B 2001, 63, 85409.
  16. Rice, S, A.; Zhao, M. Quantum Monte Carlo Simulation Studies of the Structure of the Liquid and the Liquid-Vapor of Sn and Pb. Phys. Chem. A 1999, 103, 10159-10165.
  17. Zhao, M.; Chekmarev, D.; Rice, S, A. Quantum Monte Carlo Simulations of the Structure in the Liquid-Vapor Interface of BiGa Binary Alloys. Chem. Phys. 1998, 108, 5055-5067.
  18. Rice, S, A.; Zhao, M. Self-Consistent Quantum Monte Carlo Simulations of the Structure of Liquid-Vapor Interface of a Eutectic Indium-Gallium Binary Alloy. Rev. B 1998, 57, 13501-13507.
  19. Zhao, M.; Chekmarev, D.; Cai, Z. H.; Rice, S, A. The Structure of Liquid Ga and the Liquid-Vapor Interface of Ga. Rev. E 1997, 56, 7033-7042.
  20. Rice, S.A.; Zhao, M.; Chekmarev, D. Theoretical Studies of the Structures of the Liquid-Vapor Interfaces of Metals and Binary Alloys in Microscopic Simulation of Interfacial Phenomena in Solids and Liquids, The Materials Research Society, Vol. 492, 1998, pp.3-14.
  21. Zhao, M.; Chekmarev, D.; Rice, S, A. Comparison of the Structure of the Liquid-Vapor Interfaces of Al, Ga, In and Tl. Chem. Phys. 1998, 109, 1959-1965.
  22. Chekmarev, D.; Zhao, M.; Rice, S. A. Structure of the Liquid-Vapor Interface of a Metal from a Simple Model Potential: Corresponding States of the Alkali Metals. Chem. Phys. 1998, 109, 768-778.
  23. Chekmarev, D.; Zhao, M.; Rice, S. A. Computer Simulation Study of the Structure of the Liquid-Vapor Interface of Mercury at 20, 100, and 200o Phys. Rev. E 1999, 59, 479-491.
  24. Zhao, M.; Rice, S. A. The Structure of Liquid-Vapor Interface of a Gallium-Tin Binary Alloy. Chem. Phys. 1999, 111, 2181-2189.
  25. Li, F.; Zhao, M. Structure and Liquid-Vapor Interface of a Simple Metal. Theor. Phys. 1998, 29, 167.
  26. Li, D.; Rice, S. A. Melting of quasi-two-dimensional crystalline Pb supported on liquid Ga. Rev. E 2005, 72, 41506.
  27. Li, D.; Jiang, X.; Yang, B.; Rice, S. A. Phase transitions in the liquid-vapor interface of dilute alloys of Bi in Ga: New experimental studies. Chem. Phys. 2005, 122, 224702.
  28. Yang, B.; Li, D.; Rice, S. A. Two-dimensional freezing of Tl in the liquid-vapor interface of dilute Tl in a Ga alloy. Rev. B 2003, 67, 212103.
  29. Yang, B.; Li, D.; Rice, S. A. Structure of the liquid-vapor interface of a dilute ternary alloy: Pb and In in Ga. Rev. B 2003, 67, 54203.
  30. Li, D.; Yang, B.; Rice, S. A. Structure of the liquid-vapor interface of a dilute ternary alloy: Pb and Sn in Ga. Rev. B 2002, 65, 224202.
  31. Yang, B.; Li, D.; Huang, Z.; Rice, S. A. The structure of the liquid-vapor interface of a dilute Pb in Ga alloy. Rev. B 2000, 62, 13111.
  32. Lei, N.; Huang, Z.; Rice, S. A. Structure of the liquid-vapor interface of a Sn:Ga alloy. Chem. Phys. 1997, 107, 4051.
  33. Lei, N.; Huang, Z.; Rice, S. A. Surface segregation and layering in the liquid-vapor interface of a dilute bismuth:gallium alloy. Chem. Phys. 1996, 104, 4802.
  34. Lei, N.; Huang, Z.; Rice, S. A.; Grayce, C. In‐plane structure of the liquid-vapor interfaces of dilute bismuth: gallium alloys: X‐ray‐scattering studies. Chem. Phys. 1996, 105, 9615.
