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Exploring properties of silicon-carbide nanotubes and their composites with polymers

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Content
Exploring Properties of Silicon-Carbide Nanotubes and Their
Composites with Polymers

by
Mahdi Khademi

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMICAL ENGINEERING)

May 2015

Copyright 2015 Mahdi Khademi

ii

Acknowledgments
I would like to use this opportunity to express my sincere gratitude toward my advisor,
Professor Dr. Muhammad Sahimi, for his guidance, motivational talks, and enthusiasm through
the years of my graduate studies and research. The support that I received and the opportunity to
do research and study at USC under his guidance have been priceless.
I would also like to express my appreciation for my Ph.D. committee members, Professor
Dr. Rajiv Kalia and Professor Dr. Aiichiro Nakano for their support and accepting to be members
of my committee. Professor Kalia’s suggestion for the work in Chapter 3 opened a new horizon
for my research.
Last, but not least, I would like to thank my family: my mother, Parvin, and my father,
Mostafa, as well as my brother Alireza for their priceless and continuous support that helped me
complete my graduate studies.

iii

Table of Contents
Acknowledgments........................................................................................................................... ii
List of Figures ................................................................................................................................ vi
List of Tables ................................................................................................................................. xi
Abstract ........................................................................................................................................ xii
Chapter 1 : Background .................................................................................................................. 1
1.1 Introduction ....................................................................................................................... 1
1.2 Nanotubes ......................................................................................................................... 1
1.3 Silicon-Carbide Nanotubes ............................................................................................... 3
1.4 Stability and Properties of SiC Nanotubes ........................................................................ 5
1.5 Fabrication of SiC Nanotubes ........................................................................................... 5
1.6 The Goals of the Thesis .................................................................................................... 6
1.7 Molecular Models and Simulation .................................................................................... 7
1.8 The COMPASS Force Field ............................................................................................. 8
1.9 Construction of the System for MD Simulation ............................................................. 11
1.10 The Nosé-Hoover Thermostat ....................................................................................... 11
1.11 The Parrinello-Rahman Method of Applying an External Pressure36 ......................... 12
1.12 Molecular Dynamics Simulation .................................................................................. 12
1.13 The NPT Ensemble ....................................................................................................... 14
References ..................................................................................................................................... 15
Chapter 2 : Molecular Dynamics Simulation of Pressure-Driven Water Flow ............................ 18
2.1 Introduction ..................................................................................................................... 18
2.2 SiC Nanotubes ................................................................................................................ 18
iv

2.3 Transport of Fluids in Nanotubes ................................................................................... 19
2.4 Review of the Previous Works ....................................................................................... 20
2.5 Details of Molecular Dynamics Simulation .................................................................... 21
2.6 Conclusions ..................................................................................................................... 33
References ............................................................................................................................. 34
Chapter 3 : Dynamics of Low-Temperature Water in Nanotubes ............................................... 39
3.1 Introduction ..................................................................................................................... 39
3.2 Water, Ice and Hydrogen Bonds ..................................................................................... 39
3.3 Confined Water ............................................................................................................... 40
3.3.1 Subcooled Water ................................................................................................ 41
3.3.2 Supercooled Confined Water: The Cage Effect ................................................. 42
3.4 SiCNT and Molecular Modeling..................................................................................... 42
3.5 Molecular Dynamic Simulation ...................................................................................... 44
3.6 Stretched Exponential Decay .......................................................................................... 46
3.7 Conclusions ..................................................................................................................... 51
References ............................................................................................................................. 52
Chapter 4 : Static and Dynamic Properties of Supercooled Water in Nanotubes ......................... 55
4.1 Introduction ..................................................................................................................... 55
4.2 Motivation ....................................................................................................................... 56
4.3 Molecular Dynamics Simulation .................................................................................... 56
4.4 Spatial and Temporal Correlation Functions .................................................................. 58
4.4.1 Self Space-Time Autocorrelation Function ....................................................... 59
4.4.2 Radial Distribution Function .............................................................................. 65
v

4.5 Diffusion and Velocity Autocorrelation Function .......................................................... 79
4.6 Conclusions ..................................................................................................................... 93
References ............................................................................................................................. 95
Chapter 5 : Mechanical, Transport and Sorption Properties of Polyetherimide-SiCNT .............. 98
5.1 Introduction ..................................................................................................................... 98
5.2 The Polymer Matrix ........................................................................................................ 99
5.3 Molecular Dynamics Simulation .................................................................................. 100
5.4 Diffusion of Gases in the PEI-SiCNT Composites ....................................................... 102
5.5 Gas Sorption .................................................................................................................. 105
5.6 Monte Carlo Simulations .............................................................................................. 107
5.7 Results and Discussion ................................................................................................. 108
5.8 Stress and Deformation ................................................................................................. 117
5.9 Analysis of the Stress with MD Simulation .................................................................. 120
5.10 Results and Discussions .............................................................................................. 121
5.11 Conclusions ................................................................................................................. 123
References ........................................................................................................................... 124
Appendix A: Compliance and Stiffness Factors ................................................................. 126

vi

List of Figures
Figure 1.1 Honeycomb lattice vectors ............................................................................................ 3
Figure 1.2 Schematic demonstration of the zigzag (9,0) and armchair (9,9) nanotubes ................ 4
Figure 1.3 High-resolution transmission microscopy of SiCNT.
11
................................................. 7
Figure 2.1 Axial velocity autocorrelation function in the smallest and largest SiCNTs .............. 23
Figure 2.2 Dependence of water viscosity on the diameter of SiCNTs. ....................................... 24
Figure 2.3 Dependence of mean axial water velocity on the applied pressure gradient ΔP/ L in
the four SiCNTs. ................................................................................................................... 25
Figure 2.4 Axial water concentration profile in the (9,0) SiCNT. The applied pressure gradient is
ΔP/ L = 5.7 TPs/m. ................................................................................................................ 26
Figure 2.5 Radial water concentration profiles in the (9,0) SiCNT. The applied pressure gradient
is ΔP/ L = 5.7 TPs/m. ............................................................................................................ 27
Figure 2.6 Same as in Figure 2.4, but with an applied pressure gradient of ΔP/ L = 18.9 ........... 28
Figure 2.7 Same as in Figure 2.5, but with an applied pressure gradient of ΔP/ L = 18.9 ........... 29
Figure 2.8 Same as in Figure 2.4, but for a (16,0) SiCNT with an applied pressure gradient of
ΔP/ L = 5.7 TPa/m. ............................................................................................................... 30
Figure 2.9 Same as in Figure 2.5, but for a (16,0) SiCNT with an applied pressure gradient of
ΔP/ L = 5.7 TPa/m. ............................................................................................................... 31
Figure 2.10 Dependence of flow enhancement factor on the applied pressure gradient in the four
SiCNTs. ................................................................................................................................. 32
Figure 2.11 Comparison of flow enhancement factors of SiC and carbon nanotubes for the same
applied pressure gradient. ..................................................................................................... 33
Figure 3.1 The system under study: a zigzag SiCNT immersed in water. .................................... 44
vii

Figure 3.2 Diffusion paths of water molecules inside the nanotube. ............................................ 46
Figure 3.3 The cage correlation function at two temperatures. .................................................... 47
Figure 3.4 Double logarithmic plot of the cage correlation function at 250 K ............................. 48
Figure 3.5 Same as in Figure 3.4, but at 273 K. ........................................................................... 49
Figure 3.6 The cage correlation fucntion versus t/τ, where τ is the relaxation time, at two
temperatures. ......................................................................................................................... 50
Figure 4.1 The SiCNT and water system. ..................................................................................... 57
Figure 4.2 Self space-time autocorrelation function for water at 298 K in SiCNT. ..................... 59
Figure 4.3 Self space-time autocorrelation function for water at 273 K in SiCNT. ..................... 60
Figure 4.4 Self space-time autocorrelation function for water at 250 K in SiCNT. ..................... 60
Figure 4.5 Self space-time autocorrelation function for water at 225 K in SiCNT. ..................... 61
Figure 4.6 Self space-time autocorrelation function for water at 200 K in SiCNT. ..................... 61
Figure 4.7 Self space-time autocorrelation function for water at 175 K in SiCNT. ..................... 62
Figure 4.8 Self space-time autocorrelation function for water at 150 K in SiCNT. ..................... 62
Figure 4.9 Self space-time autocorrelation function for water at 125 K in SiCNT ...................... 63
Figure 4.10 Self space-time autocorrelation function for water at 100 K in SiCNT. ................... 63
Figure 4.11 Self space-time autocorrelation function for water at 273 K in CNT. ...................... 64
Figure 4.12 Self space-time autocorrelation function for water at 250 K in CNT. ...................... 64
Figure 4.13 Radial Distribution function for water and HH in SiCNT. ....................................... 68
Figure 4.14 Radial distribution function for OO and OH in SiCNT. ........................................... 69
Figure 4.15 Radial distribution function for water and HH in SiCNT at 273 K........................... 69
Figure 4.16 Radial distribution function for OO and OH in SiCNT at 273 K. ............................. 70
Figure 4.17 Radial distribution function for water and HH in SiCNT at 250 K........................... 70
viii

Figure 4.18 Radial distribution function for OO and OH in SiCNT at 250 K. ............................. 71
Figure 4.19 Radial distribution function for water and HH in SiCNT at 225 K........................... 71
Figure 4.20 Radial distribution function for OO and OH in SiCNT at 225 K. ............................. 72
Figure 4.21 Radial distribution function for water and HH in SiCNT at 200 K........................... 72
Figure 4.22 Radial distribution function for OO and OH in SiCNT at 200 K. ............................. 73
Figure 4.23 Radial distribution function for water and HH in SiCNT at 175K............................ 73
Figure 4.24 Radial distribution functions for OO and OH in SiCNT at 175 K. ........................... 74
Figure 4.25 Radial distribution functions for water and HH in SiCNT at 150 K. ........................ 74
Figure 4.26 Radial distribution functions for OO and OH in SiCNT at 150 K. ........................... 75
Figure 4.27 Radial distribution functions for water and HH in SiCNT at 125 K. ........................ 75
Figure 4.28 Radial distribution functions for OO and OH in SiCNT at 125 K. ........................... 76
Figure 4.29 Radial distribution functions for water and HH in SiCNT at 100 K. ........................ 76
Figure 4.30 Radial distribution functions for OO and OH in SiCNT at 100 K. ........................... 77
Figure 4.31 Radial distribution functions for water and HH in CNT at 273 K. ........................... 77
Figure 4.32 Radial distribution functions for OO and OH in CNT at 273 K. .............................. 78
Figure 4.33 Radial distribution functions for water and HH in CNT at 250 K. ........................... 78
Figure 4.34 Radial distribution functions for OO and OH in CNT at 250 K. .............................. 79
Figure 4.35 Velocity autocorrelation function for water at 298 K in SiCNT ............................... 81
Figure 4.36 Velocity autocorrelation function for water at 273 K in SiCNT ............................... 82
Figure 4.37 Velocity autocorrelation function for water at 250 K in SiCNT. .............................. 82
Figure 4.38 Velocity autocorrelation function for water at 225 K in SiCNT ............................... 83
Figure 4.39 Velocity autocorrelation function for water at 200 K in SiCNT. .............................. 83
Figure 4.40 Velocity autocorrelation function for water at 175 K in SiCNT. .............................. 84
ix

Figure 4.41 Velocity autocorrelation function for water at 150 K in SiCNT ............................... 84
Figure 4.42 Velocity Autocorrelation function for water at 125 K in SiCNT .............................. 85
Figure 4.43 Velocity Autocorrelation function for water at 100 K in SiCNT. ............................. 85
Figure 4.44 Velocity autocorrelation function for water at 273 K in CNT. ................................. 86
Figure 4.45 Velocity autocorrelation function for water at 250 K in CNT. ................................. 86
Figure 4.46 Mean-square displacement of water at 298K in SiCNT. ........................................... 87
Figure 4.47 Mean-square displacements of water at 273 K in SiCNT ......................................... 88
Figure 4.48 Mean-square displacements of water at 250K in SiCNT. ......................................... 88
Figure 4.49 Mean-square displacements of water at 225 K in SiCNT. ........................................ 89
Figure 4.50 Mean-square displacements of water at 200K in SiCNT. ......................................... 89
Figure 4.51 Mean-square displacements of water at 175K in SiCNT. ......................................... 90
Figure 4.52 Mean-square displacement of water at 150 K in SiCNT. .......................................... 90
Figure 4.53 Mean-square displacements of water inside SiCNT at 125 K. .................................. 91
Figure 4.54 Mean-square displacement of water at 100 K in SiCNT. .......................................... 91
Figure 4.55 Mean-square displacements of water in CNT at 273 K. ........................................... 92
Figure 4.56 Mean-square displacements of water in CNT at 250 K. ........................................... 92
Figure 4.57 Temperature-dependence of diffusivity of water in SiCNT. ..................................... 93
Figure 5.1 The repeating units of polyetherimide. ........................................................................ 99
Figure 5.2 The (9,0) SiCNT and PEI composite. The methane molecules are on the left side. . 101
Figure 5.3 Temperature and total energy for MD simulation of H2, CH4, O2 and CO2 in the
PEI-SiCNT composite. ....................................................................................................... 102
Figure 5.4 Mean-square displacements (MSD) of the gases in the PEI-SiCNT composite
computed by MD simulation. ............................................................................................. 104
x

Figure 5.5 Self space-time correlation function of H2, CH4, O2 and CO2 in the PEI-SiCNT
composite. ........................................................................................................................... 105
Figure 5.6 Schematic diagram, dividing the boundaries as a thermodynamic state. .................. 106
Figure 5.7 Weight percent loading of the gases in the PEI-SiCNT composite. .......................... 109
Figure 5.8 Simulated and experimental MWSiCNT Hydrogen sorption. .................................. 110
Figure 5.9 Sorption of hydrogen and methane in PEI at 298 K. ................................................. 111
Figure 5.10 Sorption of hydrogen and methane in PEI at 423 K. ............................................... 112
Figure 5.11 Sorption of hydrogen and methane in the PEI and a single (6,0) SiCNT at 298 K. 112
Figure 5.12 Sorption of hydrogen and methane in PEI and a single (6,0) SiCNT at 423 K....... 113
Figure 5.13 Sorption of hydrogen and methane in PEI and two (6,0) SiCNTs at 298 K. .......... 113
Figure 5.14 Sorption of hydrogen and methane in PEI and two (6,0) SiCNTs at 423 K. .......... 114
Figure 5.15 Sorption of hydrogen and methane in PEI and three (6,0) SiCNTs at 298 K. ........ 114
Figure 5.16 Sorption of hydrogen and methane in PEI and three (6,0) SiCNTs at 423 K. ........ 115
Figure 5.17 Sorption of hydrogen and methane in PEI and four (6,0) SiCNTs at 298 K. .......... 115
Figure 5.18 Sorption of hydrogen and methane in PEI and four (6,0) SiCNTs at 423 K. .......... 116
Figure 5.19 Sorption of hydrogen and methane in PEI and five (6,0) SiCNTs at 298 K. .......... 116
Figure 5.20 Sorption of hydrogen and methane in PEI and five (6,0) SiCNTs at 423 K. .......... 117
Figure 5.21 Five (6,0) SiCNTs in the PEI matrix. ...................................................................... 121
Figure 5.22 Young’s modulus of the composite of (6,0) SiCNT and the PEI. ........................... 122

xi

List of Tables
Table 5.1 Temperature and total energy for MD simulation of H2, CH4, O2 and CO2 in the
PEI-SiCNT composite ........................................................................................................ 103
Table A.1 The stiffness tensor of the PEI ................................................................................... 126
Table A.2 The compliance tensor of the PEI .............................................................................. 127
Table A.3 The stiffness tensor of the composite of the PEI with one (6,0) SiCNT. .................. 127
Table A.4 The compliance tensor of the composite of the PEI and one SiCNT. ....................... 128
Table A.5 The stiffness tensor for the composite of the PEI and two SiCNTs .......................... 128
Table A.6 The compliance tensor of the composite of the PEI and two SiCNT. ....................... 129
Table A.7 The stiffness tensor of the composite of the PEI and three SiCNTs. ......................... 129
Table A.8 The compliance tensor of the PEI and three SiCNTs. ............................................... 130
Table A.9 The stiffness tensor of the composite of the PEI and four SiCNTs ........................... 130
Table A.10 The compliance tensor for the composite of the PEI and four SiCNTs................... 131
Table A.11 The stiffness tensor of the composite of the PEI and five SiCNTs. ........................ 131
Table A.12 The compliance tensor of the composite of the PEI and five SiCNTs. ................... 132
Table A.13 The stiffness tensor of the composite of the PEI and one CNT . ............................. 132
Table A.14 The compliance tensor of the composite of the PEI and one CNT. ......................... 132
Table A.15 The stiffness tensor of the composite of the PEI and two CNTs. ............................ 133
Table A.16 The compliance tensor of the composite of the PEI and two CNTs. ....................... 133
Table A.17 The stiffness tensor (Cij) for PEI and three CNT (Gpa). ......................................... 133
Table A.18 The compliance tensor of the composite of the PEI and three CNTs. ..................... 134
Table A.19 The stiffness tensor of the composite of the PEI and four CNTs. ........................... 134
Table A.20 The compliance tensor for the composite of the PEI and four CNTs. ..................... 134
xii

