- Email: [email protected]
Structure and correlations of a liquid metal surface: Gold
Surface Science 211/212 (1989) 55-60 North-Holland, Amsterdam
55
STRUCTURE AND CORRELATIONS SURFACE: GOLD S. IARLORI,
P. CARNEVALI
IBM European Center for Scientific 00147 Rome, Italy
F. ERCOLESSI International Received
OF A LIQUID METAL
and Engineering
Computing,
Via Giorgione 159,
and E. TOSATTI
School for Advanced
28 June 1988; accepted
Studies, Strada Costiera II, 34014 Trieste, Italy for publication
7 September
1988
We have studied the static properties (surface tension, density tions) of a model liquid metal surface. Classical molecular dynamics with a many-body force scheme (the “glue” model) optimized tendency of this liquid surface to be more short-range ordered characterized in detail.
profile, structure and correlahas been used, in conjunction to describe gold. A general than the bulk is found, and
Liquid metal surfaces constitute an interesting and chailenging problem, from both the theoretical and experimental point of view [1,2]. With the advent of new surface scattering techniques, information on the microscopic structure of a liquid metal surface is beginning to appear [3]. Computer simulation is often of great help in understanding phenomena in inhomogeneous systems at the atomistic level, and a wealth of information has been indeed obtained for surfaces of pairwise liquids - in particular of LennardJones (LJ) systems [l]. However, the many-body nature of the metallic interaction has so far prevented realistic machine simulation studies of liquid metal surfaces. This gap can now be filled, owing to the recent development (see, e.g., ref. [4]) of empirical or semi-empirical many-body force models which describe metals very reasonably and do not require significant additional expenses in terms of computational effort with respect to pair potential models. We have studied the structure and static correlations of the liquid gold surface, using molecular dynamics (MD) and the classical many-body “glue” Hamiltonian developed by our group [5]. The main result is that, although devoid of long-range order, the metal surface has more short-range order (i.e., it is closer to crystalline) than the underlying bulk at the same temperature. By contrast in a simple liquid with pair-wise forces, the surface is well known to 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
be less ordered than the bulk [l]. The drivin, 0 force behind this anomalous phenomenon is the “glue” many-body force term. mimicking effectively the nonlocal free electron forces in the metal. These forces - which also give rise to reconstructions on gold crystal surfaces [S] ~ act to increase surface density in the attempt to bring the undercoordinated surface atoms closer to an optimal bulk-like coordination. In doing so. they also bring the surface closer to crystallization than the underlying bulk. The system under study is a large slab containing 10752 particles. with in-plane periodic boundary conditions, two free surfaces, an area of about 80 X X0 A’, and a thickness of about 30 A. The slab is liquified by starting from an initially crystalline atomic arrangement, and then bringing the temperature well above the glue model melting temperature. known from previous work [6] to be II,, = 1357 + 5 K (in very good agreement with the experimental value. T,liXp= 1336 K). Subsequently, the system is carefully equilibrated at each of the desired temperatures (3000. 2000, 1525 and 1360 K), and thcreafter time averages of various physical quantities are taken over long runs of about 30000 MD time steps, corresponding to about 0.2 x lOmy s. On this time scale, even at 3000 K no atom is observed to evaporate from the slab surface during the simulation runs. This is consistent with the experimental evaporation rate of gold [7], which ranges from - 50 atoms/s A,’ at 1360 K. to - 6 x 10’ atoms/s. A’ at 3000 K. We present here results concerning four points: (i) surface tension; (ii) density profile; (iii) surface in-plane structure; (iv) surface planar pair correlations and structure factor (liquid surface versus bulk). (i) Surf& tension. By using the usual mechanical definition for the surface tension y [S]. which applies equally well in presence of glue forces, we obtain. at T = 7;,,. y = 1130 dyne/cm. in excellent agreement (no adjustable parame[9]. However, we also ters) with the experimental value, ycXp = 1140 dyne/cm K. which is an order of magnitude smaller obtain d y/dT = - 0.03 dyne/cm. than the accepted experimental value, dycXp/dT = - 0.52 dyne/cm. K [9]. At the melting point, the dimensionless ratio k,Tp”‘/y. where p is the bulk density, is about 0.25. while in a LJ liquid near the triple point it is about 0.55. This indicates that capillary fluctuations should be rather modest. as also checked by direct examination of the surface structure. (ii) Densit,~~profile. The (s. ~3) averaged atom densities are shown for various temperatures in fig. 1. A general striking feature is the density peak near the this peak is further split into two “surface surface. At low temperatures, layers”. which appear to float above the liquid bulk. This double layer is verb similar to that found on the unreconstructed Au(l11) surface while undergoing “blocked surface melting” [6]. At temperatures close to 7;,,. the density profile clearly shows an oscillatory behaviour, gradually decaying inside the bulk region. This stratification is to be contrasted with the profile found in LJ which is believed to be monotonic even when liquid&vapour interfaces,
Fig. 1. Density profiles at (a) 1360 K, (b) 1525 K, (c) 2000 K. (d) 3000 K. Note the denser surface which is due to the necessity of raising the outermost atom coordination without “ bilayer”, increasing too much that of the atoms Just underneath.
