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Analysing the stability of graphene wrinkles using variational calculus
Get PDFThe chemical vapour deposition method is widely used to synthesise high quality graphene with a large surface area. However, the cooling process leads to the formations of ripples and wrinkles in the graphene structure. When a self-adhered wrinkle achieves the maximum height, it then folds onto the surface and leads to a collapsed wrinkle. The presence of such deformations often affects the properties of graphene. In this article, we describe a novel mathematical model to understand the formation and geometry of these wrinkles. The stability of these wrinkles is examined based on variational derivations for the energy of each structure. The model provides detailed explanations for the geometry of these wrinkles which would help in tuning their properties.
References
J. Aljedani, M. J. Chen, and B. J. Cox. Variational model for collapsed graphene wrinkles. Appl. Phys. A 127.11, 886 (2021), pp. 1–13. doi: 10.1007/s00339-021-05000-y
A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau. Superior thermal conductivity of single-layer graphene. Nano Lett. 8.3 (2008), pp. 902–907. doi: 10.1021/nl0731872
S. Chen, Q. Li, Q. Zhang, Y. Qu, H. Ji, R. S. Ruoff, and W. Cai. Thermal conductivity measurements of suspended graphene with and without wrinkles by micro-Raman mapping. Nanotech. 23.36, 365701 (2012). doi: 10.1088/0957-4484/23/36/365701 on p. C85).
B. J. Cox, T. Dyer, and N. Thamwattana. A variational model for conformation of graphene wrinkles formed on a shrinking solid metal substrate. Mat. Res. Express 7.8, 085001 (2020). doi: 10.1088/2053-1591/abaa8f
A. K. Geim. Graphene: Status and prospects. Science 324.5934 (2009), pp. 1530–1534. doi: 10.1126/science.1158877 on p. C85).
K. Kostarelos and K. S. Novoselov. Graphene devices for life. Nature Nanotech. 9 (2014), pp. 744–745. doi: 10.1038/nnano.2014.224
F. Long, P. Yasaei, R. Sanoj, W. Yao, P. Král, A. Salehi-Khojin, and R. Shahbazian-Yassar. Characteristic work function variations of graphene line defects. ACS Appl. Mat. Inter. 8.28 (2016), pp. 18360–18366. doi: 10.1021/acsami.6b04853
R. Muñoz and C. Gómez-Aleixandre. Review of CVD synthesis of graphene. Chem. Vapor Dep. 19.10–12 (2013), pp. 297–322. doi: 10.1002/cvde.201300051
L. Spanu, S. Sorella, and G. Galli. Nature and strength of interlayer binding in graphite. Phys. Rev. Lett. 103.19, 196401 (2009). doi: 10.1103/PhysRevLett.103.196401
T. Verhagen, B. Pacakova, M. Bousa, U. Hübner, M. Kalbac, J. Vejpravova, and O. Frank. Superlattice in collapsed graphene wrinkles. Sci. Rep. 9.1, 9972 (2019). doi: 10.1038/s41598-019-46372-9
C. Wang, Y. Liu, L. Li, and H. Tan. Anisotropic thermal conductivity of graphene wrinkles. Nanoscale 6.11 (2014), pp. 5703–5707. doi: 10.1039/C4NR00423J
W. Wang, S. Yang, and A. Wang. Observation of the unexpected morphology of graphene wrinkle on copper substrate. Sci. Rep. 7.1 (2017), pp. 1–6. doi: 10.1038/s41598-017-08159-8
Y. Wang, R. Yang, Z. Shi, L. Zhang, D. Shi, E. Wang, and G. Zhang. Super-elastic graphene ripples for flexible strain sensors. ACS Nano 5.5 (2011), pp. 3645–3650. doi: 10.1021/nn103523t
Y. Wei, B. Wang, J. Wu, R. Yang, and M. L. Dunn. Bending rigidity and Gaussian bending stiffness of single-layered graphene. Nano Lett. 13.1 (2013), pp. 26–30. doi: 10.1021/nl303168w
Z. Xu and M. J. Buehler. Interface structure and mechanics between graphene and metal substrates: A first-principles study. J. Phys.: Cond. Mat. 22.48, 485301 (2010). doi: 10.1088/0953-8984/22/48/485301
Y. Zhang, N. Wei, J. Zhao, Y. Gong, and T. Rabczuk. Quasi-analytical solution for the stable system of the multi-layer folded graphene wrinkles. J. Appl. Phys. 114.6, 063511 (2013). doi: 10.1063/1.4817768
W. Zhu, T. Low, V. Perebeinos, A. A. Bol, Y. Zhu, H. Yan, J. Tersoff, and P. Avouris. Structure and electronic transport in graphene wrinkles. Nano Lett. 12.7 (2012), pp. 3431–3436. doi: 10.1021/nl300563
Estimating energy savings from a train driving advice system