Abstract
We have utilized atomistic modeling and extensive molecular dynamics (MD) simulation
to study and explore various properties of fluids in silicon-carbide nanotubes (SiCNTs), and in
their composites with a polymer.
First, we show that pressure-induced flow of water in the SiCNTs of various sizes is more
efficient than the same phenomenon in carbon nanotubes (CNTs), requiring a pressure drop (and
hence energy) that is at least one order of magnitude less than that in the CNTs.
Next, we study the dynamics of low-temperature water in SiCNTs, demonstrating that the
cage-cage correlation function, a measure of the water molecules’ motion in the nanotubes, follows
the Kohlrausch-Williams-Watts stretched exponential law with an exponent that is in excellent
agreement with the theoretical prediction by J.C. Phillips.
Third, using extensive MD simulations, we compute the various correlation functions for
water in SiCNTs and CNTs over the temperature range 100 K - 298 K, demonstrating that, in
agreement with our calculation of the cage-cage correlation function, due to spatial limitations and
steric hindrance inside small enough nanotubes, water inside such nanotubes does not freeze. This
has significant scientific and biological implications, potentially providing a method for preserving
microorganisms for very long time for advanced research and studies.
Finally, we develop an atomistic model of mixed-matric polymeric composite made of
polyetherimide (PEI) and SiCNTs and study, (i) the mechanical properties of the composite and
compare them with those of pure PEI, and (ii) diffusion and sorption of several light gases in the
composite in order to test its potential use as a membrane for gas separation. The results indicate
that such a membrane has excellent properties for separation of hydrogen from a gaseous mixture.
1

Chapter 1: Background

1.1 Introduction
Study of materials and their properties is as old as human history. The primary goal of studying
metals for improving their properties, for example, for creating better stronger alloys with better
properties and less prone to severe environmental conditions and corrosion has long been of
interest. Binding science and technology with human needs creates better environment for growth
of both.
In recent years, rapid progress in the computing capabilities for performing billions of
calculations in a very short period of time has paved the way for applying the fundamental
equations of change to solving complex problems. The application of statistical mechanics and
many body systems’ modeling for the evaluation of macro systems has long been pursued. Hence,
with the exponential increase in the computational capacity, molecular dynamics simulation has
become far more attractive.
1
Such simulations have become a very powerful tool for evaluating
the properties of existing materials, predicting their behavior and justifying the design of new
materials.
1.2 Nanotubes
The first discovery of carbon filaments is more than 100 years old. Multiwall carbon nanotubes
(MWCNTs) were first reported by Iijima
2
in 1991, while fabrication of single-wall carbon
nanotubes
3
was first reported in 1993. Nanotubes are cylindrical shaped pipes, and may be
considered as a single molecule by traditional definition, consisting of one or multiple atomic
components. Nanotubes have large length/diameter ratio. There are many ways by which one can
2

roll a sheet to create a nanotube with numerous micro- and macrostructures, presenting a variety
of physical and mechanical properties.
4
Figure 1.1 shows the honeycomb lattice, the primary structure of a nanotube before rolling into
a tube. The unit cells are represented with the and elements. For the construction of a
nanotube, we may convert a sheet into a tube by rolling it on a chiral vector . The
circumferential vector c, the chiral vector, is shown by . The direction of the chiral vector
is measured by the chiral angle θ, defined as the angle between and c:
. (1-1)
Tubes with an angle are called zigzag tubes with a specific pattern along the
circumference, whereas tubes with are called armchair tubes. The nominal diameter
of the tube is calculated by the length of the chiral vector:
2-4

. (1-2)
. (1-3)
In Figure 1.1 the chiral vector , with the corresponding tubes are
shown. Vectors c and a form a rectangle in the unit cell of tube, rolled into a cylinder.
2

1
a
2
a
 
2 2 1 1
a n a n c  
 
2 1
,n n
1
a
2
2 2 1
2
1
2
1
1
1 2
cos
n n n n
n
n
c a
c a
 




 
  0 , 0 n

 
  n n, 30

 
2
2 2 1
2
1
0
n n n n
a
c
d    
 
C Si
a a

 3
0
    5 , 9 5 9
2 1
called a a c  
3

Figure 1.1 Honeycomb lattice vectors

In the past decade, significant theoretical and experimental advances have been reported on the
study of the CNTs’ structure, and their electronic, transport and elastic properties. Nanotubes for
commercial and technological applications, such as membranes, composites, fluids separation and
drug delivery are the prime examples of the results of extensive research in progress.
5-9

1.3 Silicon-Carbide Nanotubes
Carbon nanotubes are well known for their superior physical, mechanical and electronic
properties, but silicon-carbide nanotubes (SiCNTs) have been studied much less extensively. In
2001 the production of SiCNTs in various internal and external dimensions were first reported,
using gas-solid reaction of CNTs and SiO2 to obtain single- and multiwall SiCNTs at temperatures
above 800
o
C.
10,11
The SiCNTs have also been studied for their structure and stability
characteristics.
12

4

Hydrogen storage is a promising demesne for utilizing SiCNTs at smaller scale, due to their
capacity that is larger than that of CNTs at higher pressure and lower temperature.
13
Separation of
gases using SiCNTs has yet to be studied as extensively as their carbon counterparts.
14
An excellent
property of SiCNTs, comparable with that of CNTs, is their mechanical strength.
15
One indication
of the increasing interests in the nanotubes is the number of research papers that have been
published, from a few in 1992 to more than 4000 in 2010.
16-18

Figure 1.2 Schematic demonstration of the zigzag (9,0) and armchair (9,9) nanotubes

One of the most important barriers to utilizing CNTs in extreme conditions is their temperature
limitations’ for applications under harsh conditions. The key to overcome this obstacle can be
SiCNTs, providing a new vision for more application of nanotubes’ fantastic capabilities under
harsh conditions and higher temperature. They can withstand temperatures as high 1000

C,
whereas CNTs are mostly limited to ~ 600
.
19
New horizons for using SiCNTs in advanced C
o
(9,9)
(9,0)
5

catalysts and molecular separations in extreme conditions, as well as other high-temperature
applications, are only a matter of time and reducing the cost for further commercialization.
1.4 Stability and Properties of SiC Nanotubes
Silicon carbide is used as semiconductors with significant thermal conductivity, high chemical
resistance, and superior physical and mechanical properties. It is well placed for use in high
frequency, high temperature electronic applications, increasing the industrial interest in building
and evaluating SiC nanostructures, such as ribbons, nanowires and nonorods.
20
SiC nano-structures
present many potential applications with mechanical strength, and optical and electron emission
properties that are superior to those of the bulk SiC, due to modified orientation.
21,22
SiC nanotubes
containing Si/C ratio of 1:1 and smaller present more stable morphologies for molecular
simulation.
23
1.5 Fabrication of SiC Nanotubes
One method to produce highly-oriented SiC nanowires has been their synthesis using the oxide-
assisted growth method by thermal evaporation of silicon monoxide in a furnace. SiO vapor
disproportionate distribution forms to Si and SiO2 clusters. The same method has been utilized for
fabrication of SiCNTs with disproportionation reaction of SiO and CNTs and SiCNTs. The SiCNT
structure is identified by high-resolution transmission microscopy. MWSiCNTs, composed of an
outer layer SiC has also been produced using a similar method at USC chemical engineering
department under Professor Muhammad Sahimi’s guidance.
11

The importance of the SiC nano-materials was noted by the production of “carbon-rich silicon-
carbide nano-ribbons hetero-structure” for enhanced microelectronics using nanosecond “pulsed
laser direct-write and doping (LDWD)”
24
noted as SiC and for the excellent electronic and physical
6

properties. Preparation of SiCNTs from CNTs for electrical applications is not favored, due to the
possibility of existence of metals in the latter. A recent method used to resolve the issue is the
applications of supercritical hydrothermal conditions’ method to construct SiCNTs with diameters
less than 10 nm. In this method, SiCNT is formed in the hydrothermal fluid environment, and then
carbon atoms are transferred into the structure utilizing diffusion to form SiCNT.
21
1.6 The Goals of the Thesis

Given the discussion above, the goal of this research is to explore the properties of SiCNTS, and
the various phenomena that occur in them. We study flow of liquids in SiCNTs and, when possible,
compare the results with similar phenomena in CNTs. We will also construct atomistic models of
mixed-matrix polymeric membranes, made of a polymer in which SiCNTs have been inserted. As
we demonstrate, the insertion of the nanotubes into the polymeric matrix strengthens its
mechanical properties, and enables it to withstand much higher temperatures than what a polymer
can tolerate by itself. We will then study gas separation by the mixed-matrix polymer membrane,
and compare the results with those in the polymer alone.
Tool of our study is molecular dynamics (MD) simulations. We use extensive MD simulations
and accurate force fields (see below), in order to construct realistic atomistic models of the SiCNTs
and the mixed-matrix polymer, and study various phenomena in them.
In what follows we describe the tools of our study.
7

Figure 1.3 High-resolution transmission microscopy of SiCNT.
11

1.7 Molecular Models and Simulation
In any MD simulation a force field (FF) that represents the total potential energy of the system,
and describes the various forces by which the atoms interact with each other. Selection of a proper
FF is critical to obtaining accurate representation of the atomistic interactions, to be utilized in the
MD simulation. We briefly describe the FF used in our MD simulations. Later on, we present a
summary of the fundamental principles regarding nanotubes, the process for constructing models
of SiCNTs in virtual environment, and the simulation methods.

8

1.8 The COMPASS Force Field
Accurate FFs that are used in MD simulations have been developed based on the application of
quantum mechanics and experimental models, utilizing mathematical functions to represent the
potential energy of the system and its constituents.
25

Most of the parameters of the FFs are derived from partial charges and valence parameters by
fitting the data for the potential energy surfaces and empirical optimization. In general, FFs have
different representation for an atom in different structures, such as carbon in methane and in CNTs.
To construct an accurate FF, the van der Waals radius, mass of the individual atom, bond length
in different hybridization, bond angles and other related values related to different types of
interactions, and specific orientations need to be defined and accounted for. For complex
molecules and structures, such as DNA, smaller organic compounds with similar structures are
utilized and studied, in order to evaluate and predict the parameters of related FFs.
Although very useful and powerful, but also complicated, each FF has its own strengths and
deficiencies. Some materials exhibit different properties and specifications as elemental, or in
similar compounds.
25
Polymers, although in similar structure, may show different physical and
mechanical properties, due to heat treatment or degree of orientation and crystallization that creates
challenges for their simulation, if not in their original state. These issues may be evaluated using
the known structures and their comparison with the MD results under different conditions.
In our MD simulations we have used a FF called the condensed phase optimized molecular
potentials for atomistic simulation studies (COMPASS), which has been extensively validated.
26

The locations of the atoms in the simulation cell, combined with the FF, create the potential energy
surface of a material. We may summarize the total potential energy in the following form,
(1-4)
bond non crossterm valence tot
E E E E

  
9

The valance interactions are represented by,
(1-5)
in which is the energy for bond stretching, represents the valance angle bending,
represents energy for dihedral angle torsion, describes inversion or out of plane
interactions and represents the Urey-Bradley term used for interactions between atom pairs
involved in 1-3 configurations. The valence cross terms are generally used to increase the accuracy
for angle or bond distortions caused by nearby atoms. Non-bond interactions include van der
Waals, electrostatic (Coulomb) and hydrogen bonds
(1-6)

The functional form of the total energy is given by

.
, (1-7)

UB oop torsion angle bond valence
E E E E E E     
bond
E
angle
E
torsion
E
oop
E
UB
E
hbond columb vdw bond non
E E E E   

              

b
e total
b b k b b k b b k E
4
0 4
3
0 3
2
0
             


     
4
0 4
3
0 3
2
0 2
k k k
             


   3 cos 1 2 cos 1 cos 1
3 2 1
k k k        
 
'
' ,
0
2
2
'
o
b b
b b b b k k


          
 
 
  
, ,
1 0 0 0
cos [
b b
k b b b b k      

    
 
cos [ ] 3 cos 2 cos
1
,
0 3 2
k k k
        

 
     
,
0
'
0
'
3 2
] 3 cos 2 cos k k k       

    
  
cos
'
0
'
, ,
0
'
k
 


























j i j i ij
ij
ij
ij
ij
ij
j i
r
r
r
r
r
q q
, ,
6
0
9
0
3 2 
   
6
1
6
0
6
0
0
2










j i
ij
r r
r
   
   











6
0
6
0
3
0
3
0
2
j i
j i
j i ij
r r
r r
  
10

The subscripts are as - - and out of place angle.
27-30
In order
to impose a given pressure in the MD simulation, we utilize the external bath method with coupling
that may also be applied to pressure and temperature control for non-equilibrium simulations. A
leap frog algorithm is applied for the integration of Newton’s equations of motion. An external
bath is utilized representing pressure for application in the equation of motion,

. (1-8)
in which
P
 represents the non-critical time constant, given by
. (1-9)

is the internal virial for pair additive potentials of the system, is the total kinetic energy, and
V is the volume. Internal virial may be evaluated by:


   
j i
ij ij
F r
2
1
. (1-10)
  
 
j i ij
r r r (1-11)
where
ij
F is force exerted on particle j due to presence of particle i using center of mass for both
particles.
31
We may then use the following for the calculation of pressure for anisotropic triclinic
systems:
. (1-12)
where represents the mass and

is the velocity vector of atom i. With application of the
Berendsen method for the arrangement of the positions of the particles and size of unit cell, the
pressure may be changed. To keep the pressure at a certain target, the compressibility and
Bond b, Angle ,  Angle torsion ,  
P bath
P P
dt
dP









0
) (
3
2
  
k
E
V
P

k
E






 
 
 j i
T
ij ij
i
T
i i i
F r v v m
V
P
1
i
m
i
v
11

relaxation time play a critical role. Cartesian coordinates of each atom are scaled in each step to
maintain the pressure. The change is only in the size of the cell, but the cell shape remains
unchanged. At each step, the positions of the atoms are scaled by the following expression,
. (1-13)
in which P is the instantaneous pressure, the target pressure, and is the time step.
31, 32

1.9 Construction of the System for MD Simulation
As already mentioned, CNTs are obtained from folding of graphene sheet into a tube. Thus, to
construct a molecular model of CNTs, a (atomistic) sheet of C is used to build the nanotubes. The
same was done for the SiCNTs,
33
with the additional step of replacing a specific number of carbon
atoms with the Si atoms such that every Si atom can only have carbon neighbors and every carbon
atom connected to only one silicon atom. In agreement with the previous studies, the Si-C bond
length of ~1.8 Å was calculated without any preexisting condition.
12
During the construction
process, geometry optimization was used to create stable silicon-carbide nanotubes. After setting
up the initial structure, the two-step process of energy evaluation and conformation adjustment
was implemented for each nanotube, in order to prepare structures for dynamic simulation.
1.10 The Nosé-Hoover Thermostat
Control of the temperature is of essential importance to MD simulations and improving the
quality of results. The Nosé-Hoover
34
method has evolved from the original Nosé method, derived
from the Hamiltonian utilizing position and moment in the canonical ensemble. Temperature
control is possible by allowing heat to be exchanged from the system to a heat bath.
 








 
0
1 P P
t
P



0
P t 
12

The temperature of the system is a macroscopic quantity, which is being calculated using the
kinetic energy of the microscopic environment. The Maxwell-Boltzmann distribution is applied
using the virial theorem in order to calculate the kinetic temperature
(1-14)
Where

is the instantaneous kinetic temperature.
35
1.11 The Parrinello-Rahman Method of Applying an External Pressure
36

In this thesis the Parrinello-Rahman, method with the Lagrangian formulation
36
for the externally
applied stress was used for the control of the pressure, applied externally to direct force in the flow
direction (see Chapter 2 for details). In this method the cell shape and size may change based on
the dynamic equations.
1.12 Molecular Dynamics Simulation
Statistical mechanics deals with macroscopic systems from microscopic point of view, in order
to predict and calculate properties of physical systems from constituent atoms and molecules. To
do so, general rules of physics and theoretical and empirical models are utilized. Statistical
mechanics analyzes physical systems at equilibrium, as well as in non-equilibrium states.
Principles of statistical mechanics have been applied to gases, liquids, solutions, electrolytic
solutions, polymers, adsorption, metals, spectroscopy, transport theory, DNA folding, electrical
properties and cell membranes.

