Nanomechanics of Carbon Nanotubes and Silicon Nanowires via Objective Molecular ModelingIlia Nikiforov, Dong-Bo Zhang, Traian Dumitrică. University of Minnesota.

12 13 14 15 16Stra

in E

nerg

y (b

LUE)

and

Der

ivat

ive

w/ R

espe

ct to

Ang

le (G

REE

N)

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 2.5Crit

ical

Rad

ius

of C

urva

ture

(nm

)

Radius (nm)

MWCNTs buckle at higher curvatures than SWCNTs of the

same diameter

SWCNTMWCNT

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 Virus1. nucleic acid (RNA)2. capsomer 3. capsid

4. Symmetry-Adapted DFT-based Tight-Binding

Xi,ζ = Xi +ζT , i = 1,...,N t

Xi,ζ1ζ 2= R2

ζ 2 R1ζ1 Xi +ζ1T1, i = 1,2

Z → T1,R1(θ1)Ch /d → R2 (θ2)

Fi = −∇X iV (Xi,ζ 1ζ 2

)

numberquantumhelicalnumberquantumangularNl a

1,...,0

→<≤−→−=

πκπ

αn, lκ ∝ζ 1 = 0

Ns −1∑ e

ζ 2 = 0

Na −1∑

ilθ 2ζ 2 + iκζ 1

αn,ζ1ζ 2

H(lκ)C( j,lκ) = Ei(lκ)S(lκ)C( j,lκ), j =1,...,Nobjele

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 ofCarbon 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

Stra

in E

nerg

y (e

V/a

tom

)

(a)

Ban

d ga

p (e

V)

(b) (c)

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 εc 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 amultiscale computational methodology based on a symmetry-adapted scheme. This objective will be achieved by pursuingthe following specific aims:Create a versatile symmetry-adapted density functionaltheory-based modeling capability by implementing the helicalboundary conditions into an existing density functional theorycomputational solver;Bridge the density functional theory description with finitedeformation continuum for the single-walled carbonnanotubes;Establish a dynamic mesoscopic model of the few-layerthick 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 Ys, and Shear modulus Gs as a function of NT diameter. Calculations were carried out on the “Helical-Angular” cell.

px ,ζ1ζ 2

px ,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.