capillary fluctuations are removed [lo]. As T is increased to 2000 K and above, the layering in our system also disappears - possibly owing to larger capillary fluctuations - and the two surface layers merge into a single broader hump. (iii) Surface in-plane structure. The sharpness in the density profile of fig. 1 a already justifies the suspicion that the outermost “surface layer” might be closer to crystalline. Fig. 2 shows a snapshot of the surface just above the melting temperature. While there is clearly no crystalline long-range order, the amount of triangular short-range order is impressive. (iv) Surface planar pair correlations and structure factor. We have calculated the planar pair correlation function g,,(r) between surface atoms and the corresponding planar structure factor S,,(k ,,), by time averaging over MD
J Fig. 2. Snapshot (instantaneous positions) of the surface. seen from the top, at T = 1360 K. The whole MD box is shown. A radius of 1.2 p\ is arbitrarily awgned to each particle. Islands of nearly triangular packing are evident.
trajectories. For this purpose, we have defined as “surface atoms” those which, at a given time, are not vertically shadowed by any other atom, once a conventional spherical radius ~~~is attached to each atom. By visually examining the behaviour of this algorithm on a few configurations, we have chosen r,) = 1 A as a good compromise between the need to catch all atoms at the surface (requiring a smaller Y”) and to reject all atoms in the liquid underneath (requiring a larger Y,,). As a consequence of this definition. g,,(r) is not defined below 2r,,, and S,,(k ,,) is not expected to be reliable above - 2a/2r;,. The results, compared with g(r) and S(k) for the bulk liquid as extracted from an independent zero-pressure bulk MD simulation using the same model, and from experimental tabulations [ll], are presented on fig. 3. g,,(r) is generally similar to the corresponding bulk g(r). except for three main differences: (1) the first peak is broader in R ,l(r) than in g(r), indicating a softening of atomic interactions at the surface; (2) oscillations decay more slowly with distance in g,,(r), indicating again an increase of short-range order. This same fact gives rise, particularly at the lower temperatures. to a stronger peak in S,,(k ,,) than in S(k), as also shown in fig. 3; (3) there is an overall compression of all structures of g,,(r) relative to g( r ). simply reflecting the higher planar surface density. Correspondingly, the peak position of
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Fig. 3. (a) Surface planar pair correlation function g,,(r) at T = 1360 K (sohd line) compared with the bulk pair correlation function g(r) from MD simulation at 1360 K (dashed line), and from experiment [ll] at 1423 K (dotted line). (b) Same as (a). at T = 2000 K (MD simulation) and 1973 K (experiment). (c) Surface planar structure factor S,, (k , ) at T = 1360 K (solid line) compared with the bulk structure factor S(k) from MD simulation at 1360 K (dashed line). and from experiment [ll] at 1423 K (dotted line). (d) Same as (c). at T = 2000 K (MD simulation) and 1973 K (experiment). r is given in A. k in A-‘. Note the strong evidence of increased surface short-range order. particularly at 1360 K.
S,,(k,,) is also slightly shifted. For small wave vectors. S,,(k ,,) is considerably larger than S(k) (whose limit for k + 0 is connected with the compressibility). In principle, a divergence like l/k: is expected for S ,,(k ,, --j 0), due to capillary wave fluctuations [12]. In our simulation, k, = 27r/(slab thickness) = 0.2 A-’ constitutes a natural cutoff where the slab behaves differently from a semi-infinite bulk. However, no tendency of S,, (k ,) to increase is observed when k ,, decreases down to k,, whereas in the liquid LJ surface a l/k: increase is already visible at comparably larger values of k ,, [13]. This fact is in line with the reduced extent of capillary fluctuations on our metal surface. In summary, we have presented first results of an MD study of the static properties of a model liquid gold surface. The anomalous tendency of this surface to exhibit more short-range order than the bulk has been described in detail. Further work on orientational correlations, diffusion and dynamics of this surface is currently in progress, and will be presented elsewhere [14].
