University of Southern California Dissertations and Theses

Molecular modeling of silicon carbide nanoporous membranes and transport and adsorption of gaseous mixtures therein

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Molecular modeling of silicon carbide nanoporous membranes and transport and adsorption of gaseous mixtures therein

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Content MOLECULAR MODELING OF SILICON CARBIDE NANOPOROUS MEMBRANES
AND TRANSPORT AND ADSORPTION OF GASEOUS MIXTURES THEREIN

by

Nafiseh Rajabbeigi

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

December 2009

Copyright 2009 Nafiseh Rajabbeigi
ii
Dedication

I would like to dedicate my thesis to my husband Dr. Bahman Elyassi for his love
and support, and to my parents for their love, patience, support and encouragement.

iii
Acknowledgements
I have been fortunate to work with many talented people. First and formost I
would like to thank my advisors Professors Muhammad Sahimi and Theodore T.
Tsotsis for their supports and guidance during my research. I thank them for their
constructive comments and their good-natured support.

I also thank to Professor Chi H. Mak for being my Ph.D. committee member, and
Dr. Katherine S. Shing and Professor Priya Vashishta for being my qualifying
committee members. I sincerely appreciate the help of Mr. Brendan Char and Ms.
Karen Woo, the staff of the Department of Chemical Engineering.

I would like to acknowledge the financial support of the National Science
Foundation and the Department of Energy, as well as the University of Southern
California Center for High-Performance Computing for the support of Parallel
computations for this work.

Finally, I would like to especially thank my husband Dr. Bahman Elyassi for
providing the experimental data for my research. Also I thank him for fulfilling
my heart with joy and love and his support in every moment of my life.
iv
Table of Contents

Dedication ii
Acknowledgements iii
List of Tables vi
List of Figures vii
Abstract xi
Chapter 1 ..................................................................................................................1
1. Introduction.......................................................................................................1
1.1 Introduction............................................................................................1
1.2 Nanoporous membranes.........................................................................5
1.3 Simulation methods ...............................................................................9
1.4 The plan of the thesis ...........................................................................12

Chapter 2 ................................................................................................................14
2. Single Pore Models.........................................................................................14
2.1 Introduction..........................................................................................14
2.2 Model of a carbon nanopore ................................................................14
2.3 The DCV-GCMD simulation...............................................................19
2.4 Configurational-bias Monte Carlo method ..........................................20
2.5 Molecular dynamics simulation...........................................................23
2.6 Results and discussions........................................................................25
2.7 Single-pore model of silicon-carbide membrane.................................48
2.8 The effect of wall structure of a SiC pore............................................56
2.9 Results and discussion .........................................................................56

Chapter 3 ................................................................................................................62
3. Molecular Pore-Network Model of Silicon Carbide Membranes:
Adsorption ..........................................................................................................62
3.1 Introduction..........................................................................................62
3.2 Atomistic pore-network model ............................................................63
3.3 MD Simulation in a pore network .......................................................75
v
3.4 Results and discussions........................................................................79
3.4.1 Adsorption of single gases...................................................................79
3.4.2 Comparison with experimental data ....................................................85
3.5 Summary..............................................................................................91

Chapter 4 ................................................................................................................99
4. Molecular Pore Network Model of Silicon Carbide Membranes:
Transport and Separation....................................................................................99
4.1 Introduction..........................................................................................99
4.2 Molecular model of the SiC membrane.............................................103
4.3 Non equilibrium molecular dynamics in pore network .....................107
4.4 Results and discussion .......................................................................116
4.4.1 Effect of temperature .........................................................................123
4.4.2 Effect of external pressure drop.........................................................125
4.4.3 Effect of porosity ...............................................................................127
4.4.4 Comparison with the experimental data ............................................134
4.5 Summary............................................................................................136

Bibliography.........................................................................................................138

vi
List of Tables

Table 2.1. Conversion factors between the dimensionless and actual
quantities ….…………………………………………………….……………...19

Table 2.2. The effect the structure of the pore wall on the separation
factor…………………………………………………………………………….61

Table 3.1.Value of L.J parameters used in the simulation ………….………….78

Table 3.2. The porosities of the three membranes, and the simulations'
parameters. N
P
, N
C
, and N
Si
represent, respectively, the number of Poisson
points, and carbon and silicon atoms.…………………………….……..……...86

Table 4.1. Dimensional and dimensionless variables …………………..…….108

Table 4.2. The molecular parameters of the various atoms and molecules
used in the nonequilibrium MD simulations of transport of gases in the
membrane….…………………...…………………………...…………………109

Table 4.3. Comparison of the computed and measured separation factors
for the two binary mixtures………………………………................................136
vii
List of Figures
Figure 2.1. Schematic of the slit pore used in the simulations...............................15
Figure 2.2 Density profiles for pore sizes of 2.5 (top), 3 (bottom) and 5 (next
page). Dashed lines indicate the boundaries of the pore region. ....................29
Figure 2.3 Density profiles between upper and lower walls for pore sizes of
2.5 (top), 3 (bottom) and 5 (next page)...........................................................31
Figure 2.4 Time-averaged temperature distributions in pore sizes of 2.5 (top),
3 (middle) and 5 (bottom). Dashed lines indicate the boundaries of the
pore region. .....................................................................................................33
Figure 2.5 The computed permeance and separation factors of CO
2
/C
6
H
14

mixtures for three pore sizes. ..........................................................................34
Figure 2.6 The computed permeance and separation factors of CO
2
/C
6
H
14

mixtures at three temperatures. .......................................................................35
Figure 2.7 Density profiles for pore sizes of 5 (top), 3 (bottom) and 2.5 (next
page). Dashed lines indicate the boundaries of the pore region. ....................36
Figure 2.8 Density profiles between upper and lower walls for pore sizes of 5
(top), 3 (bottom) and 2.5 (next page). .............................................................38
Figure 2.9 Computed permeance and separation factors of CO
2
/ C
3
H
8

mixtures for different pore sizes. ....................................................................40
Figure 2.10 Density profiles in the transport direction x. The CO
2
mole
fraction in the feed is 0.7 (top) and 0.9 (bottom). Dashed lines indicate the
boundaries of the pore region..........................................................................41
Figure 2.11 Computed permeance and separation factors of CO
2
/ C
3
H
8

mixtures versus the mole fraction of CO
2
. The dimensionless pore size is
H
*
=5. ...............................................................................................................42
Figure 2.12 Density profiles for the pore size of 5. The CO
2
mole fraction in
the feed is 0.7. The temperature is 70
o
C (top) and 100
o
C (bottom). Dashed
lines indicate the boundaries of the pore region. ............................................43
viii
Figure 2.13 Computed permeance and separation factors of CO
2
/ C
3
H
8

mixtures for different temperatures. The dimensionless pore size is H=5
and the mole fraction of CO
2
in the feed is 0.7...............................................44
Figure 2.14 The dependence of the permeance of CO
2
and C
3
H
8
, on pressure
drop ( ∆p) applied to the pore sizes of 5 (upper) and 2.5 (bottom) at
T=50
o
C. The mole fraction of CO
2
in the feed is 0.9, with the upstream
pressure of 20 atm. ..........................................................................................45
Fig 2.15 Effect of membrane thickness on its separation factor. ...........................47
Figure 2.16 The crystal structure of β-SiC.............................................................49
Figure 2.17 Computed radial distribution function for cubic SiC(top). The
computed radial distribution function g(r) of a-SiC (bottom). Arrows
indicate the locations of the peaks, corresponding to the covalent bonds. .....55
Figure 2.18 Dimensionless temperature profiles of Equimolar (top) mixture of
CH
4
/CO
2
and when mole fraction of CH
4
in the feed is 0.3 in a-SiC
membrane........................................................................................................58
Figure 2.19 Density profile of Equimolar mixture of CH
4
/CO
2
in β-SiC (top)
and a-SiC (bottom)..........................................................................................59
Figure 2.20 Density profile of mixture of CH
4
/CO
2
when the mole fraction of
CH
4
in the feed is 0.3 in β-SiC (top) and a-SiC (bottom)...............................60
Figure 3.1 Computed pore size distribution (PSD) for the system in which the
pores are selected biased (top) and randomly (bottom). .................................68
Figure 3.2 An example of 2D Voronoi pore network. Open polygons show the
pores. ...............................................................................................................71
Figure 3.3 The structure of the polyhedra and their connectivity in a small
portion of a 3D Voronoi pore network............................................................72
Figure 3.4 Schematic of 3D pore network used in our simulation........................73
Figure 3.5 The distribution of the coordination number in a 3D Voronoi pore
network with porosity of 0.5. ..........................................................................74
Figure 3.6 Adsorption isotherms for gases in SiC at porosity of φ=0.7. µ
i
and
ρ
i
are the chemical potential and loading of component i...............................80
ix
Figure 3.7 The same as Fig. 3.4, but φ=0.5...........................................................81
Figure 3.8 The same as Fig. 4.4, but φ=0.4............................................................82
Figure 3.9 The computed pore size distribution of the three SiC membranes......87
Figure 3.10 Nitrogen adsorption isotherms for our silicon carbide membrane
on support and support....................................................................................92
Figure 3.11 Our prepared membrane isotherm obtained from subtracting the
effect of support. .............................................................................................93
Figure 3.12 Comparison of the computed and measured nitrogen sorption
isotherms in membrane. ..................................................................................94
Figure 3.13 Pore size distribution of membrane II, adopted from Suda et
al.(Suda, Yamauchi et al., 2006).....................................................................95
Figure 3.14 Comparison of the computed and measured nitrogen sorption
isotherms in membrane II. ..............................................................................96
Figure 3.15 Pore size distribution of membrane III, adopted from Suda et al.
(Suda, Yamauchi et al., 2006).........................................................................97
Figure 3.16 Comparison of the computed and measured nitrogen sorption
isotherms in membrane III. .............................................................................98
Figure 4.1 Structural representation of the SiC Pore network. Spheres and
stars represent carbon and silicon atoms.......................................................106
Figure 4.2 Schematic of the pore network and two, high and low pressure,
control volumes.............................................................................................110
Figure 4.3 The average dimensionless temperature T* in the membrane (in
the middle) and the two control volumes on the left and right sides.
Numbers in the parentheses indicate the set temperatures. The porosity is
0.4, and the applied external pressure drop is 2 atm.....................................115
Figure 4.4 The dimensionless density profiles of the H
2
/CO
2
mixture in the
axial (X*) direction, in the membrane (middle) and the two control
volumes. The porosity is 0.4, and the applied external pressure drop is 2
atm.................................................................................................................119
x
Figure 4.5 The dimensionless density profiles of the H
2
/CH
4
mixture in the
axial (X*) direction, in the membrane (middle) and the two control
volumes. The porosity is 0.4, and the applied external pressure drop is 2
atm.................................................................................................................120
Figure 4.6 Effect of membrane thickness on its separation factor......................122
Figure 4.7 Effect of the temperature on the separation factor for the two
gaseous mixtures. ..........................................................................................124
Figure 4.8 Effect of the applied external pressure drop ∆P on the separation
factors for the mixture of H
2
/CH
4
. ................................................................126
Figure 4.9 Effect of the applied external pressure drop ∆P on the separation
factors for the mixture of H
2
/CO
2
. ................................................................127
Figure 4.10 Effect of the membrane’s porosity on its separation factor for the
H
2
/CH
4
mixture, for three applied external pressure drops ∆P.....................130
Figure 4. 11 Effect of the membrane’s porosity on the (dimensionless) fluxes
of H
2
and CH
4
................................................................................................131
Figure 4.12 Effect of the membrane’s porosity on its separation factor for the
H
2
/CO
2
mixture, for three applied external pressure drops ∆P.....................132
Figure 4.13 Effect of the membrane’s porosity on the (dimensionless) fluxes
of H
2
and CO
2
................................................................................................133

xi
Abstract
The goal of this work is to study transport of gas mixtures through nanoporous
membranes especially silicon carbide membranes.

In the first stage of this research, molecular dynamics simulation has been
performed to study the transport and separation of gas mixtures through a single
nanopore. Transport of binary mixtures of n-alkane and CO
2
in carbon nanopores
and the effect of various parameters such as temperature, pressure, pore size and
length of the system on the separation were investigated. For generating n-alkanes
chains Configurational-bias Monte Carlo method was used and combined with the
dual control volume grand canonical molecular dynamics simulation.

The next phase of this work is to study the transport of gas mixtures through
nanoporous silicon carbide membranes using molecular dynamics simulation. We
develop a new model for nanoporous materials and inorganic membranes, the pore
space of which consists of interconnected pores of irregular shapes and sizes. The
model is based on the Voronoi tessellation of the atomistic structure of the
crystalline or amorphous materials, of which the membrane is made. It generates
three-dimensional molecular pore networks with pore-size distributions (PSD) that
resemble those of real inorganic nanoporous materials. In addition to being
interconnected and having irregular shapes and distributed sizes, the pores also
xii
have rough internal surface, which is what one may expect to exist in most real
nanoporous materials. To test the validity of the model, we utilize it to model
adsorption in three distinct silicon-carbide (SiC) membranes. Equilibrium
molecular dynamics simulations are employed to compute adsorption isotherms of
nitrogen in the nanoporous SiC membranes. Using at most one adjustable
parameter, the simulated isotherms and the experimental data are in very good
agreement.

We also aim to simulate transport properties of our prepared SiC membranes using
the feed back that we get from our adsorption simulation. Non-equilibrium
molecular dynamics simulations used, in order to study transport and separation of
two binary mixtures, namely, H
2
/CO
2
and H
2
/CH
4
, and compare the results with
the experimental data. The model is demonstrated to provide accurate predictions,
in particular for the separation factors of the mixtures, without needing any
adjustable parameter.

1

Chapter 1
1. Introduction
1.1 Introduction

Porous materials are ubiquitous in nature. They range anywhere from porous rock,
to wood, skin, and various types of natural materials. They are also synthesized by
a wide variety of techniques, which have given rise to an amazing variety of such
materials. The range of pore sizes in porous materials includes pores of a few
angstroms – as in membranes – to tens of microns – as in porous rock. The prime
motivation for the synthesis of porous materials is the wide variety of the
applications that such porous materials have.

