Abstract
Materials with a two-dimensional (2D) structure have emerged as an important component of modern technologies. Using 2D materials in many applications results in strain being created in the material. Materials respond to strain in different ways depending on their mechanical properties. The present work examines the mechanical properties of two 2D nanosheets of TiO\(_2\), namely hexagonal nanosheet (HNS) and lepidocrocite nanosheet (LNS). In order to accomplish this, we use the stress–strain theory in combination with first-principles density functional theory (DFT) calculations to investigate the mechanical characteristics, including stiffness constants, Young’s modulus, Poisson’s ratio, ideal strength, and critical strain. We validate our calculations by obtaining the important mechanical properties of bulk rutile TiO\(_2\) and comparing them with the theoretical and experimental values reported by others. Then, we compare the mechanical properties of TiO\(_2\) LNS and HNS using the same computational approach. It appears that LNS is stiffer than HNS, so we analyze structural differences between the two in order to determine the reasons for this. It has been observed that under small tensile strains, the responses, including induced stress, transverse contraction, and bond length change, are linear for both HNSs and LNSs. In general, LNS displays anisotropic responses, whereas HNS exhibits isotropic responses under small tensile strains and nearly isotropic responses under higher levels of strain. Finally, we compare the mechanical properties of HNS and LNS with those of graphene, h-BN, phosphorene, and hexagonal antimonene.
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This manuscript has associated data in a data repository. [Authors’ comment: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.]
References
L. Wang, T. Sasaki, Titanium oxide nanosheets: Graphene analogues with versatile functionalities. Chem. Rev. 114(19), 9455–9486 (2014). https://doi.org/10.1021/cr400627u
X. Chen, S.S. Mao, Titanium dioxide nanomaterials: synthesis, properties, modifications, and applications. Chem. Rev. 107(7), 2891–2959 (2007). https://doi.org/10.1021/cr0500535
M. Xu, Y. Gao, E.M. Moreno, M. Kunst, M. Muhler, Y. Wang, H. Idriss, C. Wöll, Photocatalytic activity of bulk tio 2 anatase and rutile single crystals using infrared absorption spectroscopy. Phys. Rev. Lett. 106(13), 138302 (2011)
H.A. Eivari, S.A. Ghasemi, H. Tahmasbi, S. Rostami, S. Faraji, R. Rasoulkhani, S. Goedecker, M. Amsler, Two-dimensional hexagonal sheet of tio2. Chem. Mater. 29(20), 8594–8603 (2017). https://doi.org/10.1021/acs.chemmater.7b02031
R. Mas-Ballesté, C. Gómez-Navarro, J. Gómez-Herrero, F. Zamora, 2d materials: to graphene and beyond. Nanoscale 3(1), 20–30 (2011). https://doi.org/10.1039/c0nr00323a
H. Eivari, Z. Sohbatzadeh, P. Mele, M. Assadi, Low thermal conductivity: Fundamentals and theoretical aspects in thermoelectric applications. Mater. Today. Energy. 21, 100744 (2021)
D.V. Fakhrabad, M. Yeganeh, Investigation of the effect of lattice thermal conductivity on the thermoelectric performance of scn monolayer. Mater. Sci. Semicond. Process. 148, 106770 (2022)
