Abstract
In this paper, we analyse the melting of a spherically symmetric nanoparticle, using a continuum model which is valid down to a few nanometres. Melting point depression is accounted for by a generalised Gibbs–Thomson relation. The system of governing equations involves heat equations in the liquid and solid, a Stefan condition to determine the position of the melt boundary and the Gibbs-Thomson equation. This system is simplified systematically to a pair of first-order ordinary differential equations. Comparison with the solution of the full system shows excellent agreement. The reduced system highlights the effects that dominate the melting process and specifically that rapid melting is expected in the final stages, as the radius tends to zero. The results agree qualitatively with limited available experimental data.
Similar content being viewed by others
References
Abragall P, Nguyen NT (2009) Nanofluidics, 1st edn. Artech House,London
Ahmad F, Pandey A, Herzog A, Rose J, Gerba C, Hashsham S (2012) Environmental applications and potential health implications of quantum dots. J Nanopart Res 14(1038):doi:10.1007/s11,051-012-1038-7
Alexiades V, Solomon A (1993) Mathematical modelling of rfreezing and melting processes, 1st edn. Hemisphere Publishing Corporation, Washington
Bergese P, Colombo I, Gervasoni D, Depero L (2004) Melting of nanostructured drugs embedded into a polymeric matrix. J Phys Chem B 108:15,488–15,493
Buffat P, Borel JP (1976) Size effect on the melting temperature of gold particles. Phys Rev A 13(6):2287–2297
David TB, Lereah Y, Deutscher G, Kofman R, Cheyssac P (1995) Solid-liquid transition in ultra-fine lead particles. Philos Mag A 71(5):1135–1143
Davis S (2001) Theory of Solidification. Cambridge University Press, Cambridge
Debenedetti P (1996) Metastable liquids, concepts and principles. Princeton University Press, Princeton
Evans JD, King JR (2000) Asymptotic results for the Stefan problem with kinetic undercooling. QJl Mech App Math 53:449–473
Faivre C, Bellet D, Dolino G (1999) Phase transitions of fluids confined in porous silicon: A differential calorimetry investigation. Eur Phys J B 7(1):19–36
Ghosh P, Han G, De M, Kim C, Rotello V (2008) Gold nanoparticles in delivery applications. Adv Drug Deliv Rev 60:1307–1315
Govorov AO, Zhang W, Skeini T, Richardson H, Lee J, Kotov NA (2006) Gold nanoparticle ensembles as heaters and actuators: melting and collective plasmon resonances. Nanoscale Res Lett 1(1):84–90
Guisbiers G, Kazan M, Overschelde OV, Wautelet M, Pereira S (2008) Mechanical and thermal properties of metallic and semiconductive nanostructures. J Phys Chem C 112:4097–4103
Gulseren O, Ercolessi F, Tosatti E (1995) Premelting of thin wires. Phys Rev B 51(11):7377–7380
Hill JM (1987) One-Dimensional Stefan problems: an introduction, 1st edn. Longman Scientific & Technical, New York
Hinch E (2000) Perturbation methods. Cambridge University Press, Cambridge
Karmakar S, Kumar S, Rinaldi R, Maruccio G (2011) Nano-electronics and spintronics with nanoparticles. J Phys: Conf Ser 292(012002):doi:10.1088/1742-6596/292/1/012,002
Kofman R, Cheyssac P, Lereah Y, Stella A (1999) Melting of clusters approaching 0D. Eur Phys J D 9(1–4):441–444
Koga K, Ikeshoj T, Sugawara KI (2004) Size- and temperature-dependent structural transitions in gold nanoparticles. Phys Rev Lett 92(11):doi:10.1103/PhysRevLett.92.115,507
Kuo CL, Clancy P (2005) Melting and freezing characteristics and structural properties of supported and unsupported gold nanoclusters. J Phys Chem B 109:13,743–13,754
Lai SL, Guo JY, Petrova V, Ramanath G, Allen LH (1996) Size-dependent melting properties of small tin particles: Nanocalorimetric measurements. Phys Rev Lett 77(1):99–102