Transport and sorption in porous materials are of fundamental interest to
practically any application of such materials. One particular type of application of
porous materials, which is the subject of the present thesis, is gas separation by
porous membranes. Such membranes have recently received much attention, due
to the high permeability, low cost and thermo-chemical stability of that they
possess in harsh environments, which are favorable compared to the dense
2
membranes. Understanding of the permeation phenomena through membranes is
fundamental to designing effective membranes, or even identifying the best mode
of operation for the membranes (Kobayashi, Takami et al., 2002; Firouzi, Sahimi
et al., 2006). Such questions as, what is the optimum pore size, or what is the best
pore structure for a special membrane, can be answered only by gaining such
understanding. Although in the past some believed that fabricating a good
membrane or catalyst is an art - apparently due to the fact that many factors in the
fabrication process were, and still, are unknowns, or have been discovered by
chance and experience - the real art is creating sound and fundamental models that
can predict the effect of the phenomena involved.

One may divide nanoporous membranes into organic and inorganic membranes.
The former class of the membranes is usually made of polymers. Such membranes
have proven very useful and efficient to a variety of applications. They are not,
however, suitable for use in harsh conditions, namely, at elevated temperatures
and pressures, and possibly in reactive environments. Thus, much attention has
been focused on developing inorganic membranes that can better resist the harsh
environments. Due to the high potential for the applications of inorganic
membranes, several research groups have studied various inorganic membranes,
such as silica and carbon membranes (Takaba, Mizukami et al., 1998; Vieira-
Linhares e Seaton, 2003).
3
Knudsen diffusion, surface diffusion, capillary condensation, and molecular
sieving are the four basic transport mechanisms through inorganic porous
membranes (Koros e Fleming, 1993; Saracco e Specchia, 1994). For pores (in the
mesoposrous region) that are large relative to the molecular size of the permeating
gases, Knudsen diffusion controls the rate of transport. For permeating gases that
are strongly attracted to the membrane’s material, surface diffusion is important
(Heffelfinger e Vanswol, 1994; Pohl e Heffelfinger, 1999; Wang, Yu et al., 2006).
Molecular sieving happens when the molecules are sieved based on their size.
Usually, smaller molecules diffuse faster than the larger ones. Capillary
condensation happens if the transporting gas condenses in the pores.

Over the past several decades the development of advanced methods for molecular
simulations has made considerable progress, as the atomistic simulations have
gradually become quantitatively predictive and, therefore, have become the
method of choice for studying the transport properties of gases through
membranes or porous media. Equilibrium molecular dynamics (EMD) and non-
equilibrium molecular dynamics (NEMD) methods have been both used to study
the dynamics and transport properties of fluids in tight pores, while equilibrium
grand-canonical Monte Carlo (GCMC) technique has been utilized to study the
adsorption and structural properties of fluids and their mixtures in pores.
4
Over the past 15 years the USC research group has been using molecular
simulation of transport and separation of gas mixtures in carbon molecular-sieve
membranes, as well as other types of nanoporous materials, such as double layered
hydroxides, mixed-matrix polymeric membranes, and cationic pillared clays, using
both EMD and NEMD simulations (Sedigh, Onstot et al., 1998; Firouzi, Tsotsis et
al., 2003). Moreover, molecular simulations that combine

the configurational-bias
Monte Carlo (CBMC) method – a method for generating the molecular structure
of n-alkanes - and

the dual control-volume-nonequilibrium molecular dynamics
technique, were also utilized to investigate the

transport and separation of binary
mixtures involving CO
2
and n-alkanes in a carbon nanopore (Firouzi, Nezhad et
al., 2004).

Using the EMD simulation, the adsorption equilibria of several gases
and their binary and ternary mixtures in modeled pillared clays and carbon
molecular sieve membranes were also extensively studied in our group (Yi,
Ghassemzadeh et al., 1998; Ghassemzadeh, Xu et al., 2000; Xu, Tsotsis et al.,
2001).

The USC group was first to successfully prepare microporous silicon-carbide (SiC)
membrane (Ciora, Fayyaz et al., 2004). But, to the best of our knowledge there
have been no molecular simulation studies of transport and adsorption of fluids in
the SiC membranes. Such simulations are the subject of this thesis.

5
1.2 Nanoporous membranes
Inorganic membranes, and in particular the SiC membranes that are the subject of
this thesis, belong to a wider class of nanoporous materials that also includes
catalysts, adsorbents, and natural (biological) membranes, are of much current
interest. In particular, adsorption, flow and transport of fluids and their mixture in
such porous materials have been studied for a long time (Unger e International
Union of Pure and Applied Chemistry., 1988; Pinnavaia e Thorpe, 1995; Sahimi,
1995; Lowell e Lowell, 2004). To be used efficiently in any practical application,
it is crucial to understand how such equilibrium and nonequilibrium processes
occur in the pore space of nanoporous materials. Development of such an
understanding is greatly facilitated, if one has an accurate model of the materials'
pore space. Such understanding may then help one to address fundamental issues,
such as the design of the best mode of operation for a membrane.

Our interest in this thesis, as indicated in the Introduction, is an important class of
nanoporous materials, namely, inorganic nanoporous membranes, and in particular
the SiC membranes, that have been studied, both experimentally and by computer
simulations, for separation of gaseous and liquid mixtures into their constituent
components. Such membranes have recently received much attention, due to their
high permeability and thermochemical stability in harsh environments. Because of
their high potential for gas separation, several groups have developed various
6
inorganic membranes, such as silica, zeolite, and carbon molecular-sieve (CMS)
membranes.

The USC group (as well as other groups) has been preparing a variety of
amorphous nanoporous membranes for separation of gaseous mixtures, including
the CMS (Sedigh, Onstot et al., 1998; Sedigh, Xu et al., 1999; Sedigh, Jahangiri et
al., 2000) and SiC membranes (Suwanmethanond, Goo et al., 2000; Ciora, Fayyaz
et al., 2004; Elyassi, Sahimi et al., 2007; , 2008). Optimizing the performance of
such membranes, and in particular their selectivity, entails developing much
deeper understanding of how adsorption and transport of gaseous (or liquid)
mixtures through them is affected by the morphology of the pore space, and in
particular the pore size distribution (PSD), pore connectivity, surface roughness,
and other controlling factors.

Several approaches have already been developed for modeling of the pore space of
a membrane. One approach is phenomenological. It relies on the continuum
equations of transport of gases (or liquids) through the pore space, and ignores
important information about the porous material's morphology, including its PSD
and the pores' connectivity. An example of such an approach was described in our
recent paper (Chen, Mourhatch et al., 2008a).

7
Another approach is based on utilizing pore network models of porous materials
(Sahimi, Gavalas et al., 1990; Sahimi, 1993; Sahimi, 1995; Rieckmann e Keil,
1997; Keil, 1999; Rieckmann e Keil, 1999; Chen, Mourhatch et al., 2008b), using
the idea that any porous medium can, in principle, be mapped onto an equivalent
network of interconnected pores. The pores' sizes are distributed according to the
measured pore size distribution (PSD) of the membrane. Since the network's pores
are interconnected, the effect of their connectivity, which greatly influences
transport and separation of molecules through the pore space, is also automatically
taken into account. Thus, one has a realistic model of a nanoporous membrane that
can be used to study the effect of a variety of factors on the transport and
separation of gaseous mixtures through it.

Due to the pores' nanoscale size, molecular interactions between the guest
molecules in the pore space, and between them and the pore's walls, are important.
Therefore, the third approach to modeling of transport and separation of mixtures
through nanoporous materials has been based on atomistic simulations (Cracknell,
Nicholson et al., 1995; Gelb, Gubbins et al., 1999; Rieth e Schommers, 2006).
Most of such models are based on atomistic simulation of flow and transport of a
fluid mixture through a single nanopore. As described above, however, the pore
space of a membrane layer (as well as its support) consists of a three-dimensional
(3D), or quasi-2D, network of interconnected pores, with sizes that are distributed
8
according to a PSD. In order to accurately model any membrane and the
phenomena that take place there, one must take into account the effect of its pores'
interconnectivity. Thus, there have recently been some attempts to generate more
realistic atomistic models of nanoporous materials (Pikunic, Llewellyn et al., 2005;
Nguyen, Bhatia et al., 2006; Terzyk, Furmaniak et al., 2007) with interconnected
pores. The USC group has also carried out extensive molecular simulation of
transport and separation of gaseous mixtures using atomistic models of the CMS
membranes (Xu, Sahimi et al., 2000; Xu, Tsotsis et al., 2001; Sahimi e Tsotsis,
2003), as well as of some adsorbents (Yi, Ghassemzadeh et al., 1998;
Ghassemzadeh, Xu et al., 2000; Ghassemzadeh e Sahimi, 2004; Kim, Kim et al.,
2005; Kim, Harale et al., 2007).

In this thesis we propose and develop a new molecular pore network model for
nanoporous membranes, the pore space of which consists of pores of irregular
shapes, and sizes that follow a PSD. The approach, which is quite general,
combines a purely geometrical concept with molecular simulations, in order to
develop a 3D pore network model that has many virtues that a real nanoporous
membrane may be expected to have (see below). To test the accuracy of the model,
we utilize it in chapter 3 to compute the adsorption isotherms of nitrogen in three
nanoporous membranes, for which the isotherms have also been measured. In
chapter 4 we will utilize the adsorption-validated model to simulate the transport
9
and separation of binary gaseous mixtures in the same membranes, in order to
demonstrate the predictive power of the model.

We describe the model by developing it for a nanoporous SiC membrane. Due to
its many desirable properties, such as high thermal conductivity (Takeda,
Nakamura et al., 1987), high thermal shock resistance (Schulz e Durst, 1994),
biocompatibility (Rosenbloom, Sipe et al., 2004), resistance in acidic or alkali
environments, chemical inertness, and high mechanical strength (Zorman,
Fleischman et al., 1995; Kenawy e Nour, 2005), SiC has attracted wide attention
for various applications. As mentioned above, The USC group has been
fabricating SiC membranes for gas separation applications (Suwanmethanond,
Goo et al., 2000; Ciora, Fayyaz et al., 2004; Elyassi, Sahimi et al., 2007; , 2008),
such as hydrogen separation. Thus, the goal of this thesis is to study the same
phenomena in the SiC membranes through molecular simulations.
1.3 Simulation methods
Modeling and analyzing transport mechanisms through membranes, such as silica,
alumina and carbon, are important to the effective design of separation systems.
Adsorption in porous media has been successfully studied by Monte Carlo (MC)
methods with various ensembles. Among them is the GCMC method in which the
temperature, chemical potential and adsorption volumes are fixed, while the
10
number of fluids adsorbed could be varied. The GCMC method is well suited for
studying the physical adsorption of Lennard-Jones fluids, because it allows a
direct calculation of the equilibrium between gas phase and the adsorbent phase.
For instance, the GCMC method was used to study adsorption of hydrogen (Wang
e Johnson, 1999) and mixtures of carbon dioxide and nitrogen in carbon nanotubes
(Skoulidas, Sholl et al., 2006). In addition, the GCMC method was combined with
the CBMC approach to investigate the adsorption isotherms of alkanes and their
mixtures in zeolites (Jakobtorweihen, Hansen et al., 2005).

The EMD method is another important way to study the adsorption and
thermodynamic properties of fluids near a wall surface and inside pores. For
example, by using the iso-kinetic MD simulations, the chemical potential of
adsorbed gases on graphite has been studied (Cracknell, Nicholson et al., 1995).
Self-diffusivity, which is a key property of gases in a pore space, has also been
calculated by the EMD method (Jia e Murad, 2006).

The NEMD methods have been increasingly used for studying the transport
properties of fluids. Maginn et al. (Maginn, Bell et al., 1993) developed two
NEMD methods to study diffusion restricted to the linear response regime. The
transport diffusivity can be calculated by the NEMD method using Fick’s law
11
under the presence of a concentration gradient. However, the method is not valid
under a large driving force, where the system is close to equilibrium.

A NEMD method, called the grand-canonical molecular dynamics (GCMD)
method has been developed, in which Monte Carlo and MD simulations are
combined (Heffelfinger e Vanswol, 1994; Ford e Heffelfinger, 1998; Vieira-
Linhares e Seaton, 2003). Heffelfinger et al. (Heffelfinger e Vanswol, 1994; Ford
e Heffelfinger, 1998) employed two control volumes (CVs) corresponding to two
chemical potentials (one larger, one smaller), leading to the development of a
steady-state chemical potential gradient across the system under study, as well as a
density gradient, hence enabling a direct measurement of the transport diffusivity.

To determine the transport properties the pore size and the interactions between
the permeating molecules and the membrane’s pores’ wall are the key parameters
(Takaba, Mizukami et al., 1998; Vieira-Linhares e Seaton, 2003). Several research
groups have modeled the interaction of fluid and membranes, including silica,
carbon and zeolite membranes, using the Lennard-Jones (LJ) potential (Talu e
Myers, 2001; Takaba, Matsuda et al., 2002; Bhatia e Nicholson, 2003b; Kaganov
e Sheintuch, 2003; Huang, Nandakumar et al., 2006). We also assume the LJ
interactions between the gas molecules and SiC membrane and, as described in
12
chapter 3, fit the simulated adsorption isotherms of the gases in the SiC
membranes to the experimental data.
1.4 The plan of the thesis

The goal of this thesis is developing and testing an atomistic model of SiC
membranes, and testing it by comparing its predictions with the experimental data.
Thus, the plan of this thesis is as follows. In Chapter 2 we present the results of
preliminary molecular simulations of binary gas mixtures, involving carbon
dioxide and n-alkanes in single-pore models of nanoporous membranes. A single
pore is often inadequate for representing the complex pore space of a membrane.
However, it does provide insights into the complex phenomena of transport and
separation of mixtures through nanopores. In particular, we study binary mixtures
in single pore models of carbon and SiC membranes.

In chapter 3 we propose and develop a 3D molecular pore network model for the
SiC membranes, using atomistic simulations. We show that by adjusting at most
one parameter – the average pore size of the membrane – the model provides
highly accurate predictions for the sorption isotherms of nitrogen in three different
SiC membranes, including one that has been fabricated by the USC group.

13
Chapter 4 utilizes the model developed in chapter 3, in order to study transport and
separation of two binary gas mixtures in the SiC membranes. We show that,
without using any new adjustable parameter, the model provides accurate
estimates of the separation factors for the binary mixtures that are close to their
experimental values.