D.V. Fakhrabad, M. Yeganeh, Piezoelectric properties in two-dimensional gec and its surface functionalization by chlorination, fluorination, and chloro-fluorination. Mater. Sci. Semicond. Process. 148, 106797 (2022)
M. Mousavi, S.T. Yazdi, M.B. Mohagheghi, Magneto-transport and magneto-optical properties of cr-alloyed sno2 thin films: A correlation between structural and magnetic behaviors. Solid. State. Commun. 298, 113641 (2019)
S. Goedecker, Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems. J. Chem. Phys. 120(21), 9911–9917 (2004)
M. Amsler, S. Goedecker, Crystal structure prediction using the minima hopping method. J. Chem. Phys. 133(22), 224104 (2010). https://doi.org/10.1063/1.3512900
S.A. Ghasemi, A. Hofstetter, S. Saha, S. Goedecker, Interatomic potentials for ionic systems with density functional accuracy based on charge densities obtained by a neural network. Phys. Revi. B 92(4), 045131 (2015). https://doi.org/10.1103/physrevb.92.045131
R. Hafizi, S.A. Ghasemi, S.J. Hashemifar, H. Akbarzadeh, A neural-network potential through charge equilibration for ws2: From clusters to sheets. J. Chem. Phys. 147(23), 234306 (2017)
S. Faraji, S.A. Ghasemi, S. Rostami, R. Rasoulkhani, B. Schaefer, S. Goedecker, M. Amsler, High accuracy and transferability of a neural network potential through charge equilibration for calcium fluoride. Phys. Rev. B 95(10), 104105 (2017)
R. Rasoulkhani, H. Tahmasbi, S.A. Ghasemi, S. Faraji, S. Rostami, M. Amsler, Energy landscape of zno clusters and low-density polymorphs. Phys. Rev. B 96(6), 064108 (2017)
G. Liu, L. Wang, H.G. Yang, H.-M. Cheng, G.Q.M. Lu, Titania-based photocatalysts—crystal growth, doping and heterostructuring. J. Mater. Chem. 20(5), 831–843 (2010). https://doi.org/10.1039/b909930a
M. Osada, T. Sasaki, Two-dimensional dielectric nanosheets: Novel nanoelectronics from nanocrystal building blocks. Adv. Mater. 24(2), 210–228 (2011). https://doi.org/10.1002/adma.201103241
M. Fehse, E. Ventosa, Is TiO2(b) the future of titanium-based battery materials? Chem. Plus. Chem. 80(5), 785–795 (2015). https://doi.org/10.1002/cplu.201500038
I.M. Markus, S. Engelke, M. Shirpour, M. Asta, M. Doeff, Experimental and computational investigation of lepidocrocite anodes for sodium-ion batteries. Chem. Mater. 28(12), 4284–4291 (2016). https://doi.org/10.1021/acs.chemmater.6b01074
H.G. Yang, C.H. Sun, S.Z. Qiao, J. Zou, G. Liu, S.C. Smith, H.M. Cheng, G.Q. Lu, Anatase TiO2 single crystals with a large percentage of reactive facets. Nature 453(7195), 638–641 (2008). https://doi.org/10.1038/nature06964
D.A.H. Hanaor, M.H.N. Assadi, S. Li, A. Yu, C.C. Sorrell, Ab initio study of phase stability in doped TiO2. Comput. Mech. 50(2), 185–194 (2012). https://doi.org/10.1007/s00466-012-0728-4
M.H.N. Assadi, D.A.H. Hanaor, The effects of copper doping on photocatalytic activity at (101) planes of anatase TiO2: A theoretical study. Appl. Surf. Sci. 387, 682–689 (2016). https://doi.org/10.1016/j.apsusc.2016.06.178
H. Asnaashari Eivari, S.A. Ghasemi, Comparison between pbe and hse06 functionals for the calculation of electronic band-structure of tio2. J. Res. Many. body. Syst. 9(3), 1–15 (2019)
X. Yan, G. Liu, L. Wang, Y. Wang, X. Zhu, J. Zou, G.Q.M. Lu, Antiphotocorrosive photocatalysts containing CdS nanoparticles and exfoliated TiO2 nanosheets. J. Mater. Res. 25(1), 182–188 (2010). https://doi.org/10.1557/jmr.2010.0007
