Course NE.05 – Lecture 7
7/3/2006
Course NE.05 – Lecture 7
7/3/2006
Computational modelling of
nanomaterials
Introduction to techniques,
and application to mechanical
properties of CNTs
Dr James Elliott
1.1.1 Introduction
z
z
z
Computational modelling increasingly used as a tool to
study the physical properties of nanoscale systems
Often have poor control over experimental conditions,
and huge ‘space’ of possible solutions to search
Analytical theories of statistical mechanics and quantum
mechanics lead to equations too complicated to solve
MODELLING
The “Desirable Triangle”
THEORY
EXPT.
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
1
Course NE.05 – Lecture 7
7/3/2006
1.1.2 Introduction
z
In past 30 years, computational power (driven by Moore’s
Law) has increased by over 5 orders of magnitude
z
Computational modelling now ‘auto-catalyses’ its own
progress → exponential growth in progress!
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
1.1.3 Introduction
z
In this lecture, will briefly review range of modelling techniques that
are available, then look at applications to some specific examples
(CNT synthesis and electrically conductive composites)
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
2
Course NE.05 – Lecture 7
7/3/2006
1.2.1 Quantum mechanical modelling
z
z
z
The Schrödinger equation governs the temporal and spatial
evolution of the quantum mechanical wave function:
⎛ −= 2 2
⎞
∂Ψ (r , t )
∇ + V (r , t ) ⎟ Ψ ( r , t ) = i=
⎜
∂t
⎝ 2m
⎠
Time-dependent form
⎛ −= 2 2
⎞
∇ + V (r ) ⎟ Ψ (r ) = E Ψ (r )
⎜
⎝ 2m
⎠
Time-independent form
However, a rigorous solution is possible only for a very few
special cases (e.g. simple potential wells, hydrogen atom,
dihydrogen ion H2+, simple harmonic oscillator, rigid rotor)
Born-Oppenheimer approximation separates nuclear and
electronic degrees of freedom
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
1.2.2 Quantum mechanical modelling
z
z
For poly-electronic atoms and molecules, effects of
electron correlation and exchange interactions render an
exact solution impossible
To address this problem, there are two popular theories
that are amenable to computational treatment:
–
MOLECULAR ORBITAL THEORY
z
z
–
DENSITY FUNCTIONAL THEORY
z
z
z
Both ab initio or semi-empirical approaches
John Pople: Nobel Prize in Chemistry 1998 “for his development of
computational methods in quantum chemistry"
An ab initio treatment using approximate functionals
Walter Kohn: Nobel Prize in Chemistry 1998 "for his development of
the density-functional theory“
Often, we must resort to a classical approximation
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
3
Course NE.05 – Lecture 7
7/3/2006
1.3 Classical molecular mechanics modelling
z
z
z
z
z
So-called force field methods ignore electronic motions
and consider potential energy as function of only nuclear
co-ordinates
Can be applied to systems with hundreds of thousands of
atoms, and yield answers as accurate as even the
highest level of QM calculations
However, cannot be used in situations where properties
depend on electron distribution (e.g. bond dissociation,
electrical or magnetic properties)
Force fields constructed by parameterising potential
function using experimental data (X-ray and electron
diffraction, NMR and IR spectroscopy) or ab initio and
semi-empirical quantum mechanical calculations
Replace the true potential function with a simplified
model valid in the region being simulated
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
1.4 The Metropolis Monte Carlo algorithm
z
The Metropolis algorithm can be summarised as follows:
1.
2.
3.
4.
z
z
z
Start with a system in (an arbitrarily chosen) state µ and evaluate
the energy Eµ
Generate a new state ν by a perturbation to µ, and evaluate Eν
If Eν – Eσ < 0 then accept the new state. If Eν – Eσ > 0 then accept
the new state with probability exp[–β(Eν – Eσ )]
Return to step 2 and repeat until equilibrium is achieved
The Metropolis algorithm is characterised by having a
transition probability of unity if the new state has a lower
energy than the initial state
However, sequences of sampled states are uncorrelated
in time → no direct information about dynamics
MC methods compute thermodynamic averages
[1] Leach
R. “Molecular
modelling:
applications”
2nd ed., Prentice-Hall (2001) pp. 414
Copyright
© 2002A.
University
of Cambridge.
Not to principles
be quoted or&copied
without permission.
4
Course NE.05 – Lecture 7
7/3/2006
1.5 Continuous time molecular dynamics
z
z
z
z
z
By calculating the derivative of a macromolecular force
field, we find the forces on each atom as a function of its
position
Require a method of evolving the positions of the particles
in space and time to produce a ‘true’ dynamical trajectory
Standard technique is to solve Newton’s equations of
motion numerically, using some finite difference scheme,
which is known as integration.
