MECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR - PowerPoint PPT Presentation

MECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR DYNAMICS AND MORI − TANAKA METHODS Vinu Unnithan and J. N. Reddy US-South American Workshop: Mechanics and Advanced Materials − Research and Education August 2-6, 2004

Carbon Nanotube • CNTs can span 23,000 miles without failing due to its own weight. • CNTs are claimed to be 100 times stronger than steel. • Many times stiffer than any known material • Conducts heat better than diamond • Can be a conductor or insulator without any doping. • Lighter than feather.

Carbon Nanotubes • Carbon nanotube (CNT) is a tubular form of carbon with diameter as small as 1 nm. • CNT is equivalent to a two dimensional graphene sheet rolled into a tube. � CNT exhibits extraordinary mechanical properties • Young’s modulus over 1 Tera Pascal (as stiff as diamond) • tensile strength ~ 200 GPa. � CNT can be metallic or semiconducting, depending on chirality.

Polymer Composites Based on CNTs To make use of these extraordinary properties, CNTs are used as reinforcements in polymer based composites � CNTs can be in the form � Matrix can be � Single wall nanotubes � Polypropylene � Multi-wall nanotubes � PMMA � Powders � Polycarbonate � films � Polystyrene � paste � poly(3-octylthiophene) (P3OT)

Polymer Composites Based on CNTs What are the critical issues? • Structural and thermal properties • Bridging the scales • Load transfer and mechanical properties • Manufacturing

CONTENTS • Molecular Dynamics of CNTs • Internal Stress Tensor: Cauchy vs Virial Stress • Molecular Dynamics of CNT based Nanocomposite • Modeling • MD simulation • Micromechanics of CNT based composite • Homogenization principle using Mori-Tanaka Method • Two phase and three phase model • Conclusions

NANOMECHANICS Nanotechnology is the science and technology of precisely controlling the structure of matter at the molecular level. Carbon Nanotubes have very high modulus and are extremely light weight; hence, they find application in a variety of engineering scenarios. Single Walled Carbon Nanotube Multiple Walled Carbon Nanotube Carbon Nanorope

Nomenclature of Carbon Nanotube (CNT) Chiral vector is defined on the hexagonal lattice as Ch = n â 1 + m â 2 , where â 1 and â 2 are unit vectors, n and m are integers.

Molecular Dynamics Simulations MD simulations involve the determination of classical trajectories of atomic nuclei by integrating the Newton’s second law of motion ( F = m a ) of a system. Simulations are carried out on an N particle system Components of the Interatomic Interactions A common molecular dynamics force field has a form where the total potential energy is given by the sum of the following contributions: = + + + E U ( U U U ) vdW S B T { 1 4 2 4 4 3 4 NonBonded potential Valence interactio ns

Lennard-Jones (LJ) Potential (Non Bonded Potential) ⎡ ⎤ 12 6 ⎛ ⎞ ⎛ ⎞ r r r ≤ ⎜ ⎟ ⎜ ⎟ r ⎢ ⎥ = 0 − 0 U ( r ) 4 k ⎜ ⎟ ⎜ ⎟ ij c vdW ij ⎢ ⎥ r r ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ij ij 2 = ∑ − U K ( r r ) s s ij 0 1 , 2 Pairs k is a parameter characterizing the interaction strength r defines a molecular length scale. 0 r is the cutoff distance, and c = − r r r = r r ij i j ij ij K spring constant of bond stretching s

Stress Tensor Concept of stress extended to atomistic level, i.e., to every individual atom, we have the potential 3 N 1 ∂ W 1 α β αβ αβ = ε = −∇ = σ = ∑ Ω σ F E ( r ) W f ij r ∑∑ αβ ij i i ij ∂ ε V 2 αβ i αβ = 1 , i j Virial Stress ⎡ ⎤ N N N − 1 1 1 αβ α β σ = ⎢ + ⎥ ( t ) m v v r f i i ∑ ∑ ∑ i i ij ij * V 2 ⎣ ⎦ = + i i j i 1 ε Applied strain on the atomic bond αβ α , β Cartesian component of the stress in an atom i , is the volume of the Voronoi Cell of atom i , Ω i V * is the total volume of the system

