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Molecular Dynamics Simulation of Sintering Dynamics of Many TiO\(_{2}\) Nanoparticles


The sintering processes of many TiO\(_{2}\) nanoparticles in chains of both solid and liquid phases have been studied in detail via molecular dynamics simulation. For the liquid phase, a modified correlation for the characteristic sintering time of multi-particle chains is obtained by including a correction factor of \((N/2)^{1/3}\), where N is the number of primary particles. The temperature rise during sintering is found to be linearly proportional to \((1-N^{-1/3})\). Moreover, this study provides a way to calculate the surface energy of nanoparticles of small diameters in liquid phase, which is experimentally unattainable. For the solid phase, sintering induced nucleation is observed for \(N \ge 4\) cases both at \(T_{0 }= 1220\) and 960 K with a sharp increase in the temperature and a decrease in the potential energy. The formation of rutile from nucleation of many solid amorphous particles through sintering is observed for the first time.

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  1. 1.

    Goldstein, A.N., Echer, C.M., Alivisatos, A.P.: Melting in semiconductor nanocrystals. Science 256, 1425–1427 (1992)

    ADS  Article  Google Scholar 

  2. 2.

    Colvin, V.L., Alivisatos, A.P., Tobin, J.G.: Valence-band photoemission from a quantum-dot system. Phys. Rev. Lett. 66, 2786 (1991)

    ADS  Article  Google Scholar 

  3. 3.

    Yetter, R.A., Risha, G.A., Son, S.F.: Metal particle combustion and nanotechnology. Proc. Combust. Inst. 32, 1819–1838 (2009)

    Article  Google Scholar 

  4. 4.

    Kung, M.C., Davis, R.J., Kung, H.H.: Understanding Au-catalyzed low-temperature CO oxidation. J. Phys. Chem. C 111, 11767–11775 (2007)

    Article  Google Scholar 

  5. 5.

    Mueller, R., Mädler, L., Pratsinis, S.E.: Nanoparticle synthesis at high production rates by flame spray pyrolysis. Chem. Eng. Sci. 58, 1969–1976 (2003)

    Article  Google Scholar 

  6. 6.

    Swihart, M.T.: Vapor-phase synthesis of nanoparticles. Curr. Opin. Colloid Interface Sci. 8, 127–133 (2003)

    Article  Google Scholar 

  7. 7.

    Wang, J., Li, S., Yan, W., Tse, S.D., Yao, Q.: Synthesis of TiO\(_{2}\) nanoparticles by premixed stagnation swirl flames. Proc. Combust. Inst. 33, 1925–1932 (2011)

    Article  Google Scholar 

  8. 8.

    Pratsinis, S.E.: Flame aerosol synthesis of ceramic powders. Proc. Energy Combust. Sci. 24, 197–219 (1998)

    Article  Google Scholar 

  9. 9.

    Zachariah, M.R., Carrier, M.J.: Molecular dynamics computation of gas-phase nanoparticle sintering: a comparison with phenomenological models. J. Aerosol Sci. 30, 1139–1151 (1999)

    Article  Google Scholar 

  10. 10.

    Koch, W., Friedlander, S.K.: The effect of particle coalescence on the surface area of a coagulating aerosol. J. Colloid Interface Sci. 140, 419–427 (1990)

    Article  Google Scholar 

  11. 11.

    Frenkel, J.: On the theory of electric contacts between metallic bodies. J. Phys. 9, 385 (1945)

    Google Scholar 

  12. 12.

    Pokluda, O., Bellehumeur, C.T., Vlachopoulos, J.: Modification of Frenkel’s model for sintering. AlChE J. 43, 3253–3256 (1997)

    Article  Google Scholar 

  13. 13.

    Friedlander, S.K., Wu, M.K.: Linear rate law for the decay of the excess surface-area of a coalescing solid particle. Phys. Rev. B 49, 3622–3624 (1994)

    ADS  Article  Google Scholar 

  14. 14.

    Martínez-Herrera, J.I., Derby, J.J.: Viscous sintering of spherical particles via finite element analysis. J. Am. Ceram. Soc. 78, 645–649 (1995)

    Article  Google Scholar 

  15. 15.

    Kirchhof, M.J., Schmid, H.J., Peukert, W.: Three-dimensional simulation of viscous-flow agglomerate sintering. Phys. Rev. E 80, 026319 (2009)

    ADS  Article  Google Scholar 

  16. 16.

    Eggersdorfer, M.L., Kadau, D., Herrmann, H.J., Pratsinis, S.E.: Multiparticle sintering dynamics: from fractal-like aggregates to compact structures. Langmuir 27, 6358–6367 (2011)

    Article  Google Scholar 

  17. 17.

    Eggersdorfer, M.L., Pratsinis, S.E.: Restructuring of aggregates and their primary particle size distribution during sintering. AIChE J. 59, 1118–1126 (2013)

    Article  Google Scholar 

  18. 18.

    Buesser, B., Grohn, A.J., Pratsinis, S.E.: Sintering rate and mechanism of TiO\(_{2}\) nanoparticles by molecular dynamics. J. Phys. Chem. C 115, 11030–11035 (2011)

    Article  Google Scholar 

  19. 19.

    Koparde, V.N., Cummings, P.T.: Molecular dynamics simulation of titanium dioxide nanoparticle sintering. J. Phys. Chem. B 109, 24280–24287 (2005)

    Article  Google Scholar 

  20. 20.

    Zhu, H.: Sintering processes of two nanoparticles: a study by molecular dynamics simulations. Philos. Mag. Lett. 73, 27–33 (1996)

    ADS  Article  Google Scholar 

  21. 21.

