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

MECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR DYNAMICS AND MORI−TANAKA METHODS

US-South American Workshop: Mechanics and Advanced Materials − Research and Education August 2-6, 2004

Vinu Unnithan and J. N. Reddy

• 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 Nanotube

• 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.

Carbon Nanotubes

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)

Polymer Composites Based on CNTs

What are the critical issues?

• Structural and thermal properties
• Bridging the scales
• Load transfer and mechanical properties
• Manufacturing

Polymer Composites Based on CNTs

• 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

CONTENTS

Single Walled Carbon Nanotube Multiple Walled Carbon Nanotube Carbon Nanorope

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.

Chiral vector is defined on the hexagonal lattice as Ch = nâ1 + mâ2, where â1 and â2 are unit vectors, n and m are integers.

Nomenclature of Carbon Nanotube (CNT)

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: 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

Molecular Dynamics Simulations

{ 4 4 3 4 4 2 1

ns interactio Valence potential NonBonded

) (

T B S vdW

U U U U E + + + =

Lennard-Jones (LJ) Potential (Non Bonded Potential)

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

6 12

4 ) (

ij ij ij vdW

r r r r k r U

c ij

r r ≤

2 2 , 1

) ( r r K U

ij Pairs s s

− = ∑

ij ij

r r =

j i ij

r r r − =

k

is a parameter characterizing the interaction strength

r

defines a molecular length scale.

c

r

is the cutoff distance, and

s

K spring constant of bond stretching

Concept of stress extended to atomistic level, i.e., to every individual atom, we have the potential

α ,β Cartesian component of the stress in an atom i,

Ωi

is the volume of the Voronoi Cell of atom i,

V* is the total volume of the system

Stress Tensor

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + =

∑ ∑ ∑

− + = N i N i j ij ij i N i i i i

f r v v m V t

1 1 *

2 1 1 ) (

β α αβ

σ

αβ

ε

Applied strain on the atomic bond

Virial Stress

∑Ω

= = ∂ ∂

i i i

V W

αβ αβ αβ

σ σ ε 1

∑∑

=

=

3 1 ,

2 1

αβ αβ β α

ε

N j i ij ij r

f W

) r ( F

ij

E −∇ =

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

1 * 1

1 ( )

a a

N N i ij ij i j i

t r f V σ

− = +

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

∑ ∑

Thus using the energy equivalence and the balance

• f linear momentum we can define an equivalent

continuum representing a discrete particle system.

Elastic Properties of Carbon Nanotube by Molecular Dynamic Simulation

2 2 , 1

) ( r r K U

ij Pairs s stretch bon

− = ∑

−

{

ns interactio Valence potential NonBonded

) ( ) (

S vdW

U U E + = 3 2 1

stretch bond bonded

U U

−

=

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

6 12

4 ) (

ij ij ij

r r r r k r U

Z X Y

zz

ε

zz

ε

Molecular Dynamics Modeling

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

5E+11 1E+12 1.5E+12 2E+12 2.5E+12 3E+12 3.5E+12 0.005 0.01 0.015 0.02 0.025

Strain

Axial Modulus (Pa) C(10,10) C (10,10) + PE

800 Carbon (10,10)

Effect of Polyethylene Matrix on the Elastic Property of CNT

Elastic Properties of Carbon Nanotube by Molecular Dynamics Simulation

RVE VOLUME AVERAGE PROPERTIES The volume average of the continuum stress and the continuum strain over the entire section

1 dv V

αβ αβ

σ σ

Ω

= ∫

1 dv V

αβ αβ

ε ε

Ω

=

∫

C σ ε =

1

1

N i i

N

αβ αβ

σ σ

=

=

∑

C σ ε =

For a atomic ensemble the volume average of the discrete stress and the discrete strain is given by

1

1

N i i

N

αβ αβ

ε ε

=

=

∑

MICROMECHANICS OF CNT BASED COMPOSITES

Mori-Tanaka Method

.

Denotes a volume averaged quantity

dil r

A

Dilute strain concentration factor Effective modulus of the composite

k

C

Assume the composite is composed of N phases.

σ ε

Uniform Stress Uniform Strain

S

Eshelby Tensor for an ellipsoidal inclusion

Three Phase Dispersed Model

MICROMECHANICS OF CNT BASED COMPOSITES

dil r k

A ε ε =

[ ]

1 1 N dil dil k m k m k n n n n

S C C C A v S A

− −

⎡ ⎤ + − − = −Ι ⎣ ⎦

∑

( ) { }

1 ,... , , − = N g f n k

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − Ι =

∑

− = 1 1 N k dil k k m k

A v C C

k tot k k tot

C ε σ =

Mean Field elastic constitutive relations

MICROMECHANICS OF CNT BASED COMPOSITES

Modulus of Matrix = 8Gpa

MICROMECHANICS OF CNT BASED COMPOSITES

Effective property of two phase model

• f nanocomposite

Modulus of Matrix = 8Gpa Interphase CNT Bulk Matrix Effective property of three phase model

• f nanocomposite

MICROMECHANICS OF CNT BASED COMPOSITES

◆ 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.

CONCLUSIONS