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NMGNMG Mathematical Mathematical Modelling in Modelling in

NanotechnologyNanotechnology

Dr Ngamta (Natalie) ThamwattanaNanomechanics Group, University of Wollongong

NanomechanicsNanomechanics GroupGroup

• Mechanics of carbon nanotubes

• Bionanotechnology

• Electrorheologicalfluids

• Thermal conductivity of nanofluids

• Nanofluidics

http://research.uow.edu.au/nano/

Supported by the Discovery Project scheme of the Australian Research Council

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Developments in the resolving power of microscopes, has enabled us to see the smallest building blocks of nature.

Carbon nanostructures

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Fullerenes and Fullerenes and Carbon Carbon NanotubesNanotubes

D. Qian et al. Mechanics of carbon nanotubes, Appl. Mech. Rev. (2002) 55, 495‐533.

Carbon Carbon NanotubesNanotubes• Nanotubes are approximately 1-10

nanometers in diameter.• Comprised of single and multi walled

arrangements of carbon atoms.• Can be either metals or semi-

conductors depending on structure.• Incredibly strong and have great thermal

conductivity.• Applications include nano electronic and

mechanical components.

Carbon Nanotubes – can be thought of graphitic sheets with hexagonal lattice wrapped into a seamless cylinder

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)0,0(1a

2a

zigzag

armchair

φ

)0,4(

)3,0( )3,4(

chiral

C

21 aaC mn +=

axis tube

Structure of Carbon Nanotube

• Armchair: φ = 0o (n,n)Zigzag: φ = 30o (n,0)Chiral: 0o < φ < 30o (n,m)

• |a1| = |a2| = 0.246

• Circumference: |C| = 0.246(n2 + nm + m2)1/2

• Diameter: |C|/π

armchair zigzag

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Electrical Properties Electrical Properties

• Unique Electrical Properties• n – m = 0,3,6,9,… metallic carbon nanotubes

can carry extremely large current densities (>1013 A/m2)(household copper wire: <107 A/m2)

• otherwise semiconducting carbon nanotubescan be electrically switched on and off as field-effect transistors (~ 500 times smaller than current devices)

• Potential Applications: Nanotube-based electronics

Icosahedral fullerenes

• Goldberg fullerenes consist of twenty equilateral triangles, each specified by (n, m)

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Icosahedral fullerenes

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Symmetry of fullerenes• Number of carbon atoms N in a fullerene CN:

• Diameter of the icosahedron:

where aC-C is average carbon-carbon bond length

• Ih symmetryIh type 1: n = m (e.g. C60, C240, C540, C960, C1500 )Ih type 2: n = 0 or m = 0 (e.g. C20, C80, C180)

• I symmetry: n ≠ m

2 220( )N n nm m= + +

( )1/ 22 25 3 C Cad n nm m

π−= + +

Euler’s Formula for Polyhedra

V − E + F = 2

V: number of verticesE: number of edgesF: number of faces

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Euler’s Observation

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12

864

Faces

23012Icosahedron

23020Dodecahedron

2126Octahedron2128Cube264Tetrahedron

V−E+FEdgesVerticesPlatonic Solid

Number of each type of face

Let p be the number of pentagonal faces and h be the number of hexagonal faces

V = (5p+6h)/3, E = (5p+6h)/2, F = p+h,So

2 = V – E + F,2 = (5p+6h)/3 − (5p+6h)/2 + p+h,12 = 10p + 12h − 15p − 18h + 6p + 6h,12 = p.

Therefore, such a surface must contain exactly 12 pentagons

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Viruses tobacco mosaic virus

Adenovirus Papillomavirus

Gigahertz nano-oscillators

Double-walled nanotube oscillatorsC60-single-walled nanotube oscillators

http://tam.northwestern.edu.wkl/c60_in_tubeLegoas et al., Phys. Rev. Lett. (2003) 90, 055504.

Applications: Ultra-fast optical filters and ultra-sensitive nano-antennae

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Sir John Edward Sir John Edward LennardLennard--JonesJones

• Mathematician who held a chair of Theoretical Physics at Bristol University (1925-32)

• Proposed Lennard-Jones potential (1931)

• A chair of Theoretical Science at Cambridge University (1932-53)

• Holding the 1st chair of Theoretical Chemistry in UK

• Atomic and molecular structures, valency and intermolecular forces

(October 27, 1894 – November 1, 1954)

“Father of modern computational chemistry”

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( ) 4V rr rσ σε

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

• The term 1/r12, dominating at short distance, models the repulsion between atoms when they are very close to each other.

• The term 1/r6, dominating at large distance, constitutes the attractive part (weak interaction).

V/ε

r/σ

ε : well depth, σ : van der Waals diameter

rmin = σ 21/6, Vmin = −ε

LennardLennard--Jones potentialJones potential

dVFdr

= −

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Interaction energy between Interaction energy between two carbon moleculestwo carbon molecules

• The nonbonded interaction energy is obtained by summing the interaction potential energy for each atom pair

• In continuum models, the interaction energy is obtained by averaging over the surface of each entity.

where n1 and n2 are the mean surface atomic density for each molecule, and r is the distance between two surface elements dΣ1 and dΣ2 on two different molecules.

