Anisotropy of Melting for Cubic Metals Superheated Liquids Containing

Nature Vol. 275 21 September 1978 203

3. Patterson, A. L. & Kasper, J. S. International Tables for X-ray 0-,stallograplty 2, 342 (Kynoch Press. London, I 967). Table 1 Some thermal properties of close-packed cubic metals 4. Sanden, J. V. Acta cr;sta/logr. AU, 427 (I 968). 5. Nowick, A. S. & Mader, S. R. I.B.M. JI, September-November, 358 (1965). Bulk component Surface component 6. Visscber, W. M. & Bolsterli. M. Nature 239,504 (1972). 7 . Darraiih, P. J., Gaskin, A. J., Terrell, B. C. & Sanders, J. V. Naturr 209, 13 (1966). Debye 8. Snook, I. & van Megen, W. J. Colloid Inttrfac, Sci. 51, 47 (1976). Debye Melting temperaturet Melting point§ 9. Love, J. D. J. chem. Soc., Faraday Trans. II 73,669 (1977). temperature* pointt Bos(K) TMs (K) 10. Hou, R., Healy, T. W. & Fuerstenau, D. W. TtanJ. Fa,aday Soc. 62, 1638 (1966). Metal Boa(K) TMa(K) (110) (100) (111) (110) (100) (111) 11. Kittel, C. Introduction 111 Solid State Physics, 585 (Wiley, New York 1966). 12. Hachisu, S., Kose. A., Kobayashi, Y. & Takano, K. J. Colloid Inttrfac• Sci. 55,499 (1976). Ag 226.5 1,233 142 142 147 887 906 966 Cu 334.4 1,356 191 192 196 809 836 866 Ni 476.0 1,728 225 225 230 706 722 750 Pb 105.3 600 54 56 58 289 317 338 Anisotropy of melting for cubic metals Pt 238.4 2,042 167 166 174 1,837 1,851 2,023