This means that we advance the system by some small
time step ∆t, recalculate the forces and velocities, and then
repeat the process iteratively
Provided ∆t is small enough, this produces an acceptable
approximate solution to the continuous equations of motion
[1] Leach
R. “Molecular
modelling:
applications”
2nd ed., Prentice-Hall (2001) sec. 7.3
Copyright
© 2002A.
University
of Cambridge.
Not to principles
be quoted or&copied
without permission.
Part 2 : Mesoscale modelling of CNT synthesis
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
5
Course NE.05 – Lecture 7
7/3/2006
2.1.1 CNT synthesis by MO-CVD method
z
Experimental set-up:
Flow meter
1st Furnace
2nd Furnace
Substrate
Outlet
Ar
Flow meter
Ferrocene-Toluene
syringe pump
H2
Calcium
Chloride
z
Activated Paraffin
Carbon Bubbler
T = 550–940°C, ~10 wt.% of ferrocene in toluene
[1] Singh,
ShafferofM.S.P.,
Kinloch
Windle
A.H. without
Physicapermission.
B 323, 339-340 (2002).
Copyright
© 2002C.,
University
Cambridge.
Not toI.A.,
be quoted
or copied
[2] Li Y., Kinloch I.A. et al., Chem. Mat. 26, 5637-5643 (2004).
2.1.2 CNT synthesis by MO-CVD method
180
160
Length (microns)
Silica
140
120
100
80
60
40
20
0
0
50
100
150
200
250
300
350
400
Time (minutes)
Graphite
Mica
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
6
Course NE.05 – Lecture 7
7/3/2006
2.2.2 MC model for CNT growth on substrate
carbon
sheet
contact angle, θ
addition of
carbon
substrate
10°
addition of
carbon
catalyst
particle
h
30°
r
50°
70°
90°
Catalyst-Substrate Contact Angle
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
2.4 MC model for CNT growth on substrate
z
z
Graphene sheet represented by a triangular mesh
Mesh evolved by two-stage MMC process:
–
–
Addition
Nodes (carbon) added to mesh at base of catalyst particle
Relaxation
Mesh relaxed via potential energy function with terms involving:
Ec =
6
1
1
K c AJ 2 + ∑ K c A j J 2j
2
j =1 2
•
surface curvature
•
bond stretching Eb = K b (l − l0 )2
•
carbon-carbon interactions
•
carbon-catalyst interactions
1
2
acceptance probability = exp ⎡⎣ −β ( Enew − Eold ) ⎤⎦
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
7
Course NE.05 – Lecture 7
7/3/2006
2.5 Link between MC, classical MD, ab initio
bottom up
MC
MD (Classical)
Ab initio
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
2.6 Potentials for classical MD modelling
{
}
B * = 1 + b( N C − 1)
Eij = VR + V A
∑ f (r
N C = 1+
carbon k ( ≠ j )
)
VR: Repulsive term
NC: carbon coordinate number of metal atoms
VA: Attractive term
f(rij) : cut-off function
Co – C potential
Ni – C potential
Re(Å)
1.7628
β(1/Å)
1.8706
b
0.0688
δ
–0.5351
2
E(eV)
E(eV)
2
ab initio fitting
De(eV) 2.4673
CoC
CoC3
CoC4
ab initio fitting
De(eV) 3.7507
Re(Å)
1.6978
2
β(1/Å) 1.3513
b
0.0889
δ
–0.6256
E(eV)
ab initio fitting
NiC
NiC3
NiC4
Fe – C potential
4
4
4
ik
δ
0
FeC
FeC3
FeC4
De(eV) 3.3249
Re(Å)
β(1/Å)
1.7304
1.5284
b
0.0656
δ
–0.4279
0
0
–2
–2
–2
2 r(Å)
3 –4
2 r(Å)
3
–4
2 r(Å)
3
[1] Y.©Yamaguchi
and of
S.Cambridge.
Maruyama,
J.orD,copied
9, 385
(1999).
Copyright
2002 University
NotEur.
to bePhys.
quoted
without
permission.
[2] courtesy of Dr Y. Shibuta, analysis by B3LYP/LANL2DZ.
8
Course NE.05 – Lecture 7
7/3/2006
2.7 Simulation of free-standing metal clusters
Fe
Co
Ni
[1] courtesy
Dr Y. Shibuta,
analysis
B3LYP/LANL2DZ.
Copyright
© 2002 of
University
of Cambridge.
Not by
to be
quoted or copied without permission.