Get PDFTTG Energymiser is an in-cab system that provides real-time driving advice to train drivers with the aim of reducing energy use subject to meeting the train schedule. A survey of the efficacy of Energymiser has been undertaken, to provide evidence for marketing claims. Results from 23 different trials are analysed, where 16 of the trials were on passenger routes and 7 were on freight routes. Each trial consists of many trips, with Energymiser activated for around half, and yields an estimate of the change in energy use when Energymiser is used. A Bayesian hierarchical model is fitted to the 16 estimates from passenger routes and provides an estimate of the mean saving and the standard deviation of individual trials about the mean. The mean saving is 7.2% and the standard deviation of individual trials is estimated as 3.3%. The corresponding mean and standard deviation for freight routes are 8.4% and 5.8%, respectively.
References
A. Albrecht, P. Howlett, P. Pudney, X. Vu, and P. Zhou. The key principles of optimal train control—Part 2: Existence of an optimal strategy, the local energy minimization principle, uniqueness, computational technique. Transport. Res. B: Method. 94 (2016), pp. 509–538. doi: 10.1016/j.trb.2015.07.024.
A. Albrecht, P. Howlett, P. Pudney, X. Vu, and P. Zhou. The key principles of optimal train control—Part 1: Formulation of the model, strategies of optimal type, evolutionary lines, location of optimal switching points. Transport. Res. B: Method. 94 (2016), pp. 482–508. doi: 10.1016/j.trb.2015.07.023.
A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin. Bayesian Data Analysis. Chapman and Hall/CRC Press, 2012. doi: 10.1201/b16018
C. Röver. Bayesian random-effects meta-analysis using the bayesmeta R package. J. Stat. Software 93.6 (2020), pp. 1–51. doi: 10.18637/jss.v093.i0
Approximately multiplicative decompositions of nuclear maps
No full texthttp://dx.doi.org/10.1017/S000497271200033
Centre of Banach algebra valued Beurling algebras
No full texthttp://dx.doi.org/10.1017/S000497271200033
On the parity of the generalised Frobenius partition functions
No full texthttp://dx.doi.org/10.1017/S000497271200033
Alternating circular sums of binomial coefficients
No full texthttp://dx.doi.org/10.1017/S000497271200033
On asymptotic properties of non-linear functionals of random fields
No full texthttp://dx.doi.org/10.1017/S000497271200033
The magic of Nash social welfare in optimization: Do not sum, just multiply!
No full textWe explain some key challenges when dealing with a single- or multi-objective optimization problem in practice. To overcome these challenges, we present a mathematical program that optimizes the Nash social welfare function. We refer to this mathematical program as the Nash social welfare program (NSWP). An interesting property of the NSWP is that it can be constructed for any single- or multi-objective optimization problem. We show that solving the NSWP could result in more desirable solutions in practice than its single- or multi-objective counterpart. We also discuss several promising approaches that could be employed to solve the NSWP in practice.
doi:10.1017/S1446181122000074
The primitive Orr–Sommerfeld equation and its solution by finite elements
Get PDFThe linear stability of parallel shear flows of incompressible viscous fluids is classically described by the Orr–Sommerfeld equation in the disturbance streamfunction. This fourth-order equation is obtained by eliminating the pressure from the linearized Navier–Stokes equation. Here we consider retaining the primitive velocity-pressure formulation, as is required for general multidimensional geometries for which the streamfunction is unavailable; this affords a uniform description of one-, two-, and three-dimensional flows and their perturbations. The Orr–Sommerfeld equation is here discretized using Python and scikit- fem, in classical and primitive forms with Hermite and Mini elements, respectively. The solutions for the standard test problem of plane Poiseuille flow show the primitive formulation to be simple, clear, very accurate, and better-conditioned than the classical.
References
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