The MD simulation employs computer-based models and principles of Newtonian physics
related to equations of motion, combined with atomic force concepts. The resulting changes of
total energy of a system, including the potential and kinetic energy, with time enables us to
 


 
N
i i
i
B B
m
p
Nk Nk
K
1
2
3
1
3
2

13

calculate temperature, pressure, acceleration, viscosity, heat capacity, electric properties and
numerous other characteristics of a system. This is a very powerful tool to carry out detailed studies
and evaluate systems that either have not been yet constructed, or considered impractical to study
in laboratories. Laboratory experiments carry their own potential errors, including experiment
design and measurement tools, inconsistency in models and high cost.
Molecular dynamics starts with assigning the initial positions to all the atoms in a system. With
selection of time step, the forces acting on the atoms and molecules in the system, which are the
gradient of potential energy and acceleration, are included in Newton’s second law of motion, and
the resulting equations are integrated numerically for one time step. This yields the new positions
for all the atoms in new time slot. The time is increased by one step forward, and the new positions
are calculated again. The procedure continues based on the simulation time and computational
limits. The results are recorded and used to calculate the properties.
With continuous improvement in the calculation capabilities, MD simulation has become a
leading scientific and engineering tool for studying of complex molecular systems with increased
applications in industry. We use MD simulations to study carbon and silicon-carbide nano-
structures, various flow phenomena in them, as well as membranes made of a composite of a
polymer and the nanotubes, and to investigate their properties and potential for scientific and
industrial applications.
The basis of the MD simulation is Newton’s second law of motion which states sum of the
forces F, acting on a molecule is equal to the molecule’s mass m times its acceleration a:
. (1-15)

and the forces on the atom are computed based on the gradient of potential energy:
ma F 
14

(1-16)

In our simulations, the initial velocities may be randomly generated, and the simulation results for
the velocity and positions of the atoms are used in the next step of the simulations. For each
simulation, a stress tensor is applied with nine elements. Elements of the tensor are the forces (per
unit area) that act on the surface of the system. Pressure is calculated by the virial theorem as a
thermodynamic property.
37-39

1.13 The NPT Ensemble
The constant-temperature, constant-pressure ensemble, with a fixed total number of particles N,
referred to as the (NPT) ensemble, allows control over both the temperature and pressure. The unit
cell vectors are allowed to change, and the pressure is adjusted by adjusting the volume (which, in
some cases, is the size, and in some occasions is the shape of the unit cell).
44

2
2
t
r
m
r
V






15

References
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Chem. Phys. 31, 459 (1959).
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363 (1993).
4. S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nanotubes: Basic Concepts and Physical
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silicon (from disproportionation of silicon monoxide) with carbon nanotubes,” J. Am.
Chem. Soc. 124, 14464 (2002).
12. M. Menon et al., "Structure and stability of SiC nanotubes,” Phys. Rev B. 69, 115322
(2004).
13. G. Mpourmpakis et al., "SiC nanotubes: a novel material for hydrogen storage,” Nano Lett.
6, 1851 (2006).
14. K. Malek and M. Sahimi, "Molecular dynamics simulations of adsorption and diffusion of
gases in silicon-carbide nanotubes,” J. Chem. Phys. 132, 01430 (2010).
16

15. W. H. Moon, J. K. Ham and H. J. Hwang. "Mechanical properties of SiC nanotubes,” in
Proceedings of the 2003 Nanotechnology Conference and Trade Show 77, 035436 (2003).
16. I. J. Wu and G. Y. Guo, "Optical properties of SiC nanotubes: An ab initio study,” Phys.
Rev. Lett. 76, 035453 (2007).
17. K. M. Alam and A. K. Ray, "Hybrid density functional study of armchair SiC
nanotubes,” Phys. Rev. B 77, 035436 (2008).
18. H. Shen and K. Cheng, "Tensile properties and thermal conductivity of graphene
nanoribbons encapsulated in single-walled carbon nanotube,” Mol. Simul. 38, 922 (2012).
19. I. W. Chiang et al., "Purification and characterization of single-wall carbon nanotubes
(SWNTs) obtained from the gas-phase decomposition of CO (HiPco process),” J. Phys.
Chem. 105, 8297 (2001).
20. G. W. Ho et al., "Three-dimensional crystalline SiC nanowire flowers,”
Nanotechnology 15, 996 (2004).
21. C. Zhang et al., "Modelling of microvia drilling with a Nd: YAG laser,” J. Phys. D. Appl.
Phys. 39, 3910 (2006).
22. E. W. Wong, P. E. Sheehan, and C. M. Lieber, "Nanobeam mechanics: elasticity, strength,
and toughness of nanorods and nanotubes,” Science 277, 1971 (1997).
23. A. Mavrandonakis et al., "From pure carbon to silicon-carbon nanotubes: An ab-initio
study,” Nano Lett. 3, 1481 (2003).
24. I. A. Salama, N. R. Quick, and A. Kar, "Laser synthesis of carbon-rich SiC nanoribbons,” J.
Appl. Phys. 93, 9275 (2003).
25. http://en.wikipedia.org/wiki/Force_field_(chemistry)
26. http://www.scripps.edu/rc/softwaredocs/msi/cerius45/compass/1_Introduction.fm.html#7
65283
27. http://accelrys.com/products/datasheets/compass.pdf
28. H. Sun, P. Ren, and J. R. Fried, "The COMPASS force field: parameterization and
validation for phosphazenes,” Comput. Theor. Polym. S 8¸ 229 (1998).
29. S. W. Bunte and H. Sun, "Molecular modeling of energetic materials: the parameterization
and validation of nitrate esters in the COMPASS force field,” J. Phys. Chem. 104, 2477
(2000).
17

30. H. Sun, "COMPASS: an ab initio force-field optimized for condensed-phase applications
overview with details on alkane and benzene compounds,” J. Phys. Chem. 102, 7388
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Phys. 81, 3684 (1984).
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nd
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London, 2004)
38. S. Nosé, "A molecular dynamics method for simulations in the canonical ensemble,” Mol.
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18

Chapter 2: Molecular Dynamics Simulation of Pressure-Driven
Water Flow in Silicon-Carbide Nanotubes
2.1 Introduction
Over the past decade fabrication and characterization of one-dimensional (1D) or quasi-1D
materials, which include nanowires,
1-6
nanoparticles,
7-15
and particularly carbon nanotubes
16

(CNTs), have received considerable attention. One type of CNTs are grapheme sheets rolled into
seamless tubes,
17,18
which are usually referred to as single-wall carbon nanotubes (SWCNTs).
Their structure is characterized by two integers m and n, and are usually referred to as (m, n) CNTs.
When m = n, the C atoms that are around the circumference are in an armchair pattern. The zigzag
nanotubes correspond to n = 0, whereas when the rows of hexagons spiral along the nanotubes'
axis, one obtains the chiral structure with m = n. If many graphene sheets are rolled into
concentrically stacked tubes, one obtains a multiwall CNT
19
(MWCNT). Many properties of CNTs
have been studied.
16

2.2 SiC Nanotubes
As already pointed out in Chapter 1, silicon carbide (SiC) is an important material with many
excellent properties, such as high fracture toughness, thermal shock resistance, and the ability for
withstanding high temperatures and corrosive environments, and has been used extensively in the
electronic industry for many years. It is also a promising material for preparation of nanoporous
membranes for separation of fluid mixtures in harsh environments, where other types of
membranes fail due to severe environmental conditions. The group at USC under supervision of
Professors Muhammad Sahimi and Theodore Tsotsis has fabricated
20-22
SiC nanoporous
19

membranes with high selectivity’s, as well as
23
SiC nanotubes and nanotubes whose fabrication
has recently attracted considerable attention.
24-26

2.3 Transport of Fluids in Nanotubes
Similar to CNTs, transport of fluids through SiC nanotubes (SiCNTs), both in the gas and liquid
phases, are interesting and important. Detailed atomistic simulation of diffusion of light gases and
other mixtures in SWCNTs
27-31
yielded diffusivities that were much larger than those in other types
of nanoporous materials, such as zeolites. They were attributed to the nanotubes' highly smooth
potential energy landscape. The simulations
30,31
had not, however, taken into account the flexibility
of the nanotubes' structure, and had assumed that the structure was rigid. Subsequent simulations
32

indicated that the diffusivity of gases in the CNTs would not be as large as reported, if the
nanotubes' structure is assumed to be flexible.
The tubules formation of the SiCNTs and the existence of specific charge arrangement on their
surface make them attractive alternative materials for a variety of purposes. In addition, due to
their excellent properties, SiCNTs naturally draw comparison with the SWCNTs. Thus, for
example, hydrogen storage capacity of SiCNTs was studied
33,34
and compared with that of
CNTs.
35-36
Using molecular dynamics (MD) simulations, Malek and Sahimi examined
37

adsorption and diffusion of several gases in SiCNTs as a function of the pressure, the nanotubes'
diameter, and chirality. Whereas adsorption capacity and diffusion of gases in CNTs are not
affected much by the nanotubes' morphology,
38
the MD simulations by Malek and Sahimi
37

demonstrated the strong effect of the SiCNTs' curvature and chirality on the same properties. In
addition to their importance on their own, such differences between the two types of nanotubes
also motivate further study of various phenomena in them in order to discern the advantages and
disadvantages of each.
20

2.4 Review of the Previous Works
Majumder et al.
39
and Holt et al.
40
reported the results of experimental studies of pressure- driven
flow through CNTs of 1.6 and 7 nm in diameter. The flow rates were 2 - 5 orders of magnitude
larger than what is predicted by the Hagen-Poiseuille (HP) equation. Since water wets the surface
of the CNTs, the flow enhancement might be due to its slip on the internal surface, but the
theoretically-estimated slip length required for achieving such large enhancements based on the
HP equation are 2 - 3 orders of magnitude larger than typical slip lengths on solid boundaries.
41

Noy et al.
42
suggested that the enhancement may be due to single-file transport, identified by
Hummer et al.
43
through simulation of water flow in a CNT. If true, it would be somewhat similar
to gas transport in the CNTs described earlier. Thus, MD simulations of water flow in CNTs have
been carried out.
43-49
It appears that
47-49
if one takes into account the dependence of water viscosity
on the diameter of the nanotubes, the observed flow enhancement may still be explained based on
the HP equation down to very small CNTs. In this chapter we report the results of MD simulation
of pressure-driven flow through SiCNTs, and the effect of the nanotubes' size on the results. A
major goal of the present study is comparing the performance of SiCNTs with that of CNTs in
terms of flow enhancement and the required applied pressure gradients. The comparison should
also provide insights for another important problem, namely, the possibility of developing
composite membranes using SiCNTs to separate fluid mixtures. In the pharmaceutical industry,
for example, protein purification using nanoporous SiC membranes is gaining attention. SiC
nonporous membranes
20,22
allow
50
diffusion of proteins up to 29000 Daltons, but exclude larger
ones.

21

2.5 Details of Molecular Dynamics Simulation
The COMPASS forcefield,
51
described in Chapter 1, was used in the MD calculations. As
described in Chapter 1, in COMPASS force field the total energy is written as the sum of the
valence, cross terms, and non-bond interactions. The valence term includes contributions by bond
stretching, angle changing, dihedral angle torsion, out-of-plane interactions, and the Urey-Bradley
term representing the interactions based on the distance between atoms separated by two bonds
(1,3 interaction). The cross terms are generally used to increase the accuracy of the angle or bond
distortions caused by the nearby atoms. The non-bond term includes contributions by the van der
Waals and electrostatic (Coulomb) interactions and hydrogen bonds. The van der Waals
interactions are represented by a 6-9 Lennard-Jones type potential. The contribution of the
electrostatic interactions is computed by the atom-based method for the summation of the all the
contributions, which is much faster than Ewald summation. The latter method was also used in
limited simulations to check the accuracy of the former. The length of the Si-C bonds is about 1.8
0
A , 10% longer than that of C-C bond in CNTs. There is also out of surface (torsional) motion
and, therefore, the increase in the bond length and the dihedral angles leads to a distorted channel
wall. Moreover, the polar-polar repulsion between the π bonds of SiC and water does not allow
water molecules to come close to the surface, implying that the nanotubes' wall is not wetted by
water.
Single-wall SiC nanotubes of the zigzag type (m,0) were utilized in the MD simulations with m
= 6, 9, 12 and 16 and initial diameters of 0.595, 0.893, 1.191, and 1.588 nm, respectively. The
length of the nanotubes was 5.3 nm. The water molecules were represented by the three-site SPC/E
model.
52
The van der Waals radius of water is still larger than the SiC ring size. The Nosé-Hoover
thermostat was used to hold the temperature at 298 K (see Chapter 1). The Parrinello- Rahman
22

method
53
was used for applying the external pressure to the nanotubes, because it allows both the
cell's shape and volume to be modified. Water viscosity in the nanotubes was estimated from the
Einstein relation,
(2-1)
where T is the temperature,
B
k the Boltzmann's constant, d the diameter of water, and
z
D its axial
diffusivity that is estimated using the Green-Kubo relation:
   
1
0
1
0
N
Z i i
i
D v t v dt
N





, (2-2)
where   t v
i
is the axial velocity of molecule i at time t, and N is the total number of molecules. The
bracketed quantity represents the velocity autocorrelation function (ACF; see also Chapter 4). In
small nanotubes the diffusivity
z
D is dependent upon the tubes' diameter. We did not assume that
the atomistic structure of the nanotubes is rigid, so that the C and Si atoms could move in response
to their environment in the presence of the water molecules and the applied pressure gradient
according to the equation of motion.
,
3
z
B
dD
T k

 
23

Figure 2.1 Axial velocity autocorrelation function in the smallest and largest SiCNTs

Figure 2.1 presents the axial velocity ACF in the smallest and largest nanotubes that we have
simulated. The ACF declines sharply after a short time, and then oscillates irregularly around zero.
Figure 2.2 presents the computed viscosity µ of water and its dependence on the nanotubes'
diameter. As expected, µ increases with increasing tube size. The estimated viscosities were used
24

in the HP equation for fluid flow through tubes and its comparison with the results of the MD
simulations (see below).

Figure 2.2 Dependence of water viscosity on the diameter of SiCNTs.

Figure 2.3 presents the dependence of the mean axial velocity on the applied pressure gradient
ΔP/ L. The velocity increases with the pressure gradient, but not linearly as in laminar flow in
tubes, due to the small sizes of the tubes. Moreover, for a given pressure gradient, the velocity
does not increase with the nanotubes' radius monotonically. The lowest curve corresponds to the
(12,0) tube, the highest one to the (16,0) tube, with the other two tube sizes in between.
25

Figure 2.3 Dependence of mean axial water velocity on the applied pressure gradient ΔP/ L in
the four SiCNTs.

This effect is associated with the inertial losses that depend on the velocity, and are generated by
flow of water from the reservoir into the nanotube at one end, and vice versa at the opposite end.
For much larger nanotubes, however, such losses are insignificant. Thus, one obtains a monotonic
increase of the velocity with the tube size, and the relation between the velocity and pressure
gradient would be linear.
Figures 2.4 and 2.5 present the axial and radial concentration profiles of water molecules, as
well as the axial velocity profile in the (9,0) nanotube, for a pressure gradient of ΔP/ L = 5.7 TPa/m.
In the case of the axial profiles, the results represent averages over the length of the nanotubes
after steady-state was established. Unlike the case of CNTs, no water layer is formed by the walls,
26

because the tube is too small and water does not wet the walls. Instead, one has a water layer at
the center as a single file. The axial concentration, on the other hand, varies widely. To test this
further, we carried out limited simulations with the (10,10) SiCNT, which is also small. We found
for this case that water molecules move as 5-molecule rings.

Figure 2.4 Axial water concentration profile in the (9,0) SiCNT. The applied pressure gradient is
ΔP/ L = 5.7 TPs/m.

In the case of the axial velocity, there is a plug-like flow almost everywhere, followed by a step
drop as the wall is approached. The qualitative aspects of the results do not change if the pressure
gradient increases. For example, Figures 2.6 and 2.7 present the same profiles in the same nanotube
like figures 2.4 and 2.5, but under a pressure gradient of ΔP/ L = 18.9 TPa/m. The velocity profile
27

is still plug-like, and while the radial concentration profile is smoother, other qualitative features
of the results remain the same.

Figure 2.5 Radial water concentration profiles in the (9,0) SiCNT. The applied pressure gradient
is ΔP/ L = 5.7 TPs/m.

As the size of the nanotube increases the shape of the axial velocity profile and its dependence
on the radial position do not change, but the radial concentration profile changes significantly.
Figure 2.8 presents the profiles for the (16,0) nanotube under a pressure gradient of ΔP/ L = 5.7
TPa/m. While, as before, the axial velocity profile is still plug-like and the axial concentration
profile varies irregularly, the radial concentration profile contains two water layers, one at the
center, and a larger (with higher concentrations) midway between the center and the wall, a pattern
that persists at higher pressure gradients as well.
28

Figure 2.6 Same as in Figure 2.4, but with an applied pressure gradient of ΔP/ L = 18.9

29

Figure 2.7 Same as in Figure 2.5, but with an applied pressure gradient of ΔP/ L = 18.9

As discussed earlier, the important question is whether the water flow in the nanotubes is
described by the HP equation and its flux in such tubes is enhanced. The HP equation predicts that
the average axial velocity is given by,  
L
P
K v
HP z

 , where
   8
2
R
K
HP
 is the hydraulic
conductivity, with R being the tube's radius, and being independent of the applied pressure
gradient. The MD simulation results may also be expressed in a similar form,  
L
P
K v
MD z

 ,
where
MD
K is the apparent hydraulic conductivity computed based on the MD results. Thus, one
may define
47
a flow enhancement factor,
HP
MD
K
K
  .
30

Figure 2.8 Same as in Figure 2.4, but for a (16,0) SiCNT with an applied pressure gradient of
ΔP/ L = 5.7 TPa/m.

Figure 2.9 presents the flow enhancement factor ε in four SiCNTs as a function of the applied
pressure gradient. The smallest tube, (6,0), enhances water flow that increases sharply with the
applied pressure gradient and can be very large. For a given pressure gradient, larger nanotubes
have smaller enhancement factors, which are expected. In addition, as the nanotube size increases,
the dependence of the enhancement factor on the pressure gradient becomes weaker, because
MD
K
increasingly becomes independent of  
L
P 
. For the (16,0) nanotube, the enhancement factor is
  100  .
31

Figure 2.9 Same as in Figure 2.5, but for a (16,0) SiCNT with an applied pressure gradient of
ΔP/ L = 5.7 TPa/m.

