A kinetic 5-vertex model is used to investigate hexagon-islands formation on growing single-walled carbon nanotubes (SWCNT). In the model, carbon atoms adsorption and migration processes on the SWCNT edge are considered. These two dynamic processes are assumed to be mutually independent as well as mutually dependent as far as the whole growth of the nanotube is concerned. Key physical parameters of the model are the growth time t, the diffusion length Γ defined as the ratio of the diffusion rate D to the carbon atomic flux F and the SWCNT chiral angle. The kinetic equation that describes the nanotube edge dynamics is solved using kinetic Monte Carlo simulations with the Bortz, Kalos and Lebowitz update algorithm. The behaviors of islands density and size distribution are investigated within the growth parameters’ space. Our study revealed key mechanisms that enable the formation of a new ring of hexagons at the SWCNT edge. The growth occurs either by pre-existing steps propagation or by hexagon-islands growth and coalescence on terraces located between dislocation steps, depending on values of model parameters. This should offer a road map for edge design in nanotubes production. We also found that in appropriate growth conditions, the islands density follows Gaussian and generalized Wigner distributions whereas their size distribution at a given growth time shows a decreasing exponential trend.
References
[1]
Venables, J.A. (2000) Introduction to Surface and Thin Film Processes. Cambridge University Press. https://doi.org/10.1017/cbo9780511755651
[2]
Mo, Y.W., Kleiner, J., Webb, M.B. and Lagally, M.G. (1991) Activation Energy for Surface Diffusion of Si on Si (001): A Scanning-Tunneling-Microscopy Study. Physical Review Letters, 66, 1998-2001. https://doi.org/10.1103/physrevlett.66.1998
[3]
Ernst, H., Fabre, F. and Lapujoulade, J. (1992) Nucleation and Diffusion of Cu Adatoms on Cu (100): A Helium-Atom-Beam Scattering Study. Physical Review B, 46, 1929-1932. https://doi.org/10.1103/physrevb.46.1929
[4]
Amar, J.G. and Family, F. (1996) Characterization of Surface Morphology in Epitaxial Growth. Surface Science, 365, 177-185. https://doi.org/10.1016/0039-6028(96)00692-9
[5]
Lobo, L.S. and Carabineiro, S.A.C. (2021) Kinetics of Carbon Nanotubes and Graphene Growth on Iron and Steel: Evidencing the Mechanisms of Carbon Formation. Nanomaterials, 11, Article 143. https://doi.org/10.3390/nano11010143
[6]
Bartelt, M.C., Tringides, M.C. and Evans, J.W. (1993) Island-Size Scaling in Surface Deposition Processes. Physical Review B, 47, 13891-13894. https://doi.org/10.1103/physrevb.47.13891
[7]
Ferrando, R., Hontinfinde, F. and Levi, A.C. (1997) Morphologies in Anisotropic Cluster Growth: A Monte Carlo Study on Ag (110). Physical Review B, 56, R4406-R4409. https://doi.org/10.1103/physrevb.56.r4406
[8]
Hontinfinde, F., Videcoq, A., Montalenti, F. and Ferrando, R. (2004) Leapfrog-Induced Selective Faceting in the Growth of Missing-Row (110) Surfaces. Chemical Physics Letters, 398, 50-55. https://doi.org/10.1016/j.cplett.2004.09.008
[9]
Mottet, C., Ferrando, R., Hontinfinde, F. and Levi, A.C. (1998) A Monte Carlo Simulation of Submonolayer Homoepitaxial Growth on Ag (110) and Cu (110). Surface Science, 417, 220-237. https://doi.org/10.1016/s0039-6028(98)00611-6
[10]
Bachilo, S.M., Balzano, L., Herrera, J.E., Pompeo, F., Resasco, D.E. and Weisman, R.B. (2003) Narrow (n, m)-Distribution of Single-Walled Carbon Nanotubes Grown Using a Solid Supported Catalyst. Journal of the American Chemical Society, 125, 11186-11187. https://doi.org/10.1021/ja036622c
[11]
Maruyama, S., Miyauchi, Y., Murakami, Y. and Chiashi, S. (2003) Optical Characterization of Single-Walled Carbon Nanotubes Synthesized by Catalytic Decomposition of Alcohol. New Journal of Physics, 5, 149-149. https://doi.org/10.1088/1367-2630/5/1/149
[12]
Omrane, B. (2009) Controlled Growth and Assembly of Single-Walled Carbon Nanotubes for Nanoelectronics. Ph.D. Thesis, University of Victoria.