Virial Stress is not Cauchy Stress • The first term in the virial stress denotes the thermodynamic pressure exerted by the atoms. • The second term arises from inter atomic forces. The KE term is small compared to the inter atomic forces for solids. • The interatomic force alone and fully constitutes the Cauchy stress ⎡ ⎤ − N 1 N 1 a a = ⎢ σ ( ) t r f ⎥ ∑ ∑ i ij ij * V ⎣ ⎦ = + i j i 1 Thus using the energy equivalence and the balance of linear momentum we can define an equivalent continuum representing a discrete particle system.

Elastic Properties of Carbon Nanotube by Molecular Dynamic Simulation Y ε ε Z zz zz X = + E ( U ) ( U ) vdW S 1 2 3 { Valence interactio ns NonBonded potential = U U − bonded bond stretch ⎡ ⎤ 12 6 ⎛ ⎞ ⎛ ⎞ 2 = ∑ − U K ( r r ) r r ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 0 0 − = − bon stretch s ij 0 U ( r ) 4 k ⎜ ⎟ ⎜ ⎟ ij ⎢ ⎥ r r 1 , 2 Pairs ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ij ij

Molecular Dynamics Modeling Unified Atom Model of Poly Ethylene Polyethylene chain & CNT Amorphous Matrix (450 Random Units) Crystalline Matrix

Elastic Properties of Carbon Nanotube by Molecular Dynamics Simulation 3.5E+12 3E+12 2.5E+12 800 Carbon (10,10) Axial Modulus (Pa) 2E+12 1.5E+12 1E+12 C(10,10) 5E+11 C (10,10) + PE 0 0 0.005 0.01 0.015 0.02 0.025 Strain Effect of Polyethylene Matrix on the Elastic Property of CNT

MICROMECHANICS OF CNT BASED COMPOSITES RVE VOLUME AVERAGE PROPERTIES For a atomic ensemble the volume average of the discrete stress and the discrete strain is given by N N 1 1 i i σ = σ ε = ε σ = ε C ∑ ∑ αβ αβ αβ αβ N N = = i 1 i 1 The volume average of the continuum stress and the continuum strain over the entire section 1 1 ε = ε σ = ε dv C σ = ∫ σ dv ∫ αβ αβ αβ αβ V V Ω Ω

MICROMECHANICS OF CNT BASED COMPOSITES Mori-Tanaka Method Assume the composite is composed of N phases. Three Phase Dispersed Model σ Uniform Stress 0 ε Uniform Strain 0 . Denotes a volume averaged quantity Dilute strain concentration factor dil A r k Effective modulus of the composite C S Eshelby Tensor for an ellipsoidal inclusion

MICROMECHANICS OF CNT BASED COMPOSITES dil ε = ε A r k 0 − N 1 ⎡ ⎤ − 1 [ ] dil dil + − − = −Ι S C C C A v S A ⎣ ⎦ ∑ k m k m k n n n n ( ) { } = − k , n f , g ,... N 1 ⎡ − ⎤ − N 1 k dil = Ι C C v A ⎢ ⎥ ∑ m k k ⎣ ⎦ = k 1 Mean Field elastic constitutive relations k k k σ = ε C tot tot

MICROMECHANICS OF CNT BASED COMPOSITES Effective property of two phase model of nanocomposite Modulus of Matrix = 8Gpa

MICROMECHANICS OF CNT BASED COMPOSITES CNT Interphase Bulk Matrix Modulus of Matrix = 8Gpa Effective property of three phase model of nanocomposite

CONCLUSIONS ◆ Modeling and simulation to find the effective properties of CNT reinforced PE nanocomposite using MD simulations are carried out. ◆ Surrounding matrix molecules are found to affect the overall stiffness of the CNT. ◆ MT method has been used to ascertain the effective property of the nanocomposite RVE using two phase and three phase interphase models. ◆ The variation of the effective properties of the composite has been obtained for various volume fractions of the CNT.