14
Chapter 2
2. Single Pore Models

2.1 Introduction

The simplest model of a nanoporous membrane is a single nanopore. Although the
model is not, in general, predictive, it is still useful for gaining insights into the
transport and separation of gaseous mixtures in nanoporous membranes. In this
chapter we study such a model, and present and discuss the implications of the
results for several mixtures. We consider two types of nanopores. One is made of
carbon atoms, while the second one is a pore with SiC walls. We begin with a
simple model for carbon molecular-sieve membranes.
2.2 Model of a carbon nanopore
The simplest geometry for a nanopore is a slit-like structure, i.e., the space between
two parallel and flat plates. The schematic of the system used in the simulations is
shown below:
15
Figure 2.1. Schematic of the slit pore used in the simulations

The pore in the middle is sandwiched between two control volumes (CVs) that are
in equilibrium with two bulk phases, each at a given pressure or chemical potential.
The pore system is then exposed to an external driving force, such as a chemical
potential or pressure gradient, applied in the x-direction shown in Fig. 2.1. The
pore length is nL with n being an integer, and L being the length of the CVs. As
these are preliminary studies, we used n=7 in the simulations of transport of
propane through the pore system, and n=1 for hexane. The pore’s walls were
assumed to be structureless.

The mixture under study consisted of CO
2
(component 1) and an alkane
(component 2). The carbon dioxide molecules were represented a Lennard-Jones
(LJ) spheres with effective LJ size and energy parameters σ and ε, respectively.
The interactions between CO
2,
and between them and the CH
2
and CH
3
groups in
the n-alkanes were modeled with the cut-and-shifted LJ potential:
16
LJ LJ C C
C
U(r) = U (r) - U (r ) ,if r r
U(r) = 0 ,if r > r
≤
⎧
⎨
⎩

where U
LJ
(r) is the LJ 12-6 potential, U
LJ
(r) = 4 ε [( σ/r)
12
- ( σ/r)
6
]. Here, r is the
distance between (the centers of two) interacting molecules, and r
c
is the cut-off
distance, which was

taken to be 2.5 Å. Long-range corrections were not applied.

The Steels potential (Steele, 1974) was used to model the interactions between the
molecules of gaseous mixture and the structureless walls of the pores:
2
210 4
() 2 ( ) ( )
3
5
3 (0.61 )
iw iw iw
Uz
c
iw iw iw
zz
z
σσ σ
πρ ε σ =∆ − −
∆∆+
⎧ ⎫
⎪ ⎪
⎨ ⎬
⎪ ⎪
⎩⎭
(1)
In Eq. (1) ∆ = 0.335 nm is the space between the adjacent graphite layers of the
walls, ρ
c
= 114 nm
-3
is the number density of the carbon atoms in the graphite, z is
the distance of the center of a molecules from the wall, and σ
iw
and ε
iw
are the LJ
parameters between the walls and the molecule of type i in the mixture.

The United Atom Model (UAM) was used to represent the alkanes in the
simulations. In the UAM the methyl and methylene groups are considered as
single interaction centers. The bond length was kept fixed at 1.54 Å, and the bond-
angel changing was governed by a harmonic potential (Vanderploeg e Berendsen,
1982):
17
2
()/2
0
uk
bend
θθ
θ
=− (2)
in which the equilibrium bond angel
0
θ was set to 112° and
B
k k /
θ
was set to be
62500 K rad
-2
.
B
k is the Boltzmann’s constant. Torsion was modeled using the
cosine expansion in the dihedral angel:
[1 cos ] [1 cos(2 )] [1 cos(3 )]
12 3
uc c c
tors
φ φφ =+ + + + + (3)
with,
B
k c /
1
=355.03 K,
B
k c /
2
=-68.19 K, and
B
k c /
3
=791.32 K.
The effective values of the LJ parameters of CO
2
were taken to be σ
CO2
= 3.794 Å,
and ε
CO2
= 225.3 K For the carbon atoms the LJ parameters used were, σ
C
= 3.4 Å
and ε
C
= 28 K. The Lorentz-Berthelot mixing rules were used for computing the
interaction parameters ε
ij
and σ
ij.
, namely, ε
ij
= (

ε
i
ε
j
)
1/2
and σ
ij
= ( σ
i
+ σ
j
) /2

.

In the molecular dynamics (MD) simulations, the MD moves (i.e., integration of
the equations of motion) were used in the entire system, together with grand-
canonical Monte Carlo (GCMC) insertion and deletion in the two CVs. In the MD
simulations the Verlet leapfrog algorithm (Allen e Tildesley, 1987; Rapaport, 1995)
was used to numerically integrate the equations of motion.

The iso-kinetic condition, i.e., the constant temperature condition, was maintained
by rescaling the velocity independently in the three directions. The RATTLE
18
algorithm (Andersen, 1983) was utilized in order to satisfy the constraints, i.e., the
angles between the bonds, and the bonds’ lengths, imposed on the alkane chains.
The densities of each component in the two CVs were maintained fixed, and in
equilibrium with the two bulk phases, each at a fixed pressure and concentration.
This was achieved by carrying out a sufficient number of the GCMC insertion and
deletions of the particles in the two CVs.

We also added a small streaming velocity to all the newly inserted molecules
within each CV that were located within a distance 1.9Å from the boundaries with
the pore, in order to reduce the numerical instability caused by the discontinuity
that is caused when a molecules leaves a CV and enters the pore. The molecules
that cross the boundaries of the CVs during the MD calculations were removed.
As shown in Table 2.1, all of the parameters in the simulations made
dimensionless using the LJ parameters of the lighter component

19
Table 2.1 Conversion factors between the dimensionless and actual quantities if, for example, the
lighter component is methane.

Variable Dimensionless form
Length L L
*
=L/ σ
CH4
Energy U U
*
=U/ ε
CH4

Mss M M
*
=M/M
CH4

Density ρ ρ
*
=ρσ
3
CH4

Temperature T T
*
=k
B
T/ ε
CH4

Pressure P P
*
=P σ
3
CH4
/ ε
CH4
Time t

t
*
=t ( ε
CH4
/M
CH4
σ
2
CH4
)
0.5

Flux J J
*
=J σ
3
CH4
(M
CH4
/ σ
CH4
)
Permeability K

K
*
=K(M
CH4
σ
CH4
)
0.5
/ σ
CH4

2.3 The DCV-GCMD simulation
The dual control-volume grand canonical molecular dynamics (DCV-GCMD) has
been recently used by several groups to investigate the transport of fluids through
membranes (Ford e Heffelfinger, 1998; Vieira-Linhares e Seaton, 2003; Wang, Yu
et al., 2006). Each MD move is followed by a number of GCMC insertions and/or
deletions in the two CVs. The probability of inserting a molecule of component i
is given by
min[ exp( / ) /( 1),1]
B
pZV UkTN
ii i
+
=−∆ + (4)
where
3
exp( )
iiB i
ZkT µ =Λ is the absolute activity at temperature T,
3
i
Λ and
i
µ are the Broglie wavelength and chemical potential of component i, U ∆ is the
20
potential energy change resulting from inserting or deleting a particle, and V and
N
i
are the volume and number of molecules i. The probability of deleting a particle
is given by
min[ exp( / ) / ,1]
ii B i
pN UkTZV
−
=−∆ (5)
2.4 Configurational-bias Monte Carlo method
Inserting a large molecule, such as an alkane chain, in a tight CV is very difficult.
If care is not taken, the computations will become impossible. In order to
overcome this difficulty, we utilized the configurational-bias MC (CBMC) method.
In this method, instead of a random insertion of the

n–alkanes into the CVs that are
connected

to the pore, the CBMC technique (Harris e Rice, 1988; Depablo, Bonnin
et al., 1992; Smit, Karaborni et al., 1995; Macedonia e Maginn, 1999; Firouzi,
Nezhad et al., 2004) grows the alkane chains in the

two CVs. Thus, we combined
the CBMC method with the GCMC technique. Molecules are grown atom-by-
atom in such a way that

regions of favorable energy are identified, and overlap
with other

molecules is avoided (Harris e Rice, 1988; Frenkel, Mooij et al., 1992;
Siepmann e Frenkel, 1992).
The potential energy of an atom is written as

the sum of two contributions, the
internal energy u
int
that includes parts of the intramolecular interactions, and the
external

energy u
ext
that contains the intermolecular interactions and those
21
intramolecular

interactions that are not part of the internal energy. The

division is
arbitrary and depends

on the details of the model. To grow an n-alkane molecules
atom by atom, we used the following procedure (Smit, Karaborni et al., 1995):

(1) We insert the first atom at a random position

and calculate the energy u
1
(n)
along with a quantity

w
1
= exp[– βu
1
(n)], where β= (k
B
T)
–1
, with n being the

new
state in which the system is in.
(2) Then, k

trial orientations, denoted by {b}
k
=b
1
,b2,…,b
k
, are generated to insert
the next

atom l, with a

probability ) (
int
i
l
b p , which is a function of the internal
energy:
)] ( exp[
1
) (
int int
i l i l
b u
C
b p β − = (6)
where C is a normalization factor. For each

trial orientation, the external energy
) (
i
ext
l
b u is also computed, along

with the quantity,
)] ( exp[
1
) (
j
b
ext
l
u
k
j
n
l
w β −
=
∑ =
(7)
One orientation, out of the k

trial positions, is then selected with the probability
)] ( exp[
) (
1
) (
i
b
ext
l
u
n
l
w
i
b
ext
l
p β − = (8)
22
(3) Finally, step (2) is repeated M–1 times until the

entire molecule is grown, and
the Rosenbluth factor (Rosenbluth e Rosenbluth, 1955), ) (
1
) ( n
l
w
M
l
n W
=
∏ = is
calculated. Any particular conformation can be generated with a

probability given
by
)] ( exp[
) (
1
1
) ( ) (
int
2
) ( n U
n W
M
C
n
ext
l
p n
l
p
M
l
n P β −
−
=
=
∏ = (9)
with
)
int
(
1 1
ext
l
u
l
u
M
l l
u
M
l
U +
=
∑ =
=
∑ = .
To insert n–alkanes into the two CVs we combine the CBMC method described
above

with a grand-canonical MC. It generates

the chain configurations one atom
at a time by the

CBMC method described above. The Rosenbluth weight W is
accumulated and used in

the acceptance criterion of the GCMC method for
insertion of

the alkanes into the system.
The probability of adding a

single chain to a system of N
i
chains is given

by
( ) exp( / )
min ,1
3
(1)
B
WnV kT
i
p
i
N
ii
µ −
+
=
Λ+
⎡⎤
⎢⎥
⎢⎥
⎣⎦
(10)
23
This equation is completely similar to the probability of inserting a molecule

into a
system in a standard GCMC computation, with the

main difference being the
inclusion of the Rosenbluth weight W.

For a deletion from the system, the
Rosenbluth weight is

evaluated by pretending to grow the alkane chain into its

current position. To accomplish this, the quantities, w
1
(o) = exp[– βu
l
(o)], w
l
(o) =
∑
k
j=1
exp[– βu
l
ext
(b
′
j
)], and

W(o) =
M
l 1 =
∏ w
l
(o) are computed, using k–1 trial
orientations, together with the

actual current position of the atom l, which form the

set, where o indicates the old state of the

molecules. The probability of deletion of
a chain from the

system is then given by
3
min ,1
(0) exp( / )
B
N
ii
p
i
WV kT
i
µ
Λ
−
=
⎡⎤
⎢⎥
⎢⎥
⎣⎦
(11)
which, aside from the Rosenbluth

weight, is again similar to that of a standard
GCMC

computation.
2.5 Molecular dynamics simulation
In most cases, we set the pressures in the high- and low-pressure CVs to be 3 atm
and 1 atm, which are the typical experimental pressures used in the experiments
carried out in the USC group. We carried out the simulations for several sizes of
the slit pore, in order to understand the effect of the pore size on the results. The
24
temperature of the pore system was held at 50°C, which is above the critical
temperature of both components, using the velocity rescaling method.

For each component i, we calculated its flux J
i
by measuring the net number of its
molecules crossing a given yz plane (perpendicular to the direction of the transport)
of area A
yz
,
LRRL
NN
ii
J
i
NtA
yz
−
=
∆
(12)
where
LR
i
N and
RL
i
N
are the number of the gas molecules of type i moving from
the left to the right and vice versa, respectively, ∆t is the MD time step (we used
the MD step ∆t
*
=5×10
-3
,which is equivalent to, t 0.00685 ps) and N is the
number of MD steps over which the average was taken.

The transport process was considered to have reached steady state when the fluxes
calculated at various yz planes were within from the averaged values, after which
the fluxes were calculated at the center of the transport region. The permeability K
i

of species i was calculated using
JnLJ
ii
K
i
PnL P
ii
==
∆∆
(13)
25
where P
i
is the partial pressure for species i along the pore. The most important
property that we wish to study is the dynamic separation factor S, defined as
2
21
1
K
S
K
=
(14)

2.6 Results and discussions
Extensive equilibrium GCMC and DCV-GCMD simulations were carried out to
study transport and separation of a model binary mixture of CO
2
and C
6
H
14
in a
carbon nanopore. The effect of the pore size and temperature of the system on the
separation of mixture were studied. The pore length was taken to 155 Å. The mole
fraction of CO
2
in the feed was 0.9 and T=50ºC.

Figure 2.2 presents the density profiles of CO
2
and C
6
H
14
in a carbon nanopore in
the x-direction (the direction of the pressure drop) with dimensionless sizes of 2.5,
3, and 5. The densities in the two CVs are more or less constant, except in the pore
with a size of 3. This could be due to the fact that much longer simulations are
needed. Figure shows 2.3 the density profiles are in the z-direction (i.e., between
the upper and lower walls of the pore) for the same pores. In the two smaller pores,
only one layer of each component has formed, while in the largest pore two layers
of each components are clearly seen. Figure 2.4 exhibits the dimensionless
26
temperature profiles along the x-direction, indicating that the iso-kinetic condition
has been reached.

Figure 2.5 shows the permeance and the separation factors for the three pore sizes.
As seen, for a (dimensionless) pore size of 3 the permeance is highest, but the
separation factor is the lowest. Figure 2.6 presents the permeance and separation
factor as functions of the temperature. Increasing the temperature results in
decreasing the separation factor and permeance.

Also studied were the transport and separation of a model binary mixture of CO
2

and C
3
H
8
in the carbon nanopores. In this case the pore length was taken to be
about 1100 Å, which is close to the thickness of the membrane used in the
experiment. The mole fraction of CO
2
in the feed was 0.5 and the temperature
50ºC. Unless otherwise specified, in all the cases below, the pressures in the
upstream and downstream CVs are 3 and 1 atm, respectively.