60
S. Iarlori
et 01. / Static propertirs
of u lrquid nwtul
surf&r
References [l] [2] [3] [4] [S]
[6] [7] [8] [9] [lo] [ll] [12] [13] [14]
CA. Croxton. Statistical Mechanics of the Liquid Surface (Wiley. Chichester. 19X0). R. Evans, J. Phys. (Paris) C (1980) 775. See, e.g., B.N. Thomas, SW. Barton, F. Novak and S.A. Rice. J. Chem. Phys. 86 (1987) 1036. R.M. Nieminen, Phys. Scripta T19 (1987) 320. F. Ercolessi, A. Bartolini, M. Garofalo. M. Parrinello and E. Tosatti, Phys. Scripta T19 (1987) 399: F. Ercolessi, M. Parinello and E. Tosatti, Phil, Mag. A 58 (1988) 213. P. Carnevali. F. Ercolessi and E. Tosatti. Phys. Rev. B 36 (1987) 6701; Surface Sci. 189/190 (1987) 645. American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York. 1972) table 4k-23. O.K. Rice, Statistical Mechanics, Thermodynamics and Kinetics (Freeman, San Francisco, 1967) ch. 13. B.C. Allen. in: Liquid Metals Chemistry and Physics, Ed. S.Z. Beer (Dekker, New York, 1972). D.L. Heath and J.K. Percus, J. Statist. Phys. 49 (1937) 319. Y. Waseda, The Structure of Non-Crystalline Materials (McGraw-Hill, Toronto, 1980) appendix 8. M.S. Wertheim, J. Chem. Phys. 65 (1976) 2377. M.H. Kalos, J.K. Percus and M. Rao, J. Statist. Phys. 17 (1977) 111. S. Iarlori, P. Carnevali, F. Ercolessi and E. Tosatti, in preparation.
55
STRUCTURE AND CORRELATIONS SURFACE: GOLD S. IARLORI,
P. CARNEVALI
IBM European Center for Scientific 00147 Rome, Italy
F. ERCOLESSI International Received
OF A LIQUID METAL
and Engineering
Computing,
Via Giorgione 159,
and E. TOSATTI
School for Advanced
28 June 1988; accepted
Studies, Strada Costiera II, 34014 Trieste, Italy for publication
7 September
1988
We have studied the static properties (surface tension, density tions) of a model liquid metal surface. Classical molecular dynamics with a many-body force scheme (the “glue” model) optimized tendency of this liquid surface to be more short-range ordered characterized in detail.
profile, structure and correlahas been used, in conjunction to describe gold. A general than the bulk is found, and
Liquid metal surfaces constitute an interesting and chailenging problem, from both the theoretical and experimental point of view [1,2]. With the advent of new surface scattering techniques, information on the microscopic structure of a liquid metal surface is beginning to appear [3]. Computer simulation is often of great help in understanding phenomena in inhomogeneous systems at the atomistic level, and a wealth of information has been indeed obtained for surfaces of pairwise liquids - in particular of LennardJones (LJ) systems [l]. However, the many-body nature of the metallic interaction has so far prevented realistic machine simulation studies of liquid metal surfaces. This gap can now be filled, owing to the recent development (see, e.g., ref. [4]) of empirical or semi-empirical many-body force models which describe metals very reasonably and do not require significant additional expenses in terms of computational effort with respect to pair potential models. We have studied the structure and static correlations of the liquid gold surface, using molecular dynamics (MD) and the classical many-body “glue” Hamiltonian developed by our group [5]. The main result is that, although devoid of long-range order, the metal surface has more short-range order (i.e., it is closer to crystalline) than the underlying bulk at the same temperature. By contrast in a simple liquid with pair-wise forces, the surface is well known to 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
