Mathematical Modelling in Nanotechnology

Transcription

1 NMG Mathematical Modelling in Nanotechnology Dr Ngamta (Natalie) Thamwattana Nanomechanics Group, University of Wollongong Nanomechanics Group Supported by the Discovery Project scheme of the Australian Research Council Mechanics of carbon nanotubes Bionanotechnology Electrorheological fluids Thermal conductivity of nanofluids Nanofluidics 1

2 Developments in the resolving power of microscopes, has enabled us to see the smallest building blocks of nature. Carbon nanostructures 2

3 Fullerenes and Carbon Nanotubes D. Qian et al. Mechanics of carbon nanotubes, Appl. Mech. Rev. (2002) 55, Carbon Nanotubes Nanotubes are approximately 1-10 nanometers in diameter. Comprised of single and multi walled arrangements of carbon atoms. Can be either metals or semiconductors 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 3

4 tube axis C=n a 1 + ma 2 a 1 (0,0) (4,0) zigzag a 2 φ chiral C ( 0,3) (4,3) armchair Armchair: φ = 0 o (n,n) Zigzag: φ = 30 o (n,0) Chiral: 0 o < φ < 30 o (n,m) Structure of Carbon Nanotube a 1 = a 2 = armchair zigzag Circumference: C = 0.246(n 2 + nm + m 2 ) 1/2 Diameter: C /π 4

5 Electrical Properties Unique Electrical Properties n m = 0,3,6,9, metallic carbon nanotubes can carry extremely large current densities (>10 13 A/m 2 ) (household copper wire: <10 7 A/m 2 ) otherwise semiconducting carbon nanotubes can 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) 5

6 Icosahedral fullerenes 6

7 Symmetry of fullerenes Number of carbon atoms N in a fullerene C N : N = n + nm+ m ( ) Diameter of the icosahedron: 5 3a C d = C n + nm+ m π 2 2 ( ) 1/2 where a C-C is average carbon-carbon bond length I h symmetry I h type 1: n = m (e.g. C 60, C 240, C 540, C 960, C 1500 ) I h type 2: n = 0 or m = 0 (e.g. C 20, C 80, C 180 ) I symmetry: n m Euler s Formula for Polyhedra V E + F = 2 V: number of vertices E: number of edges F: number of faces 7

8 Euler s Observation Platonic Solid Vertices Edges Faces V E+F Tetrahedron Cube Octahedron Dodecahedron Icosahedron 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 8

9 Viruses tobacco mosaic virus Papillomavirus Adenovirus Gigahertz nano-oscillators Double-walled nanotube oscillators C 60 -single-walled nanotube oscillators Legoas et al., Phys. Rev. Lett. (2003) 90, Applications: Ultra-fast optical filters and ultra-sensitive nano-antennae 9

10 Sir John Edward Lennard- Jones Father of modern computational chemistry (October 27, 1894 November 1, 1954) Mathematician who held a chair of Theoretical Physics at Bristol University ( ) Proposed Lennard-Jones potential (1931) A chair of Theoretical Science at Cambridge University ( ) Holding the 1 st chair of Theoretical Chemistry in UK Atomic and molecular structures, valency and intermolecular forces Lennard-Jones potential 12 6 σ σ V() r 4ε = r r r min = σ 2 1/6, V min = ε The term 1/r 12, dominating at short distance, models the repulsion between atoms when they are very close to each other. V/ε r/σ ε : well depth, σ : van der Waals diameter The term 1/r 6, dominating at large distance, constitutes the attractive part (weak interaction). dv F = dr 10

11 Interaction energy between two carbon molecules The nonbonded interaction energy is obtained by summing the interaction potential energy for each atom pair E = i j V( r ) In continuum models, the interaction energy is obtained by averaging over the surface of each entity where n 1 and n 2 are the mean surface atomic density for each molecule, and r is the distance between two surface elements dσ 1 and dσ 2 on two different molecules. ij E = nn V( r) dσ dσ ( ) Oscillating C 60 in carbon nanotube Radius of C 60 = 3.55 Å (10,10) carbon nanotube Radius = Å (8,8) carbon nanotube Radius = Å 11

12 Issues 1. Acceptance Condition Will the C 60 be accepted into the nanotube? 2. Suction Energy How much energy will the C 60 gain from van der Waals interactions? 3. Oscillatory Dynamics What is the nature of the oscillatory motion? Acceptance Condition and Suction Energy C 60 will be accepted if: Z 0 FZdZ ( ) > 0 Suction energy W is the nett positive energy W = F ( Z ) dz 12

13 Acceptance Condition Local maxima define the energy level that needs to be overcome for a C 60 to be accepted Positive maxima may be overcome with initial kinetic energy (i.e. shoot the C 60 into the tube) Suction Energy Positive W when a > 6.27 Å Maximum value at a Å 13

14 Oscillatory Dynamics Van der Waals force pushes C 60 towards centre of nanotube Force acts only at nanotube ends Can be modelled with Dirac delta function For b < a << 2L, F tot z W is the pulse strength Force model [ δ ( Z + L) ( Z L) ] = W δ W = 0 F tot z ( Z) dz = F ( Z) dz 0 tot z W > 0, oscillating occurs 14

15 Velocity of C 60 Newton s second law tot dv Fz = mf dt Suction energy = Kinetic energy 2 mv W = Velocity of the oscillating C 60 f 2 v = 2W m f Oscillatory frequency C 60 travels inside the carbon nanotube at the constant speed Frequency: f = v/(4l) C 60 oscillates inside (10,10) with v = 932 m/s and f = GHz The shorter the carbon nanotube, the higher the frequency. 15

16 Double-walled carbon nanotube oscillator Legoas et al., Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators, Phys. Rev. Lett. (2003) 90, Oscillation of nanostructures Buckyball orbiting inside nanotorus Sector orbiting inside nanotorus Nanotorus oscillating along outside of nanotube 16

17 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 surface Degrade or release cap Taken up by cell Spill contents Corked or capped biodegradable? Injected, locate to target cell via chemical receptors 17

18 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) Discrete atom-atom formulation E = i j V( ρ ) ij Equivalent to: Discrete-continuum formulation E = η V( ρ ) dσ i Continuum formulation E = ηη V( ρ) dσ dσ i i Z ρ i Drug discrete a Nanotube continuous 18

19 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(NH 3 ) 2 Cl 2 (Milburn & Truter, 1966): Orientation 1 NH 3 NH 3 Pt Centre of mass Cl Cl Interaction energy a = 4.69 Å, not accepted Z (Å) Energy inside tube less than energy outside = accepted a = 4.95 Å, accepted a = 5.4 Å, accepted 19

20 Summary Application of mathematics in nanotechnology Gigahertz nano-oscillators Nanotubes for drug containers Provide overall guidelines for medical scientists and engineers Thank you! 20