( )iji j

E V r=∑∑

1 2 1 2( ) ( )E n n V r d d= Σ Σ ∗∫∫

(10,10) carbon nanotubeRadius = 6.784 Å

(8,8) carbon nanotubeRadius = 5.428 Å

http://tam.northwestern.edu/wkl/c60_in_tube

Radius of C60 = 3.55 Å

Oscillating Oscillating CC6060 in carbon in carbon nanotubenanotube

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1. Acceptance ConditionWill the C60 be accepted into the nanotube?

2. Suction EnergyHow much energy will the C60 gain from van der Waals interactions?

3. Oscillatory DynamicsWhat is the nature of the oscillatory motion?

IssuesIssues

Acceptance Condition and Acceptance Condition and Suction EnergySuction Energy

0

( ) 0Z

F Z dZ−∞

>∫

Suction energy W is the nettpositive energy

C60 will be accepted if:

( )W F Z d Z∞

−∞

= ∫

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Acceptance ConditionAcceptance Condition

• Local maxima define the energy level that needs to be overcome for a C60 to be accepted

• Positive maxima may be overcome with initial kinetic energy (i.e. shoot the C60 into the tube)

Suction EnergySuction Energy

• Positive W when a > 6.27 Å

• Maximum value at a ≈ 6.783 Å

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Oscillatory DynamicsOscillatory Dynamics

• Van der Waals force pushes C60 towards centre of nanotube

• Force acts only at nanotube ends

• Can be modelled with Dirac delta function

Force model

•• For For bb < < a a << << 22LL,,

• W is the pulse strength

• W > 0, oscillating occurs

[ ])()( LZLZWF totz −−+= δδ

∫ ∫−==∞−

∞0

0)()( dZZFdZZFW tot

ztotz

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Velocity of C60

• Newton’s second law

• Suction energy = Kinetic energy

• Velocity of the oscillating C60

totz f

dvF mdt

=

2

2fm v

W =

2

f

Wvm

=

Oscillatory frequency• C60 travels inside the

carbon nanotube at the constant speed

• Frequency: f = v/(4L)

• C60 oscillates inside (10,10) with v = 932 m/sand f = 36.13 GHz

The shorter the carbon nanotube, the higher the frequency.

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DoubleDouble--walled carbon walled carbon nanotube oscillatornanotube oscillator

Legoas et al., “Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators”, Phys. Rev. Lett. (2003) 90, 055504.

Oscillation of nanostructuresOscillation of nanostructures

• Sector orbiting inside nanotorus

• Buckyball orbiting inside nanotorus

• Nanotorus oscillating along outside of nanotube

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Nanotubes for drug delivery

• Advantages: (Martin & Kohli, 2003)– Larger inner volume– Distinct inner/outer surfaces with open ends– Readily taken up (Kam et al., 2004), enter cell nuclei

(Pantarotto et al., 2004)

• Filling techniques: (Gasparac et al., 2004)– Immerse in solution– Attach drugs to tube walls– Fill with particles (Kim et al., 2005)

• Example: test tube (Hillebrenner et al., 2006)– Convenient filling

The process

Drug encapsulated

Functionalized surfaceCorked or capped – biodegradable?

Injected, locate to target cell via

chemical receptorsTaken up

by cell

Degrade or release cap

Spill contents

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Engineered nanocapsule

• Energetically favourable for drug to be encapsulated

• Once at target site energetically favourable to be ejected from capsule

• Understand suction and expulsion characteristics– Predict whether drug will be accepted into nanotube– Radius of tube required for particular drug &

maximum intake of drug

Formulating energy

• Use discrete-continuum formulation (Hilder & Hill, 2006)

• Discrete not necessarily preferable to continuum, continuum may be “closer to reality than a set of discrete LJ centers” (Girifalco et al., 2000)

( )iji j

E V ρ=∑∑

Discrete atom-atom formulation

1 2 1 2( )E V d dηη ρ= Σ Σ∫ ∫Continuum formulation

Equivalent to:

( )ii

E V dη ρ= Σ∑∫Discrete-continuum formulation aρi

i

Z Nanotube continuous

Drug discrete

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Example: cisplatin

• Platinum-based anticancer drug (Pratt et al., 1994)

• Frequently used to treat tumours in…– Ovary, head, neck, lung, bladder, testis

• Side effects include…– Kidney damage, nerve damage, hearing loss,

nausea, vomiting• Simple structure, Pt(NH3)2Cl2 (Milburn & Truter, 1966):

Pt

Cl Cl

NH3 NH3

Centre of massOrientation 1

Interaction energy

Energy inside tube less than energy outside = acceptedZ (Å)

a = 5.4 Å, accepted

a = 4.69 Å, not accepted

a = 4.95 Å, accepted

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Summary

• Application of mathematics in nanotechnology

• Gigahertz nano-oscillators

• Nanotubes for drug containers

• Provide overall guidelines for medical scientists and engineers

Thank you!

http://research.uow.edu.au/nano/