  35. Rice, S. A. Structure of the liquid-vapor interfaces of metals and binary alloys. Non-Cryst. Solids 1996, 207, 755.
  36. Yang, B.; Gidalevitz, D.; Li, D.; Huang, Z.; Rice, S. A. Two-dimensional freezing in the liquid-vapor interface of a dilute Pb:Ga alloy. PNAS 1999, 96(23),
  37. Zhao, M.; Rice, S. A. Density Distribution in the Liquid Hg-sapphire Interface. Phys. Chem A 2011, 115, 3859–3866.
  38. Woo, C. H.; Wang, S.; Matsuura, M. Electronic structure of metals. I. Energy independent model pseudopotential formalism. Phys. F Metal Phys. 1975, 5, 1836.
  39. Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Rev. A 1965, 140, 1133.
  40. Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Rev. B 1964, 136, 864.
  41. D’Evelyn, M. P; Rice, S. A. Structure in the density profile at the liquid-metal-vapor interface. Rev. Lett. 1981, 47, 1844.
  42. D’Evelyn, M. P.; Rice, S. A. A study of the liquid-vapor interface of mercury: Computer simulation results. Chem. Phys. 1983, 78, 5081.
  43. D’Evelyn, M. P.; Rice, S. A. A Pseudoatom theory for the liquid-vapor interface of simple metals: Computer simulation studies of sodium and cesium. Chem. Phys. 1983, 78, 5225.
  44. Harris, J. G.; Gryko, J.; Rice, S. A. Self‐consistent Monte Carlo simulations of the electron and ion distributions of inhomogeneous liquid alkali metals. I. Longitudinal and transverse density distributions in the liquid-vapor interface of a one‐component system. Chem. Phys. 1987, 87, 3069.
  45. Harris, J. G.; Gryko, J.; Rice, S. A. Self-consistent Monte Carlo simulations of the electron and ion distributions of inhomogeneous liquid alkali metals. II. Longitudinal and transverse density distributions in the liquid-vapor interface of binary metallic alloys. Stat. Phys. 1987, 48, 1109.
  46. Jiang, X.; A. Rice, S. A. A theoretical study of the structure of the liquid Ga-diamond (111) interface. Chem. Phys. 2005, 123, 104703.
  47. Jiang, X.; Zhao, M.; Rice, S. A. Theoretical study of the longitudinal density distribution in the liquid-vapor interface of a dilute ternary alloy: Pb and Sn in Ga. Rev. B 2005, 72, 942011-942017.
  48. Jiang, X.; Zhao, M.; Rice, S. A. Longitudinal Density Distribution in the Liquid-Vapor Interface of a Dilute Tl in Ga Alloy. Rev. B 2005, 71, 1042031.
  49. Zhao, M; Rice, S. A. Density Distribution in the Liquid-Vapor Interface of a Dilute Pb in Ga. Rev. B 2001, 63, 85409.
  50. Rice, S, A.; Zhao, M. Quantum Monte Carlo Simulation Studies of the Structure of the Liquid and the Liquid-Vapor of Sn and Pb. Phys. Chem. A 1999, 103, 10159-10165.
  51. Zhao, M.; Chekmarev, D.; Rice, S, A. Quantum Monte Carlo Simulations of the Structure in the Liquid-Vapor Interface of BiGa Binary Alloys. Chem. Phys. 1998, 108, 5055-5067.
  52. Rice, S, A.; Zhao, M. Self-Consistent Quantum Monte Carlo Simulations of the Structure of Liquid-Vapor Interface of a Eutectic Indium-Gallium Binary Alloy. Rev. B 1998, 57, 13501-13507.
  53. Zhao, M.; Chekmarev, D.; Cai, Z. H.; Rice, S, A. The Structure of Liquid Ga and the Liquid-Vapor Interface of Ga. Rev. E 1997, 56, 7033-7042.
  54. Rice, S.A.; Zhao, M.; Chekmarev, D. Theoretical Studies of the Structures of the Liquid-Vapor Interfaces of Metals and Binary Alloys in Microscopic Simulation of Interfacial Phenomena in Solids and Liquids, The Materials Research Society, Vol. 492, 1998, pp.3-14.