E. Doustkhah, M.H.N. Assadi, K. Komaguchi, N. Tsunoji, M. Esmat, N. Fukata, O. Tomita, R. Abe, B. Ohtani, Y. Ide, In situ blue titania via band shape engineering for exceptional solar h2 production in rutile tio2. Appl. Catal. B 297, 120380 (2021)
D.R. Kripalani, A.A. Kistanov, Y. Cai, M. Xue, K. Zhou, Strain engineering of antimonene by a first-principles study: Mechanical and electronic properties. Phys. Rev. B 98(8), 085410 (2018). https://doi.org/10.1103/physrevb.98.085410
J. Kang, J. Li, F. Wu, S.-S. Li, J.-B. Xia, Elastic, electronic, and optical properties of two-dimensional graphyne sheet. J. Phys. Chem. C 115(42), 20466–20470 (2011). https://doi.org/10.1021/jp206751m
M. Topsakal, S. Cahangirov, S. Ciraci, The response of mechanical and electronic properties of graphane to the elastic strain. Appl. Phys. Lett. 96(9), 091912 (2010). https://doi.org/10.1063/1.3353968
W.-J. Yin, S. Chen, J.-H. Yang, X.-G. Gong, Y. Yan, S.-H. Wei, Effective band gap narrowing of anatase TiO2 by strain along a soft crystal direction. Appl. Phys. Lett. 96(22), 221901 (2010). https://doi.org/10.1063/1.3430005
G. Rajender, P.K. Giri, Strain induced phase formation, microstructural evolution and bandgap narrowing in strained TiO2 nanocrystals grown by ball milling. J. Alloy. Compd. 676, 591–600 (2016). https://doi.org/10.1016/j.jallcom.2016.03.154
X. Yan, Z. Wang, M. He, Z. Hou, T. Xia, G. Liu, X. Chen, Tio2 nanomaterials as anode materials for lithium-ion rechargeable batteries. Energ. Technol. 3(8), 801–814 (2015)
M.-S. Balogun, Y. Zhu, W. Qiu, Y. Luo, Y. Huang, C. Liang, X. Lu, Y. Tong, Chemically lithiated tio2 heterostructured nanosheet anode with excellent rate capability and long cycle life for high-performance lithium-ion batteries. ACS Appl. Mater. interfaces. 7(46), 25991–26003 (2015)
M.H.N. Assadi, D.A. Hanaor, Theoretical study on copper’s energetics and magnetism in tio2 polymorphs. J. Appl. Phys. 113(23), 233913 (2013)
T.-W. Fan, J.-L. Ke, L. Fu, B.-Y. Tang, L.-M. Peng, W.-J. Ding, Ideal strength of mg2x (x= si, ge, sn and pb) from first-principles. J. Magnes. Alloys. 1(2), 163–168 (2013)
G. Wang, R. Pandey, S.P. Karna, Atomically thin group v elemental films: theoretical investigations of antimonene allotropes. ACS Appl. Mater. Interfaces. 7(21), 11490–11496 (2015)
A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buscema, F. Guinea, H.S. van der Zant, G.A. Steele, Local strain engineering in atomically thin mos2. Nano Lett. 13(11), 5361–5366 (2013)
F. Liu, P. Ming, J. Li, Ab initiocalculation of ideal strength and phonon instability of graphene under tension. Phys. Rev. B 76(6), 064120 (2007). https://doi.org/10.1103/physrevb.76.064120
R.-H. Zhang, L.-P. Wang, Z.-B. Lu, Probing the intrinsic failure mechanism of fluorinated amorphous carbon film based on the first-principles calculations. Sci. Rep. 5(1), 1–9 (2015). https://doi.org/10.1038/srep09419
B. Mortazavi, O. Rahaman, M. Makaremi, A. Dianat, G. Cuniberti, T. Rabczuk, First-principles investigation of mechanical properties of silicene, germanene and stanene. Physica. E 87, 228–232 (2017). https://doi.org/10.1016/j.physe.2016.10.047