2.8 Defining carbon-catalyst interaction energy
Step1
cos θ =
γ SV − γ SL
γ LV
Step2
Step3
E of
Einteraction =
E of
-
E of
-
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
Surface area
9
Course NE.05 – Lecture 7
7/3/2006
2.9 Catalyst contact angle vs. deposition rate
1000
2500
5000 (slow)
30º
50º
70º
Number of MC relaxation steps per addition
500 (fast)
10º
90º
Catalyst-substrate contact angle
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
2.10 Contact angle vs. carbon-catalyst energy
0.01
0.1
1 (large)
30º
50º
70º
Carbon-catalyst interaction energy
0 (small)
10º
90º
Catalyst particle contact angle
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
10
Course NE.05 – Lecture 7
7/3/2006
Work in progress and the future…
Explore mechanism of bundle formation
MD
cf. Interaction for bundle
by MD calculation
(more than 1 month)
MC
link between MC and MD
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
Part 3: Mesoscale modelling of nanocomposites
L
λ = L/d
d
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
11
Course NE.05 – Lecture 7
7/3/2006
4.1 Percolation in nanofibre composites
z
Can improve conductivity of thermoplastic polymer matrix
by filling with nanofibres made from MWCNT bundles
specific electrical resistivity [Ω cm]
percolation threshold
16
10
10
10
8
0
0
10
205
40
carbon black content [wt
%]
[vol%]
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.2 Possible uses of conductive textiles
Smart-Shirt
firefly dress
Copyright © 2002 University of Cambridge. Not
(Source
to be quoted
: 15 AUGUST
or copied 2003
without
VOL
permission.
301 ‘SCIENCE’)
12
Course NE.05 – Lecture 7
7/3/2006
4.3 Classical percolation theory
MWCNT
Carbon
black
Carbon Black
CNT
Polymer-fibre interactions
Fibre-fibre interactions
Processing conditions
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.4.1 Monte Carlo modelling of percolation
Apply electric field across network
Measure impedance to current
motion by examining flux of
‘electrons’ as a function of field
⎛ ε ∆x ⎞
⎛ D⎞
pt = exp ⎜ − ⎟ exp ⎜ −
⎟
⎝ γ⎠
⎝ kBT ⎠
D : closest contact distance
γ : dielectric tunnelling length
ε : electric field strength
∆x : motion in field direction
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
13
Course NE.05 – Lecture 7
7/3/2006
4.4.2 Monte Carlo modelling of percolation
σ∝ J =
1.E-01
distance travelled
in steady-state
# MC steps
1.E-02
Flux, J
1.E-03
1.E-04
Percolation transition
1.E-05
σ ∝ (φ − φ c )t
1.E-06
1.E-07
0
0.05
0.1
0.15
0.2
0.25
0.3
Nanofibre volume fraction, φ
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.4.3 Monte Carlo modelling of percolation
Log (Flux)
-2.5
-2.6
t = 0.42
-2.7
φc =0.049 vol%
-2.8
-2.9
-3
-3.1
-3.2
-3.3
-2
-1.8
-1.6
-1.4
-1.2
-1
-0 .8
-0 .6
-0 .4
log (φ − φc )
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
14
Course NE.05 – Lecture 7
7/3/2006
4.4.4 Monte Carlo modelling of percolation
1.E+00
L/D = 1(t= 0.56)
1.E-01
L/D = 5 (t= 0.47)
L/D = 10 (t = 0.42)
1.E-02
Flux
L/D = 20 (t = 0.42)
L/D = 40 (t = 0.44)
1.E-03
L/D = 100 (t = 0.41)
L/D = 200 (t = 0.41)
1.E-04
1.E-05
1.E-06
1.E-07
0.001
0.01
0.1
1
Nanofibre volume fraction, φ
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.4.5 Monte Carlo modelling of percolation
Critical volume fraction, φc
1
Celzard [3.15 ]
Current predictions
0.1
0.01
0.001
1
10
100
1000
Aspect Ratio
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
15
Course NE.05 – Lecture 7
7/3/2006
4.5 Modelling effect of processing conditions
Temperature
Shear Stress
Viscosity
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.6 Quantifying orientational order
z
Order parameter <P2> is a measure of the quality of
alignment of the nanofibres
P2 =
3 cos 2 θ − 1
2
1.2
200 X 200 X 200
250 X 250 X 250
Order Parmeter
1
150 X 150 X 150
0.8
0.6
0.4
0.2
0
0
2000
4000
6000
8000
10000
# Timesteps
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
16
Course NE.05 – Lecture 7
7/3/2006
4.7 Effect of orientation on percolation threshold
1
Φc
0.1
0.01
0.001
0.0001
1
10
100
1000
Aspect Ratio
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
4.8 Conclusions from nanocomposites work
z
z
z
z
z
Predictions of distribution and orientation of nanofibres as
a function of aspect ratio and interfibre/matrix interactions
Predictions of electrical conductivity and critical
percolation threshold for nanofibre loading in agreement
with classical percolation theory
Will develop to predict effect of nanofibre rigidity on
critical percolation threshold
Will develop to predict effect of processing conditions
(shear, hydrostatic pressure, uniaxial extension) on
critical percolation threshold
Validate model predictions by preparing thermoplastics
and CNT composites by fibre spinning, extrusion etc.
Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.
17