  55. Zhao, M.; Chekmarev, D.; Rice, S, A. Comparison of the Structure of the Liquid-Vapor Interfaces of Al, Ga, In and Tl. Chem. Phys. 1998, 109, 1959-1965.
  56. Chekmarev, D.; Zhao, M.; Rice, S. A. Structure of the Liquid-Vapor Interface of a Metal from a Simple Model Potential: the Corresponding States of the Alkali Metals. Chem. Phys. 1998, 109, 768-778.
  57. Chekmarev, D.; Zhao, M.; Rice, S. A. Computer Simulation Study of the Structure of the Liquid-Vapor Interface of Mercury at 20, 100, and 200o Phys. Rev. E 1999, 59, 479-491.
  58. Zhao, M.; Rice, S. A. The Structure of Liquid-Vapor Interface of a Gallium-Tin Binary Alloy. Chem. Phys. 1999, 111, 2181-2189.
  59. Li, F.; Zhao, M. Structure and Liquid-Vapor Interface of a Simple Metal. Theor. Phys. 1998, 29, 167.
  60. Li, D.; Rice, S. A. Melting of quasi-two-dimensional crystalline Pb supported on liquid Ga. Rev. E 2005, 72, 41506.
  61. Li, D.; Jiang, X.; Yang, B.; Rice, S. A. Phase transitions in the liquid-vapor interface of dilute alloys of Bi in Ga: New experimental studies. Chem. Phys. 2005, 122, 224702.
  62. Yang, B.; Li, D.; Rice, S. A. Two-dimensional freezing of Tl in the liquid-vapor interface of dilute Tl in a Ga alloy. Rev. B 2003, 67, 212103.
  63. Yang, B.; Li, D.; Rice, S. A. Structure of the liquid-vapor interface of a dilute ternary alloy: Pb and In in Ga. Rev. B 2003, 67, 54203.
  64. Li, D.; Yang, B.; Rice, S. A. Structure of the liquid-vapor interface of a dilute ternary alloy: Pb and Sn in Ga. Rev. B 2002, 65, 224202.
  65. Yang, B.; Li, D.; Huang, Z.; Rice, S. A. The structure of the liquid-vapor interface of a dilute Pb in Ga alloy. Rev. B 2000, 62, 13111.
  66. Lei, N.; Huang, Z.; Rice, S. A. Structure of the liquid-vapor interface of a Sn:Ga alloy. Chem. Phys. 1997, 107, 4051.
  67. Lei, N.; Huang, Z.; Rice, S. A. Surface segregation and layering in the liquid-vapor interface of a dilute bismuth:gallium alloy. Chem. Phys. 1996, 104, 4802.
  68. Lei, N.; Huang, Z.; Rice, S. A.; Grayce, C. In‐plane structure of the liquid-vapor interfaces of dilute bismuth: gallium alloys: X‐ray‐scattering studies. Chem. Phys. 1996, 105, 9615.
  69. Zhao, M.; Rice, S. A. Density Distribution in the Liquid Hg−Sapphire Interface. Phys. Chem. A 2011, 115(16), 3859–3866.
  70. Tamam, L.; Pontoni, D.; Hofmann, T.; Ocko, B. M.; Reichert H.; Deutsch, M. Atomic-Scale Structure of a Liquid Metal−Insulator Interface. J. Phys. Chem. Lett. 2010, 1, 1041.
  71. de Jong, A. E. F.; Vonk, V.; Boćkowski, M.; Grzegory, I.; Honkimäki, V.; Vlieg, E. Complex Geometric Structure of a Simple Solid-Liquid Interface: GaN(0001)-Ga. Rev. Lett. 2020, 124(8), 086101
  72. Men, H.; Fan, Z. Atomic ordering in liquid aluminium induced by substrates with misfits. Materi. Science 2014, 85, 1-7.
  73. Vonk, V.; Cremers, M.; de Jong, A.; Pintea, S.; Vlieg, E. Atomic layering and misfit-induced densification at the Si(111)/In solid–liquid interface. Surface Science 2014, 621, 69-76.