Figure 2.7 presents the density profiles of CO
2
and C
3
H
8
in the carbon nanopores
in the x-direction (the direction of the applied pressure drop), for the
(dimensionless) pore sizes of 2.5, 3, and 5. It appears that in these pores one may
have a freezing phenomenon, whereby the density of propane becomes constant,
implying that it does not move.
27

Figure 2.8 presents the density profiles in the z-direction in the same pores. Unlike
the case of the previous mixture, only monolayers have been formed in the pores.
Figure 2.9 depicts the permeance and separation factors versus the pore sizes. The
properties of the system are similar to the previous case of hexane and CO
2
. As
seen in the figure, in the pore with a size of 3 the permeance is the highest, but the
separation factor is the lowest.

The effect of feed composition was also investigated. Two carbon dioxide mole
fractions of 0.7 and 0.9 were selected for the study of the effect of the feed
composition. In Fig. 2.10 the density profiles along the x-direction are presented.
Due to the higher density of carbon dioxide, the freezing phenomenon, which is
presumably caused by the size of the propane, has disappeared. Figure 2.11
presents the permeance and separation factors for the three different feed
compositions. As can be seen, increasing the mole fraction of CO
2
improves the
permeance and the separation factor.

The effect of pore’s temperature was studied by carrying out the simulations at
50
o
C and 90
o
C. Figure 2.12 shows the density profiles in the x-direction. Higher
temperatures prevent high amounts of adsorption and, therefore, there is no
freezing in the pore. The corresponding separation factors and permeances are
28
presented in Fig. 2.13. While the permeance of carbon dioxide remains insensitive
with respect to the temperature (a result of being at a temperature above its critical
temperature), the permeance of propane decreases. As a result, the overall
separation factor also decreases.

Also studied was the effect of the pressure gradient across the pore in pores of size
of 2.5 and 5. As can be seen in Fig. 2.14, increasing the pressure gradient across
the pore results in an increase in the permeance for both propane and carbon
dioxide, due to, perhaps, the formation of liquid-like fluids that increases the
permeance.

29
Figure 2.2 Density profiles for pore sizes of 2.5 (top), 3 (bottom) and 5 (next page). Dashed lines
indicate the boundaries of the pore region.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
040 80 120
X*
Density
CO2
C6H14

0
0.04
0.08
0.12
0.16
0.2
040 80 120
X*
Density
CO2
C6H14

30
Figure 2.2: Continued

0
0.05
0.1
0.15
0.2
0.25
040 80 120
X*
Density
CO2
C6H14

31
Figure 2.3 Density profiles between upper and lower walls for pore sizes of 2.5 (top), 3 (bottom)
and 5 (next page).

-1.25
-0.75
-0.25
0.25
0.75
1.25
0 0.1 0.2 0.3 0.4 0.5
Density
Z*
CO2
C6H14

-1.5
-1
-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Density
Z*
CO2
C6H14

32
Figure 2.3: Continued

-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Density
Z*
C6H14
CO2
33
0
1
2
3
4
040 80 120
X*
T*
0
1
2
3
4
040 80 120
X*
T*

Figure 2.4 Time-averaged temperature distributions in pore sizes of 2.5 (top), 3 (middle) and 5
(bottom). Dashed lines indicate the boundaries of the pore region.

0
1
2
3
4
0 40 80 120
X*
T*

34
Figure 2.5 The computed permeance and separation factors of CO
2
/C
6
H
14
mixtures for three pore
sizes.

0
0.2
0.4
0.6
0.8
0 1 234 56
Pore Size
Permeance
CO2
C6H14

0
10
20
30
40
50
60
70
80
01 2 3 4 5 6
Pore Size
Separation Factor

35
Figure 2.6 The computed permeance and separation factors of CO
2
/C
6
H
14
mixtures at three
temperatures.

0
0.02
0.04
0.06
0.08
0.1
0.12
20 40 60 80 100 120
Temperature (°C)
Permeance
CO2
C6H14

0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120
Temperature (°C)
Separation Factor

36
Figure 2.7 Density profiles for pore sizes of 5 (top), 3 (bottom) and 2.5 (next page). Dashed
lines indicate the boundaries of the pore region.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

0
0.05
0.1
0.15
0.2
0.25
0.3
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

37

Figure 2.7: Continued

0
0.05
0.1
0.15
0.2
0.25
0.3
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

38
Figure 2.8 Density profiles between upper and lower walls for pore sizes of 5 (top), 3 (bottom)
and 2.5 (next page).

-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Density
Z*
CO2
C3H8

-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
Density
Z*
CO2
C3H8

39

Figure 2.8: Continued

-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
0 0.2 0.4 0.6 0.8 1 1.2
Density
Z*
CO2
C3H8

40
Figure 2.9 Computed permeance and separation factors of CO
2
/C
3
H
8
mixtures for different pore
sizes.

0
0.02
0.04
0.06
0.08
0.1
0.12
01 23 4 5 6
Pore Size
Permeance
CO2
C3H8

0
2
4
6
8
10
12
012 34 56
Pore Size
Separation Factor

41
Figure 2.10 Density profiles in the transport direction x. The CO
2
mole fraction in the feed is 0.7
(top) and 0.9 (bottom). Dashed lines indicate the boundaries of the pore region.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

0
0.02
0.04
0.06
0.08
0.1
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

42
0
0.02
0.04
0.06
0.08
0.1
0.4 0.6 0.8 1
Mole Fraction of CO2
Permeance
CO2
C3H8
0
2
4
6
8
10
12
0.40.5 0.60.7 0.80.9 1
Mole Fraction of CO2
Separation Factor
Figure 2.11 Computed permeance and separation factors of CO
2
/C
3
H
8
mixtures versus the mole
fraction of CO
2
. The dimensionless pore size is H
*
=5.

43
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8
Figure 2.12 Density profiles for the pore size of 5. The CO
2
mole fraction in the feed is 0.7. The
temperature is 70
o
C (top) and 100
o
C (bottom). Dashed lines indicate the boundaries of the pore
region.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 40 80 120 160 200 240 280 320 360
X*
Density
CO2
C3H8

44
Figure 2.13 Computed permeance and separation factors of CO
2
/C
3
H
8
mixtures for different
temperatures. The dimensionless pore size is H=5 and the mole fraction of CO
2
in the feed is 0.7.
0
0.02
0.04
0.06
0.08
0 20 40 60 80 100 120
Temperature (°C)
Permeance
CO2
C3H8

0
1
2
3
4
5
6
7
8
0 2040 6080 100 120
Temperature (°C)
Separation Factor

45
Figure 2.14 The dependence of the permeance of CO
2
and C
3
H
8
, on pressure drop ( ∆p) applied to
the pore sizes of 5 (upper) and 2.5 (bottom) at T=50
o
C. The mole fraction of CO
2
in the feed is
0.9, with the upstream pressure of 20 atm.
0
0.005
0.01
0.015
0.02
0.025
0.03
0 5 10 15 20 25
∆p
Permeance
CO2
C3H8

0
0.01
0.02
0.03
0.04
0.05
0 5 10 15 20 25
∆p
Permeance
CO2
C3H8

46
We also carried out MD simulations of transport and separation of a binary
mixture of CO
2
and C
3
H
8
in a carbon nanopore, in order to study the effect of the
pore’s (membrane’s) thickness on the separation factor. Figure 2.15, where the
results are presented, indicatess an important result: the separation factor of a
membrane, as modeled by a single pore, depends on its thickness, but there is also
an optimal thickness for obtaining the highest separation factor. A membrane
thickness smaller than the optimal value, which according to the simulations is
110 nm, yields a separation factor smaller than its maximum value. Beyond a
thickness of 110 nm the same separation factor is obtained.
47
3
4
5
6
7
8
0 50 100 150 200 250
Membrane thickness (nm)
Separation factor
Fig 2.15 Effect of membrane thickness on its separation factor.

48

2.7 Single-pore model of silicon-carbide membrane
A particularly promising candidate material for a variety of inorganic membrane
applications, which is also the focus of the work in this thesis, is silicon carbide.
The material has many desirable properties, such as high thermal conductivity
(Takeda, Nakamura et al., 1987), high thermal shock resistance (Schulz e Durst,
1994) , biocompatibility (Rosenbloom, Sipe et al., 2004), resistance in acidic or
alkali environments, chemical inertness, and high mechanical strength (Zorman,
Fleischman et al., 1995; Kenawy e Nour, 2005) .

Silicon carbide can be found in at least 70 forms. However, the most commonly
encountered forms are a-SiC (amorphous) and β-SiC (crystalline). a-SiC forms at
temperatures higher that 2000
o
C and has a hexagonal structure. At lower
temperatures silicon carbide is in the form of β-SiC that has a face-centered cubic
crystal structure (similar to diamond or zinc blend). Figure 2.16 shows a unit cell
of the cubic SiC or β-SiC.

49
Figure 2.16 The crystal structure of β-SiC.

50
We have developed an atomistic model for the structure of the β-SiC. To check the
accuracy of the model, we computed the radial distribution function (see Fig. 2.17
below) at 300K, which yields a first neighbor Si-C distance of 1.88Å, in close
agreement with experimental value of 1.887Å (Ma e Garofalini, 2005).

To generate the model for a-SiC - the amorphous state of SiC - we proceeded as
follows. We began with the crystalline structure of SiC. The simulation cell
contained the unit cell of the SiC in the crystalline state, repeated a large number
of times, in order to generate a large structure. Energy minimization and MD
simulation were then utilized, in order to establish the most stable state of SiC at
the desired temperature.

To carry out the MD simulations, one must specify a force field (FF) that
accurately describes the atomistic structure of SiC. Several accurate FFs have been
developed in the past for describing SiC (M.Sahimi, 2006). Among them are those
due to Tersoff (Tersoff, 1989), Stillinger and Weber (Stillinger e Weber, 1985)
and its modification (Vashishta, Kalia et al., 2007), and Brenner (Brenner, 1990)
and its extended versions (Che, Cagin et al., 1999; Dyson e Smith, 1999). They all
provide accurate description of SiC. In the present work we utilized the extended
Brenner FF. In the extended Brenner FF, the total energy E of the material is
written as
51

() () () E Er B r Er
ij ij ij ij R A
iji
=+ ∑∑
>
⎡⎤
⎢⎥
⎣⎦
(15)

Here, subscripts R and A denote, respectively, the repulsive and attractive parts of the
total energy of the system, given by
( )
2
() () exp
1
o
DS
ij ij
o
Er f r r r
ij ij ij ij ij ij A
SS
ij ij
β =− −
−
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
(16)
( )
() () exp 2
1
o
DS
ij ij
o
Er f r S r r
ij ij ij ij ij ij ij R
S
ij
β =− −
−
⎡ ⎤
⎢ ⎥
⎣ ⎦
(17)

where
0
ij
D is the well depth, and
0
ij
r is the equilibrium distance between atoms i and
j .The quantities f
ij
(r
ij
) are cutoff functions, given by
(1)
(1) (2) (1) (1) (2)
(2)
1,
11
() cos ( )/( ) ,
22
0,
ij
ij ij ij ij ij ij
ij
rR
frrRRRRrR
rR
π
≤
⎧
⎪
⎪
=+ − − < < ⎡⎤
⎨
⎣⎦
⎪
≥ ⎪
⎩

The function B
ij
is given by

( )
1
() ( )
,
2
CSi
Bb b FN N
ij ij ji ij i i
=+ +
⎡ ⎤
⎢ ⎥
⎣ ⎦
(18)

B
ij
depends on the environment around atoms i and j, since the function F
ij
-a
correction term which is used only for carbon-carbon bonds -depends on the two
functions
) (C
i
N and
) (Si
i
N defined below. Here,
52

( ) { }
() ( ) 00
1()()exp ( )( ) ,
,
i
CSi
bgfr rRrR HNN
ij ij ij ij i i ijk ik ik ijk ik ik
ki j
δ
θα
−
∑
=+ − − − +
≠
⎡⎤ ⎡ ⎤
⎣ ⎦ ⎣⎦
(19)

where θ
ijk
is the angle between bonds ij and ik , δ
i
a fitting parameter, and the
function H
ij
is another correction term, used only for hydrocarbons. The function
g( θ
ijk
) is given by,

22
00
() 1
22 2 0
(cos )
00
cc
ga
ijk
dd h
i
ijk
θ
θ
=+ +
++
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎣⎦
(20)

where h
i
is a function that, depending on the atom i ,may depend on the
environment around i. For the C atoms, h
i
=1.The cutoff functions f
ij
(r ) are used
for defining the various quantities. Thus,

()
()
Si
Nfr
iijij
iSi
⎧⎫ ⎪⎪
⎨⎬
⎪⎪
⎩⎭
= ∑
=
(21)
()
()
C
Nfr
iijij
jC
⎧⎫ ⎪⎪
⎨⎬
⎪⎪
⎩⎭
= ∑
=
(22)

where, for example, {C} denotes the set of the carbon atoms. Here,
) ( j
i
N is the
number of atoms of j bonded to atom i .
It is clear that the extended Brenner FF has a very large number of parameters that
must be fitted to the experimental data. The fitting has already been carried out for
a large number of compounds, and the fitted values of the parameters have been
53
tabulated (Dyson e Smith, 1999). We used the fitted values of the parameters
given by (Dyson e Smith, 1999) for SiC in the MD simulations.

Thus, MD simulations were carried out to melt the SiC crystalline structure. The
initial 3D simulation box contained a large number of Si and C atoms, arranged in
the α-SiC crystalline structure. Since the final product is an amorphous material,
periodic boundary conditions cannot be used. The time step ∆t used in the
simulations was, ∆t =1.47×10
-4
ps. The melting temperature of SiC is 3103 K.
Therefore, the SiC crystalline structure was heated up to 4000 K to melt it
completely, so as to generate the amorphous SiC.

After the system reached equilibrium, the temperature was gradually lowered in
steps of 5 K back to 300 K, using MD simulations. Each step of lowering the
temperature took about 10
3
time steps to reach equilibrium. Thus, the final
amorphous SiC structure was obtained from a single heating-cooling process.

To check whether the material is in a truly amorphous state, we computed its
radial distribution function, g (r). Figure 2.17 presents the results at 300 K. The
computed g (r), and in particular the locations of the various peaks that correspond
to the three types of the covalent bonds, are in very good agreement with the
54
experimental data (M. Ishimaru, 2003), hence confirming the accuracy of the
atomistic model of the amorphous state of the SiC that was generated.