be less ordered than the bulk [l]. The drivin, 0 force behind this anomalous phenomenon is the “glue” many-body force term. mimicking effectively the nonlocal free electron forces in the metal. These forces - which also give rise to reconstructions on gold crystal surfaces [S] ~ act to increase surface density in the attempt to bring the undercoordinated surface atoms closer to an optimal bulk-like coordination. In doing so. they also bring the surface closer to crystallization than the underlying bulk. The system under study is a large slab containing 10752 particles. with in-plane periodic boundary conditions, two free surfaces, an area of about 80 X X0 A’, and a thickness of about 30 A. The slab is liquified by starting from an initially crystalline atomic arrangement, and then bringing the temperature well above the glue model melting temperature. known from previous work [6] to be II,, = 1357 + 5 K (in very good agreement with the experimental value. T,liXp= 1336 K). Subsequently, the system is carefully equilibrated at each of the desired temperatures (3000. 2000, 1525 and 1360 K), and thcreafter time averages of various physical quantities are taken over long runs of about 30000 MD time steps, corresponding to about 0.2 x lOmy s. On this time scale, even at 3000 K no atom is observed to evaporate from the slab surface during the simulation runs. This is consistent with the experimental evaporation rate of gold [7], which ranges from - 50 atoms/s A,’ at 1360 K. to - 6 x 10’ atoms/s. A’ at 3000 K. We present here results concerning four points: (i) surface tension; (ii) density profile; (iii) surface in-plane structure; (iv) surface planar pair correlations and structure factor (liquid surface versus bulk). (i) Surf& tension. By using the usual mechanical definition for the surface tension y [S]. which applies equally well in presence of glue forces, we obtain. at T = 7;,,. y = 1130 dyne/cm. in excellent agreement (no adjustable parame[9]. However, we also ters) with the experimental value, ycXp = 1140 dyne/cm K. which is an order of magnitude smaller obtain d y/dT = - 0.03 dyne/cm. than the accepted experimental value, dycXp/dT = - 0.52 dyne/cm. K [9]. At the melting point, the dimensionless ratio k,Tp”‘/y. where p is the bulk density, is about 0.25. while in a LJ liquid near the triple point it is about 0.55. This indicates that capillary fluctuations should be rather modest. as also checked by direct examination of the surface structure. (ii) Densit,~~profile. The (s. ~3) averaged atom densities are shown for various temperatures in fig. 1. A general striking feature is the density peak near the this peak is further split into two “surface surface. At low temperatures, layers”. which appear to float above the liquid bulk. This double layer is verb similar to that found on the unreconstructed Au(l11) surface while undergoing “blocked surface melting” [6]. At temperatures close to 7;,,. the density profile clearly shows an oscillatory behaviour, gradually decaying inside the bulk region. This stratification is to be contrasted with the profile found in LJ which is believed to be monotonic even when liquid&vapour interfaces,
Fig. 1. Density profiles at (a) 1360 K, (b) 1525 K, (c) 2000 K. (d) 3000 K. Note the denser surface which is due to the necessity of raising the outermost atom coordination without “ bilayer”, increasing too much that of the atoms Just underneath.
capillary fluctuations are removed [lo]. As T is increased to 2000 K and above, the layering in our system also disappears - possibly owing to larger capillary fluctuations - and the two surface layers merge into a single broader hump. (iii) Surface in-plane structure. The sharpness in the density profile of fig. 1 a already justifies the suspicion that the outermost “surface layer” might be closer to crystalline. Fig. 2 shows a snapshot of the surface just above the melting temperature. While there is clearly no crystalline long-range order, the amount of triangular short-range order is impressive. (iv) Surface planar pair correlations and structure factor. We have calculated the planar pair correlation function g,,(r) between surface atoms and the corresponding planar structure factor S,,(k ,,), by time averaging over MD
J Fig. 2. Snapshot (instantaneous positions) of the surface. seen from the top, at T = 1360 K. The whole MD box is shown. A radius of 1.2 p\ is arbitrarily awgned to each particle. Islands of nearly triangular packing are evident.