It is instructive to compare the results with those for CNTs. Figure 2.10 compares the flow
enhancement factors of the two types of nanotubes at a fixed pressure gradient, versus their sizes.
The results for CNTs were reported by Thomas and McGaughey.
47
For very small tubes, due to
single file motion, the pressure gradient in both CNT and SiCNTs to uptake water, and hence flow
enhancement, should be very close. This is indeed what Figure 2.11 indicates. But, as the size of
the nanotubes increases and multilayer formation begins, SiCNTs provide higher flow
enhancements than the CNTs, but the trend reverses at higher tube sizes, mainly because the
enhancement factor of CNTs decreases with the tube size, reaching a minimum, and then increases
again and approaches a constant value.
32

Figure 2.10 Dependence of flow enhancement factor on the applied pressure gradient in the four
SiCNTs.

If the nanotubes are not too small, the pressure in which water uptakes from CNT and SiCNT
are not clearly the same, as a result of different water-surface interaction, the flexibility of the
SiCNTs in response to the applied pressure, as well as surface wettability. Indeed, we find that for
a range of sizes (see Figure 2.11) SiCNTs produce a given flow enhancement factor of a CNT of
the same size by an applied pressure gradient that is about an order of magnitude smaller than what
is necessary for CNT. This is an important advantage of SiCNTs over their CNT counterparts.

33

Figure 2.11 Comparison of flow enhancement factors of SiC and carbon nanotubes for the same
applied pressure gradient.

2.6 Conclusions
We have presented the results of extensive MD simulations of flow of water in SiC nanotubes.
There are significant differences between flow of water in such nanotubes and their carbon
counterparts. The differences are mainly due to the wetting properties of the two types of surfaces,
atomistic structure of SiC surface, and its interaction with water and its O-H bonds. In addition,
small SiC nanotubes produce higher flow enhancements than CNTs and require smaller pressure
gradients to do so.
34

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39

Chapter 3: Dynamics of Low-Temperature Water in
Nanotubes - the Cage Correlation Function
3.1 Introduction
Water and its interaction with carbon and silicon carbide nanotubes has been extensively
investigated in ambient temperature. In Chap 2, we studied the water flow characteristic inside
SiCNTs with different diameters, presenting higher enhancement factors than CNTs. However,
micro-scale behavior of water in nanotubes at and below physical its bulk freezing point remains
a mystery. In this chapter, we utilize molecular dynamic simulation to study mobility and
molecular interactions of water at low temperatures, investigating the potential for molecular
movements in nanotubes. Clear understanding of water motion inside the nanotubes at low
temperatures has significant physical and biological importance.
3.2 Water, Ice and Hydrogen Bonds
Water, in addition to being a fundamental ingredient of life, has many unusual properties
1
,
such as the lowest density at 4 ◦C, as well as its coexistence within two distinct forms -
polymorphism
2
- and the abnormal behavior of its isothermal compressibility, isobaric heat
capacity and thermal expansion coefficient at temperatures between homogeneous nucleation (231
K) and its melting point (273 K), which have all been studied intensively. In addition, water in the
supercooled state can occur at sufficiently low temperatures, while it is still in liquid form
1
. Such
a supercooled liquid state is important from a biological standpoint because it can preserve
microorganism during freezing. In contrast to other liquids, at 277 K and lower temperatures, water
volumetric expansion is associated with reduction in the entropy due to tetrahedral symmetry of
40

the local order around each water molecule
1
, related to hydrogen bonding. Under bulk conditions
three forms of glassy water exist, namely, low density amorphous (LDA) ice, and high-density and
very high-density amorphous (HDA and VHDA, respectively) ice. The region between
homogeneous nucleation temperature (231 K) and crystallization temperature is above the glass
transition temperature of 150 K. The gradual crystallization process by decreasing the temperature
has been explained by the hypothesis that there exists a liquid-liquid critical point, based on
structural changes governed by hydrogen bonds during water clustering process, and the
development of tetrahedral coordinated network. The nature of water at 228 K, which has been
hypothesized to be a second, low temperature critical point of water, is still under investigation.
Such fascinating phenomena have all been studied over the past two decades.
3.3 Confined Water
Even more interesting phenomena occur when water is in a confined medium.
Understanding various properties of water in such media is highly important to many physical,
chemical and biological phenomena, as they appear to be fundamentally different from their bulk
counterparts. For example, diffusion of water in nanopores and nanotubes is governed by a process
that is different from the bulk phase,
3,4
because while diffusion under bulk conditions follows the
Einstein relation and only local temperature and pressure influence the transport process, the same
is not true about water diffusion in a confined medium. In such a medium the interactions of water
molecules with the solid walls, which are mainly of the van der Waals and Columbic type, affect
their mobility and usually reduce the rate of molecular diffusion.
5,6
Moreover, a water molecule in
the bulk is, on average, hydrogen-bonded to four neighboring molecules. If one hydrogen bond
breaks, the O-H-O configuration moves away from linearity by more than 25 degrees, and as three
41

or four H-bonds are broken, the molecules undergo rotational diffusion. This is not, however, the
case in a confined medium. Thus, despite considerable experimental and theoretical-computational
works to understand the possible deviations of the behavior of water molecules in confined media
from that in the bulk, no reasonably complete understanding of the properties of water in confined
media has emerged. For example, it is not yet clear whether the Stokes-Einstein relation is followed
by water molecules in nanopores and nanotubes.
3.3.1 Subcooled Water
One way of achieving sub-cooling is
7
to use pores larger than a critical size of 3 - 4 nm, in
which water freezes to a mixture of cubic crystals and amorphous ice. In smaller pores, however,
water does not crystallize. Another possibility is to study the behavior of water in nanotubes.
Kolesnikov et al.
8
investigated the behavior of encapsulated water inside carbon nanotubes
(CNTs), and reported an anomalous “soft” dynamics characterized by pliable hydrogen bonds,
anharmonic intermolecular potentials, and large-amplitude motions. They proposed that the
structure of water in nanotubes consists of a square-ice sheet wrapped into a cylinder inside the
CNT with the interior molecules being in a chainlike configuration, and that the motion of the
water molecules is enhanced along the chains. Such an enhanced motion well below the freezing
point has been verified experimentally. A key characteristic of water inside narrow nanotubes may
be fast formation of hydrogen bonds that is sustainable and lasts longer than in the bulk. Based on
molecular dynamics (MD) simulations, Mashl et al.
9
proposed 8.6 °A as the critical radius of CNTs
in which the water displays anomalous icelike behavior in both symmetry and mobility, while
retaining a liquid-like degree of water, which is hydrogen bonding. Thus, in confined media such
as nanotubes, water manifests a behavior that is intermediate between a solid and a liquid. The
42

calculated meansquare displacements (MSDs) of the water molecules inside the CNTs are
consistently smaller than those in the bulk. But, chainlike water molecules can travel up to 40 times
faster inside CNTs than in the bulk, hence providing excellent channels for molecular transfer.
10

3.3.2 Supercooled Confined Water: The Cage Effect
Using MD simulation, in this chapter, we study the dynamics of water motion at low
temperatures in silicon-carbide nanotubes (SiCNTs). In particular, we study the so-called cage
effect on water in SiCNTs. Many liquids, when cooled rapidly below the melting point, undergo a
glass transition, causing their viscosity to increase dramatically. The increase is due to a slowdown
of the dynamics often referred to as the cage effect, rather than any significant structural changes:
any of the liquid’s molecules is confined by a “cage” formed by its neighboring molecules. When
the cage rearranges, the liquid relaxes and the molecules can diffuse in the system.
11,12
The rate of
change of molecular surroundings of a molecule is characterized
12
by a cage correlation (CC)
function. Over time, the CC function decays, and the anomalous slowing of the dynamics is
characterized by the decay, as explained below. While the dynamics of the cages in liquids under
bulk conditions have been studied both experimentally and by computer simulations,
13-15
it has
not, to our knowledge, been studied in a nanostructured material, such as a nanotube that we
studied in chapter 3 and 4. In addition to its fundamental scientific importance, the phenomenon
which we study is relevant to the behavior of water in biological materials, for example, in confined
or “crowded” cellular environments.
3.4 SiCNT and Molecular Modeling
As discussed in Chapter 2, silicon carbide is an important material with many excellent
properties, such as high fracture toughness, thermal shock resistance, and the ability for
43

withstanding high temperatures and corrosive environments. It is also a promising material for
fabrication of various nanostructured materials, such as nanoporous membranes
16
with pore sizes
on the order of a few angstroms, to be used for separation of fluid mixtures in harsh environments,
in which other types of membranes fail. Needless to say, we choose to work with SiCNTs due to
their rich behavior. The tubules formation of SiCNTs, and the existence of specific charge
arrangement on their surface make them attractive materials for a variety of purposes. Moreover,
the polar-polar repulsion between the π bonds of SiCNT and water does not allow water molecules
to come too close to the surface, hence acting effectively as a “non-wetting” surface (to the extent
that the concept of wetting can be considered in such nanostructured materials). Most stable
SiCNTs have a one-to-one ratio of Si and C, while any other Si/C ratio causes their collapse onto
a nanowire or clusters with solid interior
17
. The USC group under Professor Muhammad Sahimi’s
supervision has recently fabricated SiCNTs
18
and nanofibers
19
and, using MD simulations, has
studied
20,21
a variety of phenomenon in them, both in the gas and liquid phases. In particular, our
MD simulation
21
of flow of water in SiCNTs at room temperature, described in Chapter 2,
indicated that, for a given volume flow rate of water, the pressure drop needed to induce water
flow in SiCNTs is one order of magnitude smaller than their CNT counterparts.
Similar to CNTs, there are three types of SiCNTs, but in this chapter we work with the
zigzag-type nanotubes, SiCNT (m,0), in which one has rows of alternating Si and C atoms
perpendicular to the tube’s axis. Thus, in the honeycomb lattice of SiC - its primary structure -
each Si has three C neighbors and vice versa, and only Si-C bonds are formed. Such nanotubes
may be formed by simply rolling the lattice of Si and C atoms. The length of the Si-C bond is
about 1.8 °A, 10% longer than that of the C-C bond in CNTs. To carry out the MD simulations,
we first generated the SiCNT using energy minimization and an enhanced algorithm for the
44

Newton’s method, together with the conjugate-gradient minimization. The simulations were
carried out in the canonical, (NVT), ensemble. As in Chapter 2, the forcefield used was Condensed-
phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS). The Nose´-
Hoover-Langevin dynamics was used for the thermostat, and the Ewald summation was utilized
for the calculation of the electrostatic and long-range interactions. The generated SiCNT (12,0),
with a diameter of 11.9 Å, was anchored and immersed in bulk water; see Figure 3.1.

Figure 3.1 The system under study: a zigzag SiCNT immersed in water.

The water molecules were represented by the extended simple point charge model
22
(SPC/E).
3.5 Molecular Dynamic Simulation
After the tube was immersed in water, energy minimization was performed again. Multiple
runs were then carried out in the range 200 K - 300 K, in order to evaluate the CC function by
monitoring the movement of water molecules inside the nanotube. The water molecules outside
the nanotube were also monitored. The simulations were carried out for at least 1000 ps, analyzing
45

the results in 1 ps increments. To calculate the CC function, we used a generalized neighbor list
23

to keep track of each atom’s neighbors. In liquids, such as water, the immediate neighborhood of
an atom is best described by a list of other atoms that make up the first solvation shell. If the list
of an atom’s neighbors at time t is identical to the list of the neighbors at time 0, the CC function
takes on a value of 1 for that atom.
23
But, if any of the original neighboring molecules are not in
the cage at time t, it implies that that atom (or molecule) has hoped outside the cage and, thus, the
CC function is 0 at that time. Thus,
23
if we average over all the atoms in the simulation, the CC
function provides us with a direct way of measuring the hopping times even from very short MD
simulations. We used
23
the location of the oxygen in water as the basis for calculating the CC
function, as it is the closest to the center of mass of the water molecules. By considering how each
atom’s location varies with time, we generated a list in the form of a generalized vector of
neighbors for each individual oxygen atom i of water molecules inside the tube with radial distance
list
r ,
   
,
i ij
L t f r



(3-1)
with, j = 1, 2, · · · ,N, where N is the total number of molecules (atoms) in the system, and
 


 

otherwise
r r
r f
list ij
ij
0
1
(3-2)
There are various ways of selecting
list
r
23
depending on how much computer time one can afford.
Given a suitable
list
r , the CC function at time t is then given by
 
   
  0
0
2
i
i i
t
t C
L
L L 
 , (3-3)
46

Note that, since even in the bulk the neighborhood of molecules must change completely in order
for () Ct to vanish, its decay with time is slow. Clearly, then, in a nanostructured material, the
decay of () Ct is even slower.
3.6 Stretched Exponential Decay

Figure 3.2 presents a sample of the diffusion paths for the water molecules inside the
nanotube.

Figure 3.2 Diffusion paths of water molecules inside the nanotube.

47

Figure 3.3 The cage correlation function at two temperatures.

Figure 3.3 presents the time-dependence of () Ct at 250 K and 273 K. The decay of () Ct at 273
K is faster than at 250 K, as expected, but is still slow. Our MD simulation at 220 K and lower
temperatures indicated no significant decay of () Ct over the simulation time that we could afford.
The decay is simply too slow and computing () Ct at such temperatures requires much longer
simulations. The slow decay of () Ct has motivated its representation by the Kohlrausch-Williams-
Watts
24,25
stretched exponential form,
( ) exp
t
Ct










, (3-4)
where τ is a relaxation time scale, and β is a shape factor that has to do with the topology of the
space in which the molecules move. In general
16

*
*
2
d
d
 

, (3-5)
48

where d
*
is some sort of effective dimensionality of the system. It has been proposed if only short
range forces are present, then, d
*
= 3, so that, β = 3/5. Philips proposed however, that in the
presence of both short and long range forces d
*
= d/2, where d is the spatial dimension of the
system, so that β = d/(d+4). With d = 3, one obtains, β = 3/7  0.43. To our knowledge, up to
now, Phillips’ new prediction has not been confirmed for any liquid with short and long-range
forces in any system, particularly in a confined medium, such as a nanotube.

Figure 3.4 Double logarithmic plot of the cage correlation function at 250 K
49

Figure 3.5 Same as in Figure 3.4, but at 273 K.

To estimate the exponent β, we rewrite Eq. (4) as
    ln ln lnt Ct     

. Thus, a plot
of double logarithm of () Ct versus ln t must be straight line with a slope β. Figures 3.4 and 3.5
present such plots at 250 K and 273 K. Both yield, β ≃ 0.438, in excellent agreement with Phillips’
theoretical prediction, β = 3/7. We also estimate that that the relaxation times, τ ≃ 3.0 and 0.64 for
250 K and 273 K, respectively. Thus, τ increases significantly as the system is cooled down. Our
estimate of the exponent β should be compared with those obtained under the bulk condition at
low temperatures. For Lennard-Jones fluids the estimate, β ≈ 1/2 has been reported,
23
although
values as high as β ≈ 0.8 have also been reported.
28
Phillips
27
describes a variety of supercooled
liquids for which the value of is around 3/5. Note also that if we plot () Ct versus t/τ, the CC
functions at various temperatures should collapse onto each other. This is shown in Figure 3.6.
50

Figure 3.6 The cage correlation fucntion versus t/τ, where τ is the relaxation time, at two
temperatures.

The CC function represents an average over all the atoms in the simulation. As such, it is
not capable of distinguishing between non-stretched exponential decay, which is caused by a static
distribution of local environments in the nanotube, from non-stretched exponential decrease due
to dynamically fluctuating local environments. However, because
   
0
xt
C t x e dx 




, (3-6)
where   x  is the distribution of hoping rates of the molecules into and out of the cages, the CC
function does provide information on   x  . In addition, the fact that the CC function is described
by a stretched exponential function implies that diffusion of the water molecules inside the
51

nanotubes does not follow the Stokes-Einstein relation, and one must develop a proper
generalization of the relation for nanostructured materials. Summarizing, using extensive MD
simulations, we studied the dynamics of low-temperature water in SiCNTs by computing the cage
correlation function C(t) at temperatures below water’s freezing point in the bulk.
3.7 Conclusions
Our study shows cage correlation function at time t follows the Kohlrausch-Williams-
Watts law, Eq. (3-4). For the temperature range 220 K < T ≤ 273 K we present β ≃ 0.438, in
excellent agreement with, and confirming for the first time the prediction by Phillips,
27
β = 3/7.
The stretched exponential nature of the CC function that we studied in SiCNT has important
implications for diffusion of water in nanostructured materials, conformational dynamics of
proteins in “crowded” cellular environments, and material transport in nanogeoscience, and the
validity of the Stokes-Einstein formula. Moore studies with higher computational capacity is
recommended to provide higher level of understanding of water behavior in subcooled conditions
inside nanotubes.