Clearly, if one wishes to model the crystalline structure at any temperature below
the melting temperature of SiC, the same MD simulation procedure can be used to
heat up the crystalline structure to the desired temperature.
55
Figure 2.17 Computed radial distribution function for cubic SiC(top). The computed radial
distribution function g(r) of a-SiC (bottom). Arrows indicate the locations of the peaks,
corresponding to the covalent bonds.

0
3
6
9
12
15
18
02 4 6
r (Å)
g(r)

56
2.8 The effect of wall structure of a SiC pore
Similar to the pervious section in which we studied transport and separation of
mixtures of carbon dioxide and n-alkanes, we use, as the first step of modeling a
SiC nanoporous membrane, a single pore model in the MD simulations. The
schematic representation of the system is shown in Fig. 2.1. As usual, the system
is divided into three regions. The middle region represents the pore in which the
transport of the mixture occurs, whereas the ends two control volumes (CVs) are
exposed to the bulk fluid with high and low pressures.

The dimensions of the pore used were, L= 152 Å and W=76 Å. The pore size was
H=19 Å, and the pore length about 155 Å., close to the optimal pore length that we
determined for the carbon pore. In the simulations we used the DCV-GCMD
method that we described earlier in this chapter, in order to study the transport and
separation of a model binary mixture of CO
2
and CH
4
in a silicon carbide
nanopore. The temperature of the system was set at 50°C, while the pressure
different was 2atm. We studied the effect of the compositions and type of the SiC
(a or β) on the separation of mixture.
2.9 Results and discussion
The DCV-NEMD simulations were carried out with up to 6×10
6
time steps, in
order to study the transport and separation of the mixture. Figure 2.18 shows the
57
dimensionless time-averaged temperature profiles along a pore of size H* = 5
containing an equimolar (top) mixture of CO
2
and CH
4
, as well the case in which
the mole fraction of CH
4
in the feed was 0.3 (bottom) in the a -SiC. Both profiles
are reasonably smooth, indicating the constant temperature condition was more or
less maintained. The same type of temperature profiles were obtained for the β-
SiC pore and, hence, are not shown.

Figure 2.19 presents the density profile of an equimolar mixture of CH
4
/CO
2
in the
β-SiC (top) and a-SiC (bottom) pores. The density profiles are essentially flat in
the two CVs, with numerical values of the densities that match those obtained by
the GCMC method at the same conditions, indicating that the chemical potentials
in the two CVs were properly maintained during the simulations. Figure 2.20
presents the same as in Fig. 2.19, but when the mole fraction of the CH
4
in the
feed was 0.3.

58
Figure 2.18 Dimensionless temperature profiles of Equimolar (top) mixture of CH
4
/CO
2
and
when mole fraction of CH
4
in the feed is 0.3 in a-SiC membrane.

0
1
2
3
4
0 20 406080 100 120
X*
T*

0
1
2
3
4
0 20 40 60 80 100 120
X*
T*

59
Figure 2.19 Density profile of equimolar mixture of CH
4
/CO
2
in β-SiC (top) and a-SiC (bottom).

0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
040 80 120
X*
Density
CH4
CO2

0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
040 80 120
X*
Density
CH4
CO2

60
Figure 2.20 Density profile of mixture of CH
4
/CO
2
when the mole fraction of CH
4
in the feed is
0.3 in β-SiC (top) and a-SiC (bottom).

0
0.005
0.01
0.015
0.02
0.025
0.03
040 80 120
X*
Density
CH4
CO2

0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
040 80 120
X*
Density
CH4
CO2

61
To compare the effect of the atomistic structure of the pore’s walls (amorphous
versus crystalline) on the results, we also computed the separation factors for
different compositions of the binary mixture of CO
2
and CH
4
. As shown in Table
2.2, the separation factors for both the crystalline and amorphous membrane are
mostly the same when the mole fraction of methane is 0.3. However, when the
mole fraction is increased to 0.5, the separation factor in the pore with crystalline
structure increases by a factor larger than 3, while that of the pore with amorphous
walls does not change as much, by about 30%.

Table 2.2. The effect of the structure of the pore wall on the separation factor.

S.F. (30% CH
4
) S.F. (50% CH
4
)
a -SiC
1.4 1.0
β-SiC 1.1 3.9

62
Chapter 3
3. Molecular Pore-Network Model of Silicon Carbide
Membranes: Adsorption
3.1 Introduction
In chapter two we presented the results of a preliminary study of transport and
separation of gaseous mixtures in simple models of nanoporous materials. The
model utilized was a single nanopore. However, as pointed out in Chapter 2, a
single pore cannot adequately represent the complexities of the pore space of an
actual membrane, which contains a range of pore sizes, the pore are interconnected,
their surface if rough, and their shapes are also irregular.

The interconnectivity of the pores, and their size distribution, shape, and and
surface heterogeneity all play important roles in determining the transport and
sorption properties of nanoroporous materials (Ghassemzadeh, Xu et al., 2000; Xu,
Sahimi et al., 2000; Xu, Tsotsis et al., 2001). To take these effects into account,
we develop a new atomistic model for silicon-carbide (SiC) membranes. The
model is based on the three-dimensional (3D) Voronoi tessellation of the atomistic
model of the SiC material, and generating a pore structure in the materials that
closely mimics the properties that a real membrane is expected to have.
63
3.2 Atomistic pore-network model
The first step of constructing the model consists of generating the atomistic model
of SiC, either in the crystalline or amorphous state. The procedure for doing so
was already described in Chapter 2 and, therefore, need not be repeated here.

Next, the pore network of the membrane is generated by tessellating the atomistic
model of the SiC, which has been generated in a simulation cell, by inserting in it
a given number of Poisson points, each of which is the basis for a Voronoi
polyhedron. Each polyhedron is that part of the simulation cell that is nearer to its
Poisson point than to any other Poisson point.

To carry out the tessellation, we utilized parallel computations using space
decomposition, message-passing interface strategy, and 300 processors. This
reduced the computational time by nearly two orders of magnitude. Each
processor tessellated a portion of the simulation cell. Using larger simulation cells
(with a larger number of Poisson points) increases the efficiency of the
computations

The pore space is then created by fixing its porosity at the experimental value of
the actual membrane’s porosity and selecting a number of the polyhedra in such a
way that their total volume fraction equals the membrane’s porosity. The
64
polyhedra, so chosen, are then designated as the membrane pores by removing the
carbon and silicon atoms inside them, as well as those that are connected to only
one neighboring carbon or silicon atom (the dangling atoms), since it is impossible
for such atoms to exist in the actual membrane, and be connected to the surface of
the pores. The remaining atoms constitute the membrane's solid matrix, while the
pore space consists of interconnected pores of various shapes and sizes.

The designation of the polyhedra as the pores can be done by at least two different
methods. If the pore polyhedra are selected at random, then, if the size of the
simulation cell (or the atomistic model of the SiC material) is large enough, the
size distribution of the designated pores will always be Gaussian, regardless of the
porosity of the pore space. This, however, is not realistic from a practical view
point, because the SiC membranes do not possess a Gaussian pore size distribution
(PSD).

In the second method, one designates the pore polyhedra in such a way that the
resulting PSD would mimic that of a real membrane, which is typically skewed,
sharply peaked around the average pore size, and has a tail. To obtain such PSDs,
we first sort and list the polyhedra in the simulation cell according to their sizes,
from the largest to the smallest. The size of each polyhedron is taken to be its
volume, or the radius of a sphere that has the same volume as the polyhedron. We
65
then designate the polyhedra as the pores according to their sizes, starting from the
largest ones in the list. The PSDs of the pore networks generated by the two
methods are shown in Fig. 3.1. The results were obtained using 5000 Poisson
Points. The PSDs and their average pore sizes that are generated with the bias
toward the smallest pores are, of course, dependent upon the porosity (Yi,
Ghassemzadeh et al., 1998). Note that, unlike the traditional pore networks that
are used in the simulation of flow and transport in porous media (Sahimi, 1995),
the pore networks generated in this thesis are molecular networks in which the
interaction of the gas molecules with the atoms in the network are taken into
account.

By controlling the box size and the number of the initial Poisson points, one can
independently fix the average pore size of the pore network. For example, using
simulation box of 239.8 239.8 239.8Å ×× and inserting 1000 Poisson points in
the simulation cell, the average pore size would be 34 Å, when the pores were
selected by the bias described above.

Two points are worth mentioning here. One is that, in principle, one should carry
out a second energy minimization of the membrane material and its pore space,
after the removal of the atoms inside the pore polyhedra, since the removal creates
a new environment to which the remaining atoms might respond and move. We
66
assume, however, that the SiC material in the membrane is rigid, since there are no
experimental indications that the SiC is very flexible. Thus, no further energy
minimization is carried out after the generation of the pore space.

The second point worth mentioning is that, after some of the atoms are removed,
the atoms that are connected to them have a free valence, i.e., contain some charge.
Therefore, one must take those charges into account.

Due to the disordered structure of the molecular pore network that we generate,
several important aspects of the simulations must be addressed. For example, one
must average the results over a suitable number of pore network realizations in
order to obtain representative results for the adsorption isotherms. In addition, for
any given realization, one must also average the results over a number of initial
positions of the gas molecules in the pore space in order to decrease to a
reasonable level the statistical fluctuations in the results. Finally, the initial
simulation cell must be large enough in order for the results to be independent of
its size.

Let us point out that the statistics of Voronoi tessellation have been used in the
past to characterize the porosity distribution in porous materials (Ghassemzadeh,
Xu et al., 2000; Xu, Sahimi et al., 2000; Xu, Tsotsis et al., 2001; Dominguez e
67
Rivera, 2002; Rivera e Dominguez, 2003). However, the method that we propose
in this paper is different from the previous works (Dominguez e Rivera, 2002;
Rivera e Dominguez, 2003) in that, here, we first develop the atomistic model of
the nonporous material by molecular simulations, and then generate the porosity in
it by tessellating the resulting nonporous molecular structure. In the previous
works (Dominguez e Rivera, 2002; Rivera e Dominguez, 2003) one simulates a
binary fluid mixture in which one of the components serves as a template material.
The final porous material is then generated by removing the template particles
from the mixture.

68
Figure 3.1 Computed pore size distribution (PSD) for the system in which the pores are selected
biased (top) and randomly (bottom).

Pore diameter (A°)
Pore diameter (A°)
69
Figure 3.2 presents a 2D pore network with a porosity of 0.5, in which the white
polygons represent the membrane’s matrix, while the rest are the pores. In the
simulation of the sorption isotherms described below, as well as in the next
chapter, we utilized 3D structures. Figure 3.3 depicts a small portion of a 3D pore
network obtained by this method.

An important aspect of the model, which is also a property of real porous materials,
is the interconnectivity of the pores. We characterize the pores’ interconnectivity
by the coordination number Z, defined as the number of the pores connected to a
given pore, which is a spatially distributed quantity. To demonstrate this aspect,
we computed the distribution of the coordination number for a model SiC
membrane with porosity of 0.5. The results are presented in Fig. 3.5. As can be
seen there, the coordination number varies anywhere from 0 (isolated pore) to as
high as 20. The average coordination number is about 10. Lower porosities would,
of course, result in lower average coordination numbers.

The model has several virtues that are expected for any inorganic membrane of the
type that we have been studying: (i) it generates PSDs that mimic those measured.
(ii) The pores have irregular shapes. (iii) The interconnectivity of the pores is
automatically taken into account. (iv) Removing the dangling atoms generates
pore surface roughness, which is expected to exist in any real membrane.
70

Let us point out that, although tessellating the membrane material by the Voronoi
algorithm and designating some of the resulting polyhedra as pores may seem
abstract, it is, in fact, quite natural. The pore space of many natural porous
materials, ranging from biological materials, wood, and foam (Unger e
International Union of Pure and Applied Chemistry., 1988; Gibson e Ashby, 1997),
to sandstone and other types of rock (Sahimi, 1995), can be well represented by
Voronoi-type structures. In addition, the Voronoi algorithm affords us great
flexibility for constructing disordered pore networks with many variations in the
shapes and sizes of the polyhedra. One can, in fact, modify the algorithm, in order
to generate pore polyhedra with a great variety of shapes, from completely random
to very regular shapes(Cromwell, 1997) .

71
Figure 3.2 An example of 2D Voronoi pore network. Open polygons show the pores.

72
Figure 3.3 The structure of the polyhedra and their connectivity in a small portion of a 3D
Voronoi pore network.

73
Figure 3.4 Schematic of 3D pore network used in our simulation.

74
Figure 3.5 The distribution of the coordination number in a 3D Voronoi pore network with
porosity of 0.5.

75

3.3 MD Simulation in a pore network
We first carried out some preliminary simulations in which we used equilibrium
molecular dynamics (EMD) simulations to study adsorption and diffusion of gases
through the pore networks. We used the standard cut-and-shifted Lennard-Jones
(LJ) potential to describe the interactions between the adsorbates, as well as
between them and the carbon and silicon atoms in the SiC membrane, with the
cut-off distance r
c
taken to be 19 Å.

The interaction between the gas molecules with the whole SiC pore network is the
sum of the LJ potentials between the gas molecules and each individual atom in
the network.

The EMD simulations were performed in the micro-canonical ensemble with
periodic boundary conditions in all the three directions. We used the Verlet
velocity algorithm (Haile, 1997) to solve the equations of motion and determine
the molecules' trajectories. The trajectories were collected in a typical duration of
4×10
6
time steps, after discarding the first 4×10
5
time steps for equilibration of the
system. The chemical potentials were calculated using Widom's test particle
method (Widom, 1963; Shing e Gubbins, 1982; Widom, 1982; Deitrick, Scriven et
al., 1989).
76
At the beginning of each simulation run, the simulation cell was discretized into
n
x
×n
y
×n
z
subcells. The test particle-solid (pore surface) interaction energies and
their three derivatives, associated with all the subcells, were then calculated and
recorded. They also served as the grid points values for the interpolation
calculations described below. The test particles whose interaction energies in
units of k
B
T (where k
B
is the Boltzmann's constant) were less than a certain
positive value ε (for example, 20) were considered as “active” test particles, and
those with interaction energies larger than ε were considered “idle” states, since
they make essentially no contribution to the chemical potential:
〉 − 〈 − = )] ( exp[ ln ) ln(
t
r
ti
U
i
x
ci
β ρ βµ (1)
Here, ρ is the total number density, x
i
the mole fraction of component i, U
ti
is the
potential energy of a test particle of type i, µ
ci
is the configurational chemical
potential, and <
.
> denotes an average over time and the number of active test
particles. For simplicity, in the remainder of this chapter, we will simply use µ
ci
to
denote µ
c
and refer to it as the chemical potential.

