Nanomechanics of Carbon Nanotubes and Silicon Nanowires via Objective Molecular Modeling Ilia Nikiforov, Dong-Bo Zhang, Traian Dumitrică. University of Minnesota. 12 13 14 15 16 Strain Energy (bLUE) and Derivative w/ Respect to Angle (GREEN) Bending Angle (deg) Second-order discontinuity in energy signals onset of buckling in CNTs (5,5) (10,10) (15,15) (20,20) (25,25) (30,30) 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 Critical Radius of Curvature (nm) Radius (nm) MWCNTs buckle at higher curvatures than SWCNTs of the same diameter SWCNT MWCNT 1. Goals 3. Objective Molecular Dynamics Replaces PBC over a large translational cell with Objective Boundary Conditions over a small objective cell: Repetition Rule = Translation + Rotation 2. Chiral Systems: Examples from Nanostructures and Biology SiGe/Si Nanowires Tobacco Mosaic Virus 1. nucleic acid (RNA) 2. capsomer 3. capsid 4. Symmetry-Adapted DFT-based Tight-Binding X i , ζ = X i + ζT , i = 1,..., N t X i , ζ 1 ζ 2 = R 2 ζ 2 R 1 ζ 1 X i + ζ 1 T 1 , i = 1, 2 Z → T 1 , R 1 ( θ 1 ) C h / d → R 2 ( θ 2 ) F i = −∇ X i V ( X i , ζ 1 ζ 2 ) number quantum helical number quantum angular N l a 1 ,..., 0 → < ≤ − → − = π κ π αn , lκ ∝ ζ 1 = 0 N s − 1 ∑ e ζ 2 = 0 N a − 1 ∑ il θ 2 ζ 2 + iκζ 1 αn ,ζ 1 ζ 2 H ( lκ ) C ( j , lκ ) = E i ( lκ ) S ( lκ )C ( j, lκ ), j = 1,..., N obj ele Schroedinger Equation in Matrix Form: (4,2) CNT References: • T. Dumitrica and R. D. James, Objective Molecular Dynamics, Journal of the Mechanics and Physics of Solids 55, 2206 (2007). • D.-B. Zhang, M. Hua, and T. Dumitrica, Journal of Chemical Physics 128, 084104 (2008). Carbon and Boron-Nitride Nanotubes “Translational” and “Helical-Angular” Representation of Carbon Nanotubes References: • S. S. Alexandre, M. S. C. Mazzoni, and H. Chachama, Stability, geometry, and electronic structure of the boron nitride B36N36 fullerene, Applied Physics Letters 75, 1 (1999). 6. Application: Electromechanical Characterization of Carbon Nanotubes in Torsion Strain Energy (eV/atom) Band gap (eV) The objective methodology allows us to derive the nonlinear elastic response of CNTs in torsion from a density-functional- based tight-binding model. Figures below reveal a sharply contrasting behavior in the electronic response. The critical strain ε c beyond which CNTs behave nonlinearly, the most favorable rippling morphology, and the twist- and morphology- related changes in fundamental band gap are identified. Results are assistive for experiments performed on CNT-pedal devices. In single-walled CNTs the band gap variations are dominated by rippling. Band gap of multi-walled CNTs exhibits an unexpected insensitivity. References: • D.-B. Zhang, R.D. James, and T. Dumitrica, Physical Review B (at press). The research objective of this project is to develop a multiscale computational methodology based on a symmetry- adapted scheme. This objective will be achieved by pursuing the following specific aims: Create a versatile symmetry-adapted density functional theory-based modeling capability by implementing the helical boundary conditions into an existing density functional theory computational solver; Bridge the density functional theory description with finite deformation continuum for the single-walled carbon nanotubes; Establish a dynamic mesoscopic model of the few-layer thick SiGe/Si and ZnO nanobelts. CNTs 5. Application: Linear and non-Linear Elasticity of Carbon Nanotubes Tight-Binding treatment under objective boundary conditions makes possible to compute the linear and non-linear elastic mechanical response of nanotubes Young’s modulus Y s , and Shear modulus G s as a function of NT diameter. Calculations were carried out on the “Helical-Angular” cell. p x ,ζ 1 ζ 2 p x ,00 Tight-Binding solution is represented in terms of “Helical-Angular” Adapted Bloch Sums: BNNTs The objective method allows for efficient treatment of pure bending in quasi-one-dimensional structures. Figures bellow illustrate how it can be applied to study buckling in carbon nanotubes. References: • D.-B. Zhang and T. Dumitrica, Applied Physics Letters 93, 031919 (2008). • I. Nikiforov, D-B. Zhang and T. Dumitrica, In progress.