Q. Peng, W. Ji, S. De, Mechanical properties of the hexagonal boron nitride monolayer: Ab initio study. Comput. Mater. Sci. 56, 11–17 (2012). https://doi.org/10.1016/j.commatsci.2011.12.029
Z. Sohbatzadeh, H.A. Eivari, D.V. Fakhrabad, Formation energy and some mechanical properties of hydrogenated hexagonal monolayer of GeC. Physica. B 547, 88–91 (2018). https://doi.org/10.1016/j.physb.2018.08.009
R. Ansari, M. Mirnezhad, H. Rouhi, A first principles study on the mechanical properties of hexagonal zinc oxide sheets. Superlattices. Microstruct. 79, 15–20 (2015). https://doi.org/10.1016/j.spmi.2014.12.014
Q. Yue, J. Kang, Z. Shao, X. Zhang, S. Chang, G. Wang, S. Qin, J. Li, Mechanical and electronic properties of monolayer MoS2 under elastic strain. Phys. Lett. A 376(12–13), 1166–1170 (2012). https://doi.org/10.1016/j.physleta.2012.02.029
Z. Zhang, Y. Yang, E.S. Penev, B.I. Yakobson, Elasticity, flexibility, and ideal strength of borophenes. Adv. Func. Mater. 27(9), 1605059 (2017). https://doi.org/10.1002/adfm.201605059
Q. Wei, X. Peng, Superior mechanical flexibility of phosphorene and few-layer black phosphorus. Appl. Phys. Lett. 104(25), 251915 (2014). https://doi.org/10.1063/1.4885215
H. Wang, X. Li, P. Li, J. Yang, updelta-phosphorene: a two dimensional material with a highly negative poissons ratio. Nanoscale 9(2), 850–855 (2017). https://doi.org/10.1039/c6nr08550d
X. Liu, H. Zhou, B. Yang, Y. Qu, M. Zhao, Strain-modulated electronic structure and infrared light adsorption in palladium diselenide monolayer. Sci. Rep. 7(1), 1–6 (2017). https://doi.org/10.1038/srep39995
Z. Guo, J. Zhou, C. Si, Z. Sun, Flexible two-dimensional tin-1cn(n = 1, 2 and 3) and their functionalized MXenes predicted by density functional theories. Phys. Chem. Chem. Phys. 17(23), 15348–15354 (2015). https://doi.org/10.1039/c5cp00775e
B. Mortazavi, M. Shahrokhi, M. Makaremi, T. Rabczuk, Anisotropic mechanical and optical response and negative poisson’s ratio in mo2c nanomembranes revealed by first-principles simulations. Nanotechnology 28(11), 115705 (2017). https://doi.org/10.1088/1361-6528/aa5c29
M. Topsakal, S. Ciraci, Elastic and plastic deformation of graphene, silicene, and boron nitride honeycomb nanoribbons under uniaxial tension: A first-principles density-functional theory study. Phys. Rev. B 81(2), 024107 (2010)
S.I. Lukyanov, A.V. Bandura, R.A. Evarestov, Youngs modulus and poissons ratio for tio2-based nanotubes and nanowires: modelling of temperature dependence. RSC Adv. 6(19), 16037–16045 (2016). https://doi.org/10.1039/c5ra24951a
Y. Ding, B. Xiao, Anisotropic elasticity, sound velocity and thermal conductivity of TiO2 polymorphs from first principles calculations. Comput. Mater. Sci. 82, 202–218 (2014). https://doi.org/10.1016/j.commatsci.2013.09.061
D.G. Isaak, J.D. Carnes, O.L. Anderson, H. Cynn, E. Hake, Elasticity of TiO 2 rutile to 1800 k. Phys. Chem. Miner. 26(1), 31–43 (1998). https://doi.org/10.1007/s002690050158
S. Nevhal, S. Kundalwal, Polarization in graphene nanoribbons with inherent defects using first-principles calculations. Acta Mech. 233(1), 399–411 (2022)
O. Rahaman, B. Mortazavi, A. Dianat, G. Cuniberti, T. Rabczuk, A structural insight into mechanical strength of graphene-like carbon and carbon nitride networks. Nanotechnology 28(5), 055707 (2016). https://doi.org/10.1088/1361-6528/28/5/055707