To compute the chemical potentials and the adsorption isotherms at each sampling
step, typically 1% of them were randomly selected that resulted in random
selection of about 2000 to 4000 of the active particles. Energy conservation in the
system was monitored in each simulation, and the time step was adjusted
according to the loading of the adsorbate molecules and the size of the pores, such
77
that the standard deviations of the total energy relative to the mean was about
5×10
-4
or less. The typical value of the time step selected in this way was 5×10
-3
in
the reduced unit of CH
4
, or 7.37×10
-3
ps.

To reduce the simulation time for calculating the interactions between a gas
molecule and all the carbon and silicon atoms in the molecular pore network, we
used a 3D piecewise cubic Hermite interpolation (Schultz, 1972; Kahaner, Moler
et al., 1988; Deitrick, Scriven et al., 1989) to compute the potential energy and
forces for the gas particle at any position using the previously-recorded
information at the n
x
×n
y
×n
z
grid points. The Hermite method interpolates a
function and its three first derivatives, and proved to yield accurate results using
the grid point values.

Physical quantities were expressed in reduced units; that is, the energy and size
parameters ε and σ of one component were used as the basic units of energy and
length. The conversion factors that convert the dimensionless parameters to
dimensional quantities for temperature, pressure, energy, density, and time are
listed in Table 3.1.
78
Table 3.1.Value of L.J parameters used in the simulation.
Atom σ(Å) ε/k
B
(k) mass
CH
4
4.010 142.87 16.04
CO
2
4.328 198.20 44.01
N
2
3.700 95.05 28.03
C 3.400 28
Si 3.742 23.6

The dimension of the simulation cell was 65.4 65.4 65.4 Å ×× in all three
directions. The initial number of carbon atoms and silicon atoms were equally
13500. We inserted 5000 Poisson points in the simulation cell. The schematic of
3D pore network when the porosity is 50% is shown in Fig. 3.4.

79
3.4 Results and discussions
We carried out two series of simulation of adsorption. First, we carried out
preliminary simulations of adsorption of three gases, namely, nitrogen, methane,
and carbon dioxide in the membrane, in order to gain a better understanding of
sorption phenomena in the membrane. We then carried out simulation of nitrogen
adsorption in the membranes, in order to compare the results with the
experimental data.
3.4.1 Adsorption of single gases
The adsorption simulations were carried out with membranes with porosities of
0.7, 0.5, and 0.4. Sorption isotherms of the three gases, namely, nitrogen, methane,
and carbon dioxide, were computed. The results are presented in Figs. 3.6-3.8. As
the results indicate, increasing the porosity enhances the adsorption. This is, of
course, expected. But, more importantly, the shape of the isotherms change as the
porosity varies.

80
Figure 3.6 Adsorption isotherms for gases in SiC at porosity of φ=0.7. µ
i
and ρ
i
are the chemical
potential and loading of component i.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-10 -9-8 -7-6-5-4-3 -2-1 0
βµi
ρi
N2
CH4
CO2

81

Figure 3.7 The same as Fig. 3.4, but φ=0.5.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
-10 -9-8 -7-6 -5-4 -3-2 -1
βµi
ρi
N2
CH4
CO2

82

Figure 3.8 The same as Fig. 4.4, but φ=0.4.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
-10 -9-8-7-6 -5-4-3-2-1 0
βµi
ρi
N2
CH4
CO2

83
Let us point out that the statistics of Voronoi tesselations have been used in the
past to characterize the porosity distribution in porous materials (Ghassemzadeh,
Xu et al., 2000; Xu, Sahimi et al., 2000; Xu, Tsotsis et al., 2001; Dominguez e
Rivera, 2002; Rivera e Dominguez, 2003). However, the method that we propose
in this paper is different from the previous works (Dominguez e Rivera, 2002;
Rivera e Dominguez, 2003) in that, here, we first develop the atomistic model of
the nonporous material by molecular simulations, and then generate the porosity in
it by tessellating the resulting nonporous molecular structure. In the previous
works (Dominguez e Rivera, 2002; Rivera e Dominguez, 2003) one simulates a
binary fluid mixture in which one of the components serves as a template material.
The final porous material is then generated by removing the template particles
from the mixture.

An important aspect of the model, which is also a property of real porous materials,
is the interconnectivity of the pores. We characterize the pores’ interconnectivity
by the coordination number Z, defined as the number of the pores connected to a
given pore, which is a spatially distributed quantity. To demonstrate this aspect,
we computed the distribution of the coordination number for a model SiC
membrane with porosity of 0.5. The results are presented in Fig. 3.9. As can be
seen there, the coordination number varies anywhere from 0 (isolated pore) to as
84
high as 20. The average coordination number is about 10. Lower porosities would,
of course, result in lower average coordination numbers.

The model has several virtues that are expected for any inorganic membrane of the
type that we have been studying: (i) It generates PSDs that mimic those measured.
(ii) The pores have irregular shapes. (iii) The interconnectivity of the pores is
automatically taken into account. (iv) Removing the dangling atoms generates
pore surface roughness, which is expected to exist
in any real membrane.

Let us point out that, although tessellating the membrane material by the Voronoi
algorithm and designating some of the resulting polyhedra as pores may seem as
abstract, it is, in fact, quite natural. The pore space of many natural porous
materials, ranging from biological materials, wood, and foam (Unger e
International Union of Pure and Applied Chemistry., 1988; Gibson e Ashby, 1997),
to sandstone and other types of rock (Sahimi, 1995), can be well represented by
Voronoi-type structures. In addition, the Voronoi algorithm affords one great
flexibility for constructing disordered pore networks with many variations in the
shapes and sizes of the polyhedra. One can, in fact, modify the algorithm, in order
to generate pore polyhedra with a great variety of shapes, from completely random
to very regular shapes(Cromwell, 1997) .
85
3.4.2 Comparison with experimental data
We now use the model to completely determine the structure of the membrane.
The PSD and pore connectivity of the membrane are fixed, once the porosity is
specified. In the simulations for comparing with the experiments, we fix the
porosity of the membrane at its experimental value, because the porosity can be
estimated very accurately. However, we must also fix the average pore size that
corresponds to the experimental value. Since in practice the average pore size
cannot be determined accurately, we treat the average pore size as an adjustable
parameter. To do so, we carry out simulation of sorption of nitrogen in the model
SiC membrane, and adjust the average pore size until the computed isotherm
matches the experimental one.
To compute the adsorption isotherms, we used boxes with a linear size of 65.4Å,
and between 2600 and 3000 Poisson points. The remaining atoms in the
simulation cell, which constitute the atoms in the membrane’s matrix, are
presented in Table 3.2 for the three samples. Also the calculated pore size
distributions of different samples are presented in Fig. 3.8.
86
Table 3.2. The porosities of the three membranes, and the simulations' parameters. N
P
, N
C
, and
N
Si
represent, respectively, the number of Poisson points, and carbon and silicon atoms.

Three sets of data for N
2
adsorption isotherms in three SiC membranes were used
for comparison with the computed isotherms using the model. One isotherm was
measured in our own laboratory, using the SiC membrane that we have fabricated
(Elyassi, Sahimi et al., 2007; , 2008). We refer to it as membrane I. The separation
properties of the same membrane for binary gaseous mixtures were also measured.
They will be directly compared with the computed results in the next chapter.
87
Figure 3.9 The computed pore size distribution of the three SiC membranes.

88
The other two isotherms were reported by Suda et al. (Suda, Yamauchi et al.,
2006), and are referred to as isotherms for membranes II and III. The nitrogen
adsorption isotherm with membrane I was measured using the BET method. We
first measured the isotherm in the membrane’s support (which was also SiC), and
then generated the BET data in the supported membrane. This way, we could
obtain the membrane isotherm information by subtracting the support effect.
Figure 3.10 shows the adsorption isotherms in both the support and the supported
membrane. Figure 3.11 shows the prepared membrane isotherm obtained from
subtracting the effect of support.

The membrane’s thickness was estimated to be 9 µm. Using the adsorption
isotherm and the MP-plot method (Lowell e Lowell, 2004), we estimated that the
porosity of the membrane was about 0.4. It is known that microporous region in
the adsorption isotherm mostly represents itself at relative pressures of less than
0.1. The difference in adsorption at this pressure region is attributed to micropores
region which only presents in the membrane top layer. Knowing the membrane
thickness, we subtracted the effect of support from membrane and obtained
adsorption isotherm for our membrane. Since the estimate of the average pore size
by the BET instrument is based on the Horvath-Kawazoe as an adjustable
parameter. Thus, we ran a set of EMD simulations with various average pore sizes
and porosity 0.4, using the Voronoi model. We found that the EMD simulations
89
results match the experimental isotherm almost perfectly with an average pore size
of 6.5 ˚A. This is shown in Fig. 3.12.

The PSD of membrane II (Suda, Yamauchi et al., 2006) is shown in Fig. 3.13.
Although it contains several peaks and a long tail, only its left-most side that
corresponds to the membrane layer is of interest to us.

The membrane’s porosity, estimated by the t-plot method, is about 0.43. Its
average pore size was determined by the simulation, and turned out to be 6.6 Å.
Suda et al. (Suda, Yamauchi et al., 2006) reported the average pore size of
membrane II to be about 6.4 Å, which had been, however, estimated by the HK
method. Figure 3.14 compares the computed isotherm, using the Voronoi model at
the (HK) method (Sahimi, 1995; Lowell e Lowell, 2004), which is not very
accurate, we treated the average pore size of membrane I same porosity, with the
experimental data for membrane II. The agreement is, once again, very good.

The PSD of membrane III (Suda, Yamauchi et al., 2006) is presented in Fig. 3.15,
which is similar to that shown in Fig. 3.18. The HK average pore size of the
membrane was reported to 7.2 Å. Its porosity was estimated by the MP-plot
method, and turned out to be 0.5. Using these values, the simulated isotherm did
not quantitatively agree with the data. However, fixing the porosity at 0.5 and
90
lowering the average pore size to 6.8 Å (a difference of about 5%) produced the
isotherm shown in Fig. 3.16, where it is compared with the data of Suda et al.
(Suda, Yamauchi et al., 2006). Once again the agreement is very good. Note that,
the average pore sizes of the three membranes are rather close to each other.

Let us point out that, if the porosity of a membrane is relatively large - not too
close to its percolation threshold (Sahimi, 1994) - then often a single realization of
the model produces the correct isotherm. However, if the porosity is low, or if the
system is close to a phase transition (which often happens at low temperatures),
then, one must either generate several realizations of the model and average the
results, or use a relatively large size model.

Thus, by adjusting at most one parameter, the Voronoi model for the SiC
membrane can accurately predict nitrogen sorption data that are in agreement with
the experimental data. A stringent test of the model can be made, if we use the
model of the membranes to predict, without using any additional adjustable
parameter, their separation properties. This will be taken up in the next chapter.

91
3.5 Summary

A new molecular pore network model for inorganic nanoporous membranes was
proposed. The model is based on generating an atomistic model of the membrane
material, tessellating the atomistic model using the Voronoi algorithm, and designating
the pores in a physically reasonable manner. The model was tested against experimental
data for nitrogen adsorption in three SiC membranes. Adjusting at most one parameter -
the average pore size – produced isotherms that are in very good agreement with the data.

92
Figure 3.10 Nitrogen adsorption isotherms for our silicon carbide membrane on support and
support.

93

Figure 3.11 Our prepared membrane isotherm obtained from subtracting the effect of support.

Membrane
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5 0.6
P/Po
Amount adsorbed STP (Cm3/gr)

94
Figure 3.12 Comparison of the computed and measured nitrogen sorption isotherms in membrane.

95
Figure 3.13 Pore size distribution of membrane II, adopted from Suda et al.(Suda, Yamauchi et
al., 2006).

96
Figure 3.14 Comparison of the computed and measured nitrogen sorption isotherms in membrane
II.

97
Figure 3.15 Pore size distribution of membrane III, adopted from Suda et al. (Suda, Yamauchi et
al., 2006).
98
Figure 3.16 Comparison of the computed and measured nitrogen sorption isotherms in membrane
III.

99

Chapter 4
4. Molecular Pore Network Model of Silicon Carbide
Membranes: Transport and Separation
4.1 Introduction
In chapter 3 we developed a molecular pore network model of silicon-carbide (SiC)
membranes. We showed that by adjusting at most one parameter – the average
pore size of the membrane – the model can provide predictions for the sorption
isotherms of various gases in the membrane that are in very good agreement with
the experimental data.

In the present chapter we utilize the model developed in chapter 3 in order to study
transport and separation of two binary gas mixtures in the same membranes. We
carry out non-equilibrium molecular dynamics simulations (NEMD) of the
transport and separation of the gaseous mixtures in the model nanoporous SiC
membranes, utilizing the three-dimensional (3D) molecular pore network model of
the membrane based on the Voronoi tessellation developed in chapter 3. The
separation factors of the membrane are computed for equimolar binary mixtures of
H
2
in CO
2
and in CH
4
, and are compared with the experimental results.
100
Transport of gaseous mixtures, which is usually accompanied by adsorption, in
porous materials is of fundamental practical importance. In particular, gas
separation by inorganic porous membranes has recently received much attention,
due to the high separation performance, low cost, and thermo-chemical stability of
such membranes under severe conditions required in membrane reactors. Among
such membranes SiC membranes - the subject of this thesis - is very promising,
due to many desirable properties of SiC, outlined in chapter 3. The development of
such membranes using the chemical-vapor deposition technique, the pyrolysis of
polymeric pre-ceramic precursors, and the use of sacrificial interlayers for the
preparation of the nanoporous SiC membranes has been reported recently (Elyassi,
Sahimi et al., 2007; , 2008). Thus, in order to better understand transport and
separation of mixtures in such inorganic nanoporous membranes, a study by
molecular simulation can be very useful. If the model used in the simulations is
realistic, the simulations could also be helpful to determining the optimal
conditions for separation.

Over the past two decades molecular simulations have reached the degree of
sophistication and predictive power that they have become the method of choice
for studying transport properties of gases through membranes and other porous
media. Equilibrium molecular dynamics (EMD) and NEMD methods have both
been used to study adsorption, the dynamics, and transport properties of fluids in
101
pores. Grand-canonical Monte Carlo (GCMC) technique has been utilized to study
the adsorption and structural properties of fluids and their mixtures in tight pores.
Such techniques have also been utilized by the USC group to study transport and
separation of gas mixtures in carbon molecular-sieve membranes (Sedigh, Onstot
et al., 1998; Xu, Sahimi et al., 2000; Xu, Tsotsis et al., 2001).