52

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3. S.-H. Chen et al., "Aptamers evolved from live cells as effective molecular probes for
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in single-walled aluminosilicate nanotubes," ACS Nano 3, 1548 (2009).
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8. A. I. Kolesnikov et al., "Anomalously soft dynamics of water in a nanotube: a revelation
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10. G. Hummer et al. "Water conduction through the hydrophobic channel of a carbon
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emptying kinetics of carbon nanotubes in water," J. Chem. Phys. 117, 10789 (2002); C.
Dellago, M. M. Naor, and G. Hummer, "Proton transport through water-filled carbon
nanotubes," Phys. Rev. Lett. 90, 105902 (2003).
11. C. A. Angell, "Formation of glasses from liquids and biopolymers," Science 267, 1924
(1995); F. H. Stillinger, "A topographic view of supercooled liquids and glass formation,"
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and glasses," J. Phys. Chem. 100, 13200 (1996); C. A. Angell, "Ten questions on
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glassformers, and a real spaceexcitations' model with some answers on fragility and phase
transitions," J. Phys.: Conden. Matter 12, 6463 (2000).
12. E. Rabani, J. D. Gezelter, and B. J. Berne, "Calculating the hopping rate for self-diffusion
on rough potential energy surfaces: Cage correlations," J. Chem. Phys. 107, 6867 (1997).
13. W. Kob, C. Donati, S. J. Plimpton, P. H. Poole, and S. C. Glotzer "Dynamical
heterogeneities in a supercooled Lennard-Jones liquid," Phys. Rev. Lett. 79, 2827 (1997);
C. Donati et al., "Stringlike cooperative motion in a supercooled liquid," ibid. 80, 2338
(1998); B. Doliwa and A. Heuer, A. ibid. 80, 80, 4915 (1998); E. R. Weeks and D. A.
Weitz,. "Properties of cage rearrangements observed near the colloidal glass transition,"
ibid. 89, 095704 (2002).
14. C. Donati et al., "Spatial correlations of mobility and immobility in a glass-forming
Lennard-Jones liquid,” Phys. Rev. E 60, 3107 (1999); D. N. Perera and P. Harrowell,
"Relaxation dynamics and their spatial distribution in a two-dimensional glass-forming
mixture," J. Chem. Phys. 111, 5441 (1999).
15. W. K. Kegel and A. van Blaaderen, "Direct observation of dynamical heterogeneities in
colloidal hard-sphere suspensions," Science 387, 290 (2000); E. R. Weeks et al., "Three-
dimensional direct imaging of structural relaxation near the colloidal glass transition," ibid.
287, 627 (2000).
16. B. Elyassi, M. Sahimi, and T. T. Tsotsis, "Silicon carbide membranes for gas separation
applications," J. Membr. Sci. 288, 290 (2007); B. Elyassi, M. Sahimi, and T. T. Tsotsis, "A
novel sacrificial interlayer-based method for the preparation of silicon carbide
membranes," ibid. 316, 73 (2008).
17. A. Mavrandonakis et al., "From pure carbon to silicon-carbon nanotubes: An ab-initio
study," Nano Lett. 3, 1481 (2003).
18. S. H. Barghi, T. T. Tsotsis, and M. Sahimi, "Chemisorption, physisorption and hysteresis
during hydrogen storage in carbon nanotubes," Int. J. Hydrogen Energy 39, 21107 (2014).
19. B. Elyassi, T. W. Kim, M. Sahimi, and T. T. Tsotsis, "Effect of polystyrene on the
morphology and physical properties of silicon carbide nanofibers," Mater. Chem. Phys.
118, 259 (2009).
20. K. Malek and M. Sahimi, "Molecular dynamics simulations of adsorption and diffusion of
gases in silicon-carbide nanotubes,” J. Chem. Phys. 132, 01430 (2010).
54

21. M. Khademi, M. Sahimi, "Molecular dynamics simulation of pressure-driven water flow
in silicon-carbide nanotubes," J. Chem. Phys. 135, 204509 (2011).
22. H. J. C. Berendsen, J. R. Grigeria, and T. P. Straatsma, “The missing term in effective
pair potentials,” J. Phys. Chem. 91, 6269 (1987).
23. E. Rabani, J. D. Gezelter, and B. J. Berne, "Direct observation of stretched-exponential
relaxation in low-temperature Lennard-Jones systems using the cage correlation
function," Phys. Rev. Lett. 82, 3649 (1999).
24. R. Kohlrausch, “XLVI. Theory of the electric residue in the Leyden jar," Annal. der Physik
und Chemie 91, 56 (1854).
25. G. Williams and D. C. Watts, "Further considerations of non-symmetrical dielectric
relaxation behavior arising from a simple empirical decay function," Trans. Faraday Soc.
66, 80 (1971).
26. H. Scher and M. Lax, "Stochastic transport in a disordered solid. I. Theory," Phys. Rev. B
7, 4491 (1973); P. Grassberger and I. Procaccia, "The long time properties of diffusion in
a medium with static traps," J. Chem. Phys. 77, 6281 (1982).
27. E. A. Nel et al., "Understanding biophysicochemical interactions at the nano–bio
interface,” Nature 8, 543 (2009).
28. J. C. Phillips, "Microscopic aspects of stretched exponential relaxation (SER) in
homogeneous molecular and network glasses and polymers," J. Non-Cryst. Solids 3853,
357 (2011).

55

Chapter 4: Static and Dynamic Properties of Supercooled
Water in Nanotubes
4.1 Introduction
The structure and properties of fluids in nanotubes have attracted significant attention, extending
the interests of science and industry to microscopic length scales, previously thought to be too
difficult to study. In particular, there has been much interest in studying molecular interactions of
water molecules with each other and with a nanotube’s walls at very low temperatures below its
freezing point in the bulk. The problem is partly motivated by its critical importance to biological
phenomena, and the question of how microorganisms survive at very low temperatures. The
cellular structure of such microorganisms contains nano-channels and nano-pores and, thus, what
happens to water in such nanoscale structures has significant effect on the future of preservation
of cells and live microorganisms. In Chapter 3 we studied extensively the dynamics of water inside
silicon-carbide nanotubes (SiCNTs) at and below the normal freezing temperature of water,
1
and
in particular the cage correlation function. In this chapter, we show, based on molecular dynamics
(MD) simulation, that water molecules inside nanotubes do not follow typical freezing behavior
seen in nature under the bulk conditions. We argue that due to spatial limitations and steric
hindrance inside small enough nanotubes, there is not enough room for the formation of hexagonal
ice and, therefore, water inside such tubes attains a lower energy at temperatures below the typical
freezing point without ice formation. This result has extensive consequences, which we will
describe. We also extend our simulations to carbon nanotubes in order to check whether the results
are of a general nature.

56

4.2 Motivation
Preservation of cells and especially DNA has critical importance, for both short and long
term, to scientific research. Maintaining the physical and chemical integrity of live organisms in
stable conditions are currently done in a suitable solution, in frozen form, or in dried matrix.
2,3

Cost, quality and restricted molecular mobility in an environment free of active metal catalysts
and damaging ions are also critical and represent a current challenge. Cao
4
showed that most of
the damage to proteins is resulting from the formation of small ice particle during fast freezing, as
well as change in the pH during freezing. We show by extensive MD simulations of the structure
and properties of low-temperature water that, inside nanotubes of a certain size or smaller, ice
formation does not occur. This presents a perfect environment for microorganism preservation at
low temperatures. In real life, if we can fabricate a clean pure nanotube without metal
contaminants, we would be able to freeze and preserve the basic elements of life unchanged, so
that in the future scientists can compare samples with any evolved genes, even thousands of years
from now. None of us will, of course, be present at that time, but we can still save our DNA and
our successors can compare, optimize and even analyze them at much higher scientific levels. This
will have significant scientific, ethical and financial outcomes.
4.3 Molecular Dynamics Simulation
The simulations for this study were carried out at temperatures ranging from 100K to 298K, in
order to better understand the behavior of water molecules inside SiCNTs. Anchored and
immersed in bulk water, a long (12,0) SiCNT was utilized in the MD simulation; see Fig. 4.1.
Many details of the MD simulations are the same as those described in Chapter 3 and, thus, are
not repeated here.
57

Multiple runs were carried out in the range of 100 K to 298 K, evaluating the atomic and
molecular behavior of water, as well as monitoring the movement of its molecules in the nanotube
and the large water bath that surrounds it. The same force field that was utilized in Chapters 2 and
3, namely, Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies
(COMPASS), was again used in the MD simulations. The water molecules were represented by
the three-site SPC/E model. Simulations were carried out for at least 350 ps, and we analyzed
results in picosecond increments.
5-8

Figure 4.1 The SiCNT and water system.

58

4.4 Spatial and Temporal Correlation Functions
Correlation functions represent the relation between two variables at two different points
in time or space. A classic example in statistical mechanics is the Ising model, which consists of
discrete variables (spins) that may be in one of the two states, +1 or −1, or up and down. The model
is defined on a lattice in which the spins interact with their nearest neighbors.
9-11
Thus, one can
define a variety of spatial and temporal correlation functions between two spins in the model. The
time-evolution of the spatial correlations between the atoms are described by space-time
correlation function:

, (4-1)
where < > represents an average over an ensemble of selected particles, and is the position
of particle i at time t. Double summation may be separated into a summation of the self ( ) and
distinct ( ) space-time correlation function.
12
In absence of physical data, the space-time autocorrelation function facilitates the analysis
of scattering data, and provides physical explanation for the behavior of a fluid.
13-15
The space-
time autocorrelation functions gives the probability, at a specified time that an atom will be located
at a distance r away from the location occupied by an atom at time zero. Such correlation functions
are time-dependent and over time, due to memory loss, the dependence of the correlation function
on the distance dissipates and vanishes.
9

             
 
 
    
N
i
N
j
j i d s
r t r r
N
t r G t r G t r G
1 1
0
1
, , ,
     

  t r
i

s
G
d
G
59

4.4.1 Self Space-Time Autocorrelation Function
The calculated self space-time autocorrelation function for water inside the SiCNT are
shown in Figs. 4.2 – 4.10, which compare the position of the atoms with their original locations,
monitoring changes and distances from initial starting trajectory, factoring time inside a SiCNT.
Movement of water molecules is a favorable to supporting molecular transport even at these low
temperatures, but not formation of ice. Thus, the results shown in Figures 4.2 - 4.12 provide
evidence for our initial assertion that water inside a small enough SiCNT has low mobility, but it
still breaks the cage (see Chapter 3), as expected, and predicted to be lower than 225 K (Fig. 4.5)
but higher than 200 K (Fig. 4.6), which proves that the region between homogeneous nucleation
temperature (231 K) and crystallization temperature is above the glass transition temperature of
150 K. The structure factor calculated by the space-time correlation function can be used to
evaluate the self-diffusivity from scattering in experimental studies.
15

Figure 4.2 Self space-time autocorrelation function for water at 298 K in SiCNT.
60

Figure 4.3 Self space-time autocorrelation function for water at 273 K in SiCNT.

Figure 4.4 Self space-time autocorrelation function for water at 250 K in SiCNT.

61

Figure 4.5 Self space-time autocorrelation function for water at 225 K in SiCNT.

Figure 4.6 Self space-time autocorrelation function for water at 200 K in SiCNT.

62

Figure 4.7 Self space-time autocorrelation function for water at 175 K in SiCNT.

Figure 4.8 Self space-time autocorrelation function for water at 150 K in SiCNT.

63

Figure 4.9 Self space-time autocorrelation function for water at 125 K in SiCNT

Figure 4.10 Self space-time autocorrelation function for water at 100 K in SiCNT.

For the sake of comparison, we have computed the same in a CNT of the same size under similar
conditions, the results of which are shown in Figs. 11 and 12. Qualitatively, they are similar to
64

those for SiCNT, with the main difference being in the location and magnitude of the peaks of the
functions.

Figure 4.11 Self space-time autocorrelation function for water at 273 K in CNT.

Figure 4.12 Self space-time autocorrelation function for water at 250 K in CNT.

65

As the results for the self space-time autocorrelation function for both SiCNTs and CNTs indicate,
and as our results for other correlations discussed below demonstrate, the behavior of water inside
nanotubes is less a function of the structural material of the tubes, but related more to their
dimensional characteristcs and the geometry. However, the nanotube material still affects the water
arrangement due to hydrophilicity or hydrophobicity of its walls, affecting the distance between
the tube’s wall and the first water molecule in the direction perpendicular to the wall and toward
the center.
4.4.2 Radial Distribution Function
The radial distribution function (RDF), g(r), presents the variation of the density based on the
distance from a reference particle,
10
representing the probability density at a distance r away from
the reference position. Evaluation of the RDF is significant to thermodynamic properties, as it is a
second-order correlation function that is related to the structure factor, directly linked with
experimental X-ray and neutron diffraction
16-18
(the scattering intensity is essentially the Fourier
transform of the RDF). Isothermal compressibility, pressure of the system and its potential energy
can also be calculated using the RDF properties. The RDF distribution is calculated by the
following relation:
19
, (4-2)
for a number of unique pairs (Npairs), with system’s total volume being V, r being the distance
between pairs of particles, and P(r) representing the average number of atoms between differential
distances of r and r+dr.
 
 
dr r
V
N
r P
r g
pairs dr
Lim
2
0
4











66

We computed and compared the site-site RDF for a series of temperatures ranging from 298 K to
100 K, in order to evaluate the changes arising from reducing the temperature in the SiCNT. The
results are shown in Figs. 4.13 – 4.30 for the (12,0) SiCNT. As shown, we computed the RDF for
the pairs OO, OH and HH, and the results indicate that they are distinct. The oxygen RDF is mainly
a measure of evaluation of the main contribution by the atomic radius to water molecule. The HH
interaction, however, provides us with evidence and distribution of hydrogen bonds across the
system being investigated. Although the density is slightly reduced when reducing the temperature,
theory and experimental data are systematically similar within the small truncation errors. The
peaks of the computed RDF for OO, shown in Figs 4.14 and 4.16 at 273 K and 298 K, are at
locations similar to those reported by Nilsson
20
on the structure of liquid water, typical of chainlike
behavior that we will focus on shortly.
21,22

The gradual crystallization of water in the bulk is explained by decreasing temperature and
terminating the coexistence of two phases separating the states of liquid water. Structural changes
are governed by hydrogen bonding related to the clustering process, and the development of
tetrahedral coordinated network in bulk water (related to low- density water and formation of ice
at lower temperatures).
As noted in Chapter 3, water behavior inside nanotubes has structural differences with its
bulk behavior.
23-29
Inside the nanotube, water molecules behave in a chainlike formation, not
tetrahedral. High-density water (HDW) has potentially one “distorted H-bond” that, due to single
donor nature of the hydrogen bond inside a nanotube, is considered weak. The other H-bond is
much stronger. The system needs to couple and balance bonding, leading to chainlike structure in
a “disordered network connected via weak hydrogen bonds,” Disorder in the HDW reduces the
chances of formation of a second shell. We speculate that the weaker behavior of hydrogen bond
67

is related to the spatial limitations inside the nanotube, affected by the van der Waals and quantum
effects. Currently, there are theoretical predictions, but no experimental verification, for this
arrangement to confirm that water is liberated from the expected bulk tetrahedral arrangement in
nanotubes at low temperatures.
20
The aforementioned behavior would explain why HDW inside
the nanotube is more probable. We speculate that the freedom of movement and smaller possibility
of organized crystalized arrangement provide higher flexibility and more space for other “friendly”
water molecules with different interaction intensity for the RDF. In the case of low-density water
(LDW), we expect to see a deformation (minimum) in the RDF radial at about 4.5 Å (which does
not, however, always occur) that may be attributed to the LDW and the potential continuation of
conversion of LDW and HDW. For the HDW, we expect to observe a peak around 3.5 Å or higher
in the RDF of OO. This would be a sign of higher density in the water system inside the nanotube
and formation of HDW in our simulated system. We can also further relate and explain the HDW
relation with hydrophobic behavior of the SiCNT. The water is observed to be in the center with
chainlike formation in our simulations, and there is a clear radial distance from the tube wall (the
“no man’s land”) creating another intermolecular force that will enforce the high-density state
inside the tube with water inside.
30
Our simulations provide strong evidence of HDW. There are,
however, further interactions due to the chainlike behavior and the weak H-bonds.
Figures 4.13 and 15 show the typical behavior of the HH for water. Although we observe
a small peak around 5.5 Å, the peak appears mild at 298 K and 273 K, but becomes sharper at 250
K and 225 K. We may explain this behavior as a sign of disruption of hydrogen bonds and potential
water group separation in chain-like behavior. At lower temperatures, such as 175 K, the RDF
peak for OO is more visible around 8 Å, which may be related to pulse-type behavior of water
inside the tube presented in chapter 2. This is another fascinating opportunity for future studies.
68

We also calculated the RDF for a (15,0) CNT at 250 K and 273 K; the results are shown in
Figs. 4.30 - Fig. 4.34. We observe similar behavior for water inside the CNT, providing further
evidence that the size and diameter of a nanotube are the most important factor affecting the
general behavior of water inside nanotubes.

Figure 4.13 Radial Distribution function for water and HH in SiCNT.

69

Figure 4.14 Radial distribution function for OO and OH in SiCNT.

Figure 4.15 Radial distribution function for water and HH in SiCNT at 273 K.

70

Figure 4.16 Radial distribution function for OO and OH in SiCNT at 273 K.

Figure 4.17 Radial distribution function for water and HH in SiCNT at 250 K.

71

Figure 4.18 Radial distribution function for OO and OH in SiCNT at 250 K.

Figure 4.19 Radial distribution function for water and HH in SiCNT at 225 K.

72

Figure 4.20 Radial distribution function for OO and OH in SiCNT at 225 K.

Figure 4.21 Radial distribution function for water and HH in SiCNT at 200 K.
73

Figure 4.22 Radial distribution function for OO and OH in SiCNT at 200 K.

Figure 4.23 Radial distribution function for water and HH in SiCNT at 175K
74

Figure 4.24 Radial distribution functions for OO and OH in SiCNT at 175 K.

Figure 4.25 Radial distribution functions for water and HH in SiCNT at 150 K.
75

Figure 4.26 Radial distribution functions for OO and OH in SiCNT at 150 K.