J.F. Nye. Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, ??? (1985). https://www.amazon.com/Physical-Properties-Crystals-Representation-Matrices/dp/0198511655?SubscriptionId=AKIAIOBINVZYXZQZ2U3A &tag=chimbori05-20 &linkCode=xm2 &camp=2025 &creative=165953 &creativeASIN=0198511655
P. Vannucci. Anisotropic Elasticity. Springer, ??? (2018). https://doi.org/10.1007/978-981-10-5439-6
V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, M. Scheffler, Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Commun. 180(11), 2175–2196 (2009). https://doi.org/10.1016/j.cpc.2009.06.022
V. Havu, V. Blum, P. Havu, M. Scheffler, Efficient o (n) integration for all-electron electronic structure calculation using numeric basis functions. J. Comput. Phys. 228(22), 8367–8379 (2009)
X. Ren, P. Rinke, V. Blum, J. Wieferink, A. Tkatchenko, A. Sanfilippo, K. Reuter, M. Scheffler, Resolution-of-identity approach to hartree-fock, hybrid density functionals, rpa, mp2 and gw with numeric atom-centered orbital basis functions. New J. Phys. 14(5), 053020 (2012)
J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77(18), 3865 (1996). https://doi.org/10.1103/physrevlett.77.3865
M. Iuga, G. Steinle-Neumann, J. Meinhardt, Ab-initio simulation of elastic constants for some ceramic materials. Eur. Phys. J. B 58(2), 127–133 (2007). https://doi.org/10.1140/epjb/e2007-00209-1
M.H. Manghnani, Elastic constants of single-crystal rutile under pressures to 7.5 kilobars. Journal of Geophysical Research 74(17), 4317–28 (1969)
R.M. Hazen, L.W. Finger, Bulk moduli and high-pressure crystal structures of rutile-type compounds. J. Phys. Chem. Solids 42(3), 143–151 (1981)
C. Lee, X. Wei, J.W. Kysar, J. Hone.(2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. science 321(5887):385-8. https://doi.org/10.1126/science.1157996
A. Castellanos-Gomez, M. Poot, G.A. Steele, H.S. Van Der Zant, N. Agraït, G. Rubio-Bollinger, Elastic properties of freely suspended mos2 nanosheets. Adv. Mater. 24(6), 772–775 (2012). https://doi.org/10.1002/adma.201103965
L. Song, L. Ci, H. Lu, P.B. Sorokin, C. Jin, J. Ni, A.G. Kvashnin, D.G. Kvashnin, J. Lou, B.I. Yakobson et al., Large scale growth and characterization of atomic hexagonal boron nitride layers. Nano Lett. 10(8), 3209–3215 (2010). https://doi.org/10.1021/nl1022139
J.-W. Jiang, H.S. Park, Negative poisson’s ratio in single-layer black phosphorus. Nat. Commun. 5(1), 1–7 (2014)
M. Elahi, K. Khaliji, S.M. Tabatabaei, M. Pourfath, R. Asgari, Modulation of electronic and mechanical properties of phosphorene through strain. Phys. Rev. B 91(11), 115412 (2015)
S. Kundalwal, S. Meguid, G. Weng, Strain gradient polarization in graphene. Carbon 117, 462–472 (2017)
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This work was funded by University of Zabol, Project code PR-UOZ1400-5.
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Eivari, H.A., Hafizi, R. Mechanical properties of two-dimensional sheets of TiO\(_2\): a DFT study. Eur. Phys. J. Plus 137, 1128 (2022). https://doi.org/10.1140/epjp/s13360-022-03316-z
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DOI: https://doi.org/10.1140/epjp/s13360-022-03316-z