Others have also used the same type of techniques to study complex phenomena in
tight pores. Furukawa et al. (1996) combined the GCMC technique and a
boundary-driven NEMD method to simulate gas permeation through a slit carbon
pore. Several NEMD simulation methods that use combinations of the MD with
the GCMC technique have been utilized to the simulation of molecular permeation
through porous membranes or narrow pores (Yoshioka, Miyahara et al., 1997;
Pohl e Heffelfinger, 1999; Furukawa e Nitta, 2000; Takaba, Matsuda et al., 2002).
A NEMD method called the grand-canonical molecular dynamics (GCMD)
method was developed in which Monte Carlo and MD simulations were combined
(Heffelfinger e Vanswol, 1994) Heffelfinger and co-workers (Heffelfinger e
Vanswol, 1994; Ford e Heffelfinger, 1998) employed two control volumes (CVs)
with two different chemical potentials that are in equilibrium with two bulk phases
at those chemical potentials, in order to study non-equilibrium transport of gases
under the steady-state condition in a chemical potential or density gradient
102
enabling. This technique was also used to study the diffusivity of gaseous mixtures
in graphite slit pore (Travis e Gubbins, 1999).

Transport through the slit and cylindrical pores (Mao e Sinnott, 2001; Bhatia e
Nicholson, 2003a) has also been studied widely. Moreover, Kaganov and
Sheintuch (2003) constructed a model membrane by randomly removing some of
the atoms of in the membrane material.

To our knowledge, there have been no atomistic simulations of transport and
adsorption of fluids in the SiC membranes. The goal of this chapter is, therefore,
to study the transport and separation of gaseous mixtures through the 3D
molecular pore network of SiC membranes with interconnected pores. This
enables us to study the effect of the morphology of the pore space, such as the
pores’ shapes, the pore size distribution (PSD), and pore connectivity on the
phenomena of interest. We utilize the molecular pore network of nanoporous SiC
membranes developed in chapter 3. To carry out the simulations, we employ the
dual control-volume grand canonical molecular dynamics (DCV-GCMD), and
examine the effect of pore length, pressure gradient and temperature on the
transport properties.

103
4.2 Molecular model of the SiC membrane
The development of the atomistic model of a SiC nanoporous membrane was
described in chapter 3 and, therefore, only a brief description of it is given here.
The model is applicable to both β-SiC, which is the crystalline state of SiC, and a-
SiC, which represents the amorphous state. As the first step an atomistic model of
the membrane material - SiC - is developed. This was already described in chapter
2. Here we give a brief summary of the technique.

If the model is intended for β-SiC, then, its atomistic structure is well-known: one
generates a large crystalline SiC structure by repeating the basic cell of material.
To generate the model for the amorphous a-SiC, we begin with the crystalline
structure of SiC. Energy minimization and MD simulation are then utilized, in
order to establish the most stable state of SiC at the desired temperature. To carry
out the MD simulations, one must specify a force field (FF) that accurately
describes SiC. In the present work we utilized the extended Brenner FF(Brenner,
1990), the details of which are given both in the original reference (Brenner, 1990)
and previous chapter. Utilizing the extended Brenner FF, MD simulations are
carried out to melt the SiC crystalline structure. The SiC crystalline structure is
heated up to 4000 K to melt it completely, and to generate amorphous SiC. After
the material reaches equilibrium, we gradually lower the temperature, using the
MD simulations. Complete details of the simulations are given in chapter 3.
104
In order to generate the pore network model of the membrane, we create a 3D
simulation box of the atomistic model of the membrane material that was
developed by the MD simulations described above. The dimensions of the
simulation box depend on the computation time that one can afford. Later in this
chapter we study the effect of the system's size - the thickness of the membrane
layer - on the separation properties of the membrane. We then tessellate the
simulation box by inserting in it a number of Poisson points, i.e., points that are
inserted randomly in the system. Each Poisson point is the basis for the
construction of a Voronoi polyhedron that consists of some of the atoms of the
membrane material, and is that part of the simulation cell that is closer to its
Poisson point than to any other Poisson point. To carry out the tessellation, we
utilize parallel computations using space decomposition, message-passing
interface strategy, and 300 processors. This reduces the computational time by
nearly two orders of magnitude. Each processor tessellates a certain part of the
simulation box.

The pore space is then generated by specifying the porosity of the membrane, and
selecting a number of the polyhedra in the tessellated space in such a way that
their total volume fraction equals the specified porosity. The selected polyhedra
are then designated as the membrane's pores. All the atoms that are inside such
pores are removed, including those that were dangling, i.e., connected to only one
105
other atom. Removal of the dangling atoms is necessitated by the fact that, in a
real porous material, it is physically impossible to have such atoms connected to
the internal surface of the pores. The remaining atoms in the simulation box
constitute the membrane's solid matrix, while the pore space consists of
interconnected empty pore polyhedra of various shapes and sizes. The designation
of the polyhedra as the pores is done as follows.

One designates the pore polyhedra in such a way that the resulting PSD mimics
that of a real membrane, which are typically skewed. To obtain such PSDs, we
first sort and list the polyhedra in the box according to their sizes (volumes), from
the largest to smallest. We then designate the polyhedra as the pores according to
their sizes, starting from the largest ones in the list. Figure 1 presents the PSD of
the SiC membrane, generated by the atomistic model with a porosity 0f 0.4, the
same as that of the fabricated membrane (see below). A Schematic of 3D pore
network is shown in Fig. 4.1.

The model has several virtues that any inorganic membrane of the type that we
have been studying is expected to possess: (i) It generates PSDs that mimic those
measured. (ii) The pores have irregular shapes. (iii) The connectivity of the pores
is automatically taken into account. (iv) Removal of the dangling atoms gives rise
to pore surface roughness, which is expected to exist in any real membrane.
106
Complete details and discussion of the various aspects of the model and its
generation are given in pervious chapter.

Figure 4.1 Structural representation of the SiC Pore network. Spheres and stars represent carbon
and silicon atoms.

107

4.3 Non equilibrium molecular dynamics in pore
network
We employed the NEMD simulations to study the transport and separation of the
mixtures of H
2
in CO
2
and H
2
in CH
4
in the pore networks. We used the standard
cut-and-shifted Lennard-Jones (LJ) potential to describe the interactions between
the gases, as well as between them and the carbon and silicon atoms in the SiC. the
cut-off distance r
c
was taken to be 15.2 Å.
LJ LJ c c
c
12 6
>
if r r U(r) - U(r )
()
if r r 0
() 4 [( ) - ( ) ]
rr
Ur
Ur
σσ
ε
⎧
⎨
⎩
≤
=
=

All of the physical quantities are presented in dimensionless units (Allen e
Tildesley, 1987) by using the LJ parameters of methane (see Table 4.1). The
values of size and energy parameters are presented in Table 4.2 We used the
Lorentz-Berthelot rule, ε
ij
=( ε
i
ε
j
)
0.5
and σ
ij
=0.5( σ
i
+ σ
j
), for calculating the size and
energy of the unlike molecules.

108
Table 4.1. Dimensional and dimensionless variables.

The interaction between gas molecules and entire pore network was taken into
account by calculating the sum of LJ potentials between gas molecules and each
individual carbon and silicon atoms in the network.

109
Table 4.2. The molecular parameters of the various atoms and molecules used in the
nonequilibrium MD simulations of transport of gases in the membrane.

The DCV-GCMD method was used to study the transport of fluids through the
membrane. To impose a potential or pressure gradient on the pore network, we
proceed as in chapter 2, namely, we insert two CVs on upstream and downstream
of the network. The schematic of pore network and the two CVs is shown in Fig.
4.2.

110
Figure 4.2 Schematic of the pore network and two, high and low pressure, control volumes.

Each MD move, i.e., integration of the equations of motion for one time
step, is followed by the GCMC creations and deletions of gase molecules in two
CVs, in order to keep the chemical potential or pressure constant there. The
probability of inserting a particle component i (H
2
or CO
2
or CH
4
) is given by
] 1 ), 1 /( ) / exp( min[ + ∆ − =
+
i i i
N kT U V Z p (3)
where
3
) exp(
i i i
kT Z Λ = µ is the absolute activity at temperature T,
3
i
Λ and
i
µ are the Broglie wavelength and chemical potential of component i, k is
Boltzmann’s constant, U ∆ is the potential energy change resulting from
inserting or deleting a particle, and V and N
i
are the volume and number of
particle i. The probability of deleting a particle (CO
2
or CH
4
) is given by
111
] 1 , / ) / exp( min[ V Z kT U N p
i i i
∆ − =
−
(4)

When a particle is inserted in a CV, it is assigned a thermal velocity selected from
the Maxwell-Boltzmann distribution at the given temperature. An important
parameter of the simulations is the ratio R of the number of GCMC insertions and
deletions in each CV to the number of MD steps between successive GCMC steps.
This ratio must be chosen appropriately in order to maintain the correct density
and chemical potentials in the CVs, and also reasonable transport rates at the
boundaries between the CVs and the transport region. In our simulations R=50:1
was proved to be sufficient.

During the MD calculations particles crossing the outer boundaries of the CVs
were removed. In addition, for each component we allowed for a nonzero
streaming velocity (the ratio of the flux to the concentration of each component) in
the pore region. The unrealistic assumption of a zero streaming velocity in the
transport region, used in many of the previous works, leads to severely
underestimated fluxes. Since the two CVs are assumed to be well-mixed, and in
equilibrium with the two bulk phases that are in direct contact with them, there
should be no overall nonzero streaming velocity in these regions. However, the
discontinuity of the streaming velocities at the boundaries between the CVs and
112
the transport region slows down the computations. To address this, a very small
streaming velocity was added to the thermal velocity of all the newly inserted
molecules within each CV that were located within a very small distance from the
boundaries between the CVs and the transport region in the pore (Maginn, Bell et
al., 1993; Heffelfinger e Vanswol, 1994; Kjelstrup e Hafskjold, 1996). However,
the actual streaming velocities of the molecules in the transport pore region were
still determined by the MD simulations.

The Verlet velocity algorithm was used to solve the equations of motion in the
MD simulation. To study the transport of a mixture due to a pressure gradient, the
temperature of the system must be keep constant in order to eliminate any
contribution of the temperature gradient to the transport; hence isokinetic
conditions were maintained by rescaling the velocity in all three directions. Figure
4.3 shows the dimensionless time-averaged temperature profiles along the pore
network and two CVs.

To reduce the simulation time for computing the interactions between gas
molecules and the Si and C atoms in the membrane, we discretized the simulation
box into n
x
×n
y
×n
z
grid points, with n
y
= n
z
= 149. The value of n
x
used was
dependent upon the thickness of the membrane layer. For example, for a thickness
of 13 nm (that we used in most of the MD simulations; see below), we used n
x
=
113
670. The interactions were computed at each of the grid points, and 3D piecewise
cubic Hermite interpolation was utilized for interpolating the results between the
grid points. The results were then stored and utilized in the simulations.

In addition, for each component i we also calculated its flux J
i
by computing the
net number of its particles crossing a given y
z
plane of the cross-sectional area Ayz:
yz
RL
i
LR
i
i
tA N
N N
J
∆
−
= (5)
where
LR
i
N and
RL
i
N
are the number of the gas molecules of type i moving from
the left to the right and vice versa, respectively, ∆t is the MD time step. A
dimensionless ∆t* = 5×10
−3
, i.e., ∆t = 7.3×10
−3
ps, was used, where t* is the
dimensionless time (see Table 4.2). Here, N is the number of the MD steps over
which the average was taken; we typically used, N = 3×10
4
.

The flux J
i
was computed at the entrance to and the exit from the membrane, as
well as at its central cross section, and was then averaged. The transport process
was considered to have reached steady state when the fluxes calculated at various
yz planes were within 5% from the averaged values. The equation of motion was
integrated with up to 6×10
6
time steps. The steady state was typically reached after
114
106 time steps, although the time to reach the steady state depends on the porosity
of the membrane. The permeability K
i
of gas i was calculated using:
i
i
i
i
i
P
LJ
L P
J
K
∆
=
∆
= (6)
where ∆P
i
= x
i
∆P is the partial pressure drop for species i along the membrane,
with x
i
being the mole fraction of component i, ∆P the total pressure drop imposed
on the membrane, and L the membrane’s thickness. As discussed below, the
applied pressure drop ∆P was varied, in order to study its effect on the
permeability. A most important property to compute for any membrane is the
dynamic separation factor S, defined as
1
2
21
K
K
S = (5)
where K
2
is the permeance to component 2, H
2
.

Let us point out that, if the porosity of a membrane is relatively large - not too
close to its percolation threshold (Sahimi, 1994)- then often a single realization of
the model produces the correct results. However, if the porosity is low, or if the
system is close to a phase transition (which often happens at low temperatures),
then, one must either generate several realizations of the model and average the
results, or use a model membrane with a large linear size. In the present work we
used up to 8 realizations of the model membranes, and averaged the results over
them.
115
Figure 4.3 The average dimensionless temperature T* in the membrane (in the middle) and the
two control volumes on the left and right sides. Numbers in the parentheses indicate the set
temperatures. The porosity is 0.4, and the applied external pressure drop is 2 atm.

116

4.4 Results and discussion
We carried out extensive DCV-GCMD simulations of two equimolar binary gas
mixtures in the model membranes, namely, H
2
/CO
2
and H
2
/CH
4
mixtures, under a
variety of conditions. In all the simulations, the average pore size of the membrane
is 6.5 Å, which is the same as what we obtained in chapter 3 for matching the
measured nitrogen adsorption isotherm to the experimental data. Unless specified
otherwise, the pressure drop across the membrane layer was taken to be 2 atm,
with the porosity being 0.4, the same as in our experiments (Elyassi, Sahimi et al.,
2008) for measuring the separation factor of the SiC membrane.

As the first step, we must establish that simulations have been carried out correctly
in the sense that, the pressures in the two CVs have been held fixed. Figure 4.4
presents the dimensionless density profiles ) (x
z
i
ρ for the binary mixture H
2
/CO
2
at
two temperatures. All the qualitative trends of the density profiles are the same at
the two temperatures, except that the densities are lower at the higher T, which is
expected.