Figure 4.27 Radial distribution functions for water and HH in SiCNT at 125 K.

76

Figure 4.28 Radial distribution functions for OO and OH in SiCNT at 125 K.

Figure 4.29 Radial distribution functions for water and HH in SiCNT at 100 K.
77

Figure 4.30 Radial distribution functions for OO and OH in SiCNT at 100 K.

Figure 4.31 Radial distribution functions for water and HH in CNT at 273 K.

78

Figure 4.32 Radial distribution functions for OO and OH in CNT at 273 K.

Figure 4.33 Radial distribution functions for water and HH in CNT at 250 K.

79

Figure 4.34 Radial distribution functions for OO and OH in CNT at 250 K.

There is, however, another interesting effect observed in the RDF: as the temperature is lowered
to 100 K, we still observe density changes with the distance, another evidence of slow but still
considerable movements of the water molecules, trying to find lower entropy arrangement inside
the nanotube, balancing the molecular and atomic forces, to become stable.
4.5 Diffusion and Velocity Autocorrelation Function
Albert Einstein in 1905 and Marian Smoluchowski in 1906 independently proposed what is
known as the Stokes-Einstein equation for the relation between the self-diffusivity of spherical
particles with a solution viscosity at low Reynolds number in liquids,
(4-3)
where D is self-diffusivity, represents the viscosity at temperature T, and kB is the Boltzmann’s
constant. The mean-square displacement of the diffusing molecules may be computed in order to
calculate the self-diffusion coefficient using the Einstein relation,
r
T k
D
B
 6


80

(4-4)
Where represents an average of the displacements in time over an ensemble of particles at a
given time, and is the Cartesian position vector of the molecule i at time t.
32,33

The velocity autocorrelation function is defined by
(4-5)
which represents an average over the ensemble at time t. If the VACF is flat, there may be no
interaction between the atoms in the system, based on Newton’s laws of motion, due to constant
velocity for each individual particle. However, in our MD simulation and computation of water
VACF inside the nanotubes, we observe that the VACF decreases, implying de-correlation of the
velocity with time, but does not vanish completely. If the VACF decreases rapidly, we would infer
that we have a gas or a very low- density fluid. However, in solids and liquids, due to larger
molecular forces and close proximity, the system comes into state of balance based on repulsion
and attraction, thus presenting slower damp harmonic motion.
26
We can also define the self-
diffusion coefficient through the VACF for the center of the mass in molecular motion, by the
Green-Kubo relation as noted in Chapter 2.
35
We have carried out extensive MD simulations to compute the VACF of water inside
SiCNT. The results are shown in Figs. 4.35 - 4.43 for water inside the (12,0) SiCNT. We also
computed the VACT of water in the equivalent CNT; the results are presented in Figs. 4.44 and
4.45. They represent the typical fluid liquid-like behavior inside the nanotubes, with no sign of
crystal formation or organizational order in the water molecules. The hydrophobic behavior of the
SiCNT wall affects the formation of water clusters inside and outside the shell of the nanotube,
   
2
0
6
1
i i
r t r
t
D
 
 
  t r
i

    t t v t v VACF
i i
  
0 0
81

constructing a boundary around the nanotube’s internal and external walls - the no man’s land -
due to the molecular interaction. Observation of multiple minima in the VACF below the zero line
is a strong sign of existence of liquid water inside the nanotube. Another benefit of calculating
VACF is the ability to determine the vibrational spectrum of the system by taking a Fourier
transform, which will be critical in the future experimental studies for this system.
Note that after its initial sharp decay, the VACF fluctuates around the zero line. The reason
is that the decay of the function is due to long-range memory of the molecules hoping forward.
But, if they also hope back, then, they lose the memory of the previous steps, which forces to the
VACF to decay again and hence becoming negative.

Figure 4.35 Velocity autocorrelation function for water at 298 K in SiCNT

By inspecting the VACF of water inside the SiCNT, we can describe the behavior. We see small
but gradual reduction in amplitude of the VACF, which is as another evidence for lower molecular
movement. The frequency does, however, appear to change slowly in the time period investigated
82

with longer wavelength at lower temperatures. There is a potential for a viscoelastic behavior of
water at low temperatures inside the SiCNT. Study of the Maxwell model combined with the
VACF can shed more light on this conclusion

Figure 4.36 Velocity autocorrelation function for water at 273 K in SiCNT

Figure 4.37 Velocity autocorrelation function for water at 250 K in SiCNT.

83

Figure 4.38 Velocity autocorrelation function for water at 225 K in SiCNT

Figure 4.39 Velocity autocorrelation function for water at 200 K in SiCNT.
84

Figure 4.40 Velocity autocorrelation function for water at 175 K in SiCNT.

Figure 4.41 Velocity autocorrelation function for water at 150 K in SiCNT

85

Figure 4.42 Velocity Autocorrelation function for water at 125 K in SiCNT

Figure 4.43 Velocity Autocorrelation function for water at 100 K in SiCNT.
86

Figure 4.44 Velocity autocorrelation function for water at 273 K in CNT.

Figure 4.45 Velocity autocorrelation function for water at 250 K in CNT.

87

Figure 4.46 Mean-square displacement of water at 298K in SiCNT.

The results for the mean-square displacements of the water molecules inside the nanotubes provide
two distinct insights, in addition to providing estimates of the self-diffusion coefficient. We see a
discontinuity around 250 ps in the MD data, which may be related to a potential behavioral change
in the movement of the molecules. At higher temperatures (ambient and above) the MSD curve is
continuous and increases at almost same rate. However, at lower temperatures, the MSD has a
plateau. We interpret this to be the potential exit of the water molecules from one stage to another,
which may be due to initiation of breaking the cage.
The second distinct behavior is related to the similarity of the axial MSD and the total MSD
for water molecules. It is clear that water transfer happens inside the nanotube mostly in the axial
direction. There is, however, radial diffusion, even if it is small, which is related to the travel of
the water molecules from the center toward the “no man’s land.” The dominance of the axial
diffusion is attributed to the chain-like formation, explained before, with stronger hydrogen bonds,
which we call fast sliding motion. Radial diffusion, on the other hand, is most notably related to
88

the less strong hydrogen bonds inside the tube that endows the chains the ability to move fast in
the axial direction, and explains the fast movements of the water molecules inside the tubes as we
presented in Chapter 2.

Figure 4.47 Mean-square displacements of water at 273 K in SiCNT

Figure 4.48 Mean-square displacements of water at 250K in SiCNT.
89

Figure 4.49 Mean-square displacements of water at 225 K in SiCNT.

Figure 4.50 Mean-square displacements of water at 200K in SiCNT.
90

Figure 4.51 Mean-square displacements of water at 175K in SiCNT.

Figure 4.52 Mean-square displacement of water at 150 K in SiCNT.
91

Figure 4.53 Mean-square displacements of water inside SiCNT at 125 K.

Figure 4.54 Mean-square displacement of water at 100 K in SiCNT.
92

Figure 4.55 Mean-square displacements of water in CNT at 273 K.

Figure 4.56 Mean-square displacements of water in CNT at 250 K.

93

Figure 4.57 Temperature-dependence of diffusivity of water in SiCNT.

We also calculated the self-diffusion coefficient of water in the (12,0) SiCNT; the results
are presented in Fig. 4.57. The gradual reduction of the self-diffusivity with reducing temperature
is expected, but the temperature-dependence is not linear and appears to have a slow logarithmic
dependence. This is another fascinating consequence of the supercooled water inside nanotubes,
presenting a challenge for the validity of the Stokes-Einstein equation, Eq. (4.3) that relates the
viscosity and self-diffusion. Further work in our group is in progress, aiming at evaluating the
validity of the Stokes-Einstein equation.
4.6 Conclusions
We calculated various correlation functions for low-temperature water in SiCNT and CNT,
in order to elucidate the properties of characteristics of water molecules in nano-scale materials.
94

The properties calculated included the mean-square displacement, the velocity autocorrelation
function, the radial distribution function and the self space-time autocorrelation function. The
results of our MD simulations suggest lower molecular mobility inside the nanotubes than in
ambient temperature and under the bulk conditions. Ice formation does not, however, occur in the
nanotubes. The MD simulations did not produce any evidence of ice crystal formation.

95

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98

Chapter 5: Mechanical, Transport and Sorption Properties of
Polyetherimide-SiCNT Membranes

5.1 Introduction
By now it should be clear that silicon-carbide nanotubes (SiCNTs) present new
opportunities for novel applications, such as separation of fluids and storage of gases at high
operating temperatures and pressures, at which their carbon counterparts are not as efficient or
even a viable option. The SiCNTs have superior chemical and mechanical properties, resistance to
corrosion, and are well suited for many applications under harsh conditions. In this chapter we
study the development of a mixed-matrix membrane, one that is made of SiCNTs and
polyetherimide (PEI). Our study focus includes evaluation of the mechanical properties of the
SiCNT-PEI composite at various thermodynamic conditions, as well their properties as a
membrane for gas separation. We employ molecular dynamics (MD) and Monte Carlo simulations
in order to carry out an extensive study of the composite material. Our simulations provide insights
into diffusion and sorption of hydrogen and methane in the SiCNT-PEI composite. With the
calculation of the compliance and stiffness tensors based on the MD simulation, we are able to
evaluate the mechanical properties and compare with available experimental data, whenever
possible. We show that with the high mechanical strength of the composite due to the addition of
the SiCNTs to PEI matrix, high-pressure separation of gaseous mixture by the SiCNT-PEI
composites is feasible.

99

5.2 The Polymer Matrix
The PEI with the repeating unit of (C37H24O6N2)n (see Fig. 1) is a transparent amber polymer in
amorphous state with high impact strength, temperature radiation and creep resistance. It is being
used in medical applications, as well as advanced electric devices, automobiles and aircraft
interior. It has a glass transition temperature of 216 °C and amorphous density of 1.27 g/cc at
25 °C. It is also susceptible to stress cracking in chlorinated solvents.
1,2
By carbonization of the
PEI, permanent high-selectivity membranes have been fabricated for pairs of oxygen-nitrogen,
nitrogen-helium and carbon dioxide-methane.
3
The USC group that the leadership of Professors
Muhammad Sahimi and Theodore Tsotsis has been using PEI in the fabrication of carbon
molecular-sieve membranes (CMSM), which have been commercialized.
4,5

Producing polymeric composites with carbon fibers has been an important topic in both
industrial applications and scientific innovations, although the interaction between the fiber, the
matrix and the compatibility between them have long been issues of concern, limiting the
mechanical and chemical properties that can be obtained. Cold treatment of nitrogen-oxygen
plasma for composites improves some of the properties and roughness of the fiber.
7

Figure 5.1 The repeating units of polyetherimide.
100

The PEI has also been extensively used to increase toughness for epoxy resins, although a slight
reduction in the Young’s modulus has also been reported.
6
In light of the PEI’s excellent weather
resistance and stability, as well as applications in micro-electronic devices and aerospace industry,
we study the possibility of improving its properties by formation of new composites. Nanotubes
are primary material of choice for such improvements. Significant theoretical and experimental
work has been performed using single- and multi-walled carbon nanotubes (SWCNT and
MWCNT, respectively). Nanotubes are material of choice for building cost effective, practical,
and highly reinforced composites, but the dispersion of the nanotubes and their interactions with
the matrix must still be adressed.
7
5.3 Molecular Dynamics Simulation
Similar to the previous chapters, we constructed a simulation cell with dimensions 30×30×100 Å
3

containing a single (9,0) SiCNT; see Fig. 5.2. We use a “smart” algorithm to fill the remainder of
cell with the PEI without any overlap or close contacts. In order to optimize our system and identify
its equilibrium state, we used energy minimization and multiple-step MD simulations to optimize
the structure. For all the simulations discussed in this chapter, we employed the COMPASS force
field (see Chapter 1). The Nosé-Hoover-Langevin (NHL)
10-13
dynamics was used as a consistent
thermostat. Long-range non-bond interactions of the van der Waals type were computed by
introducing a long cut-off length, and the Coulomb interactions were computed using the Ewald
summation technique.

101

Figure 5.2 The (9,0) SiCNT and PEI composite. The methane molecules are on the left side.

Using MD simulation, we created a system with initial density of at 298 K to
evaluate the transport properties of carbon dioxide, hydrogen, oxygen and methane in the
membrane. The system was initially relaxed numerous times and finally simulated for 500 ps with
500,000 time steps, analyzing results every 1000 frames. Figure 5.3 presents the simulation
temperature and energy for each system, indicating that the temperature has been fixed, and the
true minimum energy states of the composites have been reached.

3
1000
m
kg
102

Figure 5.3 Temperature and total energy for MD simulation of H2, CH4, O2 and CO2 in the
PEI-SiCNT composite.

5.4 Diffusion of Gases in the PEI-SiCNT Composites
In previous work by the USC group under Professor Sahimi’ guidance, extensive MD
simulation was performed to study the transport of CO2 and CH4 in a PEI-SWCNT composite. It
was shown that the presence of the SWCNTs creates fluctuations in the polymer cavities larger
than those in the absence of the nanotubes, giving rise to larger diffusivities in the polymer
composite than in the pure PEI.
8,9
In our work in this chapter, we evaluate transport properties of
gases in the PEI-SiCNT composite, and study the possible applications for building engineering
composites based on SiCNT-PEI material. As discussed in previous chapters, many properties of
SiCNTs are superior to those of CNTs. The purpose of the study is to evaluate performance of
103

polymeric matrix membranes by inserting SiCNTs in the polymer’s matrix and comparing the
results with their carbon counterparts. There has been no previous investigation of the diffusivity
of gases in the PEI-SiCNT composite either.
The mean-square displacements (MSD) for each individual gas inside the PEI-SiCNT
composite are shown in Fig. 5.4. The calculated diffusivities, shown in Table 5.1, indicate high
value of the self-diffusion coefficient for hydrogen in the PEI-SiCNT composite, hence
demonstrating the potential for separation of hydrogen from other gases and significant industrial
applications. The higher values of the self-diffusivities of methane and carbon dioxide, when
compared with the corresponding values in a pure PEI, provide further evidence for the validity of
the assertion

Table 5. 1 Temperature and total energy for MD simulation of H2, CH4, O2 and CO2 in the
PEI-SiCNT composite
Self-diffusion coefficients
(cm
2
/sec) ×10
9

O2 H2 CO2 CH4
PEI-(9,0) SiCNT 4.6 3500 2.56 2.1

PEI (experimental)
9
1.3-11.4 1.13

104

Figure 5.4 Mean-square displacements (MSD) of the gases in the PEI-SiCNT composite
computed by MD simulation.

We also computed the self space-time correlation function (see Chapter 4) for hydrogen,
methane, oxygen and carbon dioxide in the PEI-SiCNT; see Fig.5.5. They indicate similar patterns
of distribution and movement in the composite, as well as the mobility of the gas molecules. Once
again, hydrogen presents a significantly different behavior with much faster movement, providing
evidence regarding its fast transport in the composite.

105

Figure 5.5 Self space-time correlation function of H2, CH4, O2 and CO2 in the PEI-SiCNT
composite.

5.5 Gas Sorption
Adsorption is defined as adhesion of fluid molecules to a surface, creating adsorbate on
adsorbent as a surface-based process, with desorption being the reverse process. The total energy
is defined as:
,
(5-1)

represents the intramolecular energy between sorbate molecules, defines interaction
energy and

the total intermolecular energy between sorbate molecules. Single gases, when
Sorb
mol
IST
mol
Sorb
mol mol
U E E E   
Sorb
mol
E
IST
mol
E
Sorb
mol
U
106

adsorbed onto a sorbent, release heat, measureable by calorimetry. The heat is quantifiable by
differentiating adsorption isotherms with respect to temperature.
14,15
Figure 5.6 presents two Gibbs
phases for a gas and solid, with the solid including two phases of inert and nonvolatile sections,
plus the adsorbed phase.

Figure 5.6 Schematic diagram, dividing the boundaries as a thermodynamic state.

We define isosteric heat (heat of desorption) as the difference of partial molar enthalpy in gas
phase and excess partial molar enthalpy in adsorbed phase:
, (5-2)
With the definition of partial molar enthalpy and its relation to Gibbs energy, we conclude,
 
be
st i i
i
q h h 
107

, (5-3)
where T represents the temperature and is the chemical potential. The Isosteric heat of
adsorption at absolute pressure p is then calculated by the Clausius-Clapeyron equation
16
,
, (5-4)
Thus, using computer simulation and statistical mechanics, we calculate the isosteric heat of
adsorption. Utilizing the grand-canonical ensemble, the chemical potential is calculated as an
independent variable, while the adsorption isotherms are evaluated by simulation. We calculate
the mean number of molecules with following correlation,
j i
T
i
a
i
kT n
 











 

,
ln
(5-5)
with representing the partition function in the grand canonical ensemble.
5.6 Monte Carlo Simulations
Monte Carlo (MC) simulation reduces the number of particles in order to manage the
number of calculation and iterations in a periodic manner, which may affect surface calculation
effects. Metropolis
17
presented a modified MC method with enhancement steps, by using random
sampling in place of a regular array. It applies weighing scheme , combined with
2
e
i
e
i
n
T
hT
T
   
 








 
2
ln
1
st
q
q
p
TP
q R R
PT
T



 

  

   



















 
RT
E
exp
108

periodic boundary condition (PBC) and minimum image convention, preserving ergodicity.
Siepmann
18
proposed the configurational-bias MC method for large molecules, but in our
simulations, due to small size of the gases, we apply the Metropolis MC without configurational
bias, treating the sorbate as rigid, and incorporating rigid-body reorientations and translation to
reduce the calculations’ complexity.
19
For evaluation of the transport and sorption properties, with
application of Metropolis Monte Carlo method, we generate and transform configuration to
evaluate the sorption in each stage, with initial trial stage and transforming to a proposed
configuration to achieve target density ratio with the highest probability. Based on the sampling,
the lower energy is always accepted over higher energy configuration, unless the probability
density is significantly higher. After the random selection of each sorbate, conformation, rotation
and translational properties are monitored. Fixed pressure simulation is performed in different
pressures in order to define sorption isotherms. Each MC simulation corresponds to 100,000 steps
with at least 10,000 moves at each step (insertion, deletion and displacement).
5.7 Results and Discussion
Darkim
20
presented MC simulation of hydrogen adsorption in single-walled carbon
nanotubes (SWCNT), stacked together in parallel, presenting favorable potentials outside and
inside nanotubes with decreasing adsorption with increase in the nanotube’s diameter due to
attractive forces, reducing their effectiveness. Malek and Sahimi.
21
investigated the adsorption of
hydrogen in SiCNTs, noting the higher effectiveness of SiCNTs versus their carbon counterparts.
Our simulation provides insights into the diffusion and sorption of hydrogen, oxygen, methane and
carbon monoxide in the composite material.

