MECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR DYNAMICS AND MORI TANAKA METHODS. Vinu Unnithan and J. N. Reddy

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1 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

2 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.

3 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. Carbon Nanotubes 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.

4 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 Single wall nanotubes Multi-wall nanotubes Powders films paste Matrix can be Polypropylene PMMA Polycarbonate Polystyrene poly(3-octylthiophene) (P3OT)

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

6 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

7 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

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

9 Molecular Dynamics Simulations MD simulations involve the determination of classical trajectories of atomic nuclei by integrating the Newton s second law of motion (F = ma) 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: { vdw + ( U S + U B UT ) E = U + NonBonded potential Valenceinteractions

10 Lennard-Jones (LJ) Potential (Non Bonded Potential) U vdw ( r ) = r 4k r 0 12 r r 0 6 r r c k r U s = K 1,2 Pairs s ( r r r = r i j = 0 r ) 2 is a parameter characterizing the interaction strength r0 defines a molecular length scale. r is the cutoff distance, and c r K s spring constant of bond stretching

11 Stress Tensor Concept of stress extended to atomistic level, i.e., to every individual atom, we have the potential W = N αβ = 1 i, j α f r β ε αβ W ε αβ = σ αβ = 1 V Ω i i σ αβ i F = E(r ) Virial Stress σ αβ i ( t) N N N α β = miv iv i + r * V 2 i i j= i+ 1 f εαβ Applied strain on the atomic bond α,β Cartesian component of the stress in an atom i, Ω i is the volume of the Voronoi Cell of atom i, V * is the total volume of the system

12 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 σ Na 1 Na i() t = r f 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. 1 V *

13 Elastic Properties of Carbon Nanotube by Molecular Dynamic Simulation Y Z ε zz ε zz X E = ( U vdw ) + ( U S ) 123 NonBonded potential U = Valence { bonded U bond stretch interactio ns U ( r ) = r 4k r 0 12 r r 0 6 U bon stretch = K 1,2 Pairs s ( r r 0 ) 2

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

15 Elastic Properties of Carbon Nanotube by Molecular Dynamics Simulation 3.5E+12 3E E Carbon (10,10) Axial Modulus (Pa) 2E E+12 1E+12 5E+11 C(10,10) C (10,10) + PE Strain Effect of Polyethylene Matrix on the Elastic Property of CNT

16 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 σ αβ 1 N i σαβ i= 1 = εαβ = εαβ σ = Cε N 1 N i N i= 1 The volume average of the continuum stress and the continuum strain over the entire section 1 1 σαβ = σαβ dv V εαβ = εαβ dv V σ = C Ω Ω ε

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

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

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

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

21 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.