The density profiles are essentially at in the two CVs, i.e., in the regions 0 < X* <
30, and 60 < X* < 90. Although there are small fluctuations in the densities there,
117
the numerical values of their averages match those obtained with the GCMC
method at the same conditions, indicating that the chemical potentials in the two
CVs are properly maintained during the DCV- GCMD simulations. At the
entrance to the membrane, X* = 30, and at the exit, X* = 60, the density of CO
2

rises sharply. This is, however, an effect caused by adsorption affinity of the gases
for the SiC surface.

Since the energy parameter for CO
2
-membrane interactions is larger than that of
the CH
4
-membrane, i.e., the membrane is more attractive to the CO
2
than to CH
4
,
more CO
2
molecules accumulate close to the membrane pores’ surface than do the
CH4 molecules. This is also indicated by the heights of the peaks in the density
profiles of the two gases. On the other hand, since the energy parameter for the
H
2
-membrane interactions is weak, at least compared with the other interactions
(hydrogen has very affinity for adsorption on the SiC surface), there is not much
H
2
adsorption at the membrane’s entrance or exit. This explains why there is no
peak in the H
2
profiles.

In the transport region, 30 < X* < 60, the density profiles for both components
decrease along the X* direction, which is expected. Due to the existence of the
overall bulk pressure gradient (or an overall nonzero streaming velocity), however,
the density profiles in the membrane are not linear. The total flux is the sum of the
118
diffusive and convective parts, which result in a nonlinear profile. At the same
time, the membrane is heterogeneous with a range of the pore sizes. Thus, the
density profiles are not necessarily smooth. They vary widely in the membrane.

Similar trends are obtained for the equimolar H
2
/CH
4
mixture, for which the
results are presented in Fig. 4.5. In this case, CH
4
is the larger of the two
molecules and, therefore, it accumulates somewhat at the entrance to, and the exit
from, the membrane. Consistent with our argument above that the accumulation
effect is due to the proximity of the gases' molecular size to the pore sizes of the
membrane, the accumulation effect for CH
4
is about the as that for CO
2
, and more
severe than that for H
2
.

119
Figure 4.4 The dimensionless density profiles of the H
2
/CO
2
mixture in the axial (X*) direction,
in the membrane (middle) and the two control volumes. The porosity is 0.4, and the applied
external pressure drop is 2 atm.

120
Figure 4.5 The dimensionless density profiles of the H
2
/CH
4
mixture in the axial (X*) direction,
in the membrane (middle) and the two control volumes. The porosity is 0.4, and the applied
external pressure drop is 2 atm.

121
As is well-known, simulations results obtained with any pore network, including
molecular pore networks that are used here, depend on the network's size (Sahimi,
Gavalas et al., 1990; Sahimi, 1993; Rieckmann e Keil, 1997; , 1999; Sahimi e
Tsotsis, 2003), unless it is large enough. Thus, as the next step, we carried out a
series of simulations in which the size of the network, i.e., the thickness of the
membrane, was increased and the corresponding separation factors SF were
computed. Figure 4.6 presents the results for the H
2
/CO
2
mixture at T = 200 °C
and with ∆P = 2 atm. Completely similar trends are obtained for the H
2
/CH
4

mixture. The separation factor increases with increasing the thickness of the
membrane (holding the average pore size fixed at 6.5 Å). The membrane with a
thickness of 13.1 nm has the same separation factor as that with a thickness of 14
nm. Hence, all the results that are presented below, as well as those shown in Figs.
4.4 and 4.5, were obtained with a membrane of thickness 13 nm.

Figure 4.6 presents an important result with practical implication: the separation
factor of a membrane depends on its thickness, but there is an optimal thickness
for it. Any thickness smaller than the optimal value yields a separation factor
smaller than its maximum value, but the separation factor will not improve, and
may even decline again, if the membrane is thicker than its optimal thickness.

122
Having obtained the optimal thickness of the SiC membrane, we now study the
effect of the various parameters on the separation factor, and discuss their
implications.

Figure 4.6 Effect of membrane thickness on its separation factor.

123
4.4.1 Effect of temperature
Increasing temperature at a fixed ∆P causes the molecules to diffuse much faster,
hence resulting in increased fluxes. At high temperatures, there is also little
possibility of adsorption on the pores' surface. The combination of the two factors
gives rise to higher fluxes. Hydrogen is the lightest gas in the two binary mixtures
and, therefore, we expect higher temperatures to results in higher separation
factors for H
2
over both CO
2
and CH4. Figure 4.7 presents the results of the
computations, which confirm the trends. They indicate that S is always a strongly
increasing function of T. Note the high separation factors that the simulations
yield at 200°C, which is the temperature at which the experiments with the two
binary mixtures were also carried out. We shall come back to this point shortly.
124
Figure 4.7 Effect of the temperature on the separation factor for the two gaseous mixtures.

0
20
40
60
80
100
120
140
0 50 100 150 200 250
Temperature (ºC)
Separation Factor
H2/CO2
H2/CH4
∆P=2 atm
φ=0.5
125

4.4.2 Effect of external pressure drop
Figure 4.8 presents the dependence on the external pressure drop ∆P of the
separation factor S of the membrane for the two mixtures that we have studied,
and for two porosities. Consider, first, the H
2
/CH
4
mixture. At both porosities, the
separation factors are higher at the lower pressure differences. The reason is that,
in the tight pores of the membrane, increasing the external pressure drop creates a
mixture that is increasingly liquid-like, hence decreasing, or eliminating altogether,
the contribution of the diffusion of the lighter and faster H
2
to its overall flux. This
results in much lower H
2
fluxes, and hence smaller separation factors. The effect is
more severe at the higher porosity, because at lower external pressure drops the
separation factor is higher for larger porosities. Note that, once ∆P reaches about 4
atm, the separation factor becomes essentially constant, because the mixture is
already in its liquid-like state.

The results for the H
2
/CO
2
mixture are more or less similar to those for the H
2
/CH
4

mixture at the same porosities. There is, however, a difference between H
2
/CH
4

and H
2
/CO
2
mixtures in that, the gases in the former case do not have any
significant affinity for adsorption on the SiC surfaces, whereas, due to the
presence of the C atoms on the pores’ surface, CO
2
does have some affinity for
adsorption on the pores’ surface and, hence, surface flow in the membrane. Figure
126
0
20
40
60
80
100
120
140
01 2 3 45 67
∆P
Separation Factor
0.5
0.4
H
2
/CH
4
φ
T=200 °C
4.9 presents the results for the H
2
/CO
2
mixture. The trends with the porosity of 0.5
are similar to those for the H
2
/CH
4
mixture at the same porosity.

Figure 4.8 Effect of the applied external pressure drop ∆P on the separation factors for the
mixture of H
2
/CH
4
.

127
0
20
40
60
80
100
120
140
160
01 234 56 7
∆P
Separation Factor
0.5
0.4
H
2
/CO
2
φ
T=200 °C
Figure 4.9 Effect of the applied external pressure drop ∆P on the separation factors for the
mixture of H
2
/CO
2
.

4.4.3 Effect of porosity
Figure 4.10 presents the effect of the membrane’s porosity on the separation factor
of the H
2
/CH
4
mixture, for three external pressure differences. In all the three
cases, there is an optimal porosity, φ = 0.5, for which the separation factor is
maximum. To understand the results better, we plot in Fig. 4.11 the fluxes of the
two components as functions of the porosity and the three external pressure drops.
Higher porosities provide more accessible pores but, at the same time, decrease the
interactions between the gas molecules and the pores’ surface, because higher
128
porosities also improve the interconnectivity of the pore space, helping them to
merge and form larger pores (see Fig. 1). While the flux of H
2
reaches a maximum
with increasing porosity, and then decreases sharply, the flux of CH
4
increases
monotonically with the porosity. That the separation of the gases at high porosities
is not effective is due to the presence of very large pores that can no longer
separate the two gases by molecular sieving.

Figure 4.12 depicts the results for an equimolar mixture of H
2
/CO
2
. Once again,
there is an optimal porosity, φ = 0.5, at which the separation factor attains its
maximum value, while for larger porosities it decreases. Because H
2
has
practically no affinity for adsorption on the SiC surface at the elevated temperature,
while the CO
2
affinity for adsorption on the same surface is, relatively speaking,
larger than that of H
2
, porosity affects the flux of H
2
much more strongly than that
of CO
2
, since the main mechanisms of transport of H
2
through the pore space is
convection. This is demonstrated in Fig. 4.13, where we present the fluxes of the
two components.

The important implication of the maxima in the separation factors is that, there is
optimal pore space morphology (the porosity, the PSD, and pore connectivity) for
separation of gases in the membrane. Such effects, which are absent in the single
pore models, point to the significance of the pore space morphology on gas
129
separation. We are currently developing computationally efficient models for
determining the optimal structure of a membrane and its operating conditions,
given a target separation factor.

It is instructive to compare these results with those obtained with the same gases in
the single-pore model. The separation factors that are obtained with the single-
pore model are much smaller than the experimental values (Elyassi, Sahimi et al.,
2007; , 2008), even when the pore size is the same as the average pore size of the
membrane, whereas, as discussed below, the results presented here are much more
closely consistent with the data. Moreover, the separation factors depend on the
membrane’s porosity, the effect of which cannot be taken into account by the
single-pore model, but is accounted for by the present model. Most importantly,
for such mixtures as H
2
/CH
4
in which both the gases have no affinity for
adsorption on the pores’ surface, the single-pore model completely breaks down,
even in a qualitative sense. In contrast, because molecular sieving and the effect of
the tortuous pore space of the membrane are present in the molecular pore network
model, it provides estimates for the separation factors that are in agreement with
the data (see below).

130
0
20
40
60
80
100
120
140
0.2 0.3 0.4 0.5 0.6 0.7
Porosity
Separation Factor
6
5
2
H
2
/CH
4
∆P (atm)
T=200 °C
Figure 4.10 Effect of the membrane’s porosity on its separation factor for the H
2
/CH
4
mixture, for
three applied external pressure drops ∆P.

131
Figure 4. 11 Effect of the membrane’s porosity on the (dimensionless) fluxes of H
2
and CH
4
.

132
Figure 4.12 Effect of the membrane’s porosity on its separation factor for the H
2
/CO
2
mixture, for
three applied external pressure drops ∆P.

133
Figure 4.13 Effect of the membrane’s porosity on the (dimensionless) fluxes of H
2
and CO
2
.
134

4.4.4 Comparison with the experimental data
The nitrogen adsorption isotherm in the SiC membrane that we fabricated in our
lab was measured using the BET method. We first measured the isotherm in the
membrane’s support (which was also SiC), and then generated the BET data in the
supported membrane. We were, therefore, able to obtain the isotherm in the
membrane layer itself (because the atomistic model is intended for the membrane
layer). Using the adsorption isotherm and the t-plot method (Lowell e Lowell,
2004), we estimated the membrane’s porosity to be about 0.4. As mentioned
earlier, the average pore size of the membrane, which was estimated by matching
the measured sorption isotherm to the results of MD simulations, was estimated to
be 6.5 Å.

Table 4.3 compares the experimental values of the separation factors S for the two
mixtures with those obtained by the DCV-GCMD simulations described above. In
the case of the H
2
/CO
2
mixture, the computed separation factor is about 28%
lower than the measured value, whereas the computed S for the H
2
/CH
4
mixture is
lower than the measured value by a factor of about 1.85.

The difference between the computed and measured separation factors may be
explained in part by the structure of the fabricated membrane. The first SiC
135
membrane that was fabricated by our group yielded (Elyassi, Sahimi et al., 2007)
separation factors of 37 and 50 for the H
2
/CH
4
and H
2
/CO
2
mixtures, respectively.
These values are both lower than the computed values listed in Table 3. In an
attempt to improve the separation factor of the membrane, we applied (Elyassi,
Sahimi et al., 2008) thin polystyrene sacrificial layers to the membrane’s pore
space that enhanced the membrane’s performance considerably, resulting in the
separation factors that are listed in Table 3, both of which are larger than the
computed values. We speculate that the thin sacrificial layers prevent, on the one
hand, the infiltration of pre-ceramic polymer into the underlying support layers
and help, on the other hand, the formation of a 3D membrane structure with a high
microporous volume, which is a result of the decomposition of the polystyrene
layer.

However, the changes in the pore space of the membrane that are the result of
applying the sacrificial layers to the pore space are absent in the molecular model
that we have developed. We believe that if such changes are somehow
incorporated in the molecular model, the computed values would resemble the
experimental values much more closely. Work in this direction is in progress.
136
4.3. Comparison of the computed and measured separation factors for the two binary mixtures.

4.5 Summary
The new molecular pore network model that was developed in chapter 3 for the
SiC nanoporous membranes was utilized to study transport and separation of two
binary gaseous mixtures in a silicon-carbide membrane. The model is based on
generating an atomistic model of the membrane material, tessellating the atomistic
model using the Voronoi algorithm, and designating the pores in a physically
reasonable manner. It was demonstrated that, using the model developed in
chapter 3 by computing the sorption isotherm of nitrogen in a SiC membrane and
adjusting it to match the experimental isotherm, and utilizing no other adjustable
parameter, the model accurately predicts the separation properties of the SiC
membrane.

Thus, by adjusting at most one parameter, the new molecular model for the SiC
membrane can accurately predict not only the nitrogen sorption data that are in
agreement with the experimental data, but also its transport and separation
137
properties. We believe that the general methodology described in pervious chapter
is applicable to the modeling of a wide variety of inorganic membranes.

138

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Abstract (if available)

Abstract The goal of this work is to study transport of gas mixtures through nanoporous membranes especially silicon carbide membranes.

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Asset Metadata

Creator Rajabbeigi, Nafiseh (author)

Core Title Molecular modeling of silicon carbide nanoporous membranes and transport and adsorption of gaseous mixtures therein

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publication Date 09/28/2009

Defense Date 08/17/2009

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

Tag adsorption,molecular dynamics simulation,nanoporous,nanoscale,OAI-PMH Harvest,silicon carbide,transport of gases

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sahimi, Muhammad (committee chair), Mak, Chi Ho (committee member), Tsotsis, Theodore T. (committee member)

Creator Email n_rajabbeigi@yahoo.com,rajabbei@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2620

Unique identifier UC1152932

Identifier etd-Rajabbeigi-3225 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-256564 (legacy record id),usctheses-m2620 (legacy record id)

Legacy Identifier etd-Rajabbeigi-3225.pdf

Dmrecord 256564

Document Type Dissertation

Rights Rajabbeigi, Nafiseh

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu