Abstract
The generalized Thomson formula T m = T (∞) m (1-δ)R for the melting point of small objects T m has been analyzed from the viewpoint of the thermodynamic theory of similarity, where R is the radius of the particle and T (∞) m is the melting point of the corresponding large crystal. According to this formula, the parameter δ corresponds to the value of the radius of the T m (R -1) particle obtained by the linear extrapolation of the dependence to the melting point of the particle equal to 0 K. It has been shown that δ = αδ0, where α is the factor of the asphericity of the particle (shape factor). In turn, the redefined characteristic length δ0 is expressed through the interphase tension σ sl at the boundary of the crystal with its own melt, the specific volume of the solid phase v s and the macroscopic value of the heat of fusion λ∞:δ0 = 2σ sl v s /λ∞. If we go from the reduced radius of the particle R/δ to the redefined reduced radius R/r 1 or R/d, where r 1 is the radius of the first coordination shell and d ≈r 1 is the effective atomic diameter, then the simplex δ/r 1 or δ/d will play the role of the characteristic criterion of thermodynamic similarity. At a given value of α, this role will be played by the simplex Estimates of the parameters δ0 and δ0/d have been carried out for ten metals with different lattice types. It has been shown that the values of the characteristic length δ0 are close to 1 nm and that the simplex δ0/d is close to unity. In turn, the calculated values of the parameter δ agree on the order of magnitude with existing experimental data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. P. Filippov, Similarity of the Properties of Substances (Izd-vo Mosk. Gos. Univ., Moscow, 1978).
L. M. Shcherbakov, V. M. Samsonov, V. A. Lavrov, and O. A. Rybal’chenko, “Principles of similarity in thermodynamics of microheterogeneous systems: 1. Disperse systems,” Colloid. J. 61, 120–125 (1999).
H. Hori, T. Teranishi, M. Taki, S. Yamada, M. Miyake, and Y. Yamamoto, “Magnetic properties of nano-particles of Au, Pd and Pd/Ni alloys,” J. Magn. Magn. Mater. 226–230, 1910–1911 (2001).
Yu. I. Petrov, Physics of Small Particles (Nauka, Moscow, 1982) [in Russian].
L. M. Alabushev, V. B. Geronimus, L. M. Minkevich, and B. A. Shekhovtsov, Theories of Resemblance and Dimension. Simulation (Vysshaya Shkola, Moscow, 1968) [in Russian].
W. Thomson, “The equilibrium of vapor at a curved surface of liquid,” Philos. Mag. 42, 448–452 (1871).
K. F. Peters, J. B. Cohen, and Y.-W. Chung, “Melting of Pb nanocrystals,” Phys. Rev. B: Condens. Matter Mater. Phys. 57, 13430–13438 (1998).
A. I. Gusev, Nanomaterials, Nanostructures, Nanotechnologies (Fizmatlit, Moscow, 2005) [in Russian].
N. Eustathopoulos, “Energetics of solid/liquid interfaces of metals and alloys,” Int. Met. Rev. 28, 189–210 (1983).
M. A. Shebzukhova, Z. A. Shebzukhov, and A. A. Shebzukhov, “Interfacial tension of a crystalline nanoparticle in the liquid mother phase in a one-component metallic system.” Phys. Solid State 54, 185–193 (2012).
B. M. Patterson, K. M. Unruh, and S. I. Shah, “Melting and freezing behavior of ultrafine granular metal films,” Nanostruct. Mater. 1, 65–70 (1992).
W. H. Qi and M. P. Wang, “Size and shape dependent lattice parameters of metallic nanoparticles,” Mater. Chem. Phys. 88, 280–284 (2004).
J. H. Rose, J. Ferrente, and J. R. Smith, “Universal binding energy curves for metals and bimetallic interfaces,” Phys. Rev. Lett. 47, 675–678 (1981).
F. Guinea, J. H. Rose, J. R. Smith, and J. Ferrante, “Scaling relations in the equation of state, thermal expansion, and melting of metals,” Appl. Phys. Lett. 44, 53–55 (1984).
P. E. Strebeiko, “Effect of refinement on temperature of transition,” Doctoral Dissertation (Inst. Obshch. Neorg. Khim. Akad. Nauk SSSR, Moscow, 1939).
L. M. Shcherbakov, “On the heat of sublimation of small crystals,” Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 77–83 (1959).
Skripov, V. P. and Koverda, V.P., Spontaneous Crystallization of Supercooled Liquids (Nauka, Moscow, 1984), pp. 104–107.