KINETIC studies of superheating of metals1.2 have shown that *Ref 13. tRef. 14. tRef. 12. surfaces play an important part in initiating melting. Recent §Estimated using equation (2). work on the anisotropy of vapour pressure3 suggests a strong dependence of melting on crystallographic orientation. There apparently indicates the dependence of melting point on the 4 7 are models of melting - which indicate that surfaces may density of packing of atoms in a particular crystal plane. One can become disordered {that is, lose their long-range order) at thus conclude that a maximum vibrational instability is asso temperatures below the bulk-melting point. These temperatures ciated with the least densely packed plane. This conclusion is 5 16 7 are characteristics of the crystallographic planes of a metal. consistent with the theories of melting • · 1 • Following the work 9 Based on simple considerations, we attempt here to estimate the of Maradudin et al. , one would expect the amplitude of atomic melting temperatures of major low-index planes for close vibrations to be maximal for planes with the least atomic packing packed cubic metals. The results are discussed in terms of density. It is known that the Debye temperature is closely low-energy electron diffraction (LEED) study and current related to the amplitude of vibration. As a result, Bos is expected theories of melting. to be lower with less closely packed planes; this is apparent in 5 12 On the basis of Lindemann's melting criterion , one can Jackson's results . Because, according to Lindemann's cri express in terms of harmonic approximation the relation terion, the melting point of a metal is directly proportional to the between Debye temperature 80 , melting point TM, atomic Debye temperature (equation 1), the lowest melting point is 2 weight Mand mean-square amplitude of atomic vibration (u ) expected for a plane with the lowest surface De bye temperature, as 8 that is, with least close packing of atoms, as indeed has been indicated in the present work. On the basis of the estimated (1) values of Table 1, a satisfactory quantitative relation has been 2 where C, a constant, is equal to 9h / k 8 with h denoting the obtained between T Ms(hkl) and work function of the single reduced Planck's constant and k 8 the Boltzmann's constant. It crystal faces of a bare metal possessing close-packed cubic 18 has been well established through both experimental measure structure . 2 ments and theoretical calculations that (u ) is much larger at a B. CHATIERJEE 9 surface than in the bulk • Consequently, the degree of anhar The British Aluminium Co. Ltd, monicity should be much greater at the surface than in the bulk. Chalfont Park, From the various methods of studying surface anharmonicity1°, Gerrards Cross, the simple force constant method, which is based on anharmonic Buckinghamshire, UK perturbation theory, is used here. Only the normal component of mean-square displacement denoted by (ui) is considered, as Received 28 July; accepted 11 August 1978. it is this parameter which is generally greater than the tangential I. Cormia, R. L., Mackenzie, J. D. & Turnbull, D. J. appl. Phys. 34, 2239 (I 963). component. The ratio of this value for surface (S) and bulk (B) 2. Kass, M. & Magun. S. Z. Krista/log,. Krysrall11eom. 116, 354 (1961). 2 3. Taskaev. I. P. Izv. Akad. Nauk SSSR Met. 5, 78 (I 977). that is, (ui )5 /(u ) 8 for various crystallographic planes, is only 4. Broughton. J. 0. & Woodcock, L. V. J. phys. C: Solid State Phys. 11, 2743 (1978). known for close-packed cubic metals as 1.87, 1.83 and 1.86 for 5. Lindemann, F. A. Phys. Z. 11,609 (1910). 11 6. Gilvarry, J. J. Phys. Rev. 102, 308 (1956). (100), (110) and (111) planes, respectively • An analysis of 7. Henrion, J. & Rhead, G. E. Surface Sci. 29, 20 (1972). equation (1) in terms of surface and bulk components followed 8. Pines. D. Elementary Excitations in Solids. 34 (W. A . Benjamin, New York, 1963). would give 9. Maradudin, A . A., Montroll, E. W., Weiss, G. H. & lpatova, I. P. Theory of Lattic, by rearrangement Dynamics in the Harmonic Approximation, 59S (Academic, New York, 1971). 2 10. Lagally, M . G. in Surface Physics of Materials, Vol. 2 (ed. Blackley, J. M. B.)419 (Academic, T Ms(hkl) = [(ui)s/(u )all Bos(hkl)/ BoaJ2TMB (2) New York, I 975). I l. Allen, R. E. & de Wene, F. M. Ph ys. Rev. 188, I 320 (1969). A subsequent estimation of T Ms(hkl) for planes (hkl) is pre 12 . Jackson, D. P. Surface Sci. 43,431 (1974). sented in Table 1 for metals for which the various parameters of 13. Varma, J. K. D. & Aggarwal, M . D. J. appl. Phys. 46, 2841 (1975). 12 14 14. Smithells. C. J. Metals Reference Book, 5th edn, 186 (Butterworth, London, 1976). equation (2) are known - • IS. Couchman, P.R. & lesser, W. A. Phil. Mag. 35, 787 (1977). The present results on T Ms(hkl) are considered to be fairly 16. Kuhlmann-Wilsdorf, D. Phys. Rev. A140, 1599 (1965). melting temperature for 17. G6recki, T. Z. Mttal/kunde 65,426 (1974). reasonable. For example, the surface 18. Chatterjee, B. Phil. Mag. (submitted). Pb, which varies from (111) to (110) planes over a range of 49 K, 7 is comparable with the 40 K obtained from LEED studies • It is, of course, evident from equation (2) that T Ms values would 2 depend on the choice of the ratio (ui)s/(u ) 8 and on the Superheated liquids adequate calculation of surface Debye temperature Bos(hkl) 12 which is primarily based on summation of Morse potentials • containing suspended particles It has been pointed out15 that Lindemann's melting hypoth esis consistently predicts the absolute melting temperature too THE superheat temperature limits of particle-free liquids have high. It is argued in terms of amplitude of vibration of surface been previously investigated, notable by Skripov', Blander et 3 4 atoms that the 'critical amplitude' (for melting) will be reached al.2, Apfe1 and Porteous • Examinations of the homogeneous at the surface before the interior. This view has been confirmed nucleation of liquid pentane have found the experimental limit by LEED results 7 and molecular dynamics studies4. The results of superheat temperature to be 146.5°C, which agrees well with in Table 1, with T MB being consistently higher than T MS, there the theoretical limit of superheat of 14 7. 7°C calculated by fore agree with the above concept. It is also evident from Table 1 Blander et al. 2 using the theories of Doring5, Volmer6 and that T Ms decreases from (111) to (110) planes. Such a trend Kagan 7. Studies of superheating pure liquids have stressed the