trajectories. For this purpose, we have defined as “surface atoms” those which, at a given time, are not vertically shadowed by any other atom, once a conventional spherical radius ~~~is attached to each atom. By visually examining the behaviour of this algorithm on a few configurations, we have chosen r,) = 1 A as a good compromise between the need to catch all atoms at the surface (requiring a smaller Y”) and to reject all atoms in the liquid underneath (requiring a larger Y,,). As a consequence of this definition. g,,(r) is not defined below 2r,,, and S,,(k ,,) is not expected to be reliable above - 2a/2r;,. The results, compared with g(r) and S(k) for the bulk liquid as extracted from an independent zero-pressure bulk MD simulation using the same model, and from experimental tabulations [ll], are presented on fig. 3. g,,(r) is generally similar to the corresponding bulk g(r). except for three main differences: (1) the first peak is broader in R ,l(r) than in g(r), indicating a softening of atomic interactions at the surface; (2) oscillations decay more slowly with distance in g,,(r), indicating again an increase of short-range order. This same fact gives rise, particularly at the lower temperatures. to a stronger peak in S,,(k ,,) than in S(k), as also shown in fig. 3; (3) there is an overall compression of all structures of g,,(r) relative to g( r ). simply reflecting the higher planar surface density. Correspondingly, the peak position of
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Fig. 3. (a) Surface planar pair correlation function g,,(r) at T = 1360 K (sohd line) compared with the bulk pair correlation function g(r) from MD simulation at 1360 K (dashed line), and from experiment [ll] at 1423 K (dotted line). (b) Same as (a). at T = 2000 K (MD simulation) and 1973 K (experiment). (c) Surface planar structure factor S,, (k , ) at T = 1360 K (solid line) compared with the bulk structure factor S(k) from MD simulation at 1360 K (dashed line). and from experiment [ll] at 1423 K (dotted line). (d) Same as (c). at T = 2000 K (MD simulation) and 1973 K (experiment). r is given in A. k in A-‘. Note the strong evidence of increased surface short-range order. particularly at 1360 K.
S,,(k,,) is also slightly shifted. For small wave vectors. S,,(k ,,) is considerably larger than S(k) (whose limit for k + 0 is connected with the compressibility). In principle, a divergence like l/k: is expected for S ,,(k ,, --j 0), due to capillary wave fluctuations [12]. In our simulation, k, = 27r/(slab thickness) = 0.2 A-’ constitutes a natural cutoff where the slab behaves differently from a semi-infinite bulk. However, no tendency of S,, (k ,) to increase is observed when k ,, decreases down to k,, whereas in the liquid LJ surface a l/k: increase is already visible at comparably larger values of k ,, [13]. This fact is in line with the reduced extent of capillary fluctuations on our metal surface. In summary, we have presented first results of an MD study of the static properties of a model liquid gold surface. The anomalous tendency of this surface to exhibit more short-range order than the bulk has been described in detail. Further work on orientational correlations, diffusion and dynamics of this surface is currently in progress, and will be presented elsewhere [14].
60
S. Iarlori
et 01. / Static propertirs
of u lrquid nwtul
surf&r
References [l] [2] [3] [4] [S]
[6] [7] [8] [9] [lo] [ll] [12] [13] [14]
CA. Croxton. Statistical Mechanics of the Liquid Surface (Wiley. Chichester. 19X0). R. Evans, J. Phys. (Paris) C (1980) 775. See, e.g., B.N. Thomas, SW. Barton, F. Novak and S.A. Rice. J. Chem. Phys. 86 (1987) 1036. R.M. Nieminen, Phys. Scripta T19 (1987) 320. F. Ercolessi, A. Bartolini, M. Garofalo. M. Parrinello and E. Tosatti, Phys. Scripta T19 (1987) 399: F. Ercolessi, M. Parinello and E. Tosatti, Phil, Mag. A 58 (1988) 213. P. Carnevali. F. Ercolessi and E. Tosatti. Phys. Rev. B 36 (1987) 6701; Surface Sci. 189/190 (1987) 645. American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York. 1972) table 4k-23. O.K. Rice, Statistical Mechanics, Thermodynamics and Kinetics (Freeman, San Francisco, 1967) ch. 13. B.C. Allen. in: Liquid Metals Chemistry and Physics, Ed. S.Z. Beer (Dekker, New York, 1972). D.L. Heath and J.K. Percus, J. Statist. Phys. 49 (1937) 319. Y. Waseda, The Structure of Non-Crystalline Materials (McGraw-Hill, Toronto, 1980) appendix 8. M.S. Wertheim, J. Chem. Phys. 65 (1976) 2377. M.H. Kalos, J.K. Percus and M. Rao, J. Statist. Phys. 17 (1977) 111. S. Iarlori, P. Carnevali, F. Ercolessi and E. Tosatti, in preparation.