109

Figure 5.2 presents the schematics of the (9,0) SiCNT-PEI composite, with CH4 molecules
visible on the left side. We observe that most of the methane molecules during sorption are either
inside the nanotube with a centralized configuration and linear orientation by the tube’s axis, or
reside in the gaps and voids.
Figure 5.7 presents weight percent loading - sorption isotherms - of the four gases in the
PEI-SiCNT composite. As presented in Fig. 5.7, carbon dioxide has the highest weight percent
sorption, while hydrogen has the lowest. This is a significant finding that indicates the potential
for industrial use of this type of membrane for separation of carbon dioxide from hydrogen,
produced during steam methane reforming process at high purity and low cost.

Figure 5.7 Weight percent loading of the gases in the PEI-SiCNT composite.

110

To validate our MD results, we simulated MWSiCNT, which has been fabricated and
utilized by Professor Sahimi’s research group in large scale. We were able to compare the
experimental loadings
22
at corresponding pressures within tolerance of the measurement method
and limitations of model size (simulation calculation capabilities) presented in Fig 5.8.

Figure 5.8 Simulated and experimental MWSiCNT Hydrogen sorption.

As the second case study for adsorption, we constructed a new system with the PEI and a
(6,0) SiCNT, which was also utilized later for the evaluation of the physical and mechanical
properties (see section 5.8). We carried out MD simulations at 298 K and 423 K as the references,
in order to evaluate the loading weight fraction. This was partly intended to study the effect of the
number of nanotubes inside the composite material by increasing the number of nanotubes up to 5
111

to see how the system responds to the change. We observed that the addition of nanotubes does
not help sorption of hydrogen. Although some improvement was seen when we added the 2
nd

nanotube, the decline in the loading of hydrogen and methane is not a promising sign for this
composite. We suspect that for this PEI-SiCNT composite the voids and gaps are larger than the
optimal size to provide lower energy state for the efficient sorption of hydrogen. The results are
presented in Figs. 5.9 – 5.20.

Figure 5.9 Sorption of hydrogen and methane in PEI at 298 K.

112

Figure 5.10 Sorption of hydrogen and methane in PEI at 423 K.

Figure 5.11 Sorption of hydrogen and methane in the PEI and a single (6,0) SiCNT at 298 K.

0 500 1000 1500 2000 2500 3000
0
0.01
0.02
0.03
0.04
0.05
0
0.3
0.6
0.9
1.2
1.5
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
113

Figure 5.12 Sorption of hydrogen and methane in PEI and a single (6,0) SiCNT at 423 K.

Figure 5.13 Sorption of hydrogen and methane in PEI and two (6,0) SiCNTs at 298 K.

0 500 1000 1500 2000 2500 3000
0
0.002
0.004
0.006
0.008
0.01
0
0.3
0.6
0.9
1.2
1.5
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
114

Figure 5.14 Sorption of hydrogen and methane in PEI and two (6,0) SiCNTs at 423 K.

Figure 5.15 Sorption of hydrogen and methane in PEI and three (6,0) SiCNTs at 298 K.

0 500 1000 1500 2000 2500 3000
0
0.002
0.004
0.006
0.008
0.01
0
0.02
0.04
0.06
0.08
0.1
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
0 500 1000 1500 2000 2500 3000
0
0.01
0.02
0.03
0.04
0.05
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
115

Figure 5.16 Sorption of hydrogen and methane in PEI and three (6,0) SiCNTs at 423 K.

Figure 5.17 Sorption of hydrogen and methane in PEI and four (6,0) SiCNTs at 298 K.
0 500 1000 1500 2000 2500 3000
0
0.004
0.008
0.012
0.016
0.02
0
0.01
0.02
0.03
0.04
0.05
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
0 500 1000 1500 2000 2500 3000
0
0.01
0.02
0.03
0.04
0.05
0
0.1
0.2
0.3
0.4
0.5
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
116

Figure 5.18 Sorption of hydrogen and methane in PEI and four (6,0) SiCNTs at 423 K.

Figure 5.19 Sorption of hydrogen and methane in PEI and five (6,0) SiCNTs at 298 K.
0 500 1000 1500 2000 2500 3000
0
0.004
0.008
0.012
0.016
0.02
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
0 500 1000 1500 2000 2500 3000
0
0.01
0.02
0.03
0.04
0.05
0
0.4
0.8
1.2
1.6
2
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
117

Figure 5.20 Sorption of hydrogen and methane in PEI and five (6,0) SiCNTs at 423 K.

5.8 Stress and Deformation
“In continuum mechanics,
23
stress is a measure of the internal forces acting within a
deformable body. Quantitatively, it is a measure of the average force per unit area of a surface
within the body on which internal forces act.” Due to assumption of continuum behavior of
material, internal forces result in recoverable deformation up to a certain load. At higher loads,
external forces, or internal ones such as the residual stresses, result in permanent deformation and
failure.
23
Even before the discovery and manufacturing of nanotubes’ based on graphene sheet, by
1960s Bacon
24-25
had described a vision for the mechanical properties expected from growing
fibers similar to tensile mechanical qualities of the ideal graphene sheet. Tersoff
26
recommend a
method for utilizing properties of graphene sheet, predicting strain energy of nanotubes. In this
section we use MD simulation to predict the mechanical properties of composites constructed by
the nanotubes and the PEI. In our case study we utilize the (6,0) SiCNT for the evaluation.
0 500 1000 1500 2000 2500 3000
0
0.004
0.008
0.012
0.016
0.02
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 2500 3000
Wt % loading H2
Wt % loading CH4
Time (ps)
CH4
H2
118

We introduce the elastic stiffness tensor and write the Hook’s law (c is a fourth-rank tensor
forming major symmetries for stiffness tensor or the elasticity tensor),
, (5-6)
i and j are 1, 2 or 3. If the material follows linear elasticity, we may relate stress and strain by
Voigt notation,
27,28

, (5-7)
Taking the inverse relation between strain and stress,
,
(5-8)
S is called the compliance tensor .
29
and are the stress and strain second-
rank tensor. denotes the normal strain, while is the shear strain.
(5-9)
The physical meaning of compliance matrix is defined for solids in terms of elastic and shear
modulus, as well as the Poisson ratios.
29
 
 

3
1
3
1 k l
kl ijkl ij
c  
kl ijkl ij
C   
  : S 
 
kl ijkl ij
S   : 
ij

kl

i

ij






























































xy
zx
yz
z
y
x
z
y
x
z
y
x
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s












66 65 64 63 62 61
56 55 54 53 52 51
46 45 44 43 42 41
36 35 34 33 32 31
26 25 24 23 22 21
16 15 14 13 12 11
119

(5-10)
Ex, Ey and Ez are the effective elastic moduli in the x, y and z direction. Gyz, G xz and Gxy are the shear
moduli in the yz, xz and xy planes, while is Poisson’s ratio (ratio of strain direction in y direction
with acting stress in x direction).

is the coefficients of Chentsov characterization of shear
strain in a plane parallel to the xy plane, inducing the tangential stress in a plane parallel to xz.
is the coefficient of mutual influence for the first type of characterization of strain in the x direction
caused by shear stress acting on a plane parallel to yz.
30

The Young’s modulus is defined as the ratio of stress and the strain with a dimension of
pressure. It represents the factor of proportionality of Hook’s law,
31-33
valid only in certain
proximity of stress and strains until permanent deformation is initiated.
(5-11)
Given the elastic compliance, we can calculate the Young’s modulus.
29
Using MD or MC
simulations, and evaluating the fluctuations, we can evaluate the elastic constants. Parinello and
Rahman presented the relationship between the fluctuation and elastic constants as:




































 
 
 





















xy xz
xz xz
yz
xz xz
z
z xy
y
y xy
x
x xy
xy
xy xz
xz yz
yz xz
z
z xz
y
y xz
x
x xz
xz
xy yz
xz
xz yz
yz z
z yz
y
y yz
x
x yz
xy
xy z
xz
xz z
yz
yz z
z y
yz
x
xz
xy
xy y
xz
xz y
yz
yz y
z
yz
y x
xy
xy
xy x
zz
xy x
yz
yz x
z
zx
y
yx
x
G G G E E E
G G G E E E
G G G E E E
G G G E E E
G G G E E E
G G G E E E
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s
1
1
1
1
1
1
, , , , ,
, , , , ,
, , , , ,
, , ,
, , ,
, , ,
66 65 64 63 62 61
56 55 54 53 52 51
46 45 44 43 42 41
36 35 34 33 32 31
26 25 24 23 22 21
16 15 14 13 12 11
    
    
    
   

    
  


xy

xz xy,

x yz ,



 E
120

, (5-12)
, (5-13)
5.9 Analysis of the Stress with MD Simulation
For a comprehensive model, we constructed a 50×50×50 Å
3
cell utilizing the (6,0) SiCNTs. We
then inserted the polymer chains, filling the cell initially with no nanotube for the first series of the
MD simulation, and then added one nanotube at a time up to 5 nanotubes, randomly located in the
cell. We repeated this at least three times with the random insertion and averaged the results, in
order to improve the quality of data and repeatability for the mechanical property evaluation.
As a secondary method, in order to compare the mechanical characteristics of SiCNTs and
CNTs in the same PEI, we also created a PEI matrix structure embedded with 5 nanotubes. We
used both the (6,0) SiCNT and (6,0) CNT separately to study the potential improvements and
changes in structures, and to understand which nanotube may present higher mechanical strength.
We began from the composite with 5 nanotubes and at each step we removed one nanotube
until we had pure PEI. To avoid any potential void spaces and to relax the system, each system
was considered as new, and was individually simulated to minimize its total energy. It was then
optimized and simulated under 1 GPa pressure to form a viable configuration in multiple steps,
monitoring the structure.
Running the intermediate simulation under pressure reduces void fraction significantly. We
used a script to evaluate the compliance tensor and measure the mechanical properties for all
structures.
1 

kl ij ijkl
V
kT
C  
kl ij ijkl
kT
V
S   
121

Figure 5.21 Five (6,0) SiCNTs in the PEI matrix.

5.10 Results and Discussions
Figure 5.21 displays five SiCNTs embedded in the PEI matrix as the initial starting point
for the evaluation of the mechanical properties. The computed Young’s moduli, as a measure of
the mechanical strength of the PEI plus a number of (6,0) SiCNTs, is shown in Fig. 5.22. Although
we expected to see higher moduli with increasing the number of SiCNTs in the polymer matrix,
we do not see a significant increase. We relate this unexpected result to increased void fraction
and less favorable interaction between the SiCNTs and the PEI. In comparison, the composite with
the (6,0) CNTs possesses higher mechanical strength, due to the smaller diameter and favorable
122

interaction. The PEI chains wrap better around the CNTs, improving the strength of the structure.
The improvement of mechanical properties in the direction of the nanotubes axis is clear. The
commercial PEI modulus is reported to be about 3 GPa, which agrees with the value calculated
with our simulation.

Figure 5.22 Young’s modulus of the composite of (6,0) SiCNT and the PEI.

The nanotubes are not fixed in the matrix and during the MD simulations, they move and
deviate from the orientation in the z direction. This creates the behavior observed in the changes
of the modulus while increasing the number of nanotubes in the polymer matrix. As a case study
for one nanotube while observing the structure, the PEI chains wrap around the nanotube very
efficiently, and there is not much void left, but with 2, 3 and 4 nanotubes in the z direction, the
modulus increases, but the average does not change significantly. When 5 nanotubes are present,
especially when they are SiCNT, there are voids and the polymer does not wrap around the
nanotube as well as a single nanotube, due to limited mobility of the polymer chains. Looking at
123

MD data for four and five SiCNTs, the modulus suffers due to the increase in the void fraction and
incompatibility at higher volume ratios of SiCNTs to PEI. The MD data indicate that there is a
better compatibility between CNT and the PEI than between SiCNT and the PEI. We present all
the compliance and stiffness tensor in Appendix 5.1 at the end of this chapter.

5.11 Conclusions
This study provides, for the first time, insight into the structure of composites made of
nanotubes and a polymer, and presents the calculated values of the mechanical characteristics of
PEI-SiCNT composites. We calculated the compliance and stiffness tensors to enable further
calculation of the properties, as well as the evaluation of the system for future studies. Based on
the calculated diffusion, sorption and mechanical evaluation, the potential for high-pressure
application of polymer composite membranes for gas separation can be considered. This system
provides an excellent selective medium for gas separation.

124

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126

Appendix A: Compliance and Stiffness Factors
We note the compliance and stiffness tensors of the polyetherimide and the SiC and carbon
nanotubes in our study of Chapter 5 in the following tables.
Table A.1The stiffness tensor of the PEI
Stiffness Tensor (Cij) for PEI (Gpa) at 298K
4.2803 2.1237 1.8462 0.1656 0.0409 0.0257
2.1237 5.264 2.0538 -0.2059 0.0225 -0.1297
1.8462 2.0538 3.0481 -0.1325 0.4409 0.0836
0.1656 -0.2059 -0.1325 1.2549 0.0265 0.031
0.0409 0.0225 0.4409 0.0265 1.2358 0.0632
0.0257 -0.1297 0.0836 0.031 0.0632 0.9777
Stiffness Tensor (Cij) for PEI (Gpa) at 423K
0.9065 0.7181 1.8091 -0.2241 0.0274 0.5262
0.7181 0.935 1.6385 -0.4019 0.1265 0.6335
1.8091 1.6385 7.3891 -0.2754 0.2174 0.2516
-0.2241 -0.4019 -0.2754 -0.3629 -0.0374 0.0183
0.0274 0.1265 0.2174 -0.0374 2.0941 -0.1018
0.5262 0.6335 0.2516 0.0183 -0.1018 0.1217

127

Table A. 2 The compliance tensor of the PEI
Compliance Tensor (Sij) for pure PEI (1/TPa) at 298 K
347.5783 -77.7269 -168.896 -77.4123 52.1494 -5.929
-77.7269 282.1672 -149.56 38.582 47.5057 47.9744
-168.896 -149.56 562.9766 62.537 -191.182 -53.1574
-77.4123 38.582 62.537 821.3221 -36.945 -21.8722
52.1494 47.5057 -191.182 -36.945 877.3899 -34.3021
-5.929 47.9744 -53.1574 -21.8722 -34.3021 1036.782
Compliance Tensor (Sij) for pure PEI (1/TPa) at 423 K
2558.343 -2398.01 -112.221 1221.808 225.5872 1659.896
-2398.01 1740.947 148.5251 -503.108 -47.9984 1033.174
-112.221 148.5251 143.3042 -228.337 -55.4551 -596.43
1221.808 -503.108 -228.337 -2862.29 -102.854 -1847.18
225.5872 -47.9984 -55.4551 -102.854 471.6247 -200.867
1659.896 1033.174 -596.43 -1847.18 -200.867 -2995.49

Table A.3 The stiffness tensor of the composite of the PEI with one (6,0) SiCNT.
Stiffness Tensor (Cij) for PEI & 1SiC (Gpa) at298K
4.5741 3.8056 3.2966 0.0633 -0.151 -0.4027
3.8056 8.8574 3.6117 0.1242 0.1678 -0.1179
3.2966 3.6117 3.8218 -0.3556 -0.3402 -0.2906
0.0633 0.1242 -0.3556 2.3103 0.0974 -0.0789
-0.151 0.1678 -0.3402 0.0974 2.7804 0.0094
-0.4027 -0.1179 -0.2906 -0.0789 0.0094 0.8032
Stiffness Tensor (Cij) for PEI & 1SiC (Gpa) at 423K
4.9607 1.884 2.5801 -0.0647 -0.801 -0.1117
1.884 3.1865 2.0668 0.0982 -0.3791 -0.4415
2.5801 2.0668 9.4508 -0.3688 -0.4441 -0.6947
-0.0647 0.0982 -0.3688 1.3641 -0.0618 -0.099
-0.801 -0.3791 -0.4441 -0.0618 1.8421 -0.2844
-0.1128 -0.4276 0.1216 0.0421 0.1182 1.6236