V. P. Koverda and V. N. Skokov, “Effect of fluctuations and nonequilibrium faceting on the melting of small metallic crystals,” Fiz. Met. Metalloved. 51, 1238–1244 (1981).
M. Blackman and J. R. Sambles, “Melting of very small particles during evaporation at constant temperature,” Nature 226, 938–947 (1970).
P. Buffat and J. Borel, “Size effect on the melting temperature of gold particles,” Phys. Rev. A 13, 2287–2298 (1976).
V. N. Skokov, V. P. Koverda, and V. P. Skripov, “Liquid–crystal phase transition in gallium island films,” Fiz. Tverd. Tela 24, 562–567 (1982).
V. P. Koverda, V. N. Skokov, and V. P. Skripov, “Crystallization of small particles in island films of tin, lead, and bismuth,” Kristallografiya 27, 358–362 (1982).
A Handbook of Physical Quantities, Ed. by I. S. Grigor’ev and E. Z. Meilikhov (Energoatomizdat, Moscow, 1991), pp. 358–362 [in Russian].
I. D. Morokhov, L. I. Trusov, and V. N. Lapovok, Physical Phenomena in Ultradispersed Media (Energoatomizdat, Moscow, 1984).
S. L. Gafner, L. V. Redel’, Zh. V. Goloven’ko, Yu. Ya. Gafner, V. M. Samsonov, and S. S. Kharechkin, “Structural transitions in small nickel clusters,” J. Exp. Theor. Phys. Lett. 89, 364–369 (2009).
F. Cleri and V. Rosato, “Tight-binding potentials for transition metals and alloys,” Phys. Rev. B: Condens. Matter 40, 22–33 (1993).
N. N. Medvedev, Voronoi–Delaunay Method in Study of the Structure of Noncrystalline Systes (Ross. Akad. Nauk, Sibir. Otd., Novosibirsk, 2000) [in Russian].
W. Polak and A. Patrykiejew, Local structures in medium-sized Lennard-Jones clusters: Monte Carlo simulations,” Phys. Rev. B: Condens. Matter Mater. Phys. 67, 115402 (2003).
V. M. Samsonov, S. S. Kharechkin, and R. P. Barbasov, “Comparative molecular dynamics study of nanocrystallization processes in one-component and binary systems,” Bull. Russ. Acad. Sci.: Phys. 70, 1143–1147 (2006).
S. Sugano and H. Koizumi, Microcluster Physics (Springer-Verlag, Heidelberg, 1998).
V. I. Nizhenko, “Density of liquid metals and its temperature dependence,” in Methods of Study and Properties of Contacting Phase Interfaces (Naukova Dumka, Kiev, 1977), pp. 125–163 [in Russian].
A. B. Alchagirov, B. B. Alchagirov, T. M. Taova, and K. B. Khokonov, “Surface energy and surface tension of solid and liquid metals. Recommended Values,” Trans. of Joining and Welding Res. Inst. 30, 287–291 (2001).
H. Y. Kai, “Nanocrystalline materials. A study of their preparation and characterization,” PD Thesis (Univ. van Amsterdam, Netherlands, Amsterdam, 1993).
C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976; Nauka, Moscow, 1978).
A. Gordon and R. Ford, in The Chemist’s Companion, A Handbook of Practical Data. Techniques and References (Wiley, New York, 1972; Mir, Moscow, 1976).
K. Dick, T. Dhanasekaran, Z. Xhang, and D. Meisel, “Size-dependent melting of silica-encapsulated gold nanoparticles,” J. Am. Chem. Soc. 124, 2312–2317 (2002).
T. B. David, Y. Lereah, G. Deutsher, R. Kofman, and P. Cheyssac, “Solid–liquid transition in ultra-fine lead particles,” Philos. Mag. A 71, 1135–1143 (1995).
R. Kofman, P. Cheyssac, Y. Lereach, and A. Stella, “Melting of clusters approaching 0D,” Eur. Phys. J. D 9, 441–444 (1999).
V. P. Skripov, V. P. Koverda, and V. N. Skokov, “Size effect on melting of small particles,” Phys. Status Solidi 66, 109–118 (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.M. Samsonov, S.A. Vasilyev, A.G. Bembel, 2016, published in Fizika Metallov i Metallovedenie, 2016, Vol. 117, No. 8, pp. 775–781.
Rights and permissions
About this article
Cite this article
Samsonov, V.M., Vasilyev, S.A. & Bembel, A.G. Size dependence of the melting temperature of metallic nanoclusters from the viewpoint of the thermodynamic theory of similarity. Phys. Metals Metallogr. 117, 749–755 (2016). https://doi.org/10.1134/S0031918X16080135
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0031918X16080135