Liu X, Yangb P, Jiang Q (2007) Size effect on melting temperature of nanostructured drugs. Mat Chem Phys 103(103):1–4
McCue SW, King JR, Riley DS (2003) Extinction behaviour for two-dimensional inward-solidification problems. Proceedings of the Royal Society A 459(2032):977–999
McCue SW, Wu B, Hill JM (2009) Micro/nanoparticle melting with spherical symmetry and surface tension. IMA J Appl Math 74:439–457
Mitchell S, Vynnycky M (2009) Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. Appl Math Comput 215(4):1609–1621
Myers TG, Mitchell SL, Font F (2012) Energy conservation in the one-phase supercooled Stefan problem. Int Comm Heat Mass Trans 39:1522–1525
Nanda K (2009) Size dependent melting of nanoparticles. Pramana J Phys 72(4):617–628
Nguyen NT, Werely S (2006) Fundamentals and applications of microfluidics, 1st edn. Artech House, Boston
Petrov O, Furó I (2006) Curvature-dependent metastability of the solid phase and the freezing-melting hysteresis in pores. Phys Rev E 73:011,608
Plech A, Kotaidis V, Grésillon S, Dahmen C, von Plessen G (2004) Laser-induced heating and melting of gold nanoparticles studied by time-resolved x-ray scattering. Phys Rev B 70(19):195,423
Rana S, Bajaj A, Mout R, Rotello V (2012) Monolayer coated gold nanoparticles for delivery applications. Adv Drug Deliv Rev 64:200–216
Riley DS, Smith FT, Poots G (1974) The inward solidification of spheres and circular cylinders. Int J Heat Mass Trans 17:1507–1516
Ruan C, Murooka Y, Raman RK, Murdick RA (2007) Dynamics of size-selected gold nanoparticles studied by ultrafast electron nanocrystallography. Nano Lett 7(5):1290–1296
Salata O (2004) Applications of nanoparticles in biology and medicine. J Nanobiotech 2(3)
Samsonov V, Bazulev A, Sdobnyakovy N (2003) On applicability of Gibbs thermodynamics to nanoparticles. Central Eur J Phys 1(3):474–484
Sheng HW (1996) Superheating and melting-point depression of Pb nanoparticles embedded in Al matrices. Phil Mag Lett 73(4):179–186
Sheng HW, Ren G, Peng LM, Hu ZQ, Lu K (1996) Superheating and melting-point depression of Pb nanoparticles embedded in Al matrices. Philos Mag Lett 73(4):179–186
Shim JH, Lee BJ, Cho YW (2002) Thermal stability of unsupported gold nanoparticle: a molecular dynamics study. Surf Sci 512:262–268
Travis KP, Todd BD, Evans DJ (1997) Departure from navier-stokes hydrodynamics in confined liquids. Phys Rev E 55(4):4288–4295
Volz S, Saulnier JB, Lallemand M, Perrin B, Depondt P (1996) Transient fourier-law deviation by molecular dynamics in solid argon. Phys Rev B 54(1):340–347
Wu B, McCue SW, Tillman P, Hill JM (2009) Single phase limit for melting nanoparticles. Appl Math Model 33(5):2349–2367
Wu B, Tillman P, McCue SW, Hill JM (2009) Nanoparticle melting as a Stefan moving boundary problem. J Nanosci Nanotechnol 9(2):885–888
Wu T, Liaw HC, Chen YZ (2002) Thermal effect of surface tension on the inward solidication of spheres. International J Heat Mass Trans 45(10):2055–2065
Zhong J, Zhang LH, Jin ZH, Sui ML, Lu K (2001) Superheating of Ag nanoparticles embedded in Ni matrix. Acta Mater 49(15):2897–2904
Acknowledgments
The research of TGM was supported by a Marie Curie International Reintegration Grant Industrial applications of moving boundary problems Grant No. FP7-256417, and Ministerio de Ciencia e Innovación Grant MTM2011-23789. FF acknowledges the support of a Centre de Recerca Matemàtica PhD Grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Font, F., Myers, T.G. Spherically symmetric nanoparticle melting with a variable phase change temperature. J Nanopart Res 15, 2086 (2013). https://doi.org/10.1007/s11051-013-2086-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11051-013-2086-3