    Hawa, T., Zachariah, M.R.: Molecular dynamics simulation and continuum modeling of straight-chain aggregate sintering: development of a phenomenological scaling law. Phys. Rev. B 76, 054109 (2007)

    ADS  Article  Google Scholar 

  22. 22.

    Hawa, T., Zachariah, M.R.: Development of a phenomenological scaling law for fractal aggregate sintering from molecular dynamics simulation. J. Aerosol Sci. 38, 793–806 (2007)

    Article  Google Scholar 

  23. 23.

    Matsui, M., Akaogi, M.: Molecular dynamics simulation of the structural and physical properties of the four polymorphs of TiO\(_{2}\). Mol. Simul. 6, 239–244 (1991)

    Article  Google Scholar 

  24. 24.

    Buckingham, R.A.: The classical equation of state of gaseous helium, neon and argon. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 168, 264–283 (1938)

    ADS  Article  Google Scholar 

  25. 25.

    Traylor, J.G., Smith, H.G., Nicklow, R.M., Wilkinson, M.K.: Lattice dynamics of rutile. Phys. Rev. B 3, 3457 (1971)

    ADS  Article  Google Scholar 

  26. 26.

    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)

    ADS  Article  MATH  Google Scholar 

  27. 27.

    Koparde, V.N., Cummings, P.T.: Sintering of titanium dioxide nanoparticles: a comparison between molecular dynamics and phenomenological modeling. J. Nanoparticle Res. 10, 1169–1182 (2008)

    Article  Google Scholar 

  28. 28.

    Kazakov, A.V., Shpiro, E.S., Voskoboinikov, T.V.: Application of Debye function analysis to particle size and shape determination in Ir/SiO\(_{2}\) catalysts. J. Phys. Chem. 99, 8323–8327 (1995)

    Article  Google Scholar 

  29. 29.

    Koparde, V.N., Cummings, P.T.: Phase transformations during sintering of titania nanoparticles. ACS Nano 2, 1620–1624 (2008)

    Article  Google Scholar 

  30. 30.

    Cromer, D.T., Mann, J.B.: X-ray scattering factors computed from numerical Hartree-Fock wave functions. Acta Cryst. Sect. A 24, 321–324 (1968)

    ADS  Article  Google Scholar 

  31. 31.

    Buffat, P., Borel, J.P.: Size effect on the melting temperature of gold particles. Phys. Rev. A 13, 2287–2298 (1976)

    ADS  Article  Google Scholar 

  32. 32.

    Zhang, Y., Li, S., Yan, W., Stephen, D.T.: Effect of size-dependent grain structures on the dynamics of nanoparticle coalescence. J. Appl. Phys. 111, 124321 (2012)

    ADS  Article  Google Scholar 

  33. 33.

    Li, Y., Ishigaki, T.: Thermodynamic analysis of nucleation of anatase and rutile from TiO\(_{2}\) melt. J. Cryst. Growth 242, 511–516 (2002)

    ADS  Article  Google Scholar 

  34. 34.

    Dingwell, D.B.: The density of titanium(IV) oxide liquid. J. Am. Ceram. Soc. 74, 2718–2719 (1991)

    Article  Google Scholar 

  35. 35.

    Barnard, A.S., Zapol, P., Curtiss, L.A.: Modeling the morphology and phase stability of TiO\(_{2}\) nanocrystals in water. J. Chem. Theory Comput. 1, 107–116 (2005)

    Article  Google Scholar 

  36. 36.

    Naicker, P.K., Cummings, P.T., Zhang, H., Banfield, J.F.: Characterization of titanium dioxide nanoparticles using molecular dynamics simulations. J. Phys. Chem. B 109, 15243–15249 (2005)

    Article  Google Scholar 

  37. 37.

    Zhang, H., Peng, S., Long, X., et al.: Wall-induced phase transition controlled by layering freezing. Phys. Rev. E 89, 032412 (2014)

    ADS  Article  Google Scholar 

  38. 38.

    Wang, C., Deng, Z.X., Li, Y.: The synthesis of nanocrystalline anatase and rutile titania in mixed organic media. Inorg. Chem. 40, 5210–5214 (2001)

    Article  Google Scholar 

  39. 39.

    Zhang, H., Chen, B., Banfield, J.F.: The size dependence of the surface free energy of titania nanocrystals. PCCP 11, 2553–2558 (2009)

    ADS  Article  Google Scholar 

  40. 40.

    Zhao, B., Uchikawa, K., McCormick, J.R., Ni, C.Y., Chen, J.G., Wang, H.: Ultrafine anatase TiO\(_{2}\) nanoparticles produced in premixed ethylene stagnation flames. Proc. Combust. Inst. 30, 2569–2576 (2005)

    Article  Google Scholar 

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Support from the Major Project of the National Science Foundation of China (Grant No. 51390493) and the Center for Combustion Energy at Tsinghua University is gratefully acknowledged. The simulations were partly performed on the Tsinghua High-Performance Parallel Computer supported by the Tsinghua National Laboratory for Information Science and Technology and partly on ARCHER funded under the EPSRC Project “UK Consortium on Mesoscale Engineering Sciences (UKCOMES)” (Grant No. EP/L00030X/1).

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Mao, Q., Luo, K.H. Molecular Dynamics Simulation of Sintering Dynamics of Many TiO\(_{2}\) Nanoparticles. J Stat Phys 160, 1696–1708 (2015).

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  • Molecular dynamics simulation
  • Many nanoparticles
  • Characteristic sintering time
  • Phase transformation