[13]
Cançado, L.G., Takai, K., Enoki, T., Endo, M., Kim, Y.A., Mizusaki, H., et al. (2006) General Equation for the Determination of the Crystallite Size La of Nanographite by Raman Spectroscopy. Applied Physics Letters, 88, Article 163106. https://doi.org/10.1063/1.2196057
Lamouroux, E., Serp, P. and Kalck, P. (2007) Catalytic Routes Towards Single Wall Carbon Nanotubes. Catalysis Reviews, 49, 341-405. https://doi.org/10.1080/01614940701313200
[16]
Yu, F., Yang, M., Li, F., Su, C., Ma, B., Yuan, Z., et al. (2012) The Growth Mechanism of Single-Walled Carbon Nanotubes with a Controlled Diameter. Physica E: Low-dimensional Systems and Nanostructures, 44, 2032-2040. https://doi.org/10.1016/j.physe.2012.06.007
[17]
Zhao, J., Guo, Q., Shi, J., Liu, L., Jia, J., Liu, Y., et al. (2009) Carbon Nanotube Growth in the Pores of Expanded Graphite by Chemical Vapor Deposition. Carbon, 47, 1747-1751. https://doi.org/10.1016/j.carbon.2009.02.028
[18]
Ding, F., Harutyunyan, A.R. and Yakobson, B.I. (2009) Dislocation Theory of Chirality-Controlled Nanotube Growth. Proceedings of the National Academy of Sciences, 106, 2506-2509. https://doi.org/10.1073/pnas.0811946106
He, M., Wang, X., Zhang, S., Jiang, H., Cavalca, F., Cui, H., et al. (2019) Growth Kinetics of Single-Walled Carbon Nanotubes with a (2n, n) Chirality Selection. Science Advances, 5, eaav9668. https://doi.org/10.1126/sciadv.aav9668
[21]
Zounmenou, F., Hontinfinde, R.D. and Hontinfinde, F. (2022) Growth Kinetics of a Single-Walled Carbon Nanotube: Exact and Simulation Results. Physica A: Statistical Mechanics and Its Applications, 594, Article 127013. https://doi.org/10.1016/j.physa.2022.127013
[22]
Oke, T.D., Hontinfinde, S.I.V., Karimou, M., Zounmenou, F. and Hontinfinde, F. (2022) Atomistic Growth Model with Edge Diffusion for Chiral Carbon Nanotubes. Physica E: Low-dimensional Systems and Nanostructures, 142, Article 115298. https://doi.org/10.1016/j.physe.2022.115298
[23]
Hontinfinde, S.I.V., Kple, J., Oke, T.D., Zounmenou, F., Adda, J. and Hontinfinde, F. (2022) Nanofacet-Density Scaling on Zig-Zag Carbon Nanotubes within the Kinetic 5-Vertex Growth Model. Physica A: Statistical Mechanics and Its Applications, 608, Article 128278. https://doi.org/10.1016/j.physa.2022.128278
[24]
Wang, Y., Zhang, W., Chen, Y., Zeng, X., Huang, J., Wei, H., et al. (2023) Mechanism of Carbon Nanotube Growth in Expanded Graphite via Catalytic Pyrolysis Reaction Using Carbores P as a Carbon Source. Frontiers in Chemistry, 11, Article 1260099. https://doi.org/10.3389/fchem.2023.1260099
[25]
Moors, M., Amara, H., Visart de Bocarmé, T., Bichara, C., Ducastelle, F., Kruse, N., et al. (2009) Early Stages in the Nucleation Process of Carbon Nanotubes. ACS Nano, 3, 511-516. https://doi.org/10.1021/nn800769w
[26]
Ohta, Y., Okamoto, Y., Irle, S. and Morokuma, K. (2008) Rapid Growth of a Single-Walled Carbon Nanotube on an Iron Cluster: Density-Functional Tight-Binding Molecular Dynamics Simulations. ACS Nano, 2, 1437-1444. https://doi.org/10.1021/nn8001906
[27]
Jin, C., Suenaga, K. and Iijima, S. (2008) How Does a Carbon Nanotube Grow? An in situ Investigation on the Cap Evolution. ACS Nano, 2, 1275-1279. https://doi.org/10.1021/nn800121v
[28]
Liu, Y., Dobrinsky, A. and Yakobson, B.I. (2010) Graphene Edge from Armchair to Zigzag: The Origins of Nanotube Chirality? Physical Review Letters, 105, Article 235502. https://doi.org/10.1103/physrevlett.105.235502
[29]
Hontinfinde, F. and Touzani, M. (1995) Growth Kinetics of Bcc Crystal Surfaces. Surface Science, 338, 236-246. https://doi.org/10.1016/0039-6028(95)00545-5
[30]
Hontinfinde, F., Krug, J. and Touzani, M. (1997) Growth with Surface Diffusion in d=1+1. Physica A: Statistical Mechanics and Its Applications, 237, 363-383. https://doi.org/10.1016/s0378-4371(96)00435-9
[31]
Bortz, A.B., Kalos, M.H. and Lebowitz, J.L. (1975) A New Algorithm for Monte Carlo Simulation of Ising Spin Systems. Journal of Computational Physics, 17, 10-18. https://doi.org/10.1016/0021-9991(75)90060-1
[32]
Hontinfinde, S.I.V., Akpo, A.B. and Hontinfinde, F. (2016) Numerical Investigation of the Submonolayer Growth and Relaxation of Stepped Ag (110) and Au (110) Surfaces. The European Physical Journal B, 89, Article No. 215. https://doi.org/10.1140/epjb/e2016-70233-1
[33]
Yancey, J.A., Richards, H.L. and Einstein, T.L. (2005) Terrace Width Distributions for Vicinal Surfaces with Steps of Alternating Stiffness. Surface Science, 598, 78-87. https://doi.org/10.1016/j.susc.2005.08.028