128

Table A.4 The compliance tensor of the composite of the PEI and one SiCNT.
Compliance Tensor (Sij) for pure PEI & 1SiC (1/TPa) at 298 K
638.4177 -76.189 -478.071 -81.8995 -16.7863 128.0548
-76.189 200.4126 -133.801 -29.9971 -31.3479 -59.7591
-478.071 -133.801 824.9684 145.7806 77.7601 52.5648
-81.8995 -29.9971 145.7806 460.972 -1.1271 52.5793
-16.7863 -31.3479 77.7601 -1.1271 370.1635 10.6734
128.0548 -59.7591 52.5648 52.5793 10.6734 1324.504
Compliance Tensor (Sij) for pure PEI & 1SiC (1/TPa) at 423 K
290.3152 -134.051 -46.7009 13.3872 85.1509 -17.4522
-134.051 444.5799 -55.4367 -46.4543 30.0319 76.1002
-46.7009 -55.4367 135.3706 41.3942 7.9732 36.6802
13.3872 -46.4543 41.3942 753.6088 39.1788 49.5258
85.1509 30.0319 7.9732 39.1788 605.6926 106.051
-17.4522 76.1002 36.6802 49.5258 106.051 566.4765

Table A.5 The stiffness tensor for the composite of the PEI and two SiCNTs
Stiffness Tensor (Cij) for PEI & 2SiC (Gpa) at 298K
5.4142 2.3769 2.9855 0.1894 -0.4266 -0.2509
2.3769 4.0852 2.5759 -0.0224 -0.0153 -0.1104
2.9855 2.5759 6.9814 -0.1691 -0.3018 0.0275
0.1894 -0.0224 -0.1691 0.9961 0.033 0.2167
-0.4266 -0.0153 -0.3018 0.033 1.0017 -0.0295
-0.2509 -0.1104 0.0275 0.2167 -0.0295 0.7878
Stiffness Tensor (Cij) for PEI & 2SiC (Gpa) at 423K
15.1313 9.4957 10.5722 -0.1682 0.6394 -0.1535
9.4957 14.4028 10.6472 0.1412 0.5394 -0.0393
10.5722 10.6472 13.2184 0.4112 0.3596 0.5066
-0.1682 0.1412 0.4112 2.4619 0.2025 0.2823
0.6394 0.5394 0.3596 0.2025 2.1491 0.1221
-0.1535 -0.0393 0.5066 0.2823 0.1221 2.4589

129

Table A.6 The compliance tensor of the composite of the PEI and two SiCNT.
Compliance Tensor (Sij) for PEI & 2SiC (1/TPa) at 298K
300.0111 -119.684 -82.3967 -102.036 107.8132 113.7702
-119.684 370.0691 -88.12 16.4083 -72.1077 9.6056
-82.3967 -88.12 213.7954 62.771 24.0651 -62.434
-102.036 16.4083 62.771 1110.771 -70.8656 -340.629
107.8132 -72.1077 24.0651 -70.8656 1055.073 82.3722
113.7702 9.6056 -62.434 -340.629 82.3722 1405.992
Compliance Tensor (Sij) for PEI & 2SiC (1/TPa) at 423K
159.2731 -22.3047 -110.748 29.4576 -27.7546 30.3994
-22.3047 178.5769 -126.779 7.8277 -19.2732 27.6432
-110.748 -126.779 269.2133 -40.4794 26.9977 -61.1025
29.4576 7.8277 -40.4794 422.0791 -41.6659 -36.0786
-27.7546 -19.2732 26.9977 -41.6659 479.3185 -26.6186
30.3994 27.6432 -61.1025 -36.0786 -26.6186 427.0829

Table A.7 The stiffness tensor of the composite of the PEI and three SiCNTs.
Stiffness Tensor (Cij) for PEI & 3SiC (Gpa) at 298K
4.1882 2.102 1.7582 -0.2865 -0.3613 -0.3288
2.102 3.7145 2.2125 -0.1849 0.0677 0.0762
1.7582 2.2125 4.7183 0.1596 0.3155 -0.0198
-0.2865 -0.1849 0.1596 1.3402 0.0019 0.039
-0.3613 0.0677 0.3155 0.0019 0.7918 -0.2056
-0.3288 0.0762 -0.0198 0.039 -0.2056 1.7996
Stiffness Tensor (Cij) for PEI & 3SiC (Gpa) at 423K
4.8855 1.2356 1.4721 0.2032 -0.0996 -0.2805
1.2356 2.8317 1.7717 0.1249 -0.1106 -0.1132
1.4721 1.7717 3.1785 -0.1035 0.0477 -0.5393
0.2032 0.1249 -0.1035 -0.078 0.0546 0.0183
-0.0996 -0.1106 0.0477 0.0546 1.54 -0.1802
-0.2805 -0.1132 -0.5393 0.0183 -0.1802 1.8897

130

Table A.8 The compliance tensor of the PEI and three SiCNTs.
Compliance Tensor for PEI & 3SiC at 298K
399.5422 -179.845 -83.6785 67.0605 258.8689 107.8186
-179.845 463.3907 -146.714 44.9387 -80.0142 -64.2148
-83.6785 -146.714 325.1446 -75.9841 -160.776 -22.2194
67.0605 44.9387 -75.9841 775.6992 54.923 -1.0431
258.8689 -80.0142 -160.776 54.923 1508.998 220.1423
107.8186 -64.2148 -22.2194 -1.0431 220.1423 603.0443
Compliance Tensor for PEI & 3SiC at 423K
219.9774 -96.6714 -30.772 457.9157 -6.4804 13.0444
-96.6714 490.0462 -211.263 802.0068 0.8505 -52.9671
-30.772 -211.263 435.1958 -957.024 17.087 117.859
457.9157 802.0068 -957.024 -8786.5 424.8079 -31.726
-6.4804 0.8505 17.087 424.8079 640.5132 60.9405
13.0444 -52.9671 117.859 -31.726 60.9405 567.7039

Table A.9 The stiffness tensor of the composite of the PEI and four SiCNTs
Stiffness Tensor for PEI & 4SiC at 298
4.8962 2.5905 2.0402 -0.1247 -0.1601 -0.0635
2.5905 4.6766 2.9091 0.1997 0.2249 0.153
2.0402 2.9091 6.0154 0.4458 -0.1522 -0.0105
-0.1247 0.1997 0.4458 1.2935 0.0116 -0.0958
-0.1601 0.2249 -0.1522 0.0116 0.2379 -0.0335
-0.0635 0.153 -0.0105 -0.0958 -0.0335 1.1364
Stiffness Tensor for PEI & 4SiC at 423K
3.2288 1.784 2.6252 0.5439 -0.1119 0.4285
1.784 4.1348 2.3618 0.7876 0.1972 0.2262
2.6252 2.3618 5.5856 0.2843 0.3519 0.0809
0.5439 0.7876 0.2843 2.1958 -0.0469 -0.2564
-0.1119 0.1972 0.3519 -0.0469 0.6693 0.0917
0.4285 0.2262 0.0809 -0.2564 0.0917 -0.2081

131

Table A.10 The compliance tensor for the composite of the PEI and four SiCNTs
Compliance Tensor for PEI & 4SiC at 298K
331.4169 -203.23 -8.4838 67.2435 415.3374 63.6824
-203.23 475.17 -176.586 -33.1779 -711.43 -100.675
-8.4838 -176.586 268.1257 -66.7693 340.2552 30.1496
67.2435 -33.1779 -66.7693 813.2978 5.0184 76.3371
415.3374 -711.43 340.2552 5.0184 5411.985 281.8414
63.6824 -100.675 30.1496 76.3371 281.8414 912.1086
Compliance Tensor for PEI & 4SiC at 423K
347.5783 -77.7269 -168.896 -77.4123 52.1494 -5.929
-77.7269 282.1672 -149.56 38.582 47.5057 47.9744
-168.896 -149.56 562.9766 62.537 -191.182 -53.1574
-77.4123 38.582 62.537 821.3221 -36.945 -21.8722
52.1494 47.5057 -191.182 -36.945 877.3899 -34.3021
-5.929 47.9744 -53.1574 -21.8722 -34.3021 1036.782

Table A.11 The stiffness tensor of the composite of the PEI and five SiCNTs.
Stiffness Tensor (Cij) for PEI & 5SiC (Gpa) at 298K
6.9373 3.2292 3.0674 0.4248 -0.1979 0.1812
3.2292 6.8457 2.2019 0.2679 -0.088 -0.1561
3.0674 2.2019 9.9451 -0.9475 -0.1371 -0.4325
0.4248 0.2679 -0.9475 1.1695 0.2593 0.0063
-0.1979 -0.088 -0.1371 0.2593 0.2984 0.1049
0.1812 -0.1561 -0.4325 0.0063 0.1049 1.092
Stiffness Tensor (Cij) for PEI & 5SiC (Gpa) at 423K
5.7018 3.2016 2.7568 0.156 -0.433 -0.0183
3.2016 5.232 2.4624 0.0451 -0.1396 -0.2976
2.7568 2.4624 7.462 -0.8064 -0.3748 -0.1189
0.156 0.0451 -0.8064 -0.1345 0.0384 0.061
-0.433 -0.1396 -0.3748 0.0384 0.5303 0.0966
-0.0183 -0.2976 -0.1189 0.061 0.0966 0.9882

132

Table A.12 The compliance tensor of the composite of the PEI and five SiCNTs.
Compliance Tensor (Sij) for PEI & 5SiC (1/TPa) at 298K
237.8453 -78.7855 -75.195 -197.383 309.868 -109.122
-78.7855 191.9278 -19.9538 -34.9004 14.4693 31.4079
-75.195 -19.9538 147.5864 191.9144 -184.502 84.6898
-197.383 -34.9004 191.9144 1385.182 -1335.52 224.0343
309.868 14.4693 -184.502 -1335.52 4840.565 -579.624
-109.122 31.4079 84.6898 224.0343 -579.624 1026.294
Compliance Tensor (Sij) for PEI & 5SiC (1/TPa) at 423K
277.8989 -170.207 -5.676 310.6521 170.6603 -82.6337
-170.207 311.7246 -31.9172 109.6306 -103.893 90.2695
-5.676 -31.9172 93.175 -539.129 87.1232 26.2491
310.6521 109.6306 -539.129 -3682.26 134.1763 188.0126
170.6603 -103.893 87.1232 134.1763 2091.672 -230.351
-82.6337 90.2695 26.2491 188.0126 -230.351 1051.709

Table A.13 The stiffness tensor of the composite of the PEI and one CNT .
Stiffness Tensor (Cij) for PEI & 1CNT (Gpa) at 298K
15.7676 11.4725 10.6331 0.1718 0.0574 -0.301
11.4725 15.608 11.2827 0.3552 -0.0855 -0.1581
10.6331 11.2827 17.6529 -0.0708 0.0529 -0.2099
0.1718 0.3552 -0.0708 1.2829 -0.0322 0.0305
0.0574 -0.0855 0.0529 -0.0322 1.6202 -0.0576
-0.301 -0.1581 -0.2099 0.0305 -0.0576 2.7623

Table A.14 The compliance tensor of the composite of the PEI and one CNT.
Compliance Tensor (Sij) for pure PEI & 1CNT (1/TPa) at 298K
148.2779 -82.7027 -36.3274 0.6274 -8.1151 8.4893
-82.7027 167.136 -57.2357 -37.9606 12.7505 -3.1097
-36.3274 -57.2357 115.247 26.9205 -4.9202 1.1211
0.6274 -37.9606 26.9205 791.9235 12.5524 -8.5444
-8.1151 12.7505 -4.9202 12.5524 619.0252 12.242
8.4893 -3.1097 1.1211 -8.5444 12.242 363.193

133

Table A.15 The stiffness tensor of the composite of the PEI and two CNTs.
Stiffness Tensor (Cij) for PEI & 2CNT (Gpa) at 298K
14.4976 10.0994 10.2017 -0.1919 0.58 0.1473
10.0994 14.7314 10.3629 0.1782 0.3339 -0.2587
10.2017 10.3629 14.5548 0.2984 0.155 0.1861
-0.1919 0.1782 0.2984 2.4083 -0.011 0.2642
0.58 0.3339 0.155 -0.011 2.3896 -0.2527
0.1473 -0.2587 0.1861 0.2642 -0.2527 3.1979

Table A.16 The compliance tensor of the composite of the PEI and two CNTs.
Compliance Tensor (Sij) for PEI & 2CNT (1/TPa) at 298K
163.6837 -62.8516 -70.085 27.6801 -27.653 -13.0194
-62.8516 161.3151 -70.8482 -10.46 -0.5268 20.8901
-70.085 -70.8482 168.6481 -20.1321 14.8768 -9.4761
27.6801 -10.46 -20.1321 424.6869 -5.8566 -36.5011
-27.653 -0.5268 14.8768 -5.8566 427.938 34.6704
-13.0194 20.8901 -9.4761 -36.5011 34.6704 321.3025

Table A.17 The stiffness tensor (Cij) for PEI and three CNT (Gpa).
Stiffness Tensor (Cij) for PEI & 3CNT (Gpa) at 298K
15.8774 11.8477 10.5973 -0.0525 0.0654 -0.0998
11.8477 16.6122 11.5043 0.2285 0.2202 -0.1039
10.5973 11.5043 19.1755 0.0319 0.6296 -0.2258
-0.0525 0.2285 0.0319 2.0497 0.0675 -0.1421
0.0654 0.2202 0.6296 0.0675 1.9625 -0.0056
-0.0998 -0.1039 -0.2258 -0.1421 -0.0056 2.1126

134

Table A.18 The compliance tensor of the composite of the PEI and three CNTs.
Compliance Tensor (Sij) for pure PEI & 3CNT (1/TPa) at 298K
145.3169 -82.2107 -31.4648 12.9354 14.0315 0.3687
-82.2107 149.8806 -44.5131 -18.3184 0.8274 -2.5027
-31.4648 -44.5131 97.1479 3.9551 -25.2395 6.9074
12.9354 -18.3184 3.9551 493.0335 -16.4949 33.2535
14.0315 0.8274 -25.2395 -16.4949 517.6478 -1.7277
0.3687 -2.5027 6.9074 33.2535 -1.7277 476.2249

Table A.19 The stiffness tensor of the composite of the PEI and four CNTs.
Stiffness Tensor (Cij) for PEI & 4CNT (Gpa) at 298K
11.9322 9.5003 10.3807 -0.3144 0.4644 0.1014
9.5003 14.4838 9.6554 -0.4401 0.3804 -0.5423
10.3807 9.6554 19.1504 0.3059 -0.3776 0.1885
-0.3144 -0.4401 0.3059 1.8414 0.113 0.0772
0.4644 0.3804 -0.3776 0.113 2.2283 -0.3105
0.1014 -0.5423 0.1885 0.0772 -0.3105 2.5567

Table A.20 The compliance tensor for the composite of the PEI and four CNTs.
Compliance Tensor (Sij) for pure PEI & 4CNT (1/TPa) at 298K
235.1579 -101.395 -77.5683 33.2934 -51.0361 -32.3192
-101.395 153.0723 -23.023 21.8447 -4.8642 36.9364
-77.5683 -23.023 107.3138 -38.8581 39.7349 -3.7183
33.2934 21.8447 -38.8581 564.0705 -48.1898 -16.6992
-51.0361 -4.8642 39.7349 -48.1898 477.431 57.507
-32.3192 36.9364 -3.7183 -16.6992 57.507 408.0085

Abstract (if available)

Abstract We have utilized atomistic modeling and extensive molecular dynamics (MD) simulation to study and explore various properties of fluids in silicon‐carbide nanotubes (SiCNTs), and in their composites with a polymer. ❧ First, we show that pressure‐induced flow of water in the SiCNTs of various sizes is more efficient than the same phenomenon in carbon nanotubes (CNTs), requiring a pressure drop (and hence energy) that is at least one order of magnitude less than that in the CNTs. ❧ Next, we study the dynamics of low‐temperature water in SiCNTs, demonstrating that the cage‐cage correlation function, a measure of the water molecules’ motion in the nanotubes, follows the Kohlrausch‐Williams‐Watts stretched exponential law with an exponent that is in excellent agreement with the theoretical prediction by J.C. Phillips. ❧ Third, using extensive MD simulations, we compute the various correlation functions for water in SiCNTs and CNTs over the temperature range 100 K - 298 K, demonstrating that, in agreement with our calculation of the cage‐cage correlation function, due to spatial limitations and steric hindrance inside small enough nanotubes, water inside such nanotubes does not freeze. This has significant scientific and biological implications, potentially providing a method for preserving microorganisms for very long time for advanced research and studies. ❧ Finally, we develop an atomistic model of mixed‐matric polymeric composite made of polyetherimide (PEI) and SiCNTs and study, (i) the mechanical properties of the composite and compare them with those of pure PEI, and (ii) diffusion and sorption of several light gases in the composite in order to test its potential use as a membrane for gas separation. The results indicate that such a membrane has excellent properties for separation of hydrogen from a gaseous mixture.

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Asset Metadata

Creator Khademi, Mahdi (author)

Core Title Exploring properties of silicon-carbide nanotubes and their composites with polymers

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publication Date 04/23/2017

Defense Date 04/23/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag molecular dynamic,nanotube,OAI-PMH Harvest,silicon carbide

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sahimi, Muhammad (committee chair), Kalia, Rajiv K. (committee member), Nakano, Aiichiro (committee member)

Creator Email khademimehdi@gmail.com,mkhademi@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-560771

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Legacy Identifier etd-KhademiMah-3395.pdf

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Document Type Dissertation

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Rights Khademi, Mahdi

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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