Abstract
This study investigates the effect of size on bulk modulus and its related parameters, including melting temperature and mass density based on the ratio number of surface atoms to that of its internal. The equation of bulk modulus in the bulk state B(∞) is modified to include the related size-dependent parameters without any adjustable parameter, and is applied to Si nanocrystals. The bulk modulus B(r) decreases from 9.8 × 1010 N m2 for the bulk state to 5.93 × 1010 N m2 for a 5 nm diameter of Si nanoparticles. An inherent relation between bulk modulus and change of the lattice parameter in nanocrystals obtained from the variation in the surface to volume ratio, this leads to increase in the mean bond length. The effect of mass density and melting temperature on bulk modulus are also discussed. Calculated results for bulk modulus are verified by experimental as well as the available computer simulation data.
Similar content being viewed by others
References
Sun C Q 2009 Thermo-mechanical behavior of low-dimensional systems: The local bond average approach. Prog. Mater. Sci. 54: 179–307
Gleiter H 2000 Nanostructured materials: basic concepts and microstructure. Acta Materialia 48: 1–29
Zhao M and Jiang Q 2010 Size effect on thermal properties in low-dimensional materials. Key Engineering Materials, Trans Tech Publications, 189–218
Zhang Z, Zhao M and Jiang Q 2001 Melting temperatures of semiconductor nanocrystals in the mesoscopic size range. Semicond. Sci. Technol. 16:L33
Qi W 2005 Size effect on melting temperature of nanosolids. Physica B: Condens. Matter. 368: 46–50
Zhu Y, Lian J and Jiang Q 2009 Modeling of the melting point, Debye temperature, thermal expansion coefficient, and the specific heat of nanostructured materials. J. Phys. Chem. C 113: 16896–16900
Lu H, Li P, Cao Z and Meng X 2009 Size-, shape-, and dimensionality-dependent melting temperatures of nanocrystals. J. Phys. Chem. C 113: 7598–7602
Hou M, Elazzaoui M, Pattyn H, Verheyden J, Koops, G and Zhang G 2000 Growth and lattice dynamics of Co nanoparticles embedded in Ag: a combined molecular-dynamics simulation and Mössbauer study. Phys. Rev. B 62: 5117
Yang C, Xiao M, Li W and Jiang Q 2006 Size effects on Debye temperature, Einstein temperature, and volume thermal expansion coefficient of nanocrystals. Solid State Communications 139: 148–152
Abdullah B J, Omar M S and Jiang Q 2017 Size effects on cohesive energy, Debye temperature and lattice heat capacity from first-principles calculations of Sn nanoparticles. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 1-4
Kittel C 2005 Introduction to solid state physics. Wiley, Hoboken, 25
Koc H, Mamedov A M, Deligoz E and Ozisik H 2012 First principles prediction of the elastic, electronic, and optical properties of Sb2S3 and Sb2Se3 compounds. Solid State Sci. 14: 1211–1220
Gacem A, Doghmane A and Hadjoub Z 2011 Quantification the Effect of the Thickness of Thin Films on their Elastic Parameters. Adv. Mater. Res. Trans. Tech. Pub. 93–96
Qiao Z, Latz R and Mergel D 2004 Thickness dependence of In2O3: Sn film growth. Thin Solid Films 466: 250–258
Safaei A 2012 Size-dependent mass density of nanocrystals. Nano 7: 1250009
Lovell S and Rollinson E 1968 Density of thin films of vacuum evaporated metals. Nature 218: 1179–1180
Gu M, Sun C Q, Chen Z, Yeung T A, LI S, Tan C and Nosikn V 2007 Size, temperature, and bond nature dependence of elasticity and its derivatives on extensibility, Debye temperature, and heat capacity of nanostructures. Phys. Rev. B 75: 125403
Zhao M and Jiang Q 2004 Melting and surface melting of low-dimensional in crystals. Solid State Commun. 130: 37–39
Liang L and Li B 2006 Size-dependent thermal conductivity of nanoscale semiconducting systems. Phys. Rev. B 73: 153303
Avramov I and Michailov M 2008 Specific heat of nanocrystals. J. Phys. Condens. Matter. 20: 295224
Omar M 2012 Models for mean bonding length, melting point and lattice thermal expansion of nanoparticle materials. Mater. Res. Bull. 47: 3518–3522
Omar M 2007 Lattice thermal expansion for normal tetrahedral compound semiconductors. Mater. Res. Bull. 42: 319–326
Jiang Q, Shi H and Zhao M 1999 Melting thermodynamics of organic nanocrystals. J. Chem. Phys. 111: 2176–2180
Zhang Z, Li J and Jiang Q 2000 Modelling for size-dependent and dimension-dependent melting of nanocrystals. J. Phys. D Appl. Phys. 33: 2653
Shen T D, Zhang J and Zhao Y 2008 What is the theoretical density of a nanocrystalline material? Acta Materialia 56: 3663–3671
Abdullah B J, Jiang Q and Omar M S 2016 Effects of size on mass density and its influence on mechanical and thermal properties of ZrO2 nanoparticles in different structures. Bull. Mater. Sci. 39: 1295–1302
Lam P K, Cohen M L and Martinez G 1987 Analytic relation between bulk moduli and lattice constants. Phys. Rev. B 35: 9190–9194
Bievere P D, Valkiers S, Peiser S, Becker P, Ludicke F, Spieweck F and Stumpel J 1995 A more accurate value for the Avogadro constant. IEEE Trans. Instrum. Meas. 44: 530–532
Goldstein A N 1996 The melting of silicon nanocrystals: Submicron thin-film structures derived from nanocrystal precursors. Appl. Phys. A 62: 33–37
Kang K and Cai W 2010 Size and temperature effects on the fracture mechanisms of silicon nanowires: Molecular dynamics simulations. Int. J. Plast. 26: 1387–1401
Wolf H F 1971 Semiconductors. Wiley, Hoboken, 128
Cherian R, Gerard C, Mahadevan P, Cuong N T and Maezono R 2010 Size dependence of the bulk modulus of semiconductor nanocrystals from first-principles calculations. Phys. Rev. B 82: 235321
Zhu Y, Zheng W and Jiang Q 2009 Modeling lattice expansion and cohesive energy of nanostructured materials. Appl. Phys. Lett. 95: 083110
Siegel J, Lyutakov O, Rybka V, Kolská Z and Svorcík V 2011 Properties of gold nanostructures sputtered on glass. Nanoscale Res. Lett. 6: 96
Opalinska A, Malka I, Dzwolak W, Chudoba T, Presz A and Lojkowski W 2015 Size-dependent density of zirconia nanoparticles. Beilstein J. Nanotechnol. 6: 27–35
Banerjee R, Sperling E A, Thompson G B, Fraser H L, Bose S and Ayyub P 2003 Lattice expansion in nanocrystalline niobium thin films. Appl. Phys. Lett. 82: 4250–4252
Chattopadhyay P P, Nambissan P M G, Pabi S K and Manna I 2001 Polymorphic bcc to fcc transformation of nanocrystalline niobium studied by positron annihilation. Phys. Rev. B 63: 054107
Acknowledgement
The financial support from Ministry of Higher Education and Scientific Research- Salahaddin-Erbil University with the cooperation of Jilin University are acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdullah, B.J., Omar, M.S. & Jiang, Q. Size dependence of the bulk modulus of Si nanocrystals. Sādhanā 43, 174 (2018). https://doi.org/10.1007/s12046-018